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[
[
"Transferability of Adversarial Examples to Attack Cloud-based Image\n Classifier Service"
],
[
"Abstract In recent years, Deep Learning(DL) techniques have been extensively deployed for computer vision tasks, particularly visual classification problems, where new algorithms reported to achieve or even surpass the human performance.",
"While many recent works demonstrated that DL models are vulnerable to adversarial examples.",
"Fortunately, generating adversarial examples usually requires white-box access to the victim model, and real-world cloud-based image classification services are more complex than white-box classifier,the architecture and parameters of DL models on cloud platforms cannot be obtained by the attacker.",
"The attacker can only access the APIs opened by cloud platforms.",
"Thus, keeping models in the cloud can usually give a (false) sense of security.",
"In this paper, we mainly focus on studying the security of real-world cloud-based image classification services.",
"Specifically, (1) We propose a novel attack method, Fast Featuremap Loss PGD (FFL-PGD) attack based on Substitution model, which achieves a high bypass rate with a very limited number of queries.",
"Instead of millions of queries in previous studies, our method finds the adversarial examples using only two queries per image; and (2) we make the first attempt to conduct an extensive empirical study of black-box attacks against real-world cloud-based classification services.",
"Through evaluations on four popular cloud platforms including Amazon, Google, Microsoft, Clarifai, we demonstrate that FFL-PGD attack has a success rate over 90\\% among different classification services.",
"(3) We discuss the possible defenses to address these security challenges in cloud-based classification services.",
"Our defense technology is mainly divided into model training stage and image preprocessing stage."
],
[
"INTRODUCTION",
"In recent years, Deep Learning(DL) techniques have been extensively deployed for computer vision tasks, particularly visual classification problems, where new algorithms reported to achieve or even surpass the human performance.",
"Success of DL algorithms has led to an explosion in demand.",
"To further broaden and simplify the use of DL algorithms, cloud-based services offered by Amazonhttps://aws.amazon.com, Googlehttps://cloud.google.com, Microsofthttps://azure.microsoft.com, Clarifaihttps://www.clarifai.com/, and others to offer various computer vision related services including image auto-classification, object identification and illegal image detection.",
"Thus, users and companies can readily benefit from DL applications without having to train or host their own models.",
"[1] discovered an intriguing properties of DL models in the context of image classification for the first time.",
"They showed that despite the state-of-the-art DL models are surprisingly susceptible to adversarial attacks in the form of small perturbations to images that remain (almost) imperceptible to human vision system.",
"These perturbations are found by optimizing the input to maximize the prediction error and the images modified by these perturbations are called as adversarial example.",
"The profound implications of these results triggered a wide interest of researchers in adversarial attacks and their defenses for deep learning in general.The initially involved computer vision task is image classification.",
"For that, a variety of attacking methods have been proposed, such as L-BFGS of [1], FGSM of [2], PGD of [3],deepfool of [4] ,C&W of [5] and so on.",
"Fortunately, generating adversarial examples usually requires white-box access to the victim model, and real-world cloud-based image detection services are more complex than white-box classification and the architecture and parameters of DL models on cloud platforms cannot be obtained by the attacker.",
"The attacker can only access the APIs opened by cloud platforms[6], [7].",
"Thus, keeping models in the cloud can usually give a (false) sense of security.",
"Unfortunately, a lot of experiments have proved that attackers can successfully deceive cloud-based DL models without knowing the type, structure and parameters of the DL models[8], [9].",
"In general, in terms of applications, research of adversarial example attacks against cloud vision services can be grouped into three main categories: query-based attacks, transfer learning attacks and spatial transformation attacks.",
"Query-based attacks are typical black-box attacks, attackers do not have the prior knowledge and get inner information of DL models through hundreds of thousands of queries to successfully generate an adversarial example [10].",
"In [11], thousands of queries are required for low-resolution images.",
"For high-resolution images, it still takes tens of thousands of times.",
"For example, they achieves a 95.5% success rate with a mean of 104342 queries to the black-box classifier.",
"In a real attack, the cost of launching so many requests is very high.Transfer learning attacks are first examined by [1], which study the transferability between different models trained over the same dataset.",
"[12] propose novel ensemble-based approaches to generate adversarial example .",
"Their approaches enable a large portion of targeted adversarial example to transfer among multiple models for the first time.However, transfer learning attacks have strong limitations, depending on the collection of enough open source models, but for example, there are not enough open source models for pornographic and violent image recognition.Spatial transformation attacks are simple and effective.",
"[13] found that adding an average of 14.25% impulse noise is enough to deceive the Google’s Cloud Vision API.",
"[14] found 7 major categories of spatial transformation attacks to evade explicit content detection while still preserving their sexual appeal, even though the distortions and noise introduced are clearly observable to humans.",
"To the best of our knowledge, no extensive empirical study has yet been conducted to black-box attacks and defences against real-world cloud-based image classification services.",
"We summarize our main contributions as follows: We propose a novel attack method, Fast Featuremap Loss PGD(FFL-PGD) attack based on Substitution model ,which achieves a high bypass rate with a very limited number of queries.",
"Instead of millions of queries in previous studies, our method finds the adversarial examples using only one or twe of queries.",
"We make the first attempt to conduct an extensive empirical study of black-box attacks against real-world cloud-based image classification services.",
"Through evaluations on four popular cloud platforms including Amazon, Google, Microsoft, Clarifai, we demonstrate that our FFL-PGD attack has a success rate almost 90% among different classification services.",
"We discuss the possible defenses to address these security challenges in cloud-based classification services.",
"Our protection technology is mainly divided into model training stage and image preprocessing stage.",
"In this paper, we assume that the attacker can only access the APIs opened by cloud platforms, and get inner information of DL models through limited queries to generate an adversarial example.Without any access to the training data, model, or any other prior knowledge,is a real black-box attack."
],
[
"Criterion and Evaluation",
"The same with [15],We choose top-1 misclassification as our criterion,which means that our attack is successful if the label with the highest probability generated by the neural networks differs from the correct label.We assume the original input is $O$ ,the adversarial example is $ADV$ .",
"For an RGB image $(m \\times n \\times 3)$ , $(x,y,b)$ is a coordinate of an image for channel $b(0 \\leqslant b \\leqslant 2)$ at location $(x,y)$ .We use Peak Signal to Noise Ratio (PSNR)[16] to measure the quality of images.",
"$PSNR = 10log_{10} (MAX^2/MSE)$ where $MAX =255$ , $MSE$ is the mean square error.",
"$MSE = \\frac{1}{mn*3}*\\sum _{b=0}^2\\sum _{i=1}^n\\sum _{j=1}^m ||ADV(i,j,b)-O(i,j,b)||^2$ Usually, values for the PSNR are considered between 20 and 40 dB, (higher is better) [17].We use structural similarity (SSIM) index to measure image similarity, the details of how to compute SSIM can be found in [18].Values for the SSIM are considered good between 0.5 and 1.0, (higher is better)."
],
[
"Problem Definition",
"A real-world cloud-based image classification service is a function $F(x) = y$ that accepts an input image $x$ and produces an output $y$ .",
"$F(.",
")$ assigns the label $C(x)=\\arg \\max _{i}F(x)_i$ to the input $x$ .",
"Original input is $O$ , the adversarial example is $ADV$ and $\\epsilon $ is the perturbation.",
"Adversarial example is defined as: $ADV=O+\\epsilon $ We make a black-box untargeted attack against real-world cloud-based classification services $F(x)$ : $C(ADV)\\ne C(O)$ We also assume that we are given a suitable loss function $L( \\theta ,x,y)$ ,for instance the cross-entropy loss for a neural network.",
"As usual, $\\theta \\in \\mathbb {R}^p$ is the set of model parameters."
],
[
"Fast Featuremap Loss PGD based on Substitution model",
"[19] proposed that the attacker can train a substituted model, which approaches the target model, and then generate adversarial examples on the substituted model.",
"Their experiments showed that good transferability exists in adversarial examples.But the attack is not totally black-box.",
"They have knowledge of the training data and test the attack with the same distributed data, and they upload the training data themselves and they know the distribution of training data [19][20][21].",
"This leads us to propose the following strategy: Substitute Model Training: the attacker queries the oracle with inputs selected by manual annotation to build a model $F{}^{\\prime }(x)$ approximating the oracle model $F(x)$ decision boundaries.",
"Adversarial Sample Crafting: the attacker uses substitute network $F{}^{\\prime }(x)$ to craft adversarial samples, which are then misclassified by oracle $F(x)$ due to the transferability.We propose Fast Featuremap Loss PGD attack to improve the success rate of transfer attack.",
"Figure: Top1 vs. network.",
"Top-1 validation accuracies for top scoring single-model architectures ."
],
[
"Substitute Model Training Algorithm",
"We observe that a large number of machine vision tasks utilize feature networks as their backends.",
"For examples, Faster-RCNN [23] and SSD [24] use the same VGG-16 [25].",
"If we destroy the extracted features from the backend feature network, both of them will be influenced.",
"We can choose one of AlexNet [26], VGG [25]and ResNet [27] which pretrained on ImageNet as our substitute model.",
"Better top-1 accuracy means stronger feature extraction capability.",
"As can be seen from Figure 1, ResNet-152 has relatively good top-1 accuracy, so we choose ResNet-152 as our substitute model.",
"We fix the parameters of the feature layer and train only the full connection layer of the last layer.",
"Our Substitute Model Training Algorithm is very simple, we train substitute model and generate adversarial example with the same images.",
"[t] - Substitute Model Training: for oracle $F$ , a substitute architecture $F{}^{\\prime }$ , and an initial training set $S$ .",
"[1] $F{}^{\\prime }(x)$ ,$S$ Define architecture $F{}^{\\prime }(x)$ //Label the substitute training set with $F$ $S \\leftarrow \\left\\lbrace (x, F(x)) : x\\in S \\right\\rbrace $ // Train $F{}^{\\prime }$ on $S$ to evaluate parameters $\\theta _F{}^{\\prime }$ $\\theta _F{}^{\\prime } \\leftarrow \\mbox{train}(F{}^{\\prime },S)$ $\\theta _F$"
],
[
"Adversarial Sample Crafting Algorithm",
"Previous work [28] has shown that one-step or multi-step attack algorithm such as $FGSM$ and $FGSM^k$ , has better robustness in transfer attacks than $CW2$ , which is based on optimization.$FGSM$ is an attack for an $\\ell _{\\infty }$ -bounded adversary and computes an adversarial example as: $x+\\varepsilon \\operatorname{sgn}\\left(\\nabla _{x} L(\\theta , x, y)\\right)$ A more powerful adversary is the multi-step variant $FGSM^k$ , which is essentially projected gradient descent ($PGD$ ) on the negative loss function [3]: $x^{t+1}=\\Pi _{x+\\mathcal {S}}\\left(x^{t}+\\varepsilon \\operatorname{sgn}\\left(\\nabla _{x} L(\\theta , x, y)\\right)\\right)$ As usual, loss function $L( \\theta ,x,y)$ is cross-entropy loss for a neural network.",
"We propose Fast Featuremap Loss PGD attack which has a novel loss function to improve the success rate of transfer attack.The loss function $L$ is defined as: $L = class\\_loss+\\beta *FeatureMaps\\_loss$ Where $\\varepsilon $ and $\\beta $ are the relative importance of each loss function.",
"Next, we will introduce each component of the loss function in detail.",
"Class Loss The core goal of generating adversarial example is to make the result of classification wrong.",
"The first part of our loss function is class loss.",
"Assuming that the Logits output of the classifier can be recorded as $Z(x)$ , the output value corresponding to the classification label $i$ is $Z_{i}(x)$ , and $t$ is the label of normal image.",
"The greater the value of $Z_{i}(x)$ , the greater the confidence that $x$ is recognized as $i$ by the classifier.",
"$class\\_loss (x) = \\max (\\max \\lbrace Z(x)_i : i \\ne t\\rbrace - Z(x)_t, -\\kappa ).$ $\\kappa $ is a hyperparameter and $\\kappa $ is a positive number, the greater the $\\kappa $ is, the greater the confidence of the adversarial example is recognized as $\\kappa $ by the classifier.",
"[5] discussed the performance of various class loss in detail in CW2 algorithm.",
"We choose the class loss selected in CW2 algorithm.",
"The empirical value of $\\kappa $ is 200.",
"FeatureMap Loss Only class loss and distance loss can be used to generate adversarial example, which is what the L-BFGS algorithm of [1] does.",
"However, as a black-box attack, we have no knowledge of the parameters and structure of the attacked model.",
"We can only generate adversarial example through the known substitute model of white-box attack, and then attack target model.",
"The success rate of the attack depends entirely on the similarity between the substitute model and the attacked model.",
"We introduce Feature Maps loss, which is the output of the last convolution layer of the substitute model, representing the highest level of semantic features of the convolution layer after feature extraction layer by layer $L_n$ .",
"We assume the original input is $O$ , the adversarial example is $ADV$ , and the featuremap loss is: $FeatureMap\\_loss(ADV,O)=\\Vert L_n(ADV) - L_n(O) \\Vert _2$ zeiler2014visualizing visualizes the differences in the features extracted from each convolution layer.",
"In cat recognition, for example, Figure REF ,the first convolution layer mainly recognizes low level features such as edges and lines.",
"In the last convolution layer, it recognizes high level features such as eyes and nose.",
"In machine vision tasks, convolution layer is widely used for automatic feature extraction.",
"And a large number of models are based on the common VGG, ResNet pre-training parameters on ImageNet and finetuned the weights on the their own dataset.",
"We assume that the feature extraction part of the main stream cloud-based image classification services are based on common open source models as VGG or others.",
"The larger the feature Maps loss of the adversarial example and the original image, the greater the difference in the semantic level.",
"We define the hyperparameter $\\beta $ .",
"The larger the $\\beta $ , the better the transferability.",
"Figure: Illustration of cat recognition, the first convolution layer mainly recognizes low level features such as edges and lines.",
"In the last convolution layer, it recognizes high level features such as eyes and nose."
],
[
"Datasets and Preprocessing",
"100 cat images and 100 other animal images are selected from the ImageNet val set.",
"Because VGG19 and Resnet50 both accept input images of size $224 \\times 224 \\times 3$ , every input image is clipped to the size of $224 \\times 224 \\times 3$ , where 3 is the number of RGB channels.",
"The RGB value of the image is between 0 and 255.",
"We use these 100 images of cats as original images to generate adversarial examples and make a black-box untargeted attack against real-world cloud-based image classification services.",
"We choose top-1 misclassification as our criterion, which means that our attack is successful if the label with the highest probability generated by the cloud-based image classification service differs from the correct label \"cat\".",
"We count the number of top-1 misclassification to calculate the escape rate.",
"Table: Correct label by cloud APIsAccording to Table REF , we can learn that Amazon and Google, which label 98.5% of all images correctly, have done a better job than other cloud platforms."
],
[
"Fast Featuremap Loss PGD based on Substitution model",
"We choose ResNet-152 as our substitute model, fix the parameters of the feature layer and train only the full connection layer of the last layer.",
"We launched PGD and FFL-PGD attacks against our substitute model to generate adversarial examples.",
"PGD and FFL-PGD share the same hyperparameter of step size $\\varepsilon $ ,while the hyperparameter $\\beta $ of FFL-PGD set to $0.1$ .",
"Figure: In , we increase step size ϵ\\epsilon from 1 to 8, the figure records the escape rates of PGD and FFL-PGD attacks against cloud-based image classification services under different ϵ\\epsilon .",
"In , the figure records the PSNR of PGD and FFL-PGD attacks and In , the figure records the SSIM of PGD and FFL-PGD attacks.The escape rates of PGD and FFL-PGD attacks are shown in Figure REF .",
"From Figure REF , we know that the cloud-based image classification services of Amazon, Google, Microsoft and Clarifai are vulnerable to PGD and FFL-PGD attacks .",
"Step size $\\epsilon $ controls the escape rate.",
"Increasing this parameter can improve the escape rate.",
"When $\\epsilon $ is the same, FFL-PGD has a higher escape rate than PGD.",
"It can be seen that the FeatureMap Loss which added to the loss function is beneficial to improve the escape rate, that is, to improve the robustness of transfer attacks against different cloud-based image classification services.",
"From Figure REF , we know that PGD has a higher PSNR ,which is considered as better image quality .But both of them higher than 20dB when $\\epsilon $ from 1 to 8, which means both of them are considered acceptable for image quality.In addition, we can find that increasing $\\epsilon $ will lead to image quality degradation.",
"From Figure REF , we know that FFL-PGD has a higher SSIM ,which is considered as better image similarity .",
"FFL-PGD attack has a success rate over 90% among different cloud-based image classification services and is considered acceptable for image quality and similarity using only two queries per image.",
"[12] adopted an ensemble-based model to improve transferability of attack and successfully attack Clarifai.",
"We used their methods to attack the same cloud platforms and train our ensemble-based model with AlexNet, VGG-19, ResNet-50, ResNet-110 and ResNet-152.",
"Table: PSNR And SSIM of Ensemble-based Mode AttackAccording to Table REF , we can learn that increasing the number of iteration can increase PSNR and SSIM under ensemble-based model attack, which means better image quality and similarity .",
"We can infer that when the number of iterations continues to increase, the perturbation $l_2$ value of adversarial examples decreases and the PSNR increases.",
"Although the image quality can be improved, adversarial examples are over fitting the model , and the transferability decreases in the face of the pretreatment of cloud services.",
"In [29], they take advantage of the weakness of iteration-based white-box attack, and use the pre-processing steps of random scaling and translation to defense the adversarial examples."
],
[
"Effect of Attacks",
"Our research shows that FFL-PGD attack can reduce the accuracy of mainstream image classification services in varying degrees.",
"To make matters worse, for any image classification service, we can find a way that can be almost 90% bypassed."
],
[
"Defenses",
"Defense adversarial examples is a huge system engineering, involving at least two stages: model training and image preprocessing."
],
[
"Model Training",
"[2] proposed adversarial training to improve the robustness of deep learning model.",
"Retraining the model with new training data may be very helpful.",
"Adversarial training included adversarial examples in the training stage and generated adversarial examples in every step of training and inject them into the training set.",
"On the other hand, we can also generate adversarial samples offline, the size of adversarial samples is equal to the original data set, and then retrain the model.",
"We have developed AdvBox[30]https://github.com/advboxes/AdvBox, which is convenient for developers to generate adversarial samples quickly."
],
[
"Image Preprocessing",
"[31] evaluated the effect of JPG compression on the classification of adversarial images and their experiments demonstrate that JPG compression can reverse small adversarial perturbations.",
"However, if the adversarial perturbations are larger, JPG compression does not reverse the adversarial perturbation.",
"[28] proposed a randomization-based mechanism to mitigate adversarial effects and their experimental results show that adversarial examples rarely transfer between different randomization patterns, especially for iterative attacks.",
"In addition, the proposed randomization layers are compatible to different network structures and adversarial defense methods, which can serve as a basic module for defense against adversarial examples.",
"Although all the above efforts can only solve some problems, chatting is better than nothing."
],
[
"RELATED WORK",
"Previous works mainly study the security and privacy in DL models via white-box mode [1] [2] [3] [4].",
"In the white-box model, the attacker can obtain the adversarial examples quickly and accurately.",
"However, it is difficult for the attacker to know the inner parameters of models in the real world, so researchers have launched some black-box attacks on DL models recently.",
"In general, in terms of applications, research of adversarial example attacks against cloud vision services can be grouped into three main categories: query-based attacks, transfer learning attacks and spatial transformation attacks.",
"Query-based attacks are typical black-box attacks, attackers do not have the prior knowledge and get inner information of DL models through hundreds of thousands of queries to successfully generate an adversarial example [10].In [11], thousands of queries are required for low-resolution images.",
"For high-resolution images, it still takes tens of thousands of times.",
"But attacking an image requires thousands of queries, which is not operable in actual attacks of real-world cloud-based image classification services.",
"In order to reduce the number of queries,[19] attack strategy consists in training a local model to substitute for the target DL models, using inputs synthetically generated by an adversary and labeled by the target DL models.",
"They have knowledge of the training data and test the attack with the same distributed data, and they upload the training data themselves and they know the distribution of training data [19][20][21] .",
"Transfer learning attacks are first examined by [1], which study the transferability between different models trained over the same dataset.",
"[12] propose novel ensemble-based approaches to generate adversarial example .",
"Their approaches enable a large portion of targeted adversarial example to transfer among multiple models for the first time.",
"Spatial transformation attacks are very interesting, [13] evaluate the robustness of Google Cloud Vision API to input perturbation, they show that adding an average of 14.25% impulse noise is enough to deceive the API and when a noise filter is applied on input images, the API generates mostly the same outputs for restored images as for original images.",
"[14] report the first systematic study on the real-world adversarial images and their use in online illicit promotions.",
"They categorize their techniques into 7 major categories, such as color manipulation, rotation, noising and blurring.",
"[15] make the first attempt to conduct an extensive empirical study of black-box attacks against real-world cloud-based image detectors such as violence, politician and pornography detection.",
"Our FFL-PGD attack based on Substitution model can be classified as a combination of query-based attack and transfer learning attack."
],
[
"CONCLUSION AND FUTURE WORK",
"In this paper, (1) We propose a novel attack method, Fast Featuremap Loss PGD (FFL-PGD) attack based on Substitution model, which achieves a high bypass rate with a very limited number of queries.",
"Instead of millions of queries in previous studies, our method finds the adversarial examples using only two queries per image; and (2) we make the first attempt to conduct an extensive empirical study of black-box attacks against real-world cloud-based classification services.",
"Through evaluations on four popular cloud platforms including Amazon, Google, Microsoft, Clarifai, we demonstrate that FFL-PGD attack has a success rate almost 90% among different classification services.",
"(3) We discuss the possible defenses to address these security challenges in cloud-based classification services.",
"Our defense technology is mainly divided into model training stage and image preprocessing stage.",
"In the future, we aim to explore the space of adversarial examples with less perturbation in black-box and attempt to study target attack using FFL-PGD attack.",
"On the other hand, we will focus on the defense in the cloud environment, so that AI services in the cloud environment away from cybercrime.",
"We hope cloud service providers will not continue to forget this battlefield."
]
] | 2001.03460 |
[
[
"Dispersive instabilities in Passively Mode-Locked Integrated\n External-Cavity Surface-Emitting Lasers"
],
[
"Abstract We analyze the dynamics of passively mode-locked integrated external-cavity surface-emitting Lasers (MIXSELs) using a first-principle dynamical model based upon delay algebraic equations.",
"We show that the third order dispersion stemming from the lasing micro-cavity induces a train of decaying satellites on the leading edge of the pulse.",
"Due to the nonlinear interaction with carriers, these satellites may get amplified thereby destabilizing the mode-locked states.",
"In the long cavity regime, the localized structures that exist below the lasing threshold are found to be deeply affected by this instability.",
"As it originates from a global bifurcation of the saddle-node infinite period type, we explain why the pulses exhibit behaviors characteristic of excitable systems.",
"Using the multiple time-scale and the functional mapping methods, we derive rigorously a master equation for MIXSELs in which third order dispersion is an essential ingredient.",
"We compare the bifurcation diagram of both models and assess their good agreement."
],
[
"Introduction",
"In spite of its early discovery in 1965 [1], the passive mode-locking (PML) of lasers is still a subject of intense research, not only due to its important technological applications [2], [3], but also because it involves the self-organization of a large number of laser modes that experience an out-of-equilibrium phase transition [4], [5].",
"Vertical external-cavity surface-emitting semiconductor lasers (VECSELs) are prominent laser sources in various industrial and scientific applications that require high output power and good beam quality [6], [7], [8], [9].",
"Here, the PML phenomenon is obtained by closing the external cavity with a semiconductor saturable absorber mirror (SESAM); the intensity-dependent losses promote pulsed over continuous wave emission and lead to pulses that are typically in the picosecond range, see [10], [11] for reviews.",
"Among the wealth of dynamical regimes encountered in such complex photonic systems involving coupled cavities, the regular pulsating regimes have direct applications, e.g., for compact spectroscopy [12] and metrology [13].",
"Designs based upon the VCSEL-SESAM geometry yield output powers that evolved from $200\\,$ mW [14], [15], [16] towards $20\\,$ W [17] and, recently, transform limited pulses in the 100 fs range with peak power of 500 W [18] were obtained with repetition rate in the GHz range.",
"A new type of nonlinear cavity appeared in the last decade, the so-called Mode-Locked Integrated External-Cavity Surface-Emitting Lasers (MIXSEL) [19], [20] in which both the gain and the saturable absorber share the same micro-cavity.",
"Coupling the MIXSELs to an external mirror provides for the essential optical feedback that defines the repetition rate of the pulse train.",
"More generally, the mode-locking of VECSELs is based upon micro-cavities operated in reflection: one or two for MIXSELs and VCSEL-SESAMs, respectively.",
"The rich PML dynamics and the multi-pulses regimes can be controlled e.g., with time-delayed optical feedback [23], [24], coherent optical injection [25] or photonic crystal structures [26].",
"In addition, carrier dynamics in multi-level active materials such as quantum dots [27], [28] or sub-monolayer quantum dots [29] leads to rich behaviors.",
"The cavity geometry also proved its relevance and peculiar pulse clusters appear in V-shaped cavities [30], [31].",
"While it was commonly accepted that semiconductor mode-locked lasers could not emit pulse trains at rates well below the GHz, a regime of temporal localization allowing arbitrary low repetition rates and individual pulse addressing was disclosed [32], [33], [34], [21], [35].",
"This transition from PML towards addressable temporal localized structures (TLSs) could have applications for dense frequency combs generation [36] and all optical data processing [37].",
"To add to its technological relevance, the long cavity regime, in which the photon round-trip is longer than the semiconductor gain recovery time, is compatible with spatial confinement; stable three-dimensional light bullets were predicted in broad area micro-cavities [38], [39], [40].",
"Recently, a new kind of instability for the pulse trains obtained in the long cavity regime was experimentally observed in mode-locked VECSELs [22].",
"At its core, these unstable pulsating regimes results from the face-to-face coupling of the micro-cavities containing the gain and the saturable absorber media.",
"Operated in reflection, the gain and absorber micro-cavities behave as dispersive Gires-Tournois interferometers (GTI) [41].",
"After several round-trips, the third order dispersion (TOD) induced by the micro-cavities give rise to serrated wave forms that consists in a decaying sequence of satellites accumulating in front of the leading edge of the pulse.",
"It was shown that these satellites may become unstable leading to a low frequency modulation of the pulse envelope on a slow time scale, typically on the order of hundreds of round-trips.",
"Understanding this regime is of paramount importance in order to avoid time and amplitude jitter and spectral broadening of the lines forming the frequency comb of the PML regimes.",
"In this manuscript, we predict that such dispersive instabilities may also occurs in MIXSELs.",
"While MIXSELs micro-cavities contain both the gain and the saturable absorption, they are also operated in reflection and it stands to reason that they shall exhibit the same dispersive behavior characteristics of GTIs.",
"Finally, due to its simpler geometry, the MIXSEL proves to be a particularly instructive minimal model for our analysis.",
"We present in Sec.",
"a first principle model for the MIXSEL.",
"Following the method of [42], it consists in solving analytically the field propagation in the linear sections of the micro-cavity, while considering the gain and absorber sections as nonlinear boundary conditions.",
"The boundary conditions for the field at the micro-cavity/air interface together with the optical feedback from the external mirror impose the structure of the model as time-delayed algebraic equations (DAEs).",
"We propose to conduct what we believe to be the first bifurcation analysis of such DAEs model in Photonics.",
"We demonstrate in Sec.",
", how a series of decaying satellites on the leading edge of the pulse causes an instability of the pulse train and how a global bifurcation with features of excitability as well as more intricate oscillating dynamics can appear in Sec.",
"and , respectively.",
"Using the rigorous formalism of the multiple time-scale [43] as well as the newly developed functional mapping method [40], we complement our analysis in Sec.",
"with the derivation of a master equation governing the slow pulse evolution over the round-trip time scale, similar to the so-called Haus master equation [10], and that takes the form of a generalized complex Ginzburg-Landau partial differential equation (PDE).",
"The latter allows to identify why TOD has such an important effect in the dynamics of MIXSELs.",
"Finally, using modern continuation techniques [44], we compare the results of the PDE with that of the DAE model and find in which conditions a good agreement can be obtained."
],
[
"Model system",
"The schematic setup of a MIXSEL system is depicted in Fig.",
"REF .",
"The gain and the absorber media are enclosed into micro-cavities whose length is of the order of the lasing wavelength.",
"The two mirrors of the micro-cavity provide additional degrees of freedom for controlling the light-matter interaction.",
"The interaction strength with the active medium —that is only a few tens of nanometers long— can be dramatically increased, as the expense of the available bandwidth, by using high-Q cavities.",
"Similarly, the effective saturation of the active material can be increased (or decreased) by using resonant or anti-resonant cavity designs, respectively.",
"Because of the vast scale separation between the external cavity length and that of the micro-cavity, the natural framework for our analysis is that of time-delayed systems.",
"The latter appear not only as natural modeling approaches for PML [42], [45] but in many branches of physics.",
"Delayed systems have strong links with spatially extended systems such as the Ginzburg-Landau equation [46], [47] and lead to rich dynamical behaviors [48], [49], [50], [51], [52], see [53] for a review.",
"We consider the case of a resonant cavity and we denote by $E$ the micro-cavity field over an antinode and $Y$ the field in the external cavity.",
"The output field is denoted by $O$ , $\\tau $ is the external cavity round trip time, whereas $r_{1,2}$ are the top and bottom Distributed Bragg Reflector (DBR) reflectivities, $t_{1}$ is the transmission coefficient of the top DBR and $r_{e}$ is the external mirror reflectivity.",
"We follow the approach of [42] that consists in solving the field propagation in the linear sections of the micro-cavity.",
"That way one obtains a dynamical model linking the two fields $E$ and $Y$ .",
"Their coupling is achieved considering the transmission and reflection coefficients of the top DBR.",
"After normalization, one obtains the rate equations for the field $E$ , the gain $N_{1}$ and absorber $N_{2}$ population inversions as $\\dot{E} & =\\left[\\left(1-i\\alpha _{1}\\right)N_{1}+\\left(1-i\\alpha _{2}\\right)N_{2}-1\\right]E+hY,\\\\\\dot{N}_{1} & =\\gamma _{1}\\left(J_{1}-N_{1}\\right)-\\left|E\\right|^{2}N_{1},\\\\\\dot{N}_{2} & =\\gamma _{2}\\left(J_{2}-N_{2}\\right)-s\\left|E\\right|^{2}N_{2},\\\\Y & =O\\left(t-\\tau \\right)=\\eta \\left[E\\left(t-\\tau \\right)-Y\\left(t-\\tau \\right)\\right].$ We scaled Eqs.",
"(REF -) by the photon lifetime in the micro-cavity $\\tau _p$ , and $\\alpha _{1}$ and $\\alpha _{2}$ are the linewidth enhancement factors of the gain and absorption, respectively.",
"We set the bias and the recovery time in the gain as $\\left(J_{1},\\gamma _{1}\\right)$ and in the absorber section as $\\left(J_{2},\\gamma _{2}\\right)$ , respectively.",
"The ratio of the gain and absorber saturation intensities is $s$ .",
"The cavity enhancement due to the high reflectivity mirrors can be scaled out, making that $E$ and $Y$ are of the same order of magnitude.",
"This scaling has the additional advantage of simplifying the input-output relation of the micro-cavity; using Stokes relations, we find that it reads $O=E-Y$ .",
"The minus sign represents the $\\pi $ phase shift of the incoming field $Y$ upon reflection from the top DBR.",
"After a round-trip in the external cavity of duration $\\tau $ , the output field $O\\left(t-\\tau \\right)$ is re-injected with an attenuation factor $\\eta =r_{e}\\exp \\left(\\omega _{0}\\tau \\right)$ , with $\\omega _{0}\\tau $ the propagation phase, defining $\\omega _{0}$ as the carrier frequency of the field.",
"The coupling between $E$ and $Y$ is given in Eq.",
"() by a delayed algebraic equation (DAE), that takes into account the multiple reflections in the external cavity.",
"In the limit of a very low external mirror reflectivity $\\eta \\ll 1$ , one would truncate the infinite hierarchy generated by Eq.",
"() to obtain $Y=\\eta E\\left(t-\\tau \\right)+\\mathcal {O}\\left(\\eta ^{2}\\right)$ leading to the so-called Lang-Kobayashi model [54].",
"Yet, for mode-locked configurations $\\eta =\\mathcal {O}\\left(1\\right)$ and the multiple reflections in the external cavity must be taken into account.",
"Instead of considering an infinite number of delayed terms in Eq.",
"(REF ) with values $\\tau ,\\,2\\tau ,\\,\\cdots ,\\,n\\tau $ , the DAE given by Eq.",
"() allows for an elegant representation of the strongly coupled cavity dynamics without needing an a priori truncation.",
"The coupling efficiency of the external field $Y$ into the micro-cavity is given by the parameter $h=\\left(1+\\left|r_{2}\\right|\\right)\\left(1-\\left|r_{1}\\right|\\right)/\\left(1-\\left|r_{1}r_{2}\\right|\\right)$ .",
"There exist three instructive limit cases for the coupling parameter $h$ that correspond to certain types of devices: A non-transmitting top DBR $\\left|r_{1}\\right|=1$ yields $h=0$ , making the cavity equivalent to a perfect (linear) mirror.",
"Equal reflectivities for both DBRs $\\left|r_{1}\\right|=\\left|r_{2}\\right|$ yield $h=1$ and correspond to a symmetric Fabry-Perot cavity.",
"Finally, a fully reflecting bottom DBR $\\left|r_{2}\\right|=1$ yields $h=2$ , which corresponds to the GTI case [41].",
"Gires-Tournois interferometers are known for inducing controllable second order dispersion and they are used as optical pulse shaping elements.",
"Resonant photons transmitted into the micro-cavity will remain on average for the photon lifetime.",
"When transmitted back into the external cavity they will have collected a phase difference with respect to the off-resonance photons that are directly reflected upon the top DBR.",
"Note that this phase shift is a function of the detuning of the photons with respect to the closest micro-cavity mode.",
"The recombination of various wavelength in the external cavity leads to dispersion.",
"This process is fully captured by Eq.",
"(REF -).",
"Second order dispersion is typically the dominating effect and its amount is tunable by choosing the detuning.",
"Using red or blue detuning one can achieve either normal or anomalous dispersion while around resonance TOD becomes the leading term as the second order contribution vanishes and switches sign.",
"Gires-Tournois interferometers are designed to conserve the photon number using high reflective bottom mirrors and therefore yield purely dispersive spectrum in models such as given by Eqs.",
"(REF -), see [22] for more details.",
"In order to achieve directional emission and low losses the bottom DBRs of VCSELSs are optimized towards $\\left|r_{2}\\right|\\rightarrow 1$ , i.e., they are well approximated by the GTI regime and $h\\rightarrow 2$ .",
"We set the photon lifetime as $\\tau _p=3\\,$ ps which corresponds to a Full Width at Half Maximum (FWHM) of $(\\pi \\tau _p)^{-1}=106\\,$ GHz.",
"The gain and absorber lifetimes are $1\\,$ ns and $30\\,$ ps, respectively, while we set the round-trip time in the cavity to $3\\,$ ns, hence $\\left(\\gamma _{1},\\gamma _{2},\\tau \\right)=\\left(0.003,0.1,1000\\right).$ If not stated otherwise, the other parameters are $(J_{2},\\eta ,s,h)=(-0.5,0.7,10,2)$ ."
],
[
"Satellite instability",
"Due to TOD from the GTI-like micro-cavity pulses can have a series of decaying satellites on the leading edge that was found to cause an instability of the pulse train [22].",
"Without linewidth enhancement factors $\\alpha _{1}=\\alpha _{2}=0$ the satellites are most clearly developed because of the absence of chirp and their instability can be better understood starting from this situation.",
"In a real semiconductor medium the change in carrier density along the pulse profile causes a varying detuning with respect to the micro-cavity resonance due to the alpha factors.",
"The resulting mixture between chirp and dispersion creates a more involved dynamics that will be discussed later in this manuscript.",
"Figure: (color online)Pseudo-space-time diagram (a) and time trace (b) for the pulse train in the unstable satellite regime obtained from DNSs of Eqs. (-).",
"The pulse intensity for EE (blue) and YY (orange) fields is shown.",
"The purple crosses at the intensity peaks illustrate the creation-annihilation cycle.",
"For sufficiently large gain the largest satellite is amplified, eventually replacing its parent pulse.",
"Parameters are (J 1 ,α 1 ,α 2 )=(0.65,0,0).",
"(J_1,\\,\\alpha _1,\\,\\alpha _2)=(0.65,\\,0,\\,0).We operated in the regime of localization where the pulses are temporal localized states that appear below the lasing threshold bias defined as $J_{1}^{\\mathrm {th}}$ .",
"As detailed in [32], the TLSs appear via a saddle-node bifurcation of limit cycles.",
"Sufficiently close to the lasing threshold, the main pulses and therefore their parasitic satellites become large enough to bleach the absorber and open the net gain window prematurely.",
"As a consequence, they grow exponentially from one round-trip towards the next while the parent pulse meets an increasingly depleted gain carrier density and eventually dies out.",
"It is replaced by its satellite in front, resulting in forward leaping motion that can best be seen in a pseudo-space-time representation which is shown in Fig.",
"REF (a) where $N_{rt}$ is the number of round-trips.",
"The corresponding temporal trace obtained from direct numerical simulations (DNSs) of Eqs.",
"(REF -) is depicted in Fig.",
"REF (b).",
"It demonstrates how this cycle of creation and annihilation leads to a low frequency modulation of the pulse train (see the purple crosses).",
"Figure: (color online)(a) Bifurcation diagram in the (J 1 ,α 1 )(J_1,\\alpha _1)-plane obtained with DNSs of Eqs.",
"(-) for α 2 =0\\alpha _2=0.",
"The off, stable pulsating, and quasi-periodic pulsating regimes are shown in dark blue, light blue and green, respectively.",
"Bistable regions are orange and yellow for stable and oscillating solutions.",
"Pure satellite instability is observed for low α 1 \\alpha _1 whereas trailing edge AH instability for high α 1 \\alpha _1 values.",
"Both can mix forming the region around α 1 ≈1\\alpha _1\\approx 1 close to threshold that is bistable with stable pulses.",
"(b) Branch of a single pulse solution showing max (I)\\mathrm {max(}I) as a function of the scaled gain bias J 1 J_1, superposing results from DNS (blue points) and path-continuation for α 1 =0\\alpha _1=0.",
"Orange solid line shows the stable part of the branch which becomes unstable (green dashed line) in a fold of the limit cycle (red circles), i.e., the satellite instability does not a stem from secondary AH bifurcation.",
"After the fold on the unstable branch there is a supercritical pitchfork bifurcation (yellow star) that gives birth to another unstable branch (dashed magenta).A full bifurcation analysis of the DAE system (REF -) is out of the scope of the current manuscript.",
"The main reason being that the stability analysis of periodic solutions of DAEs is not possible at the moment, even using the most advanced continuation softwares such as DDE-BIFTOOL [55].",
"Yet, we show in Fig.",
"REF (a) a numerical two parameter bifurcation diagram in the $\\left(J_{1},\\alpha _{1}\\right)$ plane for the single pulse solution.",
"The stable region and off solution are depicted in light and dark blue, respectively, in Fig.",
"REF (a).",
"For high values of the linewidth enhancement factor of the gain $\\alpha _{1}$ , a quasi-periodic instability due to self-phase modulation is found, similar to that discussed in [56].",
"This instability was found to be a local secondary Andronov-Hopf (AH) bifurcation.",
"In addition to these regimes, that exists in the generic mode-locked ring laser model of [45], a new instability induced by TOD is found for low values of $\\alpha _{1}$ .",
"It corresponds to the regime depicted in Fig.",
"REF .",
"Both unstable regions are depicted in green.",
"In addition, around $\\alpha _{1}\\approx 1$ , there exists a bistable region close to threshold between a stable PML regime with higher intensity pulses and low frequency modulated PML; this latter quasi-periodic dynamics has the characteristics of both instabilities found for low and large $\\alpha _{1}$ ; it is discussed in more detail in Section .",
"The bistable region is depicted in orange where stable and modulated pulses are stable and yellow where only one solution is stable.",
"Note that parts along the borders of the pure instabilities are also bistable with a regular PML solution with smaller pulses.",
"We show in Fig.",
"REF (b) the branch for the single TLS solution obtained using DDE-BIFTOOL.",
"At $\\alpha _{1}=0$ , the solution branch folds three times with the second fold at a critical value of the current $J_{1}^{c}\\sim 0.86J_{1}^{\\mathrm {th}}$ that coincides with the onset of the satellite instability.",
"The results from the corresponding DNSs are superposed (blue points), indicating the all the values of the pulse maxima integrated over many round-trips.",
"Shortly after the second fold, a second unstable branch (dashed magenta) appears in a pitchfork bifurcation.",
"Above $J_{1}^{c}$ , no stable single pulse solution branch exists and one observes the low frequency periodic dynamics discussed in Fig.",
"REF .",
"Figure: (color online)The dependence of the period length on the distance to the bifurcation point J 1 c J_1^c.",
"The crosses denote the period obtained from DNSs of Eqs.",
"(-) for α 1 =0.01\\alpha _1=0.01 (blue) and α 1 =0.02\\alpha _1=0.02 (orange).",
"The straight lines correspond to the theoretically predicted scaling behavior for a homoclinic (green) as well as for a SNIPER (purple) bifurcation.It has to be noted that for $\\alpha _{1}\\ll 1$ this instability does not stem from a local Andronov-Hopf bifurcation but from a global bifurcation.",
"The limit cycle is born with infinite period at the second fold of the TLS branch in Fig.",
"REF (b).",
"The period scaling is presented in Fig.",
"REF where we show the period evolution for two exemplary values of $\\alpha _{1}$ (blue and orange crosses) and the characteristic scalings for saddle-node infinite period ($\\mu ^{-1/2}$ ) and homoclinic ($\\ln \\mu ^{-1}$ ) bifurcations as a function of the distance to the bifurcation point $J_{1}^{c}$ .",
"The results reveal that for small values of $\\alpha _{1}$ the satellite instability can be identified to be of the saddle-node infinite period (SNIPER) type.",
"However, for increasing $\\alpha _{1}$ the SNIPER bifurcation changes into a local AH bifurcation (cf.",
"the scaling for $\\alpha _{1}=0.02$ )."
],
[
"Excitability",
"Under the influence of noise the satellite instability exhibits dynamical behavior characteristics of excitable systems.",
"To trigger the replacement of the parent pulse a satellite must have sufficient intensity to bleach the absorber and open an early net-gain window.",
"Close, yet below, to the critical energy, noise can help or hinder the satellite emergence by adding or subtracting some energy, respectively.",
"As a result, the period of the pulse eruptions can become highly irregular and we present in Fig.",
"REF an exemplary pseudo-space-time diagram for a single pulse (a) and the corresponding time trace (b) in the excitable regime.",
"By all accounts, this is certainly in its irregular form that such a regime is most likely going to appear in an experimental situation, where noise is unavoidable.",
"Figure: (color online)The dependence of the mean value (a) and standard deviation (b) of the period of the satellite instability as a function of the noise parameter σ noise \\sigma _\\mathrm {noise} and the distance of J 1 J_1 to the bifurcation point J 1 c J_1^{c}.",
"Parameters are (α 1 ,α 2 )=(0,0)(\\alpha _1,\\alpha _2)=(0,0).In Fig.",
"REF we show the statistical properties of the satellite instability under the influence of gaussian noise with a standard deviation of $\\sigma _{\\mathrm {noise}}$ for varying distances of the gain bias $J_{1}$ from the critical value $J_{1}^{c}$ corresponding to the onset of the SNIPER bifurcation.",
"Panel (a) shows the mean value and panel (b) the standard deviation of the period of the satellite eruptions.",
"At each point of the diagrams hundred of consecutive eruptions were analyzed.",
"To make such calculations feasible, the simulations were performed in the long delay limit using the functional mapping approach [40].",
"One can see how noise enables eruptions below the critical pump rate.",
"The stronger the noise the less gain bias is necessary for this.",
"The cutoff visible in the data is arbitrary, as the necessary integration times grows very fast and becomes a limiting factor to the feasibility of DNSs.",
"Statistically, eruptions can happen anywhere, albeit very rarely depending on the parameters.",
"For increasingly strong noise the observed average period decreases and depends little on the actual gain bias.",
"Similarly, for strong pumping the noise hardly changes the mean period.",
"The standard deviation of the eruptions is very large at the onset below critical pumping due to noise.",
"Here the eruptions are rare and the excitability is most visible.",
"Otherwise the variance decreases with both increasing gain bias and decreasing noise level."
],
[
"Combined instabilities",
"With realistic linewidth enhancement factors for semiconductor media the satellite and self-phase modulation instabilities combine to form a dynamics of the kind presented in [22].",
"By performing parameter scans as a function of various parameters we were able to deduce some rules of thumb for this satellite instability.",
"Generally, the pulse-width is proportional to the photon lifetime and can be written as $\\tau _{p}=\\kappa ^{-1}f\\left(\\cdots \\right)$ with $f\\left(\\cdots \\right)$ a function that depends on all the other parameters of the PML setup.",
"Optimizing PML consists in finding the parameter combination for which the function $f$ is the smallest.",
"Since the temporal separation between the main pulse and its first sattelite is fixed by the cavity photon lifetime, it is in these optimal cases where, as depicted in Fig.",
"REF , that the satellites become better resolved from the main pulse and where they are prone to become unstable.",
"Hence, the satellite instability can be obtained by tuning any parameter leading to the optimal PML and shortest pulse-width, making this instability an essential limitation in the optimization procedure to find the narrowest pulses.",
"For instance, increasing the saturation $s$ or even the linewidth enhancement factor of the absorber $\\alpha _{2}$ , if it compensates for the chirp induced in the gain section, can adversely narrow the pulse which destabilizes the PML regime.",
"In \"bad conditions\" the pulse and its sattelites are smeared out into a globally broader pulse that is stable.",
"Note that in the VCSEL-SESAM setup studied in [22] the detuning between the two micro-cavities is an additional factor leading to further complexity since second order dispersion also plays a role.",
"Finally, we show in Fig.",
"REF how the satellite instability evolves for more realistic situations settubg $\\alpha _{1}=2.1$ and $\\alpha _{2}=0.5$ .",
"We note that, in addition, this regime can be bistable with the stable pulsating solution, as shown in the orange region in Fig.",
"REF (a)."
],
[
"The Dispersive Master Equation",
"Bifurcation analysis does not always allows for an intuitive interpretation of the dynamics.",
"In addition, even if the time delayed equations Eqs.",
"(REF -) contain a lot of the physics of PML, their form remains complicated.",
"For instance, their dispersive features do not appear so obvious.",
"We derive in this section a PDE for the field amplitude $E$ that approximates the dynamics of the full DADE model (REF -).",
"In the PDE representation, the field depends on a slow and a fast time, i.e.",
"$E\\equiv E\\left(\\xi ,z\\right)$ .",
"Here, the slow time $\\left(\\xi \\right)$ represents the evolution of the field profile from one round-trip to the next while the “spatial variable” $\\left(z\\right)$ describes the fast evolution of the pulse within the round-trip.",
"While this approach has been used in several time-delayed systems, see e.g., [53], [52] for a review, such PDE models are usually termed Haus master equations in the framework of mode-locking, see [10] for a review.",
"In the case of high frequency PML dynamics, i.e.",
"the regimes in which the cavity round-trip is much shorter than the gain recovery time, a PDE approximating the dynamics of a ring cavity model based upon delay differential equations [45] was proposed in [57].",
"The multiple time-scales analysis method was used and the scaling of parameters consisted in assuming low losses, low gain, and weak spectral filtering.",
"In the model of [45], these three physical effects are controlled by three independent parameters.",
"In the multiple time-scales approach one finds, at the lowest order, a periodic solution, e.g., a pulse evolving over the fast time scale $\\left(z\\right)$ that circulates in the cavity without deformation.",
"At third order in the expansion scheme, a solvability condition allows finding that the dynamics on the slow time scale $\\left(\\xi \\right)$ is governed by the weak effects of gain, loss and spectral filtering.",
"In a PDE representation, the gain filtering in the model of [45] takes the form of a diffusion over the fast time, i.e.",
"a term $d_{2}\\partial _{z}^{2}E$ with $d_{2}>0$ .",
"While the smallness of gain, losses and filtering can be, in some situations, debatable, the advantage of the approach presented in [57] is the uniform accuracy of the PDE representation that was not, e.g., limited to the vicinity of the lasing threshold.",
"For instance, no a priori conditions over the magnitude of the field were necessary.",
"It is interesting —and surprising— to notice that the aforementioned approach fails if one tries to export it to the case of the DADE model (REF -); the resulting PDE obtained similarly as a third order solvability condition does possess gain and losses, yet it is devoid of spectral filtering $\\left(d_{2}=0\\right)$ which leads to singular dynamics and unphysical pulse collapse.",
"The physical reason that underlies this mathematical phenomenon is that the filtering of the micro-cavity is not in our modeling approach independent from gain and losses.",
"It is the actual level of the population inversion in the micro-cavity that defines the breadth and hence the curvature of the resonance.",
"This effect is particularly pronounced in the Gires-Tournois regime (i.e., $h=2$ ), where the empty cavity reflectivity is unity, which corresponds to no curvature at all and $d_{2}=0$ .",
"That is, positive (resp.",
"negative) curvature induces diffusion (resp.",
"anti-diffusion).",
"Generally, the average gain experienced by the pulse must be positive to compensate for the cavity losses incurred by the mirror reflectivity $\\eta $ .",
"A proper analysis of the model given by Eqs.",
"(REF -) shall result in a PDE whose diffusion term $d_{2}$ depends on other parameters such as the cavity losses.",
"This discussion materializes by taking as the expansion point the cavity at the lasing threshold instead of the empty cavity, as done in [57].",
"At the lasing threshold, the unsaturated gain and losses exactly compensate.",
"This modification will allow to obtain the filtering induced by the cavity at threshold, instead of that of the empty cavity leading to $d_{2}=0$ .",
"The drawback of our approach is that we have to assume the pulse to be not too intense and treat the nonlinear effects pertubatively, limiting our analysis to the vicinity of the lasing threshold.",
"We start by normalizing time by the cavity round-trip $\\tau $ time as $\\sigma =t/\\tau $ and define a smallness parameter $\\varepsilon =1/\\tau $ .",
"As we operate in the long cavity limit, the carriers are not independent functions of time that can lead to resonant terms and solvability conditions.",
"Instead, the carrier evolutions depend uniquely on the initial conditions at the beginning of the round-trip, which in the long cavity limit is the equilibrium value, and on the amplitude of the field, i.e., $N_{j}=N_{j}\\left(J_{j},E\\right)$ .",
"Hence, we can concentrate solely on the field dynamics that reads $\\varepsilon \\frac{dE}{d\\sigma } & = & \\left[\\left(1-i\\alpha _{1}\\right)N_{1}+\\left(1-i\\alpha _{2}\\right)N_{2}-1\\right]E+hY,\\\\Y\\left(\\sigma \\right) & = & \\eta \\left[E\\left(\\sigma -1\\right)-Y\\left(\\sigma -1\\right)\\right]\\,.$ We assume a small deviation of the gain and absorber with respect to their equilibrium values that we scale as $N_{j} & = & J_{j}+\\varepsilon ^{3}n_{j}$ with $j\\in \\left[1,2\\right]$ .",
"We also assume $\\eta $ to be real as the feedback case is irrelevant in the long cavity regime.",
"Defining the Fourier transform of the field profiles at the $n$ -th round-trip as $\\left(E_{n},Y_{n}\\right)$ and using that $\\dfrac{d}{d\\sigma }\\rightarrow -i\\omega $ we obtain $\\left(1-G_{t}-i\\varepsilon \\omega \\right)E_{n}=hY_{n}+\\varepsilon ^{3}\\sum _{j=1}^{2}\\left(1-i\\alpha _{j}\\right)\\left(n_{j}E\\right)_{n},$ where we used the shorthand for the total complex gain $G_{t}=\\left(1-i\\alpha _{1}\\right)J_{1}+\\left(1-i\\alpha _{2}\\right)J_{2}.$ Noticing that the DAE for $Y\\left(\\sigma \\right)$ in Fourier space reads $Y_{n}+\\eta Y_{n-1}=\\eta E_{n-1}$ and, by making a linear combination of $E_{n}$ and $\\eta E_{n-1}$ , we get after simplification a functional mapping $E_{n} & = & \\eta \\frac{h-1+G_{t}+i\\varepsilon \\omega }{1-G_{t}-i\\varepsilon \\omega }E_{n-1}\\\\& + & \\varepsilon ^{3}\\frac{1}{1-G_{t}-i\\varepsilon \\omega }\\sum _{j=1}^{2}\\left(1-i\\alpha _{j}\\right)\\left[\\left(n_{j}E\\right)_{n}+\\eta \\left(n_{j}E\\right)_{n-1}\\right].\\nonumber $ Now we impose the value of $G_{t}$ to be a convenient expansion point and we set the threshold condition over the linear multiplier $\\mu $ $\\mu =\\eta \\frac{h-1+G_{t}+i\\varepsilon \\omega }{1-G_{t}-i\\varepsilon \\omega } & = & 1,$ which allows finding the lasing frequency shift at threshold as $\\varepsilon \\omega _{t}=\\alpha _{1}J_{1}+\\alpha _{2}J_{2}$ .",
"This leaves us with a real equation for the amplification factor $\\eta \\frac{h-1+N_{t}}{1-N_{t}} & = & 1\\,,$ where we defined $N_{t}=J_{1}+J_{2}$ .",
"The last relation implies that the threshold is defined by $N_{t}=J_{1}+J_{2} & = & 1-\\frac{h\\eta }{1+\\eta }\\,.$ We can now express the field multiplier $\\mu $ from one round-trip towards the next as $\\mu \\left(\\omega ,N_{t}\\right) & = & \\eta \\frac{h-1+N_{t}+i\\varepsilon \\left(\\omega -\\omega _{t}\\right)}{1-N_{t}-i\\varepsilon \\left(\\omega -\\omega _{t}\\right)}$ and the functional mapping given by Eq.",
"REF reads $E_{n} & = & \\eta \\frac{h-1+N_{t}+i\\varepsilon \\left(\\omega -\\omega _{t}\\right)}{1-N_{t}-i\\varepsilon \\left(\\omega -\\omega _{t}\\right)}E_{n-1}+\\varepsilon ^{3}\\frac{1}{1-N_{t}-i\\varepsilon \\left(\\omega -\\omega _{t}\\right)}\\sum _{j=1}^{2}\\left(1-i\\alpha _{j}\\right)\\left[\\left(n_{j}E\\right)_{n}+\\eta \\left(n_{j}E\\right)_{n-1}\\right]\\,.$ One only needs Eq.",
"REF to be accurate up to third order in $\\varepsilon $ in order to obtain the proper expression of the diffusion, TOD and nonlinear terms.",
"As such, we can simplify the last term of Eq.",
"REF by replacing the value of $E_{n}$ at the lowest order, i.e., we can set $E_{n}=\\eta \\frac{h-1+N_{t}}{1-N_{t}}E_{n-1}+\\mathcal {O}\\left(\\varepsilon \\right)=E_{n-1}+\\mathcal {O}\\left(\\varepsilon \\right),$ where we used the threshold definition given by Eq.",
"REF .",
"Using that $E_{n}=E_{n-1}$ in the nonlinear term of REF , replacing the expression of the threshold and noticing that all the frequencies are relative to that of the lasing threshold, so that one can set $\\tilde{\\omega }=\\omega -\\omega _{t}$ , yields the expression $E_{n} & = & \\eta \\frac{h-1+N_{t}+i\\varepsilon \\tilde{\\omega }}{1-N_{t}-i\\varepsilon \\tilde{\\omega }}E_{n-1}\\\\& + & \\varepsilon ^{3}\\frac{\\left(1+\\eta \\right)^{2}}{h\\eta }\\sum _{j=1}^{2}\\left(1-i\\alpha _{j}\\right)\\left(n_{j}E\\right)_{n-1}+\\mathcal {O}\\left(\\varepsilon ^{4}\\right)\\,.\\nonumber $ Now let us assume there exists a PDE for the field $E\\left(\\xi ,z\\right)$ with $\\xi $ and $z$ the slow and fast times, respectively.",
"In Fourier space for the variable $z$ , one obtains $\\partial _{\\xi }E & = & \\mathcal {L}\\left(\\tilde{\\omega }\\right)E+\\varepsilon ^{3}\\mathcal {N}\\left(\\xi ,\\tilde{\\omega }\\right)\\,.$ The form of Eq.",
"REF consists naturally of a linear operator $\\mathcal {L}\\left(\\tilde{\\omega }\\right)$ that should correspond to the linear multiplier of the mapping in Eq.",
"REF , while $\\mathcal {N}\\left(\\sigma ,\\tilde{\\omega }\\right)$ accounts for nonlinear gain and absorber effects.",
"We assume that $\\mathcal {L}\\left(\\tilde{\\omega }\\right)$ is small, i.e, $\\mathcal {L}\\left(\\omega \\right)\\sim 0+\\mathcal {O}\\left(\\varepsilon \\right)$ .",
"Such a scaling is consistent with the definition of the lasing threshold and will be checked a posteriori.",
"Integrating exactly Eq.",
"REF over a round-trip yields $E_{n} & = & e^{\\mathcal {L}}E_{n-1}+\\varepsilon ^{3}\\int _{n-1}^{n}e^{\\left(n-\\xi \\right)\\mathcal {L}}\\mathcal {N}\\left(\\xi ,\\tilde{\\omega }\\right)d\\xi \\,.$ Because the integral term in Eq.",
"REF is already at third order in $\\varepsilon $ , we can approximate $e^{\\left(n-\\xi \\right)\\mathcal {L}}=1+\\mathcal {O}\\left(\\varepsilon \\right)$ and evaluate the nonlinear operator using the Euler explicit method.",
"Indeed, since $\\mathcal {N}$ depends on the field, the error in the integration will be proportional to the slow evolution of the field from one round-trip towards the next, i.e., $\\partial _{\\xi }\\mathcal {N}\\sim \\partial _{\\xi }E\\sim \\mathcal {O}\\left(\\varepsilon \\right)$ .",
"That is, we find $E_{n} & = & e^{\\mathcal {L}}E_{n-1}+\\varepsilon ^{3}\\mathcal {N}\\left(n-1,\\tilde{\\omega }\\right)+\\mathcal {O}\\left(\\varepsilon ^{4}\\right).$ Comparing Eq.",
"REF and Eq.",
"REF we deduce that $\\mathcal {L} & = & \\ln \\left(\\eta \\frac{h-1+N_{t}+i\\varepsilon \\tilde{\\omega }}{1-N_{t}-i\\varepsilon \\tilde{\\omega }}\\right)\\,.$ Using Eq.",
"REF the last expression can be simplifies as $\\mathcal {L} & = & \\ln \\left(\\frac{1+i\\varepsilon \\eta \\frac{1+\\eta }{h\\eta }\\tilde{\\omega }}{1-i\\varepsilon \\frac{1+\\eta }{h\\eta }\\tilde{\\omega }}\\right)\\,.$ We can verify easily that $\\mathcal {L}=\\mathcal {O}\\left(\\tilde{\\varepsilon }\\right)$ which allows to check a posteriori our approximation regarding the order of the operator $\\mathcal {L}$ .",
"One can also expand Eq.",
"REF in $\\tilde{\\omega }$ up to third order will yields the drift, diffusion and TOD coefficients $d_{1},d_{2}$ and $d_{3}$ as $\\mathcal {L}=d_{1}\\left(-i\\varepsilon \\tilde{\\omega }\\right)+d_{2}\\left(-\\varepsilon ^{2}\\tilde{\\omega }^{2}\\right)+d_{3}\\left(i\\varepsilon ^{3}\\tilde{\\omega }^{3}\\right)+\\mathcal {O}\\left(\\varepsilon ^{4}\\right)$ with $d_{1} & = & -\\frac{\\left(\\eta +1\\right)^{2}}{h\\eta }\\,,\\\\d_{2} & = & \\frac{1-\\eta ^{2}}{2}\\left(\\frac{\\eta +1}{h\\eta }\\right)^{2}\\,,\\\\d_{3} & = & -\\frac{\\eta ^{3}+1}{3}\\left(\\frac{\\eta +1}{h\\eta }\\right)^{3}\\,.$ The values of the coefficient $d_{j}$ is particularly instructive and, in particular, how they deviate from the expression one can find easily in the case of an empty Gires-Tournois micro-cavity coupled to an external mirror: $(d_{1},d_{2},d_{3})=(-2,0,-2/3)$ .",
"Here, we notice that although $d_{1}$ and $d_{3}$ are somehow modified by the value of the cavity losses, the value of $d_{2}$ is not vanishing.",
"In the good cavity limit $\\eta \\rightarrow 1$ , we have $d_{2}\\sim 1-\\eta \\rightarrow 0$ , which explains why TOD is important as it becomes the leading order term.",
"Further, we identify the nonlinear operator $\\mathcal {N}$ as $\\mathcal {N}\\left(n-1,\\tilde{\\omega }\\right) & = & \\frac{\\left(1+\\eta \\right)^{2}}{h\\eta }\\sum _{j=1}^{2}\\left(1-i\\alpha _{j}\\right)\\left(n_{j}E\\right)_{n-1}$ Finally, reverting Eq.",
"REF to direct space using that $-i\\varepsilon \\tilde{\\omega }\\rightarrow \\partial _{z}$ , the sought PDE for the field $E$ reads $\\partial _{\\xi }E & = & \\left(d_{1}\\partial _{z}+d_{2}\\partial _{z}^{2}+d_{3}\\partial _{z}^{3}\\right)E\\\\& + & \\frac{\\left(1+\\eta \\right)^{2}}{h\\eta }\\sum _{j=1}^{2}\\left(1-i\\alpha _{j}\\right)\\left(N_{j}-J_{j}\\right)E\\,.\\nonumber $ By using the definition of the lasing threshold and of the carrier frequency, we find that the dispersive master equation for the field $E$ in the long cavity limit reads $\\partial _{\\xi }E & = & \\left(d_{1}\\partial _{z}+d_{2}\\partial _{z}^{2}+d_{3}\\partial _{z}^{3}\\right)E+\\frac{\\left(1+\\eta \\right)^{2}}{h\\eta }\\left\\lbrace \\left(1-i\\alpha _{1}\\right)N_{1}+\\left(1-i\\alpha _{2}\\right)N_{2}-1+\\frac{h\\eta }{1+\\eta }-i\\omega _{t}\\right\\rbrace E\\,,$ whereas the equations for the carriers take the form $\\frac{\\partial N_{1}}{\\partial z} & = & \\gamma _{1}(J_{1}-N_{1})-|E|^{2}N_{1}\\,,\\\\\\frac{\\partial N_{2}}{\\partial z} & = & \\gamma _{2}(J_{2}-N_{2})-s|E|^{2}N_{2}\\,.$ Note that the rotation term $i\\omega _{t}$ in Eq.",
"(REF ) is immaterial and can be removed by setting $\\tilde{E}\\left(\\xi ,z\\right)=E\\left(\\xi ,z\\right)\\exp \\left(i\\omega _{t}\\xi \\right)$ .",
"Figure: (color online)Space-time diagram of the satellite instability found in the DNS of the master PDE (-).",
"The pulse intensity I=|E| 2 I=|E|^2 is shown.",
"Parameters are (J 1 ,J 2 ,α 1 ,α 2 ,η,s)=(0.119,-0.1,0,0,0.9,15)(J_1,\\,J_2,\\,\\alpha _1,\\,\\alpha _2,\\,\\eta ,\\,s)=(0.119,\\,-0.1,\\,0,\\,0,\\,0.9,\\,15).First we studied the PDE model (REF )-() numerically to demonstrate the existence of the satellite instability in the parameter space and the resulting space-time diagram obtained for zero linewidth enhancement factors $\\alpha _{1}=\\alpha _{2}=0$ is shown in Fig.",
"REF .",
"Again, we observe the clean cut satellite instability in this parameter range, but also that our PDE approximates very well the dynamics of the underlying time delayed model.",
"To compare the PDE (REF )-() with the DAE (REF -) in detail we performed bifurcation analysis of the PDE by using pseudo-arclength continuation methods within the pde2path framework [44].",
"To this aim, first we seek for the steady localized pulse solutions of Eqs.",
"(REF )-() that can be found by setting $E(z,\\xi )=A(z-\\upsilon \\xi )e^{-i\\varpi \\xi }$ leading to the following equation for the stationary field $A$ $0 & =\\upsilon \\,\\frac{\\partial A}{\\partial z}+d_{2}\\frac{\\partial ^{2}A}{\\partial z^{2}}+d_{3}\\frac{\\partial ^{3}A}{\\partial z^{3}}+i\\varpi \\,A\\\\& \\quad +\\frac{(1+\\eta )^{2}}{h\\eta }\\left((1-i\\alpha )N_{1}+(1-i\\beta )N_{2}-1+\\frac{h\\eta }{1+\\eta }\\right)A\\,.\\nonumber $ Note that both the spectral parameter $\\varpi $ and the drift velocity $\\upsilon $ become free parameters that can be found by imposing additional auxiliary integral conditions.",
"In addition we set the following boundary conditions for the domain $z\\in [0,L]$ $\\frac{\\partial A}{dz}\\Big |_{z=0,L} & = & 0\\,,\\\\N_{1}\\big |_{z=0} & = & J_{1}\\,,\\quad -\\frac{\\partial N_{1}}{\\partial z}\\Big |_{z=L}+\\gamma _{1}\\left(J_{1}-N_{1}\\big |_{z=L}\\right)=0\\,,\\\\N_{2}\\big |_{z=0} & = & J_{2}\\,,\\quad -\\frac{\\partial N_{1}}{\\partial z}\\Big |_{z=L}+\\gamma _{2}\\left(J_{2}-N_{2}\\big |_{z=L}\\right)=0\\,.$ Now we can follow the TLS of the PDE (REF )-() in parameter space and in Fig.",
"REF we present two branches of TLSs for different values of the linewidth enhancement factors.",
"Panel (a) shows the intensity of the TLS as a function of the normalized pump rate for the case of $\\alpha _{1}=\\alpha _{2}=0$ .",
"Like in the DAE case (cf.",
"Fig.REF ), one can see that the branch folds three times (points $\\mathrm {F}_{i}$ ) when continuing in the pump rate $J_{1}$ and the second fold $\\mathrm {F}_{2}$ is responsible for the SNIPER bifurcation after the first leading satellite becoming sufficiently large to saturate the absorber (cf.",
"Fig.",
"REF ).",
"Note that an additional unstable branch connects to the main one in a branching point $\\mathrm {{BP}}$ .",
"Panel (b) shows the same gain interval for non-vanishing linewidth enhancement factors $\\alpha _{1}=1.5$ and $\\alpha _{2}=0.5$ .",
"Here, the branch continues without the additional folds $\\mathrm {F}_{2}$ and $\\mathrm {F}_{3}$ and only becomes Andronov-Hopf unstable at large gain value.",
"Indeed, the second fold and the branching point have merged while the part of the branch with the third fold has detached, giving a qualitatively different scenario.",
"The chirp induced by the linewidth enhancement factors smears out the TOD effect responsible for the satellites.",
"Figure: (color online)Branches of the TLSs in the master PDE ()-().",
"The maximum of the field intensity I=|E| 2 I=|E|^2 as a function of the normalized pump current is shown.",
"(a) For (α 1 ,α 2 )=(0,0)(\\alpha _1,\\,\\alpha _2)=(0,0) the pulse profiles show defined satellites.",
"The branch has three folds F i \\mathrm {F}_i and a branching point BP \\mathrm {{BP}}.",
"The second fold F 2 \\mathrm {F}_2 coincides with an infinite period limit cycle in a global SNIPER bifurcation.",
"An unstable branch emerges from BP \\mathrm {{BP}} in a pitchfork bifurcation.",
"(b) For (α 1 ,α 2 )=(1.5,0.5)(\\alpha _1,\\,\\alpha _2)=(1.5,0.5), F 2 \\mathrm {F}_2 and BP \\mathrm {{BP}} have merged into an AH bifurcation point H\\mathrm {H}.",
"The profiles only show intensity bumps on the leading edge of the pulse and there is no longer a SNIPER.",
"Other parameters are (J 2 ,η,s)=(-0.1,0.9,15)(J_2,\\,\\eta ,\\,s)=(-0.1,\\,0.9,\\,15).Finally, in Fig.",
"REF we superpose the results on top of data obtained through DNSs of the DAE model (REF -) in the long delay limit using the functional mapping approach [40].",
"Pulses are fully localized TLSs in this regime.",
"In panel (a) the standard deviation of the pulse energy is shown as a function of the gain bias $J_{1}$ normalized to threshold $J_{1}^{th}$ and the gain linewidth enhancement factor $\\alpha _{1}$ along with the bifurcation curves from continuation.",
"For small $\\alpha _{1}$ one can see the satellite unstable region which is similar to the previous parameter set, i.e., close to $\\alpha _{1}=0$ there is an additional fold presented in solid blue and the satellite instability sets in as a global SNIPER bifurcation after it.",
"Panel (b) shows a zoom-in on this area where the fold merges with a branching point (dotted blue) thereby forming an AH bifurcation depicted in dash dotted red.",
"Both models quantitatively agree in this area.",
"The principal fold of the subcritical TLS branch in solid red is also reproduced correctly.",
"For higher $\\alpha _{1}$ the stable pulse region is limited by another AH bifurcation corresponding to self-phase modulation, shown as well in dash dotted red.",
"In contrast to the DAE, for the PDE model the bifurcation curve slopes down in $\\alpha _{1}$ for increasing gain.",
"Both the nature of the instability and the discrepancy found in the equivalent PDE are somewhat similar to the bifurcation structure in the Vladimirov-Turaev model for passive mode-locking in a unidirectional ring laser [45], [56].",
"For the parameters of Fig REF (a) the DAE system exhibits a region close to threshold at high values of $\\alpha _{1}$ that is partly stable on the high $\\alpha _{1}$ edge.",
"The corresponding fold and AH curves shown in dotted orange or indicated by red crosses, respectively, are found in the PDE with a qualitatively similar shape but the position of this area is shifted significantly towards lower gain as compared to the DAE.",
"This region corresponds to the bistable region for the previous parameter set.",
"Indeed we found that generally it moves and changes shape significantly as a function of the other parameters and so do the principal pulse and satellite instability regions.",
"For example increasing $\\alpha _{2}$ makes the stable pulses follow by moving up in $\\alpha _{1}$ by roughly the same amount, while the satellite instability moves down.",
"The second region moves up in $\\alpha _{1}$ much quicker and completely detaches, at least when constricting one's view at the area below threshold (see Fig.",
"REF ).",
"Figure: (color online)(a) Bifurcation diagram in the (J 1 ,α 1 )(J_1, \\alpha _1) plane of the DAE model (-) in the long delay limit superposed with the bifurcation diagram of the equivalent PDE ()-().",
"The color coding shows the standard deviation of the pulse energy obtained by DNSs of the DAE.",
"The evolution of the fold F 1 \\mathrm {F}_1 is marked by a solid red line, the satellite instability around α 1 =0\\alpha _1=0 with the corresponding fold F 3 \\mathrm {F}_3 (branching point) is in solid (dotted) blue and the AH part in dash dotted red.",
"The other AH bifurcation corresponding to self-phase modulation slopes down for the PDE case in contrast to the DAE.",
"The high α 1 \\alpha _1 region in the PDE is significantly shifted with respect to the DAE.",
"Its fold branch is depicted in dotted orange and AH bifurcations are indicated by red crosses.",
"Parameters are (J 2 ,α 2 ,η,s)=(-0.1,0,0.9,15)(J_2,\\,\\alpha _2,\\,\\eta ,\\,s)=(-0.1,\\,0,\\,0.9,\\,15).",
"(b) Zoom-in on the SNIPER region where a fold and a branching point merge into an AH bifurcation.Figure: (color online)For larger α 2 \\alpha _2 the different regions move and deform.",
"The stable pulse region moves up in α 1 \\alpha _1 by a similar amount while the satellite region moves down.",
"The high α 1 \\alpha _1 region completely detaches in the area below threshold.",
"Results from bifurcation analysis of the PDE strongly differs from the DAE in this region.",
"Parameters are (J 2 ,α 2 ,η,s)=(-0.1,0.5,0.9,15)(J_2,\\,\\alpha _2,\\,\\eta ,\\,s)=(-0.1,\\,0.5,\\,0.9,\\,15)."
],
[
"Conclusion",
"In conclusion, we discussed the dynamics of MIXSELs using a first-principle dynamical model based upon delay algebraic equations.",
"We have found that the third order dispersion induced by the micro-cavity induces satellites on the leading edge of the pulses and shown that the latter can become unstable.",
"Using a combination of direct time simulations and a path-continuation methods we reconstructed the branch of a single pulse solution.",
"In the limit of vanishing line enhancement factors we show that the onset of the satellite instability is associated with a global bifurcation of the saddle-node infinite period type.",
"The influences of noise as well as features of excitability are discussed.",
"In the case of non-vanishing linewidth enhancement factors, we showed that the satellite and self-phase modulation instabilities can combine leading to a intricate oscillating dynamics which shed further light on the results obtained in a VCSEL-SESAM setup [22].",
"Finally, we derived an approximate dispersive master PDE model and performed a full bifurcation analysis.",
"We demonstrated that this PDE reproduces the satellite as well as the low $\\alpha _{1}$ SNIPER and AH bifurcation structure but it becomes increasingly inaccurate at high $\\alpha _{1}$ values."
],
[
"Acknowledgements",
"C.S.",
"and J.J. acknowledge the financial support of the MINECO Project MOVELIGHT (PGC2018-099637-B-100 AEI/FEDER UE).",
"S.G. acknowledges the PRIME program of the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF)."
],
[
"PDE via multiple time scales",
"We consider in this appendix the derivation of the PDE Eq.",
"REF by the use of multiple time scale analysis.",
"Our starting point is still Eqs.",
"(REF ,) $\\varepsilon \\frac{dE}{d\\sigma } & = & \\left[\\left(1-i\\alpha _{1}\\right)N_{1}+\\left(1-i\\alpha _{2}\\right)N_{2}-1\\right]E+hY,\\\\Y\\left(\\sigma \\right) & = & \\eta \\left[E\\left(\\sigma -1\\right)-Y\\left(\\sigma -1\\right)\\right],$ For the sake of simplicity, we set $\\alpha _{j}=0$ and assume that $\\eta $ is real.",
"It is also convenient for our analysis to transform the system composed of an ODE and a DAE as given by Eqs.",
"(REF ,) into a single Neutral Differential Delay Equation (NDDE) $\\varepsilon \\left[\\frac{dE}{d\\sigma }\\left(\\sigma \\right)+\\eta \\frac{dE}{d\\sigma }\\left(\\sigma -1\\right)\\right]=\\left[\\left(N_{1}+N_{2}\\right)E\\right]\\left(\\sigma \\right)-E\\left(\\sigma \\right)+\\eta \\left[\\left(N_{1}+N_{2}\\right)E\\right]\\left(\\sigma -1\\right)+\\left(h-1\\right)\\eta E\\left(\\sigma -1\\right),$ We can define small deviations of the carriers as $N_{j}=J_{j}+\\varepsilon ^{3}n_{j}$ with $j\\in \\left[1,2\\right]$ and scale the field, for convenience, as $E=\\varepsilon ^{\\frac{3}{2}}A$ .",
"Inserting these scaling relations in Eq.",
"REF , we find $A\\left(\\sigma \\right)=\\eta \\frac{h-1+N_{t}}{1-N_{t}}A\\left(\\sigma -1\\right)+\\frac{\\varepsilon ^{3}\\bar{\\mathcal {N}}-\\varepsilon \\bar{\\mathcal {L}}}{1-N_{t}},$ with $N_{t}=J_{1}+J_{2}$ the threshold inversion given in Eq.",
"REF and the filtering $\\bar{\\mathcal {L}}$ and nonlinear operator $\\bar{\\mathcal {N}}$ defined by $\\bar{\\mathcal {L}} & = & \\frac{d}{d\\sigma }\\left[A\\left(\\sigma \\right)+\\eta A\\left(\\sigma -1\\right)\\right]\\\\\\bar{\\mathcal {N}} & = & \\sum _{j=1}^{j=2}\\left(n_j A\\right)\\left(\\sigma \\right)+\\eta \\left(n_j A\\right)\\left(\\sigma -1\\right)$ We can now use the definition of threshold given in Eq.",
"REF to find the system upon which one can perform the multi-scale analysis $A\\left(\\sigma \\right)=A\\left(\\sigma -1\\right)+\\frac{1+\\eta }{h\\eta }\\left(\\varepsilon ^{3}\\bar{\\mathcal {N}}-\\varepsilon \\bar{\\mathcal {L}}\\right),$ By inspecting Equation REF , one notices that the solutions are weakly perturbed period one (P1) orbits.",
"Because we assume that the nonlinear term scales as $\\varepsilon ^{3}$ , we can safely restrict our analysis to the multi-scale analysis on the linear part of Eq.",
"REF , that is $A\\left(\\sigma \\right)=A\\left(\\sigma -1\\right)-\\tilde{\\varepsilon }\\frac{d}{d\\sigma }\\left[A\\left(\\sigma \\right)+\\eta A\\left(\\sigma -1\\right)\\right],$ where we defined the short-hand $\\tilde{\\varepsilon }=\\varepsilon \\left(1+\\eta \\right)/\\left(h\\eta \\right)$ .",
"The linear NDDE as given by Eq.",
"REF only depends on one parameter, which are the cavity losses, and the smallness parameter.",
"It is also an excellent toy model to test various multi-scale schemes.",
"Notice that such a linear NDDE can be solved in the Fourier domain directly.",
"We, however, solve the dynamics by defining a multi-scale expansion.",
"The fast time is $\\sigma _{0}=\\sigma /T$ with $T$ the natural period of the solution.",
"The period $T$ shall be close to unity and its deviation leads to the slow drift in the PDE representation.",
"We set $T=1+\\tilde{\\varepsilon }a$ where $a$ can be chosen to cancel resonant terms in the multi-scale expansion at first order.",
"This approach avoids introducing altogether the intermediate time scale $\\sigma _{1}=\\varepsilon \\sigma $ .",
"However, one can also find analytically the period of the solution using the functional mapping method which yields $a=1+\\eta $ .",
"Finally, we define the slow times $\\sigma _{2}=\\tilde{\\varepsilon }^{2}\\sigma $ and $\\sigma _{3}=\\tilde{\\varepsilon }^{3}\\sigma $ in order to take into account the effect of nonlinearity, diffusion and third order dispersion.",
"The chain rule yields $\\frac{d}{d\\sigma }\\rightarrow \\frac{1}{T}\\frac{\\partial }{\\partial \\sigma _{0}}+\\tilde{\\varepsilon }^{2}\\frac{\\partial }{\\partial \\sigma _{2}}+\\tilde{\\varepsilon }^{3}\\frac{\\partial }{\\partial \\sigma _{3}}$ while for the conjugated Fourier variables, we have $\\omega _{\\sigma }\\rightarrow \\frac{1}{T}\\omega _{0}+\\tilde{\\varepsilon }^{2}\\omega _{2}+\\tilde{\\varepsilon }^{3}\\omega _{3}$ We expand the solution as $A\\left(\\sigma _{0},\\sigma _{2},\\sigma _{3}\\right)=\\sum _{j=0}^{\\infty }\\varepsilon ^{j}A_{j}\\left(\\sigma _{0},\\sigma _{2},\\sigma _{3}\\right)$ The various solvability conditions lead to several conditions of the form $\\bar{L}\\tilde{A}\\left(\\omega _{0},\\omega _{2},\\omega _{3}\\right)=0$ where $\\tilde{A}\\left(\\omega _{0},\\omega _{2},\\omega _{3}\\right)$ is the (triple) Fourier transform of $A\\left(\\sigma _{0},\\sigma _{2},\\sigma _{3}\\right)$ .",
"The expression of $\\bar{L}$ is $\\bar{L}\\left(\\omega _{\\sigma }\\right)=\\exp \\left(i\\omega _{\\sigma }\\right)-1+i\\tilde{\\varepsilon }\\omega _{\\sigma }\\left[1+\\eta \\exp \\left(i\\omega _{\\sigma }\\right)\\right]$ The zeroth order operator reads naturally $\\bar{L}_{0}=e^{i\\omega _{0}}-1$ hence showing that P1 solutions on the fast scale $\\sigma _{0}$ , i.e.",
"$A_{0}\\left(\\sigma _{0}-1,\\sigma _{2},\\sigma _{3}\\right)=A_{0}\\left(\\sigma _{0},\\sigma _{2},\\sigma _{3}\\right)$ , belong to the kernel of $\\bar{L}_{0}$ .",
"The first order solvability is trivially solved since $\\bar{L}_{1}=0$ due to our adequate choice of the period $T$ .",
"The other operators found at second and third order are more complex, yet they simplify when acting on P1 solutions, $\\bar{L}_{2} & = & \\left(\\eta ^{2}-1\\right)\\omega _{0}^{2}+2i\\omega _{2},\\\\\\bar{L}_{3} & = & -i\\frac{\\eta ^{3}+1}{3}\\omega _{0}^{3}-\\frac{\\eta -1}{2}\\left(\\eta +1\\right)^{2}\\omega _{0}^{2}+i\\omega _{3}.$ One can then build the conjugated variable to the slow time $\\omega _{\\xi }$ as $\\omega _{\\xi }=\\left(\\frac{1}{T}-1\\right)\\omega _{0}+\\omega _{2}\\tilde{\\varepsilon }^{2}+\\omega _{3}\\tilde{\\varepsilon }^{3}$ We note that adding the $-\\omega _{0}$ corresponds to the stroboscopic effect that transforms a T-periodic solution into a slowly drifting steady state from one round-trip to the other.",
"Upon simplification, we find $-i\\omega _{\\xi }=i\\left(1+\\eta \\right)\\tilde{\\omega }+\\frac{\\eta ^{2}-1}{2}\\tilde{\\omega }^{2}-i\\frac{\\eta ^{3}+1}{3}\\tilde{\\omega }^{3}+\\mathcal {O}\\left(\\varepsilon \\right)$ where we defined $\\tilde{\\varepsilon }\\omega _{0}=\\tilde{\\omega }$ .",
"Using the definition of $\\tilde{\\varepsilon }$ and going back to the original variable, i.e.",
"$\\varepsilon \\omega _{0}\\rightarrow \\omega $ , gives the following expression for the slow evolution $\\omega _{\\xi }=d_{1}\\omega -id_{2}\\omega ^{2}-d_{3}\\omega ^{3}$ with the coefficients $d_{j}$ defined in Eq.",
"REF .",
"Finally, using that $-i\\omega \\rightarrow \\partial _{z}$ we get $\\partial _{\\xi }=d_{1}\\partial _{z}+d_{2}\\partial _{z}^{2}+d_{3}\\partial _{z}^{3}$ As we assumed that the nonlinear term scales as $\\varepsilon ^{3}$ , it can simply be added to the highest order solvability condition found for the operator $\\bar{L}_{3}$ .",
"By using that the carriers are also P1 solutions on the fast time scale $\\left(\\sigma _{0}\\right)$ we get that $\\left(n_{j}A\\right)\\left(\\sigma _{0}-1,\\sigma _{2},\\sigma _{3}\\right)=\\left(n_{j}A\\right)\\left(\\sigma _{0},\\sigma _{2},\\sigma _{3}\\right)$ with $j\\in \\left[1,2\\right]$ which allows finding the PDE in the main manuscript."
]
] | 2001.03446 |
[
[
"Convergence of Likelihood Ratios and Estimators for Selection in\n non-neutral Wright-Fisher Diffusions"
],
[
"Abstract A number of discrete time, finite population size models in genetics describing the dynamics of allele frequencies are known to converge (subject to suitable scaling) to a diffusion process in the infinite population limit, termed the Wright-Fisher diffusion.",
"In this article we show that the diffusion is ergodic uniformly in the selection and mutation parameters, and that the measures induced by the solution to the stochastic differential equation are uniformly locally asymptotically normal.",
"Subsequently these two results are used to analyse the statistical properties of the Maximum Likelihood and Bayesian estimators for the selection parameter, when both selection and mutation are acting on the population.",
"In particular, it is shown that these estimators are uniformly over compact sets consistent, display uniform in the selection parameter asymptotic normality and convergence of moments over compact sets, and are asymptotically efficient for a suitable class of loss functions."
],
[
"Introduction",
"Mathematical population genetics is concerned with the study of how populations evolve over time, offering viable models to study how various biological phenomena such as selection and mutation affect the genetic profile of the population they act upon.",
"Many models have been proposed over the years, but perhaps the most popular is the Wright–Fisher model (see for instance [12]).",
"Under a suitable scaling of both space and time, a diffusion limit exists for the Wright–Fisher model, which is referred to as the Wright–Fisher diffusion, (REF ), and is the main focus of this article.",
"The Wright–Fisher diffusion is robust in the sense that a broad class of Cannings [2] models converge to it when suitably scaled.",
"Furthermore, it has the neat property that the only contribution to the diffusion coefficient comes from random mating whilst other features such as selection and mutation appear solely in the drift coefficient.",
"This facilitates inference as one can concentrate on estimating the drift, treating the diffusion coefficient as a known expression.",
"In this article we focus on a continuously observed Wright–Fisher diffusion describing the allele frequency dynamics in a two-allele, haploid population undergoing both selection and mutation.",
"In Section 2 we start by considering a general scalar diffusion taking values in an arbitrary bounded interval $[l,r]$ (with $-\\infty < l < r < \\infty $ ) whose boundary points are either regular or entrance.",
"In this setting, we derive verifiable criteria to establish $\\vartheta $ -uniform ergodicity (see Definition REF ), and further extend these conditions to allow for $\\vartheta $ -uniform ergodicity with respect to a specific class of unbounded functions.",
"Subsequently we introduce the Wright–Fisher diffusion, and by making use of the previously derived criteria, we show that the diffusion is ergodic uniformly in both the selection and mutation parameters, and that the associated family of measures induced by the solution to the stochastic differential equation (SDE) is uniformly locally asymptotically normal (provided the mutation parameters are greater than 1).",
"In Section 3 we then shift our focus onto the properties of the maximum likelihood (ML) and Bayesian estimators for the selection parameter $s\\in \\mathcal {S}\\subset \\mathbb {R}$ (which measures how much more favourable one allele is over the other), under the assumption that the mutation parameters are a priori known.",
"We briefly discuss some technical issues associated with conducting joint inference for the selection and mutation parameters in Remark REF .",
"We point out here that by observing the path continuously through time without error, one can establish and analyse explicitly the statistical error produced by an estimator based on the whole sample path, which then sets a benchmark for the statistical performance of alternative estimators based on less informative (e.g.",
"discrete) observations.",
"In a discrete observation setting, in addition to the above mentioned statistical error, one also has to deal with observational error.",
"One certainly cannot hope for an estimator that performs better in a discrete setting than in a continuous one, so our analysis may be viewed as the `best possible' performance for inference from a discretely observed model.",
"Inference for scalar diffusions, particularly proving consistency of estimators under specific observational schemes, has generated considerable interest over the past few years [6], [13], [16], [17], [18], [21], [22], [24].",
"However, most of the work so far has considered classes of diffusions which directly preclude the Wright–Fisher diffusion, for instance by imposing periodic boundary conditions on the drift coefficients or by requiring the diffusion coefficient to be strictly positive everywhere.",
"The asymptotic study of a variety of estimators for continuously observed ergodic scalar diffusions has been entertained in great depth in [13]; see in particular Theorems 2.8 and 2.13 in [13], which are respectively adaptations of Theorems I.5.1, I.10.1 and I.5.2, I.10.2 in [9].",
"However Theorems 2.8 and 2.13 in [13] cannot be applied directly to the Wright–Fisher diffusion as certain conditions do not hold, namely the reciprocal of the diffusion coefficient does not have a polynomial majorant.",
"This discrepancy makes replicating the results for the Wright–Fisher diffusion with selection and mutation highly non-trivial.",
"Instead we exploit the explicit nature of (REF ), below, to prove, in our main result Theorem REF , uniform in the selection parameter over compact sets consistency, asymptotic normality and convergence of moments, as well as asymptotic efficiency for both the ML and Bayesian estimators.",
"We achieve this by showing that the conditions of Theorems I.5.1, I.10.1 and I.5.2, I.10.2 in [9] still hold for the Wright–Fisher diffusion and that this diffusion is ergodic uniformly in the selection and mutation parameters (a term we define in Section ).",
"Note that the uniformity in our results for the ML and Bayesian estimators is particularly useful as it controls the lowest rate (over the true parameters) at which the parameters of interest are being learned by the inferential scheme.",
"We further highlight that the conditions derived in Theorem REF provide a straightforward way to verify whether a positively recurrent diffusion on an arbitrary interval with regular or entrance boundaries is uniformly in the parameter ergodic, whilst Theorem REF illustrates additional conditions necessary to extend this notion for a specific class of unbounded functions when the diffusion has solely entrance boundaries.",
"The Wright–Fisher diffusion with selection but without mutation was tackled specifically by Watterson in [24], where the author makes use of a frequentist framework.",
"Having no mutation ensures that the diffusion is absorbed at either boundary point 0 or 1 in finite time almost surely, and by conditioning on absorption Watterson computes the moment generating function, proves asymptotic normality, and derives hypothesis tests based on the Maximum Likelihood Estimator (MLE).",
"Watterson's work however does not address the Bayesian estimator, nor does it readily extend to the case when mutation is present because the diffusion is no longer absorbed at the boundaries.",
"In this sense the results obtained in Theorem REF are complementary to those obtained by Watterson under the assumption that the mutation parameters are known.",
"Although this is a restriction, we are observing the path continuously over the interval $[0,T]$ and subsequently sending $T\\rightarrow \\infty $ , so these parameters could be inferred by considering the boundary behaviour of the diffusion.",
"More precise details about this can be found in Remark REF in Section , whilst a related argument applied to the squared Bessel process can be found in [19].",
"The rest of this article is organised as follows: in Section we start by considering a general diffusion on a bounded interval with entrance or regular boundaries, and proceed to provide verifiable criteria which ensure that the diffusion is $\\vartheta $ -uniformly ergodic in Theorem REF .",
"We then proceed to extend this notion to a specific class of unbounded functions for diffusions with entrance boundaries in Theorem REF , before moving on to introduce the Wright–Fisher diffusion and proving that this diffusion is ergodic uniformly in both the selection and mutation parameters, as well as uniformly locally asymptotically normal by making use of the general results we derive.",
"Section then focuses on the ML and Bayesian estimators for the selection parameter, proving that these estimators have a set of desirable properties in Theorem REF .",
"The results are then supported via numerical simulations in Section , whilst Section concludes with a discussion.",
"The proofs of Theorems REF and REF can be found in Appendices and respectively."
],
[
"$\\vartheta $ -uniform ergodicity for scalar diffusions",
"We start by considering an arbitrary fixed interval $[l,r]$ on which we define the SDE $dY_t = \\mu (\\vartheta ,Y_t)dt + \\sigma (Y_t) dW_t, & & Y_0 \\sim \\nu , & & \\vartheta \\in \\Theta \\subseteq \\mathbb {R}^{d}, d \\ge 1$ where $\\nu $ is an arbitrary initial distribution on $[l,r]$ , $(W_{t})_{t\\ge 0}$ a standard Wiener process defined on a given filtered probability space, $\\mu $ and $\\sigma $ are such that the SDE admits a unique strong solution which we denote by $Y:= (Y_t)_{t\\ge 0}$ , $-\\infty < l < r < \\infty $ are both either entrance or regular boundaries for $Y$ , and the observation interval is fixed to $[0,T]$ .",
"We denote by $\\mathbb {P}_{\\nu }^{(\\vartheta )}$ the law induced on the space of continuous functions mapping $[0,T]$ into $[l,r]$ (henceforth denoted $C_{T}([l,r])$ ) by the solution to (REF ) when the true diffusion parameter is set to $\\vartheta $ , and $Y_0 \\sim \\nu $ (with dependence on $T$ being implicit).",
"Furthermore we denote taking expectation with respect to $\\mathbb {P}^{(\\vartheta )}_{\\nu }$ by $\\mathbb {E}^{(\\vartheta )}_{\\nu }$ .",
"Assume further that $Y$ is positive recurrent, then using standard one-dimensional diffusion theory (see Theorem 1.16 in [13]), we get that the unique invariant density is given by $f^{Y}_{\\vartheta }(x) = \\frac{1}{G^{Y}_{\\vartheta }}\\frac{2}{\\sigma ^{2}(x)}e^{2\\int ^{x}\\frac{\\mu (\\vartheta ,z)}{\\sigma ^{2}(z)}dz}, & & x\\in [l,r], & & G^{Y}_{\\vartheta } := \\int _{l}^{r}\\frac{2}{\\sigma ^{2}(x)}e^{2\\int ^{x}\\frac{\\mu (\\vartheta ,z)}{\\sigma ^{2}(z)}dz}dx.$ In what follows, we denote taking expectation with respect to $f^{Y}_{\\vartheta }$ by $\\mathbb {E}^{(\\vartheta )}$ , where the omission of the subscript will indicate that we start from stationarity, and henceforth always assume that $\\xi \\sim f^{Y}_{\\vartheta }$ .",
"In order to derive the results in Section , we will need a slightly stronger notion of ergodicity which we now define.",
"The idea here is that we can extend pointwise ergodicity in the parameter $\\vartheta $ to any compact set $\\mathcal {K}\\subset \\Theta $ by finding the slowest rate of convergence which works within that compact set.",
"More rigorously, we introduce the following definition.",
"Definition 2.1 A process $Y$ is said to be ergodic uniformly in the parameter $\\vartheta $ (or $\\vartheta $ -uniformly ergodic), if $\\forall \\varepsilon >0$ we have that $\\lim \\limits _{T\\rightarrow \\infty }\\sup \\limits _{\\vartheta \\in \\mathcal {K}}\\mathbb {P}^{(\\vartheta )}_{\\nu }\\Bigg [\\left|\\frac{1}{T}\\int _{0}^{T}h(Y_{t})dt - \\mathbb {E}^{(\\vartheta )}\\big [h\\left(\\xi \\right)\\big ]\\right| > \\varepsilon \\Bigg ] = 0$ holds for any $\\mathcal {K}$ compact subset of the parameter space, and for any function $h:[l,r]\\rightarrow \\mathbb {R}$ bounded and measurable, where $\\xi \\sim f_{\\vartheta }^{Y}$ .",
"In the context of scalar diffusions defined on a bounded interval $[l,r]$ with $-\\infty < l < r < \\infty $ , where both boundaries are either regular or entrance we have the following theorem: Theorem 2.2 Let $Y$ be defined as above as the solution to (REF ), with boundary points $l$ and $r$ either entrance or regular, and that the expressions $\\kappa ^{l}_{\\vartheta }(a,b) &:= \\int _{a}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{l}^{\\xi }\\frac{2}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}d\\eta d\\xi , \\nonumber \\\\\\kappa ^{r}_{\\vartheta }(a,b) &:= \\int _{a}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{\\xi }^{r}\\frac{2}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}d\\eta d\\xi ,$ are bounded away from 0 on any compact set $\\mathcal {K}\\subset \\Theta $ , and any $l<a<b<r$ .",
"Then $Y$ is $\\vartheta $ -uniformly ergodic for any initial distribution $\\nu $ .",
"We postpone the proof to Appendix .",
"Note that the definition of $\\vartheta $ -uniform ergodicity involves only bounded functions $h$ , however the result above can be extended to a specific class of unbounded functions if one restricts their attention to diffusions on $[l,r]$ where $-\\infty < l < r < \\infty $ are both entrance boundaries.",
"Theorem 2.3 Let $Y$ be as in Theorem REF , and suppose that all the conditions stated there hold, but that both $l$ and $r$ are now entrance boundaries.",
"Assume further that the function $h$ is integrable with respect to the invariant density $f^{Y}_{\\vartheta }$ but possibly unbounded, that for any $l<a<b<r$ , $\\sup _{y\\in [a,b]}h(y) <\\infty $ , and that for any $x<b$ the following hold $&\\sup _{\\vartheta \\in \\mathcal {K}}\\int _{x}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{l}^{\\xi }\\frac{2h(\\eta )}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}d\\eta d\\xi < \\infty , \\\\&\\sup _{\\vartheta \\in \\mathcal {K}}\\int _{x}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{l}^{\\xi }\\frac{2h(\\eta )}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\mathbb {E}^{(\\vartheta )}_{\\eta }\\left[\\int _{0}^{T_{b}}h(Y_t)dt\\right]d\\eta d\\xi < \\infty , \\\\&\\sup _{\\vartheta \\in \\mathcal {K}}\\int _{l}^{r}\\mathbb {E}^{(\\vartheta )}_{x}\\left[\\int _{0}^{T_{b}}h(Y_t)dt\\right]\\nu (\\vartheta ,dx) < \\infty ,$ for any compact set $\\mathcal {K}\\subset \\Theta $ , and $T_{b} := \\inf \\lbrace t\\ge 0 : Y_{t} = b \\rbrace $ .",
"Then (REF ) holds for the function $h$ .",
"The proof can be found in Appendix .",
"Remark 2.4 Note that the above conditions imply that $h$ is only unbounded at the end points (because the supremum between $a$ and $b$ of $h$ is finite for any $l<a<b<r$ ), which in particular ensures that all integrals of the form $\\int _{0}^{T}h(Y_t)dt$ above are well-defined as both boundary points are unattainable (in view of them being entrance)."
],
[
"The Wright–Fisher diffusion",
"We now give a brief overview of the Wright–Fisher diffusion before showing that the diffusion is ergodic uniformly in the selection and mutation parameters, and subsequently use this to prove the uniform local asymptotic normality (LAN) of the family of measures associated to the solution of the SDE.",
"Consider an infinite haploid population undergoing selection and mutation, where we are interested in two alleles $A_{1}$ and $A_{2}$ .",
"Suppose that $\\vartheta =(s,\\theta _1,\\theta _2)\\in \\Theta =\\mathbb {R}\\times (0,\\infty )^{2}$ are the selection and mutation parameters respectively, where $s$ describes the extent to which allele $A_{2}$ is favoured over $A_{1}$ , alleles of type $A_{1}$ mutate to $A_{2}$ at rate $\\theta _{1}/2$ , and those of type $A_{2}$ mutate to $A_{1}$ at rate $\\theta _{2}/2$ .",
"Let $X_{t}$ denote the frequency of $A_{2}$ in the population at time $t$ .",
"Then the dynamics of $X_{t}$ can be described by a diffusion process on $[0,1]$ , which, after expressing the parameters on an appropriate timescale, satisfies the SDE $dX_{t} &= \\mu _{\\textnormal {WF}}(\\vartheta ,X_{t})dt + \\sigma _{\\textnormal {WF}}(X_{t})dW_{t} \\nonumber \\\\&:= \\frac{1}{2}\\left(sX_{t}(1-X_{t}) - \\theta _{2}X_{t} + \\theta _{1}(1-X_{t})\\right)dt + \\sqrt{X_{t}(1-X_{t})}dW_{t} ,$ with $X_0\\sim \\nu $ for some initial distribution $\\nu $ , and $[0,T]$ the observation interval.",
"We point out that (REF ) with $s=0$ is commonly referred to as the neutral Wright–Fisher diffusion, whilst $s\\ne 0$ is known as the non-neutral case.",
"A strong solution to (REF ) exists by the Yamada–Watanabe condition (see Theorem 3.2, Chapter IV in [10]), but weak uniqueness suffices for the results in Section .",
"In abuse of notation, we redefine $\\mathbb {P}^{(\\vartheta )}_{\\nu }$ to be the law induced on $C_{T}([0,1])$ by the solution to (REF ) when the true diffusion parameters are $\\vartheta =(s,\\theta _1,\\theta _2)$ , and $X_0 \\sim \\nu $ , and similarly for the expectation with respect to $\\mathbb {P}^{(\\vartheta )}_{\\nu }$ , $\\mathbb {E}^{(\\vartheta )}_{\\nu }$ , and with respect to the stationary distribution, $\\mathbb {E}^{(\\vartheta )}$ (the existence of which we discuss below).",
"We assume that $\\theta _{1}, \\theta _{2} > 0$ , for if at least one is 0 then the diffusion is absorbed in finite time and we are back in the regime studied by Watterson [24].",
"The boundary behaviour depends on whether the mutation parameters are either less than or greater or equal to 1, but in either case the diffusion is ergodic as long as $\\theta _{1}, \\theta _{2} > 0$ (see Lemma 2.1, Chapter 10 in [5]).",
"Substituting $\\mu _{\\textnormal {WF}}$ and $\\sigma _{\\textnormal {WF}}$ into (REF ) and simplifying terms leads to the following density for the stationary distribution of the Wright–Fisher diffusion (REF ) $f_{\\vartheta }(x) = \\frac{1}{G_{\\vartheta }}e^{sx}x^{\\theta _{1}-1}(1-x)^{\\theta _{2}-1}, \\qquad x \\in (0,1),$ where $G_{\\vartheta }$ is the normalising constant $G_{\\vartheta } = \\int _0^1 e^{sx}x^{\\theta _{1}-1}(1-x)^{\\theta _{2}-1}dx \\le \\max \\lbrace e^{s},1\\rbrace B(\\theta _{1},\\theta _{2}) < \\infty ,$ with $B(\\theta _{1},\\theta _{2}) := \\int _{0}^{1}x^{\\theta _{1}-1}(1-x)^{\\theta _{2}-1}dx$ the beta function.",
"As above we will always assume that $\\xi \\sim f_{\\vartheta }$ .",
"To the best of our knowledge, it has not been shown that the Wright–Fisher diffusion is ergodic uniformly in its parameters, which motivates the following corollary to Theorem REF .",
"Corollary 2.5 The Wright–Fisher diffusion with mutation and selection parameters $\\vartheta =(s,\\theta _1,\\theta _2)$ is $\\vartheta $ -uniformly ergodic for any initial distribution $\\nu $ .",
"We show that the conditions of Theorem REF hold for the Wright–Fisher diffusion.",
"Positive recurrence follows immediately from (REF ), whilst the existence of a unique strong solution is guaranteed by the Yamada–Watanabe condition.",
"That the boundary points 0 and 1 are either entrance or regular is a consequence of the fact that the mutation parameters are assumed to be strictly positive (see (6.18) and (6.19) in [12]).",
"It remains to show that both expressions in (REF ) are bounded away from 0 for any $\\mathcal {K}\\subset \\mathbb {R}\\times (0,\\infty )^{2}$ .",
"To this end let $\\bar{\\theta }_1 := \\sup _{\\vartheta \\in \\mathcal {K}} \\theta _1$ , $\\bar{\\theta }_2 := \\sup _{\\vartheta \\in \\mathcal {K}} \\theta _2$ .",
"Then $&\\int _{a}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu _{\\textnormal {WF}}(\\vartheta ,y)}{\\sigma ^{2}_{\\textnormal {WF}}(y)}dy}\\int _{0}^{\\xi }\\frac{1}{\\sigma ^{2}_{\\textnormal {WF}}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu _{\\textnormal {WF}}(\\vartheta ,y)}{\\sigma ^{2}_{\\textnormal {WF}}(y)}dy}d\\eta d\\xi \\nonumber \\\\&\\qquad {}= \\int _{a}^{b}2e^{-s\\xi }\\xi ^{-\\theta _{1}}(1-\\xi )^{-\\theta _{2}}\\int _{0}^{\\xi }e^{s\\eta }\\eta ^{\\theta _{1}-1}(1-\\eta )^{\\theta _{2}-1}d\\eta d\\xi \\nonumber \\\\&\\qquad {}\\ge 2\\min \\lbrace e^{-s},1 \\rbrace \\int _{a}^{b} \\xi ^{-\\theta _{1}}(1-\\xi )^{-\\theta _{2}}d\\xi \\int _{0}^{a}\\eta ^{\\theta _{1}-1}(1-\\eta )^{\\theta _{2}-1}d\\eta \\nonumber \\\\&\\qquad {}\\ge 2\\min \\lbrace e^{-s},1\\rbrace (b-a)\\frac{a^{\\theta _1}}{\\theta _1}(1-a)^{\\bar{\\theta }_2-1}, \\\\&\\int _{a}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu _{\\textnormal {WF}}(\\vartheta ,y)}{\\sigma ^{2}_{\\textnormal {WF}}(y)}dy}\\int _{\\xi }^{1}\\frac{1}{\\sigma ^{2}_{\\textnormal {WF}}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu _{\\textnormal {WF}}(\\vartheta ,y)}{\\sigma ^{2}_{\\textnormal {WF}}(y)}dy}d\\eta d\\xi \\nonumber \\\\&\\qquad {}= \\int _{a}^{b}2e^{-s\\xi }\\xi ^{-\\theta _{1}}(1-\\xi )^{-\\theta _{2}}\\int _{\\xi }^{1}e^{s\\eta }\\eta ^{\\theta _{1}-1}(1-\\eta )^{\\theta _{2}-1}d\\eta d\\xi \\nonumber \\\\&\\qquad {}\\ge 2\\min \\lbrace e^{s},1 \\rbrace \\int _{a}^{b} \\xi ^{-\\theta _{1}}(1-\\xi )^{-\\theta _{2}}d\\xi \\int _{b}^{1}\\eta ^{\\theta _{1}-1}(1-\\eta )^{\\theta _{2}-1}d\\eta \\nonumber \\\\&\\qquad {}\\ge 2\\min \\lbrace e^{s},1\\rbrace (b-a)\\frac{(1-b)^{\\theta _2}}{\\theta _2}b^{\\bar{\\theta }_1-1},$ which follows by observing that $\\xi ^{-\\theta _1}(1-\\xi )^{-\\theta _2} &> 1 & \\forall \\xi &\\in (a,b), \\forall \\theta _1,\\theta _2>0, \\\\(1-\\eta )^{\\theta _2 -1} &\\ge (1-a)^{\\bar{\\theta }_{2}-1} & \\forall \\eta &\\in (0,a), \\\\\\eta ^{\\theta _1 -1} &\\ge b^{\\bar{\\theta }_{1}-1} & \\forall \\eta &\\in (b,1).$ As the RHS of both (REF ) and () are bounded away from 0 on $\\mathcal {K}$ , the result follows by applying Theorem REF .",
"For the remainder of this section we restrict our attention to the parameter space $\\Theta \\subset \\mathbb {R}\\times [1,\\infty )^{2}$ , where $\\Theta $ is open and bounded, for if either of the mutation parameters were less than 1 then the measures $\\mathbb {P}^{(\\vartheta )}_{\\nu }$ within this region would be mutually singular with respect to one another and thus their Radon–Nikodym derivative undefined.",
"Restricting our attention to mutation parameters within the range $[1,\\infty )^{2}$ thus ensures that the family of measures $\\lbrace \\mathbb {P}^{(\\vartheta )}_{\\nu },\\vartheta \\in \\Theta \\rbrace $ are equivalent, and we have that $\\frac{d\\mathbb {P}^{(\\vartheta ^{\\prime })}_{\\nu }}{d\\mathbb {P}^{(\\vartheta )}_{\\nu }}(X^T) = \\frac{\\nu (\\vartheta ^{\\prime },X_0)}{\\nu (\\vartheta ,X_0)}\\exp \\Bigg \\lbrace &\\int _{0}^{T}\\left(\\frac{\\mu _{\\textnormal {WF}}(\\vartheta ^{\\prime },X_t)-\\mu _{\\textnormal {WF}}(\\vartheta ,X_t)}{\\sigma _{\\textnormal {WF}}(X_t)}\\right)dW_{t} \\nonumber \\\\&\\quad - \\frac{1}{2}\\int _{0}^{T}\\left(\\frac{\\mu _{\\textnormal {WF}}(\\vartheta ^{\\prime },X_t)-\\mu _{\\textnormal {WF}}(\\vartheta ,X_t)}{\\sigma _{\\textnormal {WF}}(X_t)}\\right)^{2}dt\\Bigg \\rbrace $ with $\\mathbb {P}^{(\\vartheta )}_{\\nu }$ -probability 1, where we assume the initial distributions $\\lbrace \\nu (\\vartheta ,\\cdot )\\rbrace _{\\vartheta \\in \\Theta }$ are mutually equivalent and admit a density with respect to some common dominating measure $\\lambda (\\cdot )$ , which in abuse of notation we denote by $\\nu (\\vartheta ,\\cdot )$ .",
"Proofs of the above claims regarding the equivalence of the Wright–Fisher diffusion and the form of the Radon–Nikodym derivative can be found in [3], Lemma 7.2.2 and Section 10.1.1.",
"We emphasise here that we have allowed the starting distribution $\\nu $ to depend on the parameters, as is evident from the first ratio in (REF ).",
"However if there is no such dependence then this ratio is equal to 1 and our results still apply.",
"Furthermore, restricting to mutations greater than or equal to 1 ensures that the diffusion boundaries now become entrance (see equations (6.18) and (6.19) in [12]), and as done in Theorem REF , (REF ) can be extended for a particular class of unbounded functions.",
"We focus on two such functions for this class of diffusions, as they turn out to be an essential ingredient necessary to prove local asymptotic normality.",
"Corollary 2.6 For the Wright–Fisher diffusion with mutation and selection parameters $\\vartheta \\in \\Theta \\subset \\mathbb {R}\\times [1,\\infty )^{2}$ (for $\\Theta $ an open bounded set) with initial distribution $\\nu $ satisfying (REF ), $\\vartheta $ -uniform ergodicity (REF ) holds also for the functions $h(x) = (1-x)x^{-1}$ and $h(x) = (1-x)^{-1}x$ .",
"The result holds in particular for the case $\\nu = f_{\\vartheta }$ .",
"The result follows immediately if we show that all the conditions of Theorem REF are satisfied for the above functions.",
"The conditions of Theorem REF have already been shown to hold in Corollary REF , whilst integrability with respect to the invariant density is guaranteed as we are considering mutation rates $(\\theta _1,\\theta _2)\\in (1,\\infty )^{2}$ .",
"That $\\sup _{y\\in [a,b]}h(y) <\\infty $ for any pair $0<a<b<1$ is immediate, so it remains to show (REF ), (), and ().",
"Observe that $\\mathbb {E}^{(\\vartheta )}_{x}\\left[\\int _{0}^{T_b}\\frac{1-X_t}{X_t}dt\\right] &= 2\\int _{x}^{b}e^{-s\\xi }\\xi ^{-\\theta _1}(1-\\xi )^{-\\theta _2}\\int _{0}^{\\xi }e^{s\\eta }\\eta ^{\\theta _1 -2}(1-\\eta )^{\\theta _2}d\\eta d\\xi \\nonumber \\\\&\\le 2\\max \\lbrace e^{-s},1 \\rbrace \\int _{x}^{b}\\xi ^{-\\theta _1}(1-\\xi )^{-\\theta _2}\\int _{0}^{\\xi }\\eta ^{\\theta _1 -2}d\\eta d\\xi \\nonumber \\\\&= 2\\max \\lbrace e^{-s},1 \\rbrace \\frac{1}{\\theta _1 -1}\\int _{x}^{b}\\xi ^{-1}(1-\\xi )^{-\\theta _2} d\\xi ,$ so (REF ) holds as the RHS is continuous in $\\vartheta $ and thus can be bounded from above in $\\vartheta $ over any compact set $\\mathcal {K}\\subset \\Theta $ .",
"For $x>b$ $\\mathbb {E}^{(\\vartheta )}_{x}\\left[\\int _{0}^{T_b}\\frac{1-X_t}{X_t}dt\\right] &= 2\\int _{b}^{x}e^{-s\\xi }\\xi ^{-\\theta _1}(1-\\xi )^{-\\theta _2}\\int _{\\xi }^{1}e^{s\\eta }\\eta ^{\\theta _1 -2}(1-\\eta )^{\\theta _2}d\\eta d\\xi \\nonumber \\\\&\\le 2\\max \\lbrace e^{s},1 \\rbrace \\int _{b}^{x}\\xi ^{-\\max \\lbrace \\theta _1,2\\rbrace }(1-\\xi )^{-\\theta _2}\\int _{\\xi }^{1}(1-\\eta )^{\\theta _2}d\\eta d\\xi \\nonumber \\\\&= 2\\max \\lbrace e^{s},1 \\rbrace \\frac{1}{\\theta _2 +1}\\int _{b}^{x}\\xi ^{-\\max \\lbrace \\theta _1,2\\rbrace }(1-\\xi ) d\\xi \\nonumber \\\\&\\le 2\\max \\lbrace e^{s},1\\rbrace \\frac{1}{\\theta _2 + 1}\\int _{b}^{x}\\xi ^{-\\max \\lbrace \\theta _1,2\\rbrace }d\\xi ,$ and thus () holds in view of condition (REF ).",
"In the case when $\\nu =f_{\\vartheta }$ , we get that $\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\int _{0}^{T_b}\\frac{1-X_t}{X_t}dt\\right] \\le {}& 2\\max \\lbrace e^{-s},1 \\rbrace \\frac{1}{\\theta _1 -1}\\int _{0}^{b}\\int _{x}^{b}\\xi ^{-1}(1-\\xi )^{-\\theta _2} d\\xi f_{\\vartheta }(x)dx\\nonumber \\\\&{}+ 2\\max \\lbrace e^{s},1 \\rbrace \\frac{1}{\\theta _2 +1}\\int _{b}^{1}\\int _{b}^{x}\\xi ^{-\\max \\lbrace \\theta _1,2\\rbrace } d\\xi f_{\\vartheta }(x)dx \\nonumber \\\\\\le {}& 2\\max \\lbrace e^{s},1 \\rbrace \\frac{1}{\\theta _1 (\\theta _1 -1)}\\frac{1}{G_{\\vartheta }}\\int _{0}^{b}(1-\\xi )^{-\\theta _2}d\\xi \\nonumber \\\\&{}+2\\max \\lbrace e^{s},1 \\rbrace \\frac{1}{(\\theta _2 +1)}\\int _{b}^{1}\\xi ^{-\\max \\lbrace \\theta _1,2\\rbrace }d\\xi ,$ which follows from $\\int _{0}^{b}\\int _{x}^{b}\\xi ^{-1}(1-\\xi )^{-\\theta _2}x^{\\theta _1 -1}(1-x)^{\\theta _2 -1}d\\xi dx &= \\int _{0}^{b}\\int _{0}^{\\xi }\\xi ^{-1}(1-\\xi )^{-\\theta _2}x^{\\theta _1 -1}(1-x)^{\\theta _2 -1} dx d\\xi \\\\&\\le \\frac{1}{\\theta _1}\\int _{0}^{b}\\xi ^{\\theta _1 -1}(1-\\xi )^{-\\theta _2}d\\xi \\\\&\\le \\frac{1}{\\theta _1}\\int _{0}^{b}(1-\\xi )^{-\\theta _2}d\\xi $ because $\\theta _1,\\theta _2 > 1$ , and $\\int _{b}^{1}\\int _{b}^{x}\\xi ^{-\\max \\lbrace \\theta _1,2\\rbrace }f_{\\vartheta }(x)d\\xi dx = \\int _{b}^{1}\\int _{\\xi }^{1}\\xi ^{-\\max \\lbrace \\theta _1,2\\rbrace }f_{\\vartheta }(x) dx d\\xi \\le \\int _{b}^{1}\\xi ^{-\\max \\lbrace \\theta _1,2\\rbrace }d\\xi .$ Finally, using the recursions in (REF ) and (REF ), we get that for $x<b$ , $\\mathbb {E}^{(\\vartheta )}_{x}\\left[\\left(\\int _{0}^{T_b}\\frac{1-X_t}{X_t}dt\\right)^{2}\\right] &\\le \\frac{2(2\\max \\lbrace e^{-s},1 \\rbrace )^{2}}{\\theta _1 -1}\\int _{0}^{b}\\gamma ^{\\theta _1 -2}(1-\\gamma )^{-\\theta _2} d\\gamma \\nonumber \\\\&\\quad \\times \\int _{x}^{b}\\xi ^{-\\theta _1}(1-\\xi )^{-\\theta _2}d\\xi \\nonumber \\\\&\\le \\frac{2(2\\max \\lbrace e^{-s},1 \\rbrace )^{2}}{\\theta _1 -1}(1-b)^{-\\theta _2}\\int _{0}^{b}\\gamma ^{\\theta _1 -2} d\\gamma \\nonumber \\\\&\\quad \\times \\int _{x}^{b}\\xi ^{-\\theta _1}(1-\\xi )^{-\\theta _2}d\\xi $ which follows from $\\int _{0}^{\\xi }\\eta ^{\\theta _1 -2}(1-\\eta )^{\\theta _2}\\int _{\\eta }^{b}\\gamma ^{-1}(1-\\gamma )^{-\\theta _2}d\\gamma d\\eta &\\le \\int _{0}^{b}\\eta ^{\\theta _1 -2}(1-\\eta )^{\\theta _2}\\int _{\\eta }^{b}\\gamma ^{-1}(1-\\gamma )^{-\\theta _2}d\\gamma d\\eta \\\\&\\le \\int _{0}^{b}\\gamma ^{\\theta _1 -2}(1-\\gamma )^{-\\theta _2} d\\gamma ,$ and again the corresponding RHS can be bounded from above over any compact set $\\mathcal {K}\\subset \\Theta $ using continuity in $\\vartheta $ , such that () holds and so the result follows by Theorem REF .",
"We end this section by introducing the concept of local asymptotic normality (LAN) and show that the Wright–Fisher diffusion is uniformly LAN, which will be essential in the next section.",
"Definition 2.7 (Special case of Definition 2.1 in [13]) The family of measures $\\lbrace \\mathbb {P}^{(\\vartheta )}_{\\nu },\\vartheta \\in \\Theta \\rbrace $ is said to be locally asymptotically normal (LAN) at a point $\\vartheta _{0}\\in \\Theta $ at rate $T^{-1/2}$ if for any $\\mathbf {u}\\in \\mathbb {R}^{3}$ , the likelihood ratio function admits the representation $\\hspace*{-8.53581pt}Z_{T,\\vartheta _{0}}(\\mathbf {u}) &:= \\frac{d\\mathbb {P}^{(\\vartheta _0 + \\frac{\\mathbf {u}}{\\sqrt{T}})}_{\\nu }}{d\\mathbb {P}^{(\\vartheta _0)}_{\\nu }}(X^T) \\\\&\\phantom{:}= \\exp \\left\\lbrace \\left<\\mathbf {u},\\Delta _{T}(\\vartheta _{0},X^{T})\\right> - \\frac{1}{2}\\left<\\mathbf {I}(\\vartheta _{0})\\mathbf {u},\\mathbf {u}\\right> + r_{T}(\\vartheta _{0},\\mathbf {u},X^{T})\\right\\rbrace ,$ where $\\langle \\cdot ,\\cdot \\rangle $ denotes the Euclidean inner product on $\\mathbb {R}^{3}$ , and $\\Delta _{T}(\\vartheta _{0},X^{T})$ is a random variable such that $\\Delta _{T}(\\vartheta _{0},X^{T}) \\stackrel{{\\normalfont \\mbox{\\scriptsize {d}}}}{\\rightarrow } N(\\mathbf {0},\\mathbf {I}(\\vartheta _{0})),$ with $\\mathbf {I}(\\vartheta _{0})$ the Fisher information matrix evaluated at $\\vartheta _{0}$ , i.e.",
"$\\mathbf {I}(\\vartheta _{0}) &:= \\mathbb {E}^{(\\vartheta _{0})}\\left[\\frac{\\dot{\\mu }(\\vartheta _{0},\\xi )\\dot{\\mu }(\\vartheta _{0},\\xi )^{T}}{\\sigma ^{2}(\\xi )}\\right],$ where $\\dot{\\mu }(\\vartheta ,\\xi )^{T}$ is the transpose of the vector of derivatives of $\\mu (\\vartheta ,x)$ with respect to $\\vartheta $ .",
"Moreover, the function $r_{T}(\\vartheta _{0},\\mathbf {u},X^{T})$ satisfies $\\lim \\limits _{T\\rightarrow \\infty }r_{T}(\\vartheta _{0},\\mathbf {u},X^{T}) = 0 \\hspace*{5.69054pt}\\textnormal { in \\mathbb {P}^{(\\vartheta _{0})}_{\\nu }-probability}$ The family of measures is said to be LAN on $\\Theta $ if it is LAN at every point $\\vartheta _{0}\\in \\Theta $ , and further it is said to be uniformly LAN on $\\Theta $ if both convergence (REF ) and (REF ) are uniform in $\\vartheta \\in \\mathcal {K}$ for every compact $\\mathcal {K}\\subset \\Theta $ .",
"Theorem 2.8 The family of measures $\\lbrace \\mathbb {P}^{(\\vartheta )}_{\\nu },\\vartheta \\in \\Theta \\rbrace $ induced by the weak solution to (REF ) with initial distribution satisfying $\\lim _{|\\varepsilon |\\rightarrow 0}\\frac{\\nu (\\vartheta +\\varepsilon ,x)}{\\nu (\\vartheta ,x)} = 1, \\hspace*{8.53581pt} \\forall x\\in [0,1],$ $\\sup _{\\vartheta \\in \\mathcal {K}}\\Bigg \\lbrace \\int _{0}^{b}&\\frac{\\max \\lbrace e^{-s},1 \\rbrace }{\\theta _1-1}\\int _{x}^{b}\\xi ^{-1}(1-\\xi )^{-\\theta _2}d\\xi \\nu (\\vartheta ,dx) \\nonumber \\\\&+ \\int _{b}^{1}\\frac{\\max \\lbrace e^{s},1 \\rbrace }{\\theta _2+1}\\int _{b}^{x}\\xi ^{-\\max \\lbrace \\theta _1,2\\rbrace }d\\xi \\nu (\\vartheta ,dx) \\Bigg \\rbrace \\le C_{\\mathcal {K}}$ on any compact set $\\mathcal {K}\\subset \\Theta $ with $C_{\\mathcal {K}}>0$ constant, is uniformly LAN on $\\Theta $ , with the likelihood ratio function $Z_{T,\\vartheta }(\\mathbf {u})$ admitting the representation $Z_{T,\\vartheta }(\\mathbf {u}) = \\exp \\left\\lbrace \\left<\\mathbf {u},\\Delta _{T}(\\vartheta ,X^{T})\\right> - \\frac{1}{2}\\left<\\mathbf {I}(\\vartheta )\\mathbf {u},\\mathbf {u}\\right> + r_{T}(\\vartheta ,\\mathbf {u},X^{T}) \\right\\rbrace $ for $\\mathbf {u}\\in \\mathcal {U}_{T,\\vartheta }=\\lbrace \\mathbf {u} : \\vartheta +\\frac{\\mathbf {u}}{\\sqrt{T}}\\in \\Theta \\rbrace $ , where $\\Delta _{T}(\\vartheta ,X^{T}) = \\frac{1}{\\sqrt{T}}\\int _{0}^{T}\\frac{\\dot{\\mu }_{\\textnormal {WF}}(\\vartheta ,X_t)}{\\sigma _{\\textnormal {WF}}(X_t)}dW_t.",
"$ In particular the result holds for $\\nu = f_{\\vartheta }$ .",
"From (REF ), we have that the log-likelihood ratio is given by $\\log {Z_{T,\\vartheta }(\\mathbf {u})} = {}& \\log \\frac{\\nu (\\vartheta +\\frac{\\mathbf {u}}{\\sqrt{T}},X_0)}{\\nu (\\vartheta ,X_0)} \\nonumber \\\\&{}+ \\int _{0}^{T}\\frac{1}{2}\\left(\\frac{u_1}{\\sqrt{T}}\\sqrt{X_t (1-X_t)} + \\frac{u_2}{\\sqrt{T}}\\sqrt{\\frac{1-X_t}{X_t}} - \\frac{u_{3}}{\\sqrt{T}}\\sqrt{\\frac{X_t}{1-X_t}}\\right)dW_t \\nonumber \\\\&{}-\\frac{1}{2}\\int _{0}^{T}\\frac{1}{4}\\left(\\frac{u_1}{\\sqrt{T}}\\sqrt{X_t (1-X_t)} + \\frac{u_2}{\\sqrt{T}}\\sqrt{\\frac{1-X_t}{X_t}} - \\frac{u_{3}}{\\sqrt{T}}\\sqrt{\\frac{X_t}{1-X_t}}\\right)^{2}dt \\nonumber \\\\={}& \\log \\frac{\\nu (\\vartheta +\\frac{\\mathbf {u}}{\\sqrt{T}},X_0)}{\\nu (\\vartheta ,X_0)} + \\left<\\mathbf {u},\\Delta _{T}(\\vartheta ,X^T)\\right> - \\frac{1}{2}\\left<\\mathbf {I}(\\vartheta )\\mathbf {u},\\mathbf {u}\\right> \\nonumber \\\\&{}+\\frac{1}{2}\\left<\\mathbf {I}(\\vartheta )\\mathbf {u},\\mathbf {u}\\right> - \\frac{1}{2T}\\int _{0}^{T}\\frac{\\left<\\mathbf {u},\\dot{\\mu }_{\\textnormal {WF}}(\\vartheta ,X_t)\\right>^{2}}{\\sigma ^{2}_{\\textnormal {WF}}(X_t)}dt,$ where $\\mathbf {I}(\\vartheta ) = \\mathbb {E}^{(\\vartheta )}\\left[\\frac{1}{4}\\begin{pmatrix}\\xi (1-\\xi ) & 1-\\xi & -\\xi \\\\ 1-\\xi & \\frac{1-\\xi }{\\xi } & -1 \\\\ -\\xi & -1 & \\frac{\\xi }{1-\\xi }\\end{pmatrix}\\right].$ Setting $r_T (\\vartheta ,\\mathbf {u},X^{T}) := \\log \\frac{\\nu (\\vartheta +\\frac{\\mathbf {u}}{\\sqrt{T}},X_0)}{\\nu (\\vartheta ,X_0)} + \\frac{1}{2}\\left<\\mathbf {I}(\\vartheta )\\mathbf {u},\\mathbf {u}\\right> - \\frac{1}{2T}\\int _{0}^{T}\\frac{\\left<\\mathbf {u},\\dot{\\mu }_{\\textnormal {WF}}(\\vartheta ,X_t)\\right>^{2}}{\\sigma ^{2}_{\\textnormal {WF}}(X_t)}dt,$ we show that (REF ) holds.",
"The first term goes to 0 as $T\\rightarrow \\infty $ by (REF ), and in particular $\\nu = f_{\\vartheta }$ as given in (REF ) is continuous in $\\vartheta $ .",
"Thus we deduce that (REF ) follows if we can prove that for any $\\varepsilon >0$ $\\lim \\limits _{T\\rightarrow \\infty }\\sup _{\\vartheta \\in \\mathcal {K}}\\mathbb {P}^{(\\vartheta )}_{\\nu }\\left[\\left|\\frac{1}{T}\\int _{0}^{T}\\frac{\\left<\\mathbf {u},\\dot{\\mu }_{\\textnormal {WF}}(\\vartheta ,X_t)\\right>^{2}}{\\sigma ^{2}_{\\textnormal {WF}}(X_t)}dt -\\left<\\mathbf {I}(\\vartheta )\\mathbf {u},\\mathbf {u}\\right> \\right|>\\varepsilon \\right] = 0.$ Observe that the expression inside the probability in (REF ) is made up of six distinct differences between the averages of the six distinct entries of the Fisher information matrix with respect to time and the stationary density.",
"Thus if we are able to show that each individual difference displays the same convergence as in (REF ), (REF ) follows.",
"Now, as $\\frac{\\left<\\mathbf {u},\\dot{\\mu }_{\\textnormal {WF}}(\\vartheta ,x)\\right>^{2}}{\\sigma ^{2}_{\\textnormal {WF}}(x)} &= \\frac{1}{4}\\left( u_1\\sqrt{x(1-x)} + u_2\\sqrt{\\frac{1-x}{x}} -u_3\\sqrt{\\frac{x}{1-x}} \\right)^{2} \\\\&= \\frac{1}{4}\\left(u_{1}^{2}x(1-x) + 2u_1 u_2 (1-x) - 2u_1 u_3 x - 2u_2 u_3 + u_{2}^{2}\\frac{1-x}{x} + u_{3}^{2}\\frac{x}{1-x}\\right)$ using (REF ), we can apply Corollary REF to the first four terms directly.",
"The remaining two differences involve the unbounded functions $(1-x)x^{-1}$ and $x(1-x)^{-1}$ , for which (REF ) has been shown to hold in Corollary REF with $\\nu $ satisfying (REF ).",
"Thus (REF ) holds (we also show in Corollary REF that (REF ) holds in the case $\\nu =f_{\\vartheta }$ ), and (REF ) follows from Proposition 1.20 in [13] which we can invoke in view of the above proved (REF ) and the fact that $\\sup \\limits _{\\vartheta \\in \\mathcal {K}} \\sqrt{\\left<\\mathbf {I}(\\vartheta )\\mathbf {u},\\mathbf {u}\\right>} < \\infty .$ We point out here that if the mutation parameters are known, condition (REF ) becomes redundant and Theorem REF holds for any initial distribution satisfying $\\lim _{\\varepsilon \\rightarrow 0}\\nu (s+\\varepsilon ,x)/\\nu (s,x) = 1$ for any $x\\in [0,1]$ ."
],
[
"Properties of the ML & Bayesian Estimators for the Wright–Fisher diffusion",
"We henceforth assume that the mutation parameters $\\theta _1, \\theta _2 >0$ are known, and thus focus on conducting inference solely on the selection parameter $s\\in \\mathcal {S}\\subset \\mathbb {R}$ with $\\mathcal {S}$ open and bounded.",
"Remark 3.1 The continuous observation regime entertained here would enable one to infer the mutation parameters: on $\\vartheta \\in \\mathbb {R}\\times (0,1)^{2}$ this is immediate as the family of measures $\\lbrace \\mathbb {P}_{\\nu }^{(\\vartheta )} : \\vartheta \\in \\mathbb {R}\\times (0,1)^{2} \\rbrace $ are mutually singular.",
"In particular, when either mutation parameter is less than 1, the diffusion hits the corresponding boundary in finite time almost surely, and as it does so, the diffusion coefficient (i.e.",
"noise) vanishes sufficiently quickly allowing the mutation parameters to be inferred without error.",
"Indeed, by looking at the integrands on the RHS of (REF ), we observe that as the path approaches either boundary, the likelihood ratio explodes.",
"On $\\vartheta \\in \\mathbb {R}\\times [1,\\infty )^2$ the family of measures $\\lbrace \\mathbb {P}_{\\nu }^{(\\vartheta )} : \\vartheta \\in \\mathbb {R}\\times [1,\\infty )^2 \\rbrace $ are now mutually absolutely continuous, with both boundary points unattainable.",
"However, the process can get arbitrarily close to either boundary as $T\\rightarrow \\infty $ , and again the noise vanishes sufficiently quickly that the corresponding mutation parameters can be inferred to any required precision.",
"In the case when one mutation parameter is less than 1 and the other is greater than or equal to 1, similar arguments apply.",
"Actually incorporating inference of the mutation parameter into the inferential setup below leads to some technical difficulties which we discuss in Section , so for simplicity we assume them to be known.",
"Nonetheless all the notation introduced above and definitions carry through by replacing $\\vartheta $ by $s$ .",
"We start by defining the MLE $\\hat{s}_{T}$ of $s$ in (REF ) as $\\hat{s}_{T} = \\arg \\sup _{s\\in \\mathcal {S}}\\frac{d\\mathbb {P}^{(s)}_{\\nu }}{d\\mathbb {P}^{(s_{0})}_{\\nu }}(X^T)$ where $s_{0}\\in \\mathcal {S}$ is arbitrary and its only role is to specify a reference measure whose exact value does not matter.",
"Observe that now (REF ) simplifies to $\\frac{d\\mathbb {P}^{(s^{\\prime })}_{\\nu }}{d\\mathbb {P}^{(s)}_{\\nu }}(X^T) = \\frac{\\nu (s^{\\prime },X_0)}{\\nu (s,X_0)}\\exp \\Bigg \\lbrace &\\int _{0}^{T}\\left(s^{\\prime }-s\\right)\\sqrt{X_t(1-X_t)}dW_{t} \\nonumber \\\\&\\quad - \\frac{1}{2}\\int _{0}^{T}\\left(s^{\\prime }-s\\right)^{2}X_t(1-X_t)dt\\Bigg \\rbrace ,$ with initial distributions $\\lbrace \\nu (s,\\cdot )\\rbrace _{s\\in \\mathcal {S}}$ admitting a density (which we denote $\\nu (s,\\cdot )$ ) with respect to some common dominating measure $\\lambda (\\cdot )$ .",
"In order to be able to define the Bayesian estimator, we introduce the class ${W}_{p}$ of loss functions $\\ell : \\mathcal {S} \\rightarrow \\mathbb {R_{+}}$ for which the following stipulations are satisfied: $\\ell (\\cdot )$ is even, non-negative, and continuous at 0 with $\\ell (0) = 0$ but not identically zero.",
"The sets $\\lbrace u\\in \\mathcal {S}:\\:\\ell (u) < c \\rbrace $ are convex $\\forall c>0$ (and thus $\\ell (\\cdot )$ is non-decreasing).",
"$\\ell (\\cdot )$ has a polynomial majorant, i.e.",
"there exist strictly positive constants $A$ and $b$ such that for any $u\\in \\mathcal {S}$ , $|\\ell (u)| \\le A(1+|u|^{b})$ For any $H > 0$ sufficiently large and for sufficiently small $\\gamma $ , it holds that $\\inf \\limits _{|u|>H}\\ell (u) - \\sup \\limits _{|u|\\le H^{\\gamma }} \\ell (u) \\ge 0.$ As remarked above, we assume that $\\mathcal {S}$ is an open and bounded subset of $\\mathbb {R}$ , and we denote by $p(\\cdot )$ the prior density on $\\mathcal {S}$ , which we assume belongs to ${P}_{c} := \\left\\lbrace p(\\cdot ) \\in C(\\bar{\\mathcal {S}},\\mathbb {R}_{+}) \\hspace*{2.84526pt} : \\hspace*{2.84526pt} p(u) \\le A(1+|u|^{b}) \\hspace*{5.69054pt} \\forall \\hspace*{2.84526pt}u\\in \\bar{\\mathcal {S}},\\hspace*{14.22636pt} \\int _{\\bar{\\mathcal {S}}}p(u)du=1 \\right\\rbrace ,$ where $A$ and $b$ are some strictly positive constants, and $\\bar{\\mathcal {S}}$ denotes the closure of $\\mathcal {S}$ .",
"With $p(\\cdot ) \\in {P}_{c}$ and $\\ell (\\cdot )\\in {W}_{p}$ , we define the Bayesian estimator $\\tilde{s}_{T}$ of $s$ in (REF ) as $\\tilde{s}_{T} = \\arg \\min \\limits _{\\bar{s}_{T}} \\int _{\\mathcal {S}}\\mathbb {E}^{(s)}_{\\nu }\\left[\\ell \\left(\\sqrt{T}\\left(\\bar{s}_{T}-s\\right)\\right)\\right]p(s)ds,$ where the minimization is over estimators $\\bar{s}_{T} = \\bar{s}_{T}(X^T)$ .",
"We introduce the last class of functions we will need, namely denote by ${G}$ the class of functions satisfying the following two conditions: For a fixed $T>0$ , $g_{T}(\\cdot )$ is a monotonically increasing function on $[0,\\infty )$ , with $g_{T}(y)\\rightarrow \\infty $ as $y\\rightarrow \\infty $ .",
"For any $N>0$ , $\\lim \\limits _{\\begin{array}{c}T\\rightarrow \\infty \\\\ y\\rightarrow \\infty \\end{array}}y^{N}e^{-g_{T}(y)} = 0.$ Observe that the likelihood ratio function is now given by $Z_{T,s}(u) :&= \\frac{d\\mathbb {P}^{(s+\\frac{u}{\\sqrt{T}})}_{\\nu }}{d\\mathbb {P}^{(s)}_{\\nu }}(X^T) \\nonumber \\\\&=\\frac{\\nu (s+\\frac{u}{\\sqrt{T}},X_0)}{\\nu (s,X_0)}\\exp \\Bigg \\lbrace \\left(\\frac{u}{2\\sqrt{T}}\\right)\\int _{0}^{T}\\sqrt{X_t(1-X_t)}dW_{t} \\nonumber \\\\&\\qquad {}\\qquad {}\\qquad {}\\qquad {}\\qquad {}- \\frac{1}{2}\\left(\\frac{u}{2\\sqrt{T}}\\right)^2\\int _{0}^{T}X_t(1-X_t)dt\\Bigg \\rbrace $ for $u\\in \\mathcal {U}_{T,s}:=\\left\\lbrace u\\in \\mathbb {R} : s + \\frac{u}{\\sqrt{T}} \\in \\mathcal {S} \\right\\rbrace .$ We now present the main result of this article which states that the ML and Bayesian estimators for $s$ have a set of desirable properties.",
"We prove this by showing that the conditions of Theorems I.5.1, I.5.2, I.10.1, and I.10.2 in [9] are satisfied for the Wright–Fisher diffusion.",
"A similar formulation of the result below for the general case of a continuously observed diffusion on $\\mathbb {R}$ can be found in Theorems 2.8 and 2.13 in [13], where the author proves that the conditions necessary to invoke Theorems I.5.1, I.5.2, I.10.1, and I.10.2 in [9] hold for a certain class of diffusions.",
"However, this class includes only scalar diffusions for which the inverse of the diffusion coefficient has a polynomial majorant.",
"This fails to hold in our case, forcing us to seek alternative ways to prove that the conditions of the above mentioned theorems hold.",
"Theorem 3.2 Let $\\bar{s}_{T}$ be either the ML or Bayesian estimator for the selection parameter $s\\in \\mathcal {S}$ (for open bounded $\\mathcal {S}\\subset \\mathbb {R}$ ) in the neutral or non-neutral Wright–Fisher diffusion (REF ) with initial distribution satisfying $\\lim _{\\varepsilon \\rightarrow 0}\\frac{\\nu (s+\\varepsilon ,x)}{\\nu (s,x)} = 1, \\hspace*{8.53581pt} \\forall x \\in [0,1],$ and such that for any $M\\ge 2$ and $u\\in \\mathcal {U}_{T,s}$ , $\\mathbb {P}_{\\nu }^{(s)}\\left[\\left|\\log \\left(\\frac{\\nu (s+\\frac{u}{\\sqrt{T}},X_0)}{\\nu (s,X_0)}\\right)\\right| > \\frac{1}{48}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]\\left|u\\right|^{2}\\right] \\le \\frac{C_{1}}{|u|^{M}}$ and for any $R>0$ and $u,v \\in \\mathcal {U}_{T,s}$ with $|u| < R, |v| < R$ $\\int _{0}^{1}\\left| \\nu \\left(s+\\frac{u}{\\sqrt{T}},x\\right)^{\\frac{1}{2}} - \\nu \\left(s+\\frac{v}{\\sqrt{T}},x\\right)^{\\frac{1}{2}} \\right|^{2}\\lambda (dx) \\le C_{2}\\left|u-v\\right|^{2}$ for some constants $C_{1}, C_{2} >0$ , and $\\lambda (\\cdot )$ common dominating measure introduced below (REF ) (in particular these conditions hold for the case $\\nu =f_{s}$ , the stationary density).",
"Then $\\bar{s}_{T}$ is uniformly over compact sets $\\mathcal {K}\\subset \\mathcal {S}$ consistent, i.e.",
"for any $\\varepsilon >0$ $\\lim \\limits _{T\\rightarrow \\infty }\\sup \\limits _{s\\in \\mathcal {K}}\\mathbb {P}^{(s)}_{\\nu }\\big [\\left|\\bar{s}_{T}-s\\right|>\\varepsilon \\big ] = 0;$ it converges in distribution to a normal random variable $\\sqrt{T}\\left(\\bar{s}_{T}-s\\right) \\stackrel{{\\normalfont \\mbox{\\scriptsize {d}}}}{\\rightarrow } N(0,I(s)^{-1}),$ uniformly in $s\\in \\mathcal {K}$ , where $I(s) = \\frac{1}{4}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right];$ and it displays moment convergence for any $p>0$ $\\lim \\limits _{T\\rightarrow \\infty }\\mathbb {E}^{(s)}_{\\nu }\\bigg [\\left|\\sqrt{T}\\left(\\bar{s}_{T}-s\\right)\\right|^{p}\\bigg ] = \\mathbb {E}\\bigg [\\left|I(s)^{-\\frac{1}{2}}\\zeta \\right|^{p}\\bigg ]$ uniformly in $s\\in \\mathcal {K}$ , where $\\zeta \\sim N(0,1)$ , for any compact set $\\mathcal {K}\\subset \\mathcal {S}$ .",
"Furthermore, if the loss function $\\ell (\\cdot )\\in {W}_{p}$ , then $\\bar{s}_{T}$ is also asymptotically efficient, i.e.",
"$\\lim \\limits _{\\delta \\rightarrow 0}\\lim \\limits _{T\\rightarrow \\infty }\\sup \\limits _{s:|s-s_{0}|<\\delta }\\mathbb {E}^{(s)}_{\\nu }\\left[\\ell \\left(\\sqrt{T}\\left(\\bar{s}_{T}-s\\right)\\right)\\right] = \\mathbb {E}\\left[\\ell \\left(I(s_{0})^{-\\frac{1}{2}}\\zeta \\right)\\right]$ holds for all $s_{0}\\in \\mathcal {S}$ , where $\\zeta \\sim N(0,1)$ .",
"As mentioned above, the proof relies on Theorems I.5.1, I.5.2, I.10.1, and I.10.2 in [9], which for reference we combine together in our notation into Theorem REF below.",
"Establishing that the conditions of Theorem REF hold for the Wright–Fisher diffusion is non-trivial as the standard arguments found in [13] no longer hold, and will thus be the main focus of this section.",
"The conclusions of Theorems I.5.1 and I.5.2 guarantee the uniform over compact sets consistency for the MLE and Bayesian estimator respectively, and also give that for any $\\varepsilon >0$ and for sufficiently large $T$ $\\sup \\limits _{s\\in \\mathcal {K}}\\mathbb {P}_{\\nu }^{(s)}\\left[\\left|\\sqrt{T}\\left(\\bar{s}_{T}-s\\right)\\right| > \\varepsilon \\right] \\le \\alpha e^{-\\beta g_{T}(\\varepsilon )}$ with $\\alpha , \\beta $ strictly positive constants, and $g_{T} \\in {G}$ .",
"On the other hand, Theorems I.10.1 and I.10.2 provide the necessary conditions to deduce the uniform in $s\\in \\mathcal {K}$ asymptotic normality and convergence of moments for compact $\\mathcal {K}\\subset \\mathcal {S}$ , as well as asymptotic efficiency.",
"Theorem 3.3 (Ibragimov–Has'minskii) Let $\\bar{s}_{T}$ denote either the ML or Bayesian estimator for the parameter $s\\in \\mathcal {S}$ , for open bounded $\\mathcal {S}\\subset \\mathbb {R}$ , in (REF ), with prior density $p(\\cdot )\\in {P}_{c}$ , and loss function $\\ell (\\cdot )\\in {W}_{p}$ .",
"Suppose further that the following conditions are satisfied by the likelihood ratio function $Z_{T,s}(u)$ as defined in (REF ): $\\forall \\mathcal {K}\\subset \\mathcal {S}$ compact, we can find constants $a$ and $B$ , and functions $g_{T}(\\cdot )\\in {G}$ (all of which depend on $\\mathcal {K}$ ) such that the following two conditions hold: $\\forall R>0$ , $\\forall u, v \\in \\mathcal {U}_{T,s}$ as defined in (REF ) satisfying $|u|<R$ , $|v|<R$ , and for some $m\\ge q>1$ $\\sup \\limits _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}_{\\nu }\\left[\\left|Z_{T,s}(u)^{\\frac{1}{m}} - Z_{T,s}(v)^{\\frac{1}{m}}\\right|^{m}\\right] \\le B(1+R^{a})|u-v|^{q}.$ $\\forall u\\in \\mathcal {U}_{T,s}$ $\\sup \\limits _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}_{\\nu }\\left[Z_{T,s}(u)^{\\frac{1}{2}}\\right] \\le e^{-g_{T}(|u|)}.$ The random functions $Z_{T,s}(u)$ have marginal distributions which converge uniformly in $s\\in \\mathcal {K}$ as $T\\rightarrow \\infty $ to those of the random function $Z_{s}(u)\\in C_{0}(\\mathbb {R})$ , where $C_{0}(\\mathbb {R})$ denotes the space of continuous functions on $\\mathbb {R}$ vanishing at infinity, equipped with the supremum norm and the Borel $\\sigma $ -algebra.",
"The limit function $Z_{s}(u)$ attains its maximum at the unique point $\\hat{u}(s)=u$ with probability 1, and the random function $\\psi (v) = \\int _{\\mathbb {R}}\\ell (v-u)\\frac{Z_{s}(u)}{\\int _{\\mathbb {R}}Z_{s}(y)dy}du$ attains its minimum value at a unique point $\\tilde{u}(s) = u$ with probability 1.",
"Then we have that $\\bar{s}_{T}$ is: uniformly in $s\\in \\mathcal {K}$ consistent, i.e.",
"for any $\\varepsilon >0$ $\\lim \\limits _{T\\rightarrow \\infty }\\sup \\limits _{s\\in \\mathcal {K}}\\mathbb {P}^{(s)}_{\\nu }\\big [\\left|\\bar{s}_{T}-s\\right|>\\varepsilon \\big ] = 0,$ the distributions of the random variables $\\bar{u}_{T}=\\sqrt{T}\\left(\\bar{s}_{T}-s\\right)$ converge uniformly in $s\\in \\mathcal {K}$ to the distribution of $\\bar{u}$ , and for any loss function $\\ell \\in {W}_{p}$ uniformly in $s\\in \\mathcal {K}$ $\\lim \\limits _{T\\rightarrow \\infty }\\mathbb {E}^{(s)}_{\\nu }\\left[\\ell \\left(\\sqrt{T}\\left(\\bar{s}_{T}-s\\right)\\right)\\right] = \\mathbb {E}^{(s)}_{\\nu }\\left[\\ell (\\bar{u})\\right].$ For the Bayesian estimator, the requirements for inequality (REF ) can be weakened as it suffices to show that (REF ) holds for $m=2$ and any $q>0$ .",
"[Proof of Theorem REF ] Our aim will be to prove that Conditions 1, 2, and 3 in Theorem REF hold for the Wright–Fisher diffusion, for then the ML and Bayesian estimator are uniformly on compact sets consistent.",
"Below, Condition 1 is shown to hold in Propositions REF and REF ; Condition 2 is shown in Corollary REF ; and Condition 3 is shown in Proposition REF .",
"It remains to show how uniform in $s\\in \\mathcal {K}$ asymptotic normality and convergence of moments, as well as asymptotic efficiency (under the right choice of loss function) follow.",
"Given Conditions 1, 2, and 3 of Theorem 3.2, uniform in $s\\in \\mathcal {K}$ asymptotic normality follows immediately from Proposition REF ; $\\bar{u} = I(s)^{-1}\\Delta (s)$ , $\\Delta (s) \\sim N(0,I(s))$ , and $\\bar{u}_{T}$ converges uniformly in distribution to $\\bar{u}$ .",
"Moreover, as stated in Remark I.5.1 in [9], the Ibragimov–Has'minskii conditions also give us a bound on the tails of the likelihood ratio, which can be translated into bounds on the tails of $|\\hat{u}_{T}|^{p}$ for any $p>0$ (see the display just below (2.27) in [13]).",
"Similar bounds on the tails of $|\\tilde{u}_{T}|^{p}$ hold for the Bayesian estimator by Theorem I.5.7 in [9], and thus we have that the random variables $|\\bar{u}_{T}|^{p}$ are uniformly integrable for any $p>0$ , uniformly in $s\\in \\mathcal {K}$ for any compact $\\mathcal {K}\\subset \\mathcal {S}$ .",
"Uniform convergence of the moments of the estimators follows from this and the uniform convergence in distribution (by applying a truncation argument).",
"For loss functions satisfying $\\ell (\\cdot )\\in {W}_{p}$ , observe that the uniform convergence in (REF ) allows us to deduce that $\\lim \\limits _{T\\rightarrow \\infty }\\sup \\limits _{s:|s-s_{0}|<\\delta }\\mathbb {E}^{(s)}_{\\nu }\\left[\\ell \\left(\\sqrt{T}\\left(\\bar{s}_{T}-s\\right)\\right)\\right] = \\sup \\limits _{s:|s-s_{0}|<\\delta }\\mathbb {E}\\left[\\ell \\left(I(s)^{-\\frac{1}{2}}\\zeta \\right)\\right]$ for $\\zeta \\sim N(0,1)$ .",
"As $I(s)$ is continuous in $s$ , we have that $\\lim \\limits _{\\delta \\rightarrow 0}\\sup \\limits _{s:|s-s_{0}|<\\delta }\\mathbb {E}\\left[\\ell \\left(I(s)^{-\\frac{1}{2}}\\zeta \\right)\\right] = \\mathbb {E}\\left[\\ell \\left(I(s_{0})^{-\\frac{1}{2}}\\zeta \\right)\\right],$ giving asymptotic efficiency.",
"We proceed to show that Conditions 1, 2, and 3 in Theorem REF hold for the Wright–Fisher diffusion.",
"Theorem REF gives us that the Wright–Fisher diffusion is uniformly LAN, which immediately gives the required marginal convergence of the $Z_{T,s}(u)$ in Condition 2.",
"Corollary 3.4 For any initial distribution satisfying $\\lim _{\\varepsilon \\rightarrow 0}\\frac{\\nu (s+\\varepsilon ,x)}{\\nu (s,x)} = 1 \\hspace*{8.53581pt} \\forall x \\in [0,1],$ the random functions $Z_{T,s}(u)$ given by $Z_{T,s}(u) &= \\exp \\left\\lbrace \\frac{u}{2\\sqrt{T}}\\int _{0}^{T}\\sqrt{X_{t}(1-X_{t})}dW_{t} - \\frac{u^{2}}{8}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right] + r_{T}(s,u,X^{T}) \\right\\rbrace \\\\&=: \\exp \\left\\lbrace u\\Delta _{T}(s) - \\frac{u^{2}}{2}I(s) + r_{T}(s,u,X^{T})\\right\\rbrace ,$ where $r_{T}(s,u,X^{T}) := \\log \\left(\\frac{\\nu (s+\\frac{u}{\\sqrt{T}},X_0)}{\\nu (s,X_0)}\\right)+\\frac{u^2}{8}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right] - \\frac{1}{2}\\left(\\frac{u}{2\\sqrt{T}}\\right)^{2}\\int _{0}^{T}X_t(1-X_t)dt,$ have marginal distributions which converge uniformly in $s\\in \\mathcal {K}$ as $T\\rightarrow \\infty $ to those of the random function $Z_{s}(u)\\in C_{0}(\\mathbb {R})$ given by $Z_{s}(u) &:= \\exp \\left\\lbrace u\\Delta (s) - \\frac{u^{2}}{2}I(s)\\right\\rbrace ,$ where $\\Delta (s) := \\lim \\limits _{T\\rightarrow \\infty }\\frac{1}{2\\sqrt{T}}\\int _{0}^{T}\\sqrt{X_{t}(1-X_{t})}dW_{t} \\sim N\\left(0,I(s)\\right).$ The result follows immediately from the uniform LAN of the family of measures as shown in Theorem REF ; see for illustration the display just before Lemma 2.10 in [13].",
"It is clear that $Z_{s}(u)$ vanishes at infinity and thus is an element of $C_{0}(\\mathbb {R})$ .",
"The next two results allow us to control the Hellinger distance of the likelihood ratio function as required by Condition 1 in Theorem REF .",
"Proposition 3.5 Let the initial distribution be such that for any $R>0$ and for $u,v \\in \\mathcal {U}_{T,s}$ as defined in (REF ) with $|u|<R, |v|<R$ $\\int _{0}^{1}\\left| \\nu \\left(s+\\frac{u}{\\sqrt{T}},x\\right)^{\\frac{1}{2}} - \\nu \\left(s+\\frac{v}{\\sqrt{T}},x\\right)^{\\frac{1}{2}} \\right|^{2} \\lambda (dx)\\le c\\left|u-v\\right|^{2}$ for some constant $c>0$ with dominating measure $\\lambda (\\cdot )$ as specified below (REF ).",
"Then for any $\\mathcal {K}\\subset \\mathcal {S}$ compact, we can find a constant $C$ such that for any $R>0$ , and for any $u,v\\in \\mathcal {U}_{T,s}$ as defined in (REF ) satisfying $|u|<R$ , $|v|<R$ , the following holds $\\sup \\limits _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}_{\\nu }\\left[\\Big |Z_{T,s}(u)^{\\frac{1}{2}}-Z_{T,s}(v)^{\\frac{1}{2}}\\Big |^{2}\\right] \\le C(1+R^{2}){|u-v|^{2}}.$ In particular the result holds for $\\nu = f_{s}$ .",
"In what follows we denote by $C_{i}$ , for $i\\in \\mathbb {N}$ , constants which do not depend on $u$ , $v$ , $s$ , or $T$ .",
"Observe that for any $s^{\\prime }, s^* \\in \\mathcal {S}$ it holds that $\\mathbb {E}^{(s^{\\prime })}_{\\nu }\\left[\\int _{0}^{T}\\left|\\frac{\\mu _{\\textnormal {WF}}(s^{\\prime },X_{t})-\\mu _{\\textnormal {WF}}(s^*,X_{t})}{\\sigma (X_{t})}\\right|^{4}dt\\right] &= \\mathbb {E}^{(s^{\\prime })}_{\\nu }\\left[\\int _{0}^{T}\\left|\\frac{(s^{\\prime }-s^*)}{2}\\sqrt{X_{t}(1-X_{t})}\\right|^{4}dt\\right] \\nonumber \\\\&\\le \\left(\\frac{s^{\\prime }-s^*}{4}\\right)^{4}T < \\infty ,$ and so we can use Lemma 1.13 and Remark 1.14 from [13] (as done in Lemma 2.10 there) to split the expectation in (REF ) into three $\\mathbb {E}^{(s)}_{\\nu }\\left[\\Big |Z_{T,s}^{\\frac{1}{2}}(u)-Z_{T,s}^{\\frac{1}{2}}(v)\\Big |^{2}\\right] &\\le C_{1}\\int _{0}^{1}\\left|\\nu (s_{u},x)^{\\frac{1}{2}} - \\nu (s_{v},x)^{\\frac{1}{2}}\\right|^{2}\\lambda (dx) \\nonumber \\\\&\\quad {} + C_{2} \\int _{0}^{T}\\mathbb {E}^{(s_{v})}_{\\nu }\\left[\\left(\\frac{\\mu _{\\textnormal {WF}}(s_{u},X_t)-\\mu _{\\textnormal {WF}}(s_{v},X_t)}{\\sigma (X_t)}\\right)^{2}\\right]dt \\nonumber \\\\&\\quad {} + C_{3} T \\int _{0}^{T} \\mathbb {E}^{(s_{v})}_{\\nu }\\left[\\left(\\frac{\\mu _{\\textnormal {WF}}(s_{u},X_t)-\\mu _{\\textnormal {WF}}(s_{v},X_t)}{\\sigma (X_t)}\\right)^{4}\\right]dt,$ where we denote $s_{u} = s + {u}/{\\sqrt{T}}$ and $s_{v}=s+{v}/{\\sqrt{T}}$ .",
"The first term on the RHS of (REF ) can be dealt with using (REF ), whilst for the second term observe that $\\int _{0}^{T} \\mathbb {E}^{(s_{v})}_{\\nu }\\left[\\left(\\frac{\\mu _{\\textnormal {WF}}(s_{u},X_t)-\\mu _{\\textnormal {WF}}(s_{v},X_t)}{\\sigma (X_t)}\\right)^{2}\\right]dt &= \\frac{|u-v|^{2}}{4T}\\int _{0}^{T}\\mathbb {E}^{(s_{v})}_{\\nu }\\left[X_t (1-X_t)\\right]dt \\nonumber \\\\&\\le \\frac{1}{16}|u-v|^{2}.$ Therefore $C_{2} \\int _{0}^{T}\\mathbb {E}^{(s_{v})}_{\\nu }\\left[\\left(\\frac{\\mu _{\\textnormal {WF}}(s_{u},X_t)-\\mu _{\\textnormal {WF}}(s_{v},X_t)}{\\sigma (X_t)}\\right)^{2}\\right]dt &\\le C_{4}|u-v|^{2}.$ A similar calculation can be performed for the third term in (REF ) to get $C_{3} T \\int _{0}^{T}\\mathbb {E}^{(s_{v})}_{\\nu }\\left[\\left(\\frac{\\mu _{\\textnormal {WF}}(s_{u},X_t)-\\mu _{\\textnormal {WF}}(s_{v},X_t)}{\\sigma (X_t)}\\right)^{4}\\right]dt \\le C_{5}|u-v|^{4},$ and thus the result holds in view of the fact that $|u|, |v|<R$ .",
"It remains to show that (REF ) holds for $\\nu =f_{s}$ .",
"To this end, observe that $\\int _{0}^{1}\\left|f_{s_{u}}(x)^{\\frac{1}{2}}-f_{s_{v}}(x)^{\\frac{1}{2}}\\right|^{2}dx = \\int _{0}^{1}x^{\\theta _{1}-1}\\left(1-x\\right)^{\\theta _{2}-1}e^{sx}\\Big |\\frac{1}{\\sqrt{G_{s_{u}}}}e^{\\frac{ux}{2\\sqrt{T}}}-\\frac{1}{\\sqrt{G_{s_{v}}}}e^{\\frac{vx}{2\\sqrt{T}}}\\Big |^{2}dx.$ Now we have that $C_{6}\\min \\lbrace e^{s},1\\rbrace \\le G_{s_{u}} &:= \\int _{0}^{1}x^{\\theta _{1}-1}(1-x)^{\\theta _{2}-1}e^{\\left(s+\\frac{u}{\\sqrt{T}}\\right)x}dx \\le C_{7}\\max \\lbrace e^{s},1\\rbrace ,$ where $C_6=B(\\theta _1,\\theta _2)e^{-\\textnormal {diam}(\\mathcal {S})}$ , $C_7 = B(\\theta _1,\\theta _2)e^{\\textnormal {diam}(\\mathcal {S})}$ are non-zero, positive, and independent of $s$ and $T$ , since we constrain $u,v\\in \\mathcal {U}_{T,s}$ .",
"This allows us to deduce that $G \\mapsto 1/\\sqrt{G}$ is Lipschitz on $[C_{6}\\inf _{s\\in \\mathcal {K}}\\min \\lbrace e^{s},1\\rbrace ,C_{7}\\sup _{s\\in \\mathcal {K}}\\max \\lbrace e^{s},1\\rbrace ]$ with some constant $C_{8}>0$ , i.e.",
"$\\Bigg |\\frac{1}{\\sqrt{G_{s_{u}}}}-\\frac{1}{\\sqrt{G_{s_{v}}}}\\Bigg | &\\le C_{8}\\Big |G_{s_{u}} - G_{s_{v}}\\Big | \\\\&= C_{8}\\int _{0}^{1}x^{\\theta _{1}-1}\\left(1-x\\right)^{\\theta _{2}-1}e^{sx}\\left|e^{\\frac{ux}{2\\sqrt{T}}}-e^{\\frac{vx}{2\\sqrt{T}}}\\right|dx \\\\&\\le C_{8}C_{9}\\int _{0}^{1}x^{\\theta _{1}-1}\\left(1-x\\right)^{\\theta _{2}-1}e^{sx}\\left|\\frac{ux}{2\\sqrt{T}}-\\frac{vx}{2\\sqrt{T}}\\right|dx \\\\&= \\frac{C_{8}C_{9}}{2\\sqrt{T}}\\left|u-v\\right|\\int _{0}^{1}x^{\\theta _{1}}\\left(1-x\\right)^{\\theta _{2}-1}e^{sx}dx \\\\&\\le \\frac{C_{10}}{\\sqrt{T}}\\max \\lbrace e^{s},1 \\rbrace \\left|u-v\\right|,$ where in the second inequality we have made use of the fact that $e^{z}$ is Lipschitz in $z$ on $[-\\textnormal {diam}(\\mathcal {S}),\\textnormal {diam}(\\mathcal {S})]$ with some constant $C_{9}>0$ .",
"Thus we deduce that $\\Big |\\frac{1}{\\sqrt{G_{s_{u}}}}e^{\\frac{ux}{2\\sqrt{T}}}-\\frac{1}{\\sqrt{G_{s_{v}}}}e^{\\frac{vx}{2\\sqrt{T}}}\\Big |^{2} &= \\Bigg |\\frac{1}{\\sqrt{G_{s_{u}}}}\\left(e^{\\frac{ux}{2\\sqrt{T}}}-e^{\\frac{vx}{2\\sqrt{T}}}\\right)\\Bigg |^{2} +\\Bigg |e^{\\frac{vx}{2\\sqrt{T}}}\\left(\\frac{1}{\\sqrt{G_{s_{u}}}}-\\frac{1}{\\sqrt{G_{s_{v}}}}\\right)\\Bigg |^{2} \\nonumber \\\\&\\quad {} + 2\\frac{e^{\\frac{vx}{2\\sqrt{T}}}}{\\sqrt{G_{s_{u}}}}\\Big |e^{\\frac{ux}{2\\sqrt{T}}}-e^{\\frac{vx}{2\\sqrt{T}}}\\Big |\\Big |\\frac{1}{\\sqrt{G_{s_{u}}}}-\\frac{1}{\\sqrt{G_{s_{v}}}}\\Big |\\nonumber \\\\&\\le \\frac{C_{9}^{2}x^{2}}{4T}\\frac{1}{C_{6}\\min \\lbrace e^{s},1\\rbrace }\\big |u-v\\big |^{2}\\nonumber \\\\&\\quad {}+ e^{\\textnormal {diam}(\\mathcal {S})x}\\frac{C_{10}^{2}}{T}\\max \\lbrace e^{2s},1\\rbrace \\big |u-v\\big |^{2} \\nonumber \\\\&\\quad {} + \\frac{e^{\\textnormal {diam}(\\mathcal {S})x}C_{9}C_{10}x}{T\\sqrt{C_{6}}}\\frac{\\max \\lbrace e^{s},1\\rbrace }{\\min \\lbrace e^{s/2},1\\rbrace }\\big |u-v\\big |^{2}.$ Putting (REF ) into (REF ) gives us the result $\\int _{0}^{1}x^{\\theta _{1}-1}\\left(1-x\\right)^{\\theta _{2}-1}e^{sx}\\Big |\\frac{1}{\\sqrt{G_{s_{u}}}}e^{\\frac{ux}{\\sqrt{T}}}-\\frac{1}{\\sqrt{G_{s_{v}}}}e^{\\frac{vx}{\\sqrt{T}}}\\Big |^{2}dx &\\le \\frac{C_s}{T}|u-v|^2,$ as $C_{s} := C_{11}e^{|s|} + C_{12}\\max \\lbrace e^{3s},1\\rbrace + C_{13}\\frac{\\max \\lbrace e^{2s},1\\rbrace }{\\min \\lbrace e^{s/2},1\\rbrace },$ is continuous in $s$ over any compact set $\\mathcal {K}\\subset \\mathcal {S}$ .",
"Proposition 3.6 Let the initial distribution be such that for any $M\\ge 2$ and for $u\\in \\mathcal {U}_{T,s}$ , $\\mathbb {P}_{\\nu }^{(s)}\\left[\\left|\\log \\left(\\frac{\\nu (s+\\frac{u}{\\sqrt{T}},X_0)}{\\nu (s,X_0)}\\right)\\right| > \\frac{1}{48}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]\\left|u\\right|^{2}\\right] \\le \\frac{C}{|u|^{M}}$ for some constant $C>0$ .",
"Then for $\\mathcal {K}\\subset \\mathcal {S}$ compact, there exists a function $g_{T}(\\cdot )\\in {G}$ such that for any $u\\in \\mathcal {U}_{T,s}$ as defined in (REF ) we have that $\\sup _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}_{\\nu }\\left[Z_{T,s}(u)^{\\frac{1}{2}}\\right] \\le e^{-g_{T}(|u|)}.$ The result holds in particular when $\\nu = f_{s}$ .",
"Assume for now that for any $M\\ge 2$ we have that $\\mathbb {P}^{(s)}_{\\nu }\\left[Z_{T,s}(u) > \\exp \\left\\lbrace -\\frac{1}{16}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|u|^{2}\\right\\rbrace \\right] \\le \\frac{C_{s,M}}{|u|^{M}}$ for some constant $C_{s,M}>0$ depending on $s$ and $M$ .",
"We show that if (REF ) holds, then (REF ) follows.",
"Indeed $\\mathbb {E}^{(s)}_{\\nu }\\left[Z_{T,s}(u)^{\\frac{1}{2}}\\right] &= \\mathbb {E}^{(s)}_{\\nu }\\left[Z_{T,s}(u)^{\\frac{1}{2}}1_{\\left\\lbrace Z_{T,s}(u) \\le \\exp \\left\\lbrace -\\frac{1}{16}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|u|^{2}\\right\\rbrace \\right\\rbrace }\\right] \\\\&\\quad {} + \\mathbb {E}^{(s)}_{\\nu }\\left[Z_{T,s}(u)^{\\frac{1}{2}}1_{\\left\\lbrace Z_{T,s}(u) > \\exp \\left\\lbrace -\\frac{1}{16}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|u|^{2}\\right\\rbrace \\right\\rbrace }\\right] \\\\&\\le \\exp \\left\\lbrace -\\frac{1}{32}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|u|^{2}\\right\\rbrace \\\\&\\quad {}+ \\mathbb {E}^{(s)}_{\\nu }\\left[Z_{T,s}(u)\\right]^{\\frac{1}{2}}\\mathbb {P}^{(s)}_{\\nu }\\left[Z_{T,s}(u) > \\exp \\left\\lbrace -\\frac{1}{16}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|u|^{2}\\right\\rbrace \\right]^{\\frac{1}{2}} \\\\&\\le \\exp \\left\\lbrace -\\frac{1}{32}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|u|^{2}\\right\\rbrace + \\frac{C_{s,M}}{|u|^{\\frac{M}{2}}}$ where in the first inequality we have made use of Cauchy-Schwarz, and for the second inequality we have used (REF ).",
"Therefore, $\\sup _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}_{\\nu }\\left[Z_{T,s}(u)^{\\frac{1}{2}}\\right] &\\le \\sup _{s\\in \\mathcal {K}}\\left\\lbrace \\exp \\left\\lbrace -\\frac{1}{32}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|u|^{2}\\right\\rbrace + \\frac{C_{s,M}}{|u|^{\\frac{M}{2}}} \\right\\rbrace \\\\&= \\exp \\left\\lbrace -\\frac{1}{32}\\inf _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|u|^{2}\\right\\rbrace + \\frac{\\sup _{s\\in \\mathcal {K}}C_{s,M}}{|u|^{\\frac{M}{2}}} \\\\&=: \\exp \\left\\lbrace -g_{T}(|u|)\\right\\rbrace .$ It remains to ensure that $g_{T}(\\cdot )\\in {G}$ , that $\\inf _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}[\\xi (1-\\xi )] \\ge \\kappa > 0$ for some constant $\\kappa $ , and that for any $M\\ge 2$ it holds that $\\sup _{s\\in \\mathcal {K}}C_{s,M}<\\infty $ .",
"Observe that $\\min \\left\\lbrace \\inf _{s\\in \\mathcal {K}}e^{s},1\\right\\rbrace B(\\theta _{1},\\theta _{2}) \\le G_{s} \\le \\max \\left\\lbrace \\sup _{s\\in \\mathcal {K}}e^{s},1\\right\\rbrace B(\\theta _{1},\\theta _{2}).$ Thus $\\inf _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right] &= \\inf _{s\\in \\mathcal {K}}\\left\\lbrace \\int _{0}^{1}\\frac{1}{G_{s}}e^{s\\xi }\\xi ^{\\theta _{1}}(1-\\xi )^{\\theta _{2}}d\\xi \\right\\rbrace \\nonumber \\\\&\\ge \\frac{\\inf _{s\\in \\mathcal {K}}\\left\\lbrace \\int _{0}^{1}e^{s\\xi }\\xi ^{\\theta _{1}}(1-\\xi )^{\\theta _{2}}d\\xi \\right\\rbrace }{\\max \\left\\lbrace \\sup _{s\\in \\mathcal {K}}e^{s},1\\right\\rbrace B(\\theta _{1},\\theta _{2})} \\nonumber \\\\&\\ge \\frac{\\min \\left\\lbrace \\inf _{s\\in \\mathcal {K}}e^{s},1\\right\\rbrace B(\\theta _{1}+1,\\theta _{2}+1)}{\\max \\left\\lbrace \\sup _{s\\in \\mathcal {K}}e^{s},1\\right\\rbrace B(\\theta _{1},\\theta _{2})} =: \\kappa $ and $\\kappa >0$ because $\\mathcal {K}$ is bounded, and thus both $\\sup _{s\\in \\mathcal {K}}e^{s}$ and $\\inf _{s\\in \\mathcal {K}}e^{s}$ are finite and non-zero.",
"We show that $\\sup _{s\\in \\mathcal {K}}C_{s,M}$ is finite $\\forall M\\ge 2$ in what follows.",
"We now check that $g_{T}(|u|)$ as defined above is in the class of functions ${G}$ .",
"To this end, observe that $g_{T}(|u|) = -\\log \\left(\\exp \\left\\lbrace -\\frac{1}{32}\\inf _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|u|^{2}\\right\\rbrace + \\frac{{\\sup _{s\\in \\mathcal {K}}C_{s,M}}}{|u|^{\\frac{M}{2}}}\\right).$ Indeed, for a fixed $T>0$ , $g_{T}(|u|)\\rightarrow \\infty $ as $|u|\\rightarrow \\infty $ , because $\\inf _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}[\\xi (1-\\xi )]>0$ , and furthermore given any fixed $N$ , we can choose $M$ large enough (note the way we phrased (REF ) allows us to choose our $M$ arbitrarily large, say $M>2N$ ) such that $\\lim \\limits _{\\begin{array}{c}T\\rightarrow \\infty \\\\y\\rightarrow \\infty \\end{array}}y^{N}e^{-g_{T}(y)} = \\lim \\limits _{\\begin{array}{c}T\\rightarrow \\infty \\\\y\\rightarrow \\infty \\end{array}}y^{N}\\left(\\exp \\left\\lbrace -\\frac{1}{32}\\inf _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|y|^{2}\\right\\rbrace + \\frac{\\sup _{s\\in \\mathcal {K}}C_{s,M}}{|y|^{\\frac{M}{2}}}\\right) = 0,$ where the order in which limits are taken is immaterial since our choice of $g_{T}(|u|)$ is independent of $T$ .",
"Thus we have proved that if (REF ) holds, then $\\sup _{s\\in \\mathcal {K}}\\mathbb {E}^{(s)}_{\\nu }\\left[Z_{T,s}(u)^{\\frac{1}{2}}\\right] \\le e^{-g_{T}(|u|)}, \\hspace*{28.45274pt}g_{T}(\\cdot )\\in {G}.$ To show that (REF ) holds, we make use of Markov's inequality as well as Theorem 3.2 in [14].",
"Indeed, observe that $\\mathbb {P}^{(s)}_{\\nu }&\\left[Z_{T,s}(u) \\ge \\exp \\left\\lbrace -\\frac{1}{16}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]|u|^{2}\\right\\rbrace \\right] \\\\&= \\mathbb {P}^{(s)}_{\\nu }\\Bigg [\\frac{\\nu (s+\\frac{u}{\\sqrt{T}},X_0)}{\\nu \\left(s,X_0\\right)}\\exp \\bigg \\lbrace \\frac{u}{2\\sqrt{T}}\\int _{0}^{T}\\sqrt{X_{t}(1-X_{t})}dW_{t} \\\\&\\quad {}- \\frac{|u|^{2}}{8}\\left(\\frac{1}{T}\\int _{0}^{T}X_{t}(1-X_{t})dt - \\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]\\right)\\bigg \\rbrace > \\exp \\left\\lbrace \\frac{1}{16}\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|^{2}\\right\\rbrace \\Bigg ] \\\\&\\le \\mathbb {P}^{(s)}_{\\nu }\\left[\\left|\\log \\left(\\frac{\\nu (s+\\frac{u}{\\sqrt{T}},X_0)}{\\nu \\left(s,X_0\\right)}\\right) \\right| > \\frac{1}{48}\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|^{2} \\right] \\\\&\\quad {} + \\mathbb {P}^{(s)}_{\\nu }\\left[\\left|\\frac{u}{2\\sqrt{T}}\\int _{0}^{T}\\sqrt{X_{t}(1-X_{t})}dW_{t}\\right| > \\frac{1}{48}\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|^{2} \\right] \\\\&\\quad {} + \\mathbb {P}^{(s)}_{\\nu }\\left[\\frac{|u|^{2}}{8}\\left|\\frac{1}{T}\\int _{0}^{T}{X_{t}(1-X_{t})}dt - \\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]\\right| > \\frac{1}{48}\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|^{2} \\right] \\\\& =: A_{1} + A_{2} + A_{3}.$ The bound for $A_{1}$ follows immediately from (REF ).",
"For the particular case when $\\nu =f_{s}$ , we use Markov's inequality: $A_{1} &= \\mathbb {P}^{(s)}_{\\nu }\\left[\\left|\\log \\left(\\frac{G_{s}}{G_{s+\\frac{u}{\\sqrt{T}}}}\\right) + \\frac{u}{\\sqrt{T}}X_{0}\\right| > \\frac{1}{48}\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|^{2}\\right] \\\\&\\le \\left(\\frac{48}{\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|^{2}}\\right)^{M}\\mathbb {E}^{(s)}_{\\nu }\\left[\\left|\\log \\left(\\frac{G_{s}}{G_{s+\\frac{u}{\\sqrt{T}}}}\\right) + \\frac{u}{\\sqrt{T}}X_{0}\\right|^{M}\\right].$ But $\\log \\left(\\frac{G_{s}}{G_{s+\\frac{u}{\\sqrt{T}}}}\\right) = \\log \\left(\\frac{\\int _{0}^{1}x^{\\theta _{1}-1}(1-x)^{\\theta _{2}-1}e^{sx}dx}{\\int _{0}^{1}x^{\\theta _{1}-1}(1-x)^{\\theta _{2}-1}e^{(s+\\frac{u}{\\sqrt{T}})x}dx}\\right) \\le {\\frac{|u|}{\\sqrt{T}}},$ so we have $A_{1} &\\le \\left(\\frac{48}{\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|^{2}}\\right)^{M}\\mathbb {E}^{(s)}_{\\nu }\\left[\\left|\\frac{u}{\\sqrt{T}}\\right|^{M}\\left|1 + X_{0}\\right|^{M}\\right] \\\\&= \\left(\\frac{48}{\\mathbb {E}^{(s)}[\\xi (1-\\xi )]\\sqrt{T}|u|}\\right)^{M}\\mathbb {E}^{(s)}\\left[\\left|1 + \\xi \\right|^{M}\\right] \\\\&\\le \\left(\\frac{48d_{s}}{\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|^{2}}\\right)^{M}\\mathbb {E}^{(s)}\\left[\\left|1 + \\xi \\right|^{M}\\right] =: \\frac{C^{(1)}_{s,M}}{|u|^{2M}},$ where in the second inequality we made use of the fact that $u\\in \\mathcal {U}_{T,s}$ , and thus $|u|\\le d_{s}\\sqrt{T}$ where we define $d_{s}:=\\sup _{w\\in \\mathcal {\\partial \\mathcal {S}}}|s-w|$ (which is strictly positive and bounded as $\\mathcal {S}$ is open and bounded).",
"To see that $\\sup _{s\\in \\mathcal {K}}C_{s,M}^{(1)}$ is bounded, observe that $\\sup _{s\\in \\mathcal {K}}C_{s,M}^{(1)} &= \\sup _{s\\in \\mathcal {K}}\\left\\lbrace \\left(\\frac{48d_{s}}{\\mathbb {E}^{(s)}[\\xi (1-\\xi )]}\\right)^{M}\\mathbb {E}^{(s)}\\left[\\left|1+\\xi \\right|^{M}\\right]\\right\\rbrace \\\\&\\le \\left(96\\frac{B(\\theta _{1},\\theta _{2})}{B(\\theta _{1}+1,\\theta _{2}+1) }\\sup _{s\\in \\mathcal {K}}d_{s}\\frac{\\max \\lbrace e^{s},1\\rbrace }{\\min \\lbrace e^{s},1\\rbrace }\\right)^{M},$ which is clearly finite because $\\mathcal {K}$ is bounded.",
"For $A_{2}$ we use a similar argument, but now use the fact that we have a stochastic integral: $A_{2} &\\le \\left(\\frac{48}{\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|^{2}}\\right)^{M}\\mathbb {E}^{(s)}_{\\nu }\\left[\\left|\\frac{u}{2\\sqrt{T}}\\int _{0}^{T}\\sqrt{X_{t}(1-X_{t})}dW_{t}\\right|^{M}\\right]\\\\&\\le \\left(\\frac{24}{\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|}\\right)^{M}\\left(\\frac{M}{2}(M-1)\\right)^{\\frac{M}{2}}T^{-1}\\mathbb {E}^{(s)}_{\\nu }\\left[\\int _{0}^{T}\\left|X_{t}(1-X_{t})\\right|^{\\frac{M}{2}}dt\\right] \\\\&\\le \\left(\\frac{12}{\\mathbb {E}^{(s)}[\\xi (1-\\xi )]|u|}\\right)^{M}\\left(\\frac{M}{2}(M-1)\\right)^{\\frac{M}{2}}=: \\frac{C^{(2)}_{s,M}}{|u|^{M}},$ where the first line uses Markov's inequality and the second inequality uses Lemma 1.1 (equation (1.3)) in [13].",
"That $\\sup _{s\\in \\mathcal {K}}C_{s,M}^{(2)}$ is finite follows from arguments similar to those used for the respective term in $A_1$ .",
"For $A_{3}$ we make use of Theorem 3.2 in [14], which gives us that for $M\\ge 2$ $\\mathbb {P}^{(s)}_{\\nu }\\Bigg [\\left|\\frac{1}{T}\\int _{0}^{T}{X_{t}(1-X_{t})}dt - \\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]\\right| &\\ge {\\frac{1}{6}\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]} \\Bigg ] \\nonumber \\\\&\\le K(s,X,M)\\frac{\\Vert x(1-x)\\Vert ^{M}_{\\infty }}{\\left(\\frac{\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]}{6}\\sqrt{T}\\right)^{M}}.$ For the RHS of (REF ), we have that $K(s,X,M)\\frac{\\Vert x(1-x)\\Vert ^{M}_{\\infty }}{\\left(\\frac{\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]}{6}\\sqrt{T}\\right)^{M}} \\le K(s,X,M)\\left(\\frac{6\\Vert x(1-x)\\Vert _{\\infty }d_{s}}{{\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]}|u|}\\right)^{M} =: \\frac{C^{(3)}_{s,M}}{|u|^{M}},$ where $K(s,X,M)$ is a function that depends on $M$ and on the moments of the hitting times of $X$ .",
"Finally we deduce that $\\sup _{s\\in \\mathcal {K}}C_{s,M}^{(3)}$ is finite by observing that $\\sup _{s\\in \\mathcal {K}}C_{s,M}^{(3)} &= \\sup _{s\\in \\mathcal {K}}K(s,X,M)\\left(\\frac{6\\Vert x(1-x)\\Vert _{\\infty }d_{s}}{{\\mathbb {E}^{(s)}\\left[\\xi (1-\\xi )\\right]}}\\right)^{M} \\\\&\\le \\sup _{s\\in \\mathcal {K}}K(s,X,M)\\left(\\frac{3}{2}\\frac{B(\\theta _{1},\\theta _{2})}{B(\\theta _{1}+1,\\theta _{2}+1) }\\sup _{s\\in \\mathcal {K}}d_{s}\\frac{\\max \\lbrace e^{s},1\\rbrace }{\\min \\lbrace e^{s},1\\rbrace }\\right)^{M},$ which is finite since $\\Vert x(1-x)\\Vert _{\\infty }=1/4$ , $\\mathcal {K}$ is compact, and $K(s,X,M)$ is bounded by a function which is continuous in $s$ (see Appendix for the corresponding details).",
"Finally, we present the result which guarantees that Condition 3 in Theorem REF holds, and thus that the Ibragimov–Has'minskii conditions hold for the Wright–Fisher diffusion.",
"Proposition 3.7 The random functions $Z_{s}(u)$ and $\\psi (v) := \\int _{\\mathbb {R}}\\ell \\left(v-u\\right)\\frac{Z_{s}(u)}{\\int _{\\mathbb {R}}Z_{s}(y)dy}du$ attain their maximum and minimum respectively at the unique point $\\bar{u} = \\bar{u}_{s} = I(s)^{-1}\\Delta (s)$ with probability 1.",
"The first assertion follows immediately from Corollary REF , whilst for the second we direct the interested reader to Theorem III.2.1 in [9], which relies on two results: Anderson's Lemma (Lemma II.10.1 in [9]), and Lemma II.10.2 in [9]."
],
[
"Numerical Simulations",
"We illustrate the results proved in Section by showing consistency, convergence in distribution and convergence of moments for the MLE when applied to data simulated from the Wright–Fisher diffusion.",
"By making use of the `exact algorithm' (see [11] for full details), we obtain exact draws from the Wright–Fisher diffusion.",
"The generated paths are then used to calculate the MLE, and subsequently kernel smoothed density estimates for the rescaled MLE for various terminal times $T$ are plotted against the density of the limiting distribution.",
"Using the definition in (REF ), the MLE for the selection parameter is given by $\\hat{s}_{T} = \\frac{X_{T}-X_{0} - \\int _{0}^{T}\\left(-\\theta _2 X_{t} + \\theta _{1}(1-X_{t})\\right)dt}{\\int _{0}^{T}X_{t}(1-X_{t})dt},$ which is impossible to calculate exactly in view of the infinite-dimensional path involved in the integral.",
"Instead we approximate the MLE by using Riemann sums instead of Lebesgue integrals, which gives rise to the approximation of $\\hat{s}_{T}$ given by $\\check{s}_{T} = \\frac{X_{T}-X_{0} - \\sum _{i=1}^{N}\\left(-\\theta _2 X_{t_i} + \\theta _{1}(1-X_{t_i})\\right)\\Delta _i}{\\sum _{i=1}^{N}X_{t_i}(1-X_{t_i})\\Delta _i}$ where $\\Delta _i := t_{i}-t_{i-1}$ for a time discretisation grid $\\lbrace t_i\\rbrace _{i=0}^{N}$ where $t_0 = 0$ and $t_{N} = T$ , and $N\\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace $ .",
"In particular, $\\lbrace X_{t_i}\\rbrace _{i=0}^{N}$ denotes the values of the Wright–Fisher path at the times $\\lbrace t_i\\rbrace _{i=0}^{N}$ , which corresponds to the output generated by the exact algorithm.",
"To simulate the Wright–Fisher paths, we set the selection parameter $s=4$ , the mutation parameters $\\theta _1, \\theta _2 =2$ (to ensure that we are within the regime of Theorem REF ), $\\Delta _i = 0.001$ , $X_0=0.25$ and varied the terminal time $T\\in \\lbrace 1,2,10,50\\rbrace $ .",
"For each of the 10,000 simulated paths, we computed (REF ), and subsequently for each $T$ we obtained kernel smoothed estimates of the density of $\\sqrt{T}(\\check{s}_{T}-s)$ which are plotted against the limiting $N(0,\\frac{1}{4}\\mathbb {E}^{(s)}[\\xi (1-\\xi )]^{-1})$ density in Figure REF .",
"Figure: Plots of the kernel smoothed density estimates for T(s ˇ T -s)\\sqrt{T}(\\check{s}_{T}-s) for T=1,2,10,50T=1,2,10,50, and of the limiting N(0,I(s) -1 )N(0,I(s)^{-1}) density."
],
[
"Discussion",
"In this article we have provided criteria which determine whether a general diffusion defined over a bounded interval $[l,r]$ with either boundary point being either entrance or regular is $\\vartheta $ -uniformly ergodic (Theorem REF ), together with a set of additional conditions which allow for an extension of the result to a specific class of unbounded functions for diffusions possessing solely entrance boundaries (Theorem REF ).",
"Using the criteria in Theorem REF , we have shown in Corollary REF that the Wright–Fisher diffusion is $\\vartheta $ -uniformly ergodic for $\\vartheta =(s,\\theta _1,\\theta _2)\\in \\Theta \\subset \\mathbb {R}\\times (0,\\infty )^{2}$ , extending the well-known pointwise in $\\vartheta =(s,\\theta _1,\\theta _2)$ ergodicity of the Wright–Fisher diffusion over any compact set $\\mathcal {K}\\subset \\mathbb {R}\\times (0,\\infty )^{2}$ for bounded functions.",
"We have also proved that the family of measures $\\lbrace \\mathbb {P}^{(\\vartheta )}_{\\nu } : \\vartheta \\in \\Theta \\rbrace $ induced by the solution to the SDE (REF ) are uniformly LAN when $\\Theta \\subset \\mathbb {R}\\times [1,\\infty )^{2}$ in Theorem REF (by making use of Corollary REF which uses the conditions derived in Theorem REF ), where the extra restriction on the mutation rates ensures that the likelihood ratio function is defined.",
"In Section 3 we then considered inference for the selection parameter $s$ when the diffusion is observed continuously through time and the mutation rates are known.",
"Under these assumptions, we proved that the ML and Bayesian estimators for $s\\in \\mathcal {S}$ ($\\mathcal {S}$ an open bounded subset of $\\mathbb {R}$ ) in the non-neutral Wright–Fisher diffusion started from a broad class of initial distributions which includes its stationary distribution, are uniformly over compact sets consistent and display uniform in $s\\in \\mathcal {K}$ asymptotic normality and convergence of moments, for any compact $\\mathcal {K}\\subset \\mathcal {S}$ .",
"Furthermore, for the right choice of loss function we also have asymptotic efficiency of the two estimators.",
"The uniformity in these results is particularly useful as it guarantees a lower bound on the rate at which the inferential parameters are being learned.",
"Such properties have been shown to hold for a wide class of SDEs in [13] by making use of the general theorems of Ibragimov and Has'minskii (Theorems I.5.1, I.5.2, I.10.1 and I.10.2 in [9]), however they do not hold for the Wright–Fisher diffusion as they require the diffusion coefficient to be non-zero everywhere and to have an inverse that has a polynomial majorant.",
"Both conditions fail for (REF ), forcing us to find an alternative way of proving that the Ibragimov–Has'minskii conditions still hold.",
"We emphasise here that the aim of this study is to investigate the properties of the estimators in the “ideal\" continuous observation scenario when the whole path is known to the observer.",
"Assuming that the mutation rates are known is a limitation to this study, however we emphasise that in the regime considered here these can be inferred directly from the path once the diffusion gets arbitrarily close to either boundary (see Remark REF for the corresponding details).",
"Nonetheless, extending this work to include mutation parameters greater than 1 would be of great interest.",
"This proves to be rather challenging as now the likelihood ratio function involves expressions of the form $(1-x){x}^{-1}$ and $x(1-x)^{-1}$ (as witnessed in Theorem REF ) which require much more delicate arguments in order to establish the same conclusions as in Theorem REF .",
"The main issue here is in showing that Condition 1 in the Ibragimov–Has'minskii conditions holds, for the other two conditions follow from Theorem REF and Proposition REF .",
"In particular, the fact that the functions $(1-x){x}^{-1}$ and $x(1-x)^{-1}$ are unbounded in $x$ and have only finitely many moments with respect to the stationary distribution means that the strategies used in the proofs of Propositions REF and REF cannot be used.",
"Recent advances in genome sequencing technology have led to an increase in the availability and analysis of genetic time series data.",
"Inference for selection has traditionally been conducted using techniques for and data coming from a single point in time.",
"However, having a time series of data points allows one to track the changes in allele frequencies over time, to better understand and infer the presence and effect of selection.",
"Several inferential techniques have already been developed for such a setting (see for instance [1], [15], [20], [8], [7], as well as [4] for a review on the subject), and although the techniques provide ostensibly reasonable estimation, there are not always theoretical guarantees on the statistical properties of the estimators being used.",
"The results presented in this paper offer a baseline in this regard, and prove that in the absence of observational error one is guaranteed that the ML and Bayesian estimators are uniform over compact sets consistent, asymptotically normal, and display moment convergence, besides being asymptotically efficient for the right choice of loss function."
],
[
"Acknowledgements",
"This work was supported by the EPSRC and the MASDOC DTC (under grant EP/HO23364/1), by The Alan Turing Institute under the EPSRC grant EP/N510129/1, and by the EPSRC under grant EP/R044732/1."
],
[
"Proof of Theorem ",
"We show $\\vartheta $ -uniform ergodicity for scalar diffusions on the bounded interval $[l,r]$ having entrance or regular boundary points by making use of Theorem 3.2 in [14], which allows us to bound the LHS of (REF ) in terms of the moments of the hitting times of the process.",
"That result requires the diffusion coefficient to be positive everywhere, and the drift and diffusion coefficients to be locally Lipschitz and to satisfy a linear growth condition.",
"These conditions however, are only used to guarantee the existence of a unique strong non-exploding solution to the SDE in Theorem 3.2, which we are guaranteeing explicitly in the statement of the theorem.",
"None of these requirements on the drift and diffusion coefficients are used in the proof of Theorem 3.2 in [14] when $p\\in \\lbrace 2,3,\\dots \\rbrace $ , which allows us to employ this theorem for such $p$ .",
"All that remains to prove then is that these moments can be bounded in $\\vartheta $ over compact sets in the parameter space, for then (REF ) holds.",
"To this end, we introduce some notation from [14], namely let $a,b\\in (l,r)$ be arbitrary fixed points such that $a<b$ .",
"Define $S_{0}=0$ , $R_{0}=0$ , and $S_{1} &:= \\inf \\left\\lbrace t\\ge 0 : Y_{t} = b \\right\\rbrace \\\\R_{1} &:= \\inf \\left\\lbrace t \\ge S_{1} : Y_{t} = a \\right\\rbrace \\\\S_{n+1} &:= \\inf \\left\\lbrace t \\ge R_{n} : Y_{t} = b \\right\\rbrace \\\\R_{n+1} &:= \\inf \\left\\lbrace t \\ge S_{n+1} : Y_{t} = a \\right\\rbrace $ for $n\\in \\mathbb {N}$ .",
"By the strong Markov property, $(R_{k}-R_{k-1})_{k\\in \\mathbb {N}\\setminus \\lbrace 0\\rbrace }$ is an i.i.d.",
"sequence with law under $\\mathbb {P}^{(\\vartheta )}_{\\nu }$ equal to the law of $R_{1}$ under $\\mathbb {P}^{(\\vartheta )}_{a}$ , where $\\mathbb {P}^{(\\vartheta )}_{\\nu }$ and $\\mathbb {E}^{(\\vartheta )}_{\\nu }$ are as defined in Section 2, and $\\mathbb {P}^{(\\vartheta )}_{a}$ denotes the law of the process started from $a$ .",
"Related to the process $(R_{n})_{n\\in \\mathbb {N}}$ we have the process $(N_{t})_{t\\ge 0}$ which we define as $N_{t} := \\sup \\left\\lbrace n : R_{n} \\le t \\right\\rbrace $ and for which we observe that $\\lbrace N_{t} \\ge n\\rbrace = \\lbrace R_{n} \\le t \\rbrace $ .",
"We also denote by $T_{b} := \\inf \\lbrace t\\ge 0 : Y_{t} = b \\rbrace $ the hitting time of $b$ .",
"Furthermore, let $\\ell _{\\vartheta } := \\mathbb {E}^{(\\vartheta )}[N_{1}] = {\\mathbb {E}^{(\\vartheta )}_{a}[R_{1}]}^{-1}$ (see Lemma 2.7 in [14]), and $\\bar{\\eta }_{1} := -(R_{2}-R_{1}-\\ell _{\\vartheta }^{-1})$ .",
"Then Theorem 3.2 in [14] gives us that for $p\\in \\lbrace 2,3,\\dots \\rbrace $ $\\mathbb {P}^{(\\vartheta )}_{\\nu }\\left[\\left|\\frac{1}{T}\\int _{0}^{T}h(Y_{t})dt - \\mathbb {E}^{(\\vartheta )}\\left[h(\\xi )\\right]\\right| > \\varepsilon \\right] \\le K(\\vartheta ,Y,p)\\varepsilon ^{-p}\\Vert h\\Vert _{\\infty }^{p}T^{-\\frac{p}{2}},$ where $K(\\vartheta ,Y,p) &:= {}6^{\\frac{p}{2}}\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[R_{1}^{\\frac{p}{2}}\\right] + 12^{p}C_{p}\\ell _{\\vartheta }^{\\frac{p}{2}}\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|R_{2}-R_{1}\\right|^{p}\\right]+2(6^p)\\ell _{\\vartheta }\\mathbb {E}^{(\\vartheta )}_{a}\\left[R_{1}^{p}\\right] \\nonumber \\\\&\\quad {}+ 2^{\\frac{p}{2}}\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|R_{1}-{\\ell _{\\vartheta }}^{-1}\\right|^{\\frac{p}{2}}\\right] + 2^{\\frac{3p}{2}}C_{p}\\ell _{\\vartheta }^{\\frac{p}{2}}\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|\\bar{\\eta }_{1}\\right|^{p}\\right],$ and $C_{p}$ is a constant depending only on $p$ .",
"We point out here that Theorem 3.2 in [14] holds $\\forall p\\in (1,\\infty )$ under additional assumptions, but for our case we need only $p\\in \\lbrace 2,3,\\dots \\rbrace $ .",
"Thus we are left with showing these moments can be bounded from above in $\\vartheta $ over compact sets, for then (REF ) follows.",
"Now the only terms above that depend on $\\vartheta $ are $\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[R_{1}^{\\frac{p}{2}}\\right], & & \\ell _{\\vartheta }^{\\frac{p}{2}}\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|R_{2}-R_{1}\\right|^{{p}}\\right], & & \\ell _{\\vartheta }\\mathbb {E}^{(\\vartheta )}_{a}\\left[R_{1}^{{p}}\\right], & & \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|R_{1}-\\ell _{\\vartheta }^{-1}\\right|^{\\frac{p}{2}}\\right], & & \\ell _{\\vartheta }^{\\frac{p}{2}}\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|\\bar{\\eta }_{1}\\right|^{{p}}\\right]$ and in light of the following inequalities $\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|\\bar{\\eta }_{1}\\right|^{{p}}\\right] &\\le 2^{p-1}\\left(\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|R_{2}-R_{1}\\right|^{{p}}\\right] + \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\ell _{\\vartheta }^{{-p}}\\right]\\right) \\nonumber \\\\&= 2^{p-1}\\left(\\mathbb {E}^{(\\vartheta )}_{a}\\left[R_{1}^{{p}}\\right] + \\mathbb {E}^{(\\vartheta )}_{a}\\left[R_{1}\\right]^{{p}}\\right), \\\\\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|R_{1}-\\ell _{\\vartheta }^{-1}\\right|^{\\frac{p}{2}}\\right] &\\le 2^{\\frac{p}{2}-1}\\left(\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[R_{1}^{\\frac{p}{2}}\\right] + \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\ell _{\\vartheta }^{-\\frac{p}{2}}\\right]\\right) \\\\&= 2^{\\frac{p}{2}-1}\\left(\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[R_{1}^{\\frac{p}{2}}\\right] + \\mathbb {E}^{(\\vartheta )}_{a}\\left[R_{1}\\right]^{\\frac{p}{2}}\\right), \\\\\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|R_{2}-R_{1}\\right|^{{p}}\\right] &= \\mathbb {E}^{(\\vartheta )}_{a}\\left[R_{1}^{p}\\right] \\le 2^{p-1}\\left(\\mathbb {E}^{(\\vartheta )}_{a}\\left[T_{b}^{{p}}\\right] + \\mathbb {E}^{(\\vartheta )}_{b}\\left[T_{a}^{p}\\right]\\right), \\\\\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[R_{1}^{\\frac{p}{2}}\\right] &\\le 2^{\\frac{p}{2}-1}\\left(\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[T_{b}^{\\frac{p}{2}}\\right] + \\mathbb {E}^{(\\vartheta )}_{b}\\left[T_{a}^{\\frac{p}{2}}\\right]\\right), \\\\\\mathbb {E}^{(\\vartheta )}_{a}\\left[R_{1}\\right] &= \\mathbb {E}^{(\\vartheta )}_{a}\\left[T_{b}\\right] + \\mathbb {E}^{(\\vartheta )}_{b}\\left[T_{a}\\right],$ it suffices to consider only the terms $\\ell _{\\vartheta }$ and $\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[T_{b}^{p}\\right]$ .",
"Thus we are left with showing that these two terms can be bounded from above in $\\vartheta $ over any compact set $\\mathcal {K}\\subset \\Theta $ .",
"We further point out that we can reduce our considerations in the expressions above to integer moments, for if this is not the case then $\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[T_{b}^{p}\\right] \\le \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[T_{b}^{\\lceil p \\rceil }\\right] + \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[T_{b}^{\\lfloor p \\rfloor }\\right]$ where $\\lceil \\cdot \\rceil $ and $\\lfloor \\cdot \\rfloor $ denote the ceiling and floor functions respectively.",
"We make use of the backward equation for the quantity $U_{q,b}(x) := \\mathbb {E}^{(\\vartheta )}_{x}[T_{b}^{q}]$ for $q\\in \\lbrace 1,2,\\dots \\rbrace $ , to derive the ODE (as can be found in [12] p. 203 and 210, and [23]) $\\frac{\\sigma ^{2}(x)}{2}U_{q,b}^{\\prime \\prime }(x) + \\mu (\\vartheta ,x)U_{q,b}^{\\prime }(x) + qU_{q-1,b}(x) = 0$ with boundary conditions $U_{q,b}(b) = 0$ and $\\lim _{y\\rightarrow l}S^{\\prime }(y)^{-1}\\frac{\\partial }{\\partial y}U_{q,b}(y) = 0$ when $x<b$ , or $\\lim _{y\\rightarrow r}S^{\\prime }(y)^{-1}\\frac{\\partial }{\\partial y}U_{q,b}(y) = 0$ when $x>b$ , where $S(x) := \\int ^{x}e^{-\\int ^{y}\\frac{2\\mu (z)}{\\sigma ^{2}(z)}dz}dy.$ Solving (REF ) for $x<b$ leads to $\\mathbb {E}^{(\\vartheta )}_{x}[T_{b}^{q}] = \\int _{x}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{l}^{\\xi }\\frac{2}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}qU_{q-1,b}(\\eta )d\\eta d\\xi ,$ whilst for $x>b$ we have that $\\mathbb {E}^{(\\vartheta )}_{x}[T_{b}^{q}] = \\int _{b}^{x}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{\\xi }^{r}\\frac{2}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}qU_{q-1,b}(\\eta )d\\eta d\\xi .$ We claim that for any $x<b$ and any $q\\in \\lbrace 1,2,\\dots \\rbrace $ , $\\mathbb {E}^{(\\vartheta )}_{x}[T_{b}^{q}] &\\le q!\\left(\\int _{l}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{l}^{\\xi }\\frac{2}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}d\\eta d\\xi \\right)^{q} \\nonumber \\\\&= q!\\kappa ^{l}_{\\vartheta }(l,b)^{q} < \\infty .$ To see this, observe that $\\mathbb {E}^{(\\vartheta )}_{x}[T_{b}] &= \\int _{x}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{l}^{\\xi }\\frac{2}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}d\\eta d\\xi \\nonumber \\\\&\\le \\int _{l}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{l}^{\\xi }\\frac{2}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}d\\eta d\\xi \\nonumber \\\\&= \\kappa ^{l}_{\\vartheta }(l,b),$ and we observe that $\\kappa ^{l}_{\\vartheta }(l,b)$ is finite for all $\\vartheta \\in \\Theta $ in virtue of $l$ being either an entrance or regular boundary (see Table 6.2 in [12] p. 234, and note that $\\kappa ^{l}_{\\vartheta }(l,b)$ here corresponds to $N(l)$ as defined in (6.19) there).",
"Observe that the RHS of (REF ) is independent of $x$ , so we can use the recursion in (REF ) to conclude by induction that (REF ) holds for $q\\in \\lbrace 1,2,\\dots \\rbrace $ as required.",
"Similar arguments to those presented above coupled with the requirement that the boundary point at $r$ is either entrance or regular, allows us to conclude that for $x>b$ and $q\\in \\lbrace 1,2,\\dots \\rbrace $ , $\\mathbb {E}^{(\\vartheta )}_{x}[T_{b}^{q}] &\\le q!\\left(\\int _{b}^{r}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{\\xi }^{r}\\frac{2}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}d\\eta d\\xi \\right)^{q} \\nonumber \\\\&= q!\\kappa ^{r}_{\\vartheta }(b,r)^{q} < \\infty .$ Both RHS of (REF ) and (REF ) are independent of $x$ , so trivially $\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[T_{b}^{q}\\right] &\\le q!\\left(\\kappa ^{l}_{\\vartheta }(l,b)^{q} + \\kappa ^{r}_{\\vartheta }(b,r)^{q}\\right).$ All the terms on the RHS of (REF ), (REF ) and (REF ) are finite for $\\vartheta \\in \\Theta $ , so we have our required bound when taking the supremum over a compact set $\\mathcal {K}\\subset \\Theta $ for $\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[T_{b}^{q}\\right]$ .",
"It remains to show that we can bound $\\ell _{\\vartheta }$ from above.",
"Observe that by definition $\\ell _{\\vartheta } = \\mathbb {E}^{(\\vartheta )}_{a}\\left[R_{1}\\right]^{-1} = \\left(\\mathbb {E}^{(\\vartheta )}_{a}\\left[T_{b}\\right] + \\mathbb {E}^{(\\vartheta )}_{b}\\left[T_{a}\\right]\\right)^{-1},$ and recall that we will take the supremum in $\\vartheta $ over a given compact set $\\mathcal {K}$ .",
"Using (REF ) and (REF ) with $q=1$ , coupled with (REF ), we deduce that $\\mathbb {E}^{(\\vartheta )}_{a}\\left[T_{b}\\right]$ and $\\mathbb {E}^{(\\vartheta )}_{b}\\left[T_{a}\\right]$ are bounded away from 0 for any compact $\\mathcal {K}\\subset \\Theta $ , and thus we have the required upper bound on $\\ell _{\\vartheta }$ ."
],
[
"Proof of Theorem ",
"Recall the notation introduced in Appendix , namely the regeneration times $\\lbrace S_n, R_n \\rbrace _{n=0}^{\\infty }$ and the number of upcrossings up to time $t$ , $\\lbrace N_t \\rbrace _{t\\ge 0}$ .",
"We want to prove that $\\lim \\limits _{T\\rightarrow \\infty }\\sup \\limits _{\\vartheta \\in \\mathcal {K}}\\mathbb {P}^{(\\vartheta )}_{\\nu }\\left[\\left|\\frac{1}{T}\\int _{0}^{T}h(Y_t)dt - \\mathbb {E}^{(\\vartheta )}\\left[h(\\xi )\\right]\\right| > \\varepsilon \\right] = 0$ holds for any compact set $\\mathcal {K}\\subset \\Theta $ , with $h$ as defined in the statement of the theorem.",
"The strategy here will be to decompose the sample path of the diffusion into i.i.d.",
"blocks of excursions as done in Theorem 3.5 in [14].",
"However, we will deal with the resulting expectations in a different way, namely by applying the ODE approach used in Appendix to bound these quantities from above in $\\vartheta $ over a compact set $\\mathcal {K}$ .",
"To this end, fix $\\varepsilon \\in (0,\\mathbb {E}^{(\\vartheta )}[h(\\xi )])$ and choose $\\delta \\in (0,1)$ such that $\\varepsilon =\\delta \\mathbb {E}^{(\\vartheta )}[h(\\xi )]$ , and set $\\Omega _T := \\lbrace |N_T T^{-1} - \\ell _{\\vartheta }|\\le \\ell _{\\vartheta }\\delta /4 \\rbrace $ for $\\ell _{{\\vartheta }}=\\mathbb {E}_{a}^{(\\vartheta )}[R_1]^{-1}$ .",
"Then as in the proof of Theorem 3.5 in [14], we get the following decomposition $\\mathbb {P}_{\\nu }^{(\\vartheta )}&\\left[\\left|\\frac{1}{T}\\int _{0}^{T}h(Y_t)dt - \\mathbb {E}^{(\\vartheta )}\\left[h(\\xi )\\right]\\right| >\\varepsilon \\right] \\\\\\le {} & \\mathbb {P}_{\\nu }^{(\\vartheta )}\\left[\\left|\\int _{0}^{R_1}h(Y_t)dt \\right|>\\frac{T\\varepsilon }{4}\\right] \\\\&{}+ \\mathbb {P}_{\\nu }^{(\\vartheta )}\\left[\\left|\\int _{R_1}^{R_{N_T}+1}h(Y_t)dt - N_T \\mathbb {E}^{(\\vartheta )}\\left[h(\\xi )\\right]\\mathbb {E}_{a}^{(\\vartheta )}\\left[R_1\\right]\\right|>\\frac{T\\varepsilon }{4} ; \\Omega _T\\right] \\\\&{}+ \\mathbb {P}_{\\nu }^{(\\vartheta )}\\left[\\left|N_T \\mathbb {E}^{(\\vartheta )}\\left[h(\\xi )\\right]\\mathbb {E}_{a}^{(\\vartheta )}\\left[R_1\\right]-T\\mathbb {E}^{(\\vartheta )}\\left[h(\\xi )\\right]\\right|>\\frac{T\\varepsilon }{4} ; \\Omega _T\\right] \\\\&{}+ \\mathbb {P}_{\\nu }^{(\\vartheta )}\\left[\\left|\\int _{T}^{R_{N_T}+1}h(Y_t)dt \\right|>\\frac{T\\varepsilon }{4} ; \\Omega _T\\right] + \\mathbb {P}_{\\nu }^{(\\vartheta )}\\left[\\Omega _T ^{c}\\right] =: A+B+E+C+D$ Dealing with $E$ and $D$ can be achieved as in equations (3.10) and (3.14) in [14], to deduce that $E=0$ and $D \\le \\frac{1}{T\\varepsilon ^{2}}\\mathbb {E}^{(\\vartheta )}\\left[h(\\xi )\\right]^{2}\\left(2\\mathbb {E}_{\\nu }^{(\\vartheta )}\\left[\\left|R_1 - \\ell _{\\vartheta }^{-1}\\right|\\right] + 2^{3}C_{1}^{2}\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left|\\bar{\\eta }_1\\right|^{2}\\right]\\ell _{\\vartheta }\\right),$ for $C_1$ the constant from the Burkholder-Davis-Gundy inequality.",
"All the above expressions are either constant or have been shown to be bounded in $\\vartheta $ over compact sets in the parameter space in Appendix , so it remains to deal with terms $A$ , $B$ and $C$ above.",
"Applying Markov's inequality to $A$ gives $A \\le \\frac{4}{T\\varepsilon }\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\int _{0}^{R_1}h(Y_t)dt\\right]$ and we can decompose the above integral $\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\int _{0}^{R_1}h(Y_t)dt\\right] &= \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\int _{0}^{S_1}h(Y_t)dt\\right] + \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\int _{S_1}^{R_1}h(Y_t)dt\\right] \\nonumber \\\\&\\le \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\int _{0}^{T_b}h(Y_t)dt\\right] + \\sup _{y\\in [a,b]}h(y)\\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[R_1\\right].$ So it remains to prove that the first term on the RHS can be bounded in $\\vartheta $ .",
"It turns out that $B$ and $C$ can be bounded by similar quantities, so we do this first and subsequently show that the resulting quantities can be bounded in $\\vartheta $ too.",
"Indeed, set $\\xi _k := \\int _{R_k}^{R_{k+1}}h(Y_t)dt$ , $M_0=0$ , and $M_n := \\sum _{k=1}^{n}\\left(\\xi _k - \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\xi _k\\right]\\right).$ Then $B = \\mathbb {P}^{(\\vartheta )}_{\\nu }\\left[\\left|M_{N_T}\\right| > \\frac{T\\varepsilon }{4} ; \\Omega _T\\right] &\\le \\mathbb {P}^{(\\vartheta )}_{\\nu }\\left[\\sup \\limits _{n\\le \\lfloor {T\\ell _{\\vartheta }(1+\\delta /4)}\\rfloor }\\left|M_{n}\\right| > \\frac{T\\varepsilon }{4} \\right] \\\\&\\le \\left(\\frac{4}{T\\varepsilon }\\right)^{2}\\mathbb {V}^{(\\vartheta )}_{\\nu }\\left[M_{\\lfloor {T\\ell _{\\vartheta }(1+\\delta /4)}\\rfloor }\\right]$ by the Kolmogorov inequality where $\\mathbb {V}^{(\\vartheta )}_{\\nu }$ denotes the variance with respect to the measure $\\mathbb {P}_{\\nu }^{(\\vartheta )}$ .",
"Now observe that $\\mathbb {V}^{(\\vartheta )}_{\\nu }\\left[M_{\\lfloor {T\\ell _{\\vartheta }(1+\\delta /4)}\\rfloor }\\right] &= \\sum _{k=1}^{\\lfloor {T\\ell _{\\vartheta }(1+\\delta /4)}\\rfloor }\\mathbb {V}_{\\nu }^{\\vartheta }\\left[\\left(\\xi _{k} - \\mathbb {E}_{\\nu }^{(\\vartheta )}\\left[\\xi _k\\right]\\right)^{2}\\right]\\\\&= \\lfloor {T\\ell _{\\vartheta }(1+\\delta /4)}\\rfloor \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left(\\xi _1 - \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\xi _1\\right]\\right)^{2}\\right] \\\\&\\le \\lfloor {T\\ell _{\\vartheta }(1+\\delta /4)}\\rfloor 2\\left(\\mathbb {E}^{(\\vartheta )}_{a}\\left[\\xi _0 ^{2}\\right] + \\mathbb {E}^{(\\vartheta )}_{a}\\left[\\xi _0\\right]^{2}\\right).$ because the $\\lbrace \\xi _k\\rbrace _{k=1}^{\\infty }$ are i.i.d., and moreover we have that under $\\mathbb {P}^{(\\vartheta )}_{\\nu }$ they are equal in distribution to $\\xi _0$ under $\\mathbb {P}^{(\\vartheta )}_{a}$ .",
"So $B &\\le \\frac{4^{2}\\lfloor {\\ell _{\\vartheta }(1+\\delta /4)}\\rfloor }{T\\varepsilon ^{2}}2\\left(\\mathbb {E}^{(\\vartheta )}_{a}\\left[\\xi _0 ^{2}\\right] + \\mathbb {E}^{(\\vartheta )}_{a}\\left[\\xi _0\\right]^{2}\\right).$ The second term of (REF ) can be bounded in the same way as in (REF ), whilst for the first term we can use a similar decomposition to get $\\mathbb {E}^{(\\vartheta )}_{a}\\left[\\xi _0 ^{2}\\right] \\le 2\\left(\\mathbb {E}^{(\\vartheta )}_{a}\\left[\\left(\\int _{0}^{T_b}h(Y_t)dt\\right)^{2}\\right] + \\sup _{y\\in [a,b]}h(y)^{2}\\mathbb {E}^{(\\vartheta )}_{a}\\left[R_{1}^{2}\\right]\\right).$ Finally, for $C$ we use the same arguments as in [14] (just before equation (3.13)) to get that $C &\\le \\sum _{k=1}^{\\lfloor T\\ell _{\\vartheta }(1+\\delta /4)\\rfloor }\\mathbb {P}^{(\\vartheta )}_{\\nu }\\left[\\int _{R_k}^{R_{k}+1}h(Y_t)dt>\\frac{T\\varepsilon }{4}\\right] \\\\&\\le \\frac{\\lfloor T\\ell _{\\vartheta }(1+\\delta /4)\\rfloor }{T^2 \\varepsilon ^2} \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\left(\\int _{R_1}^{R_2}h(Y_t)dt\\right)^2\\right]\\\\&\\le \\frac{\\ell _{\\vartheta }(1+\\delta /4)}{T \\varepsilon ^2} \\mathbb {E}^{(\\vartheta )}_{a}\\left[\\left(\\int _{0}^{R_1}h(Y_t)dt\\right)^2\\right],$ and we can apply the same reasoning as in (REF ).",
"It remains to show that the terms $\\mathbb {E}^{(\\vartheta )}_{a}\\left[\\int _{0}^{T_b}h(Y_t)dt\\right], & & \\mathbb {E}^{(\\vartheta )}_{\\nu }\\left[\\int _{0}^{T_b}h(Y_t)dt\\right], & & \\mathbb {E}^{(\\vartheta )}_{a}\\left[\\left(\\int _{0}^{T_b}h(Y_t)dt\\right)^2\\right]$ can be bounded in $\\vartheta $ .",
"The same arguments used to derive the ODEs in Appendix can be used here to derive an ODE for $U_n(x) := \\mathbb {E}^{(\\vartheta )}_{x}[(\\int _{0}^{T_b}h(Y_t)dt)^{n}]$ for the cases when $x<b$ and $x>b$ with the same boundary conditions as in Appendix .",
"Thus the following recursion holds for $U_n(x)$ when $x<b$ $U_n(x) &= n\\int _{x}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{l}^{\\xi }\\frac{2h(\\eta )}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}U_{n-1}(\\eta )d\\eta d\\xi , & n&=1,2,\\dots ,$ and for $x>b$ we have $U_n(x) &= n\\int _{b}^{x}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{\\xi }^{r}\\frac{2h(\\eta )}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}U_{n-1}(\\eta )d\\eta d\\xi ,& n&=1,2,\\dots .$ Now for $n=1$ , we get that for $x<b$ , $\\mathbb {E}^{(\\vartheta )}_{x}\\left[\\int _{0}^{T_b}h(Y_t)dt\\right] &= \\int _{x}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{l}^{\\xi }\\frac{2h(\\eta )}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}d\\eta d\\xi $ which is bounded over any compact set $\\mathcal {K}\\subset \\Theta $ by (REF ).",
"In view of condition (), we get that $\\mathbb {E}^{(\\vartheta )}_{\\nu }[\\int _{0}^{T_b}h(Y_t)dt]$ is also bounded from above in $\\vartheta $ over compact sets $\\mathcal {K}\\subset \\Theta $ , and finally, using the recursions in (REF ) and (REF ), we get that for $x<b$ , $\\mathbb {E}^{(\\vartheta )}_{x}\\left[\\left(\\int _{0}^{T_b}h(Y_t)dt\\right)^{2}\\right] &= 2\\int _{x}^{b}e^{-\\int ^{\\xi }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\int _{l}^{\\xi }\\frac{2h(\\eta )}{\\sigma ^{2}(\\eta )}e^{\\int ^{\\eta }\\frac{2\\mu (\\vartheta ,y)}{\\sigma ^{2}(y)}dy}\\mathbb {E}^{(\\vartheta )}_{\\eta }\\left[\\int _{0}^{T_b}h(Y_t)dt\\right]d\\eta d\\xi $ which is bounded from above in $\\vartheta $ over a given compact set $\\mathcal {K}\\subset \\Theta $ by (), giving the required bounds for the quantities $A$ , $B$ , and $C$ .",
"Combining these with the bounds for $D$ and $E$ we conclude that (REF ) holds."
]
] | 2001.03527 |
[
[
"Equivalence of Linear Tree Transducers with Output in the Free Group"
],
[
"Abstract We show that equivalence of deterministic linear tree transducers can be decided in polynomial time when their outputs are interpreted over the free group.",
"Due to the cancellation properties offered by the free group, the required constructions are not only more general, but also simpler than the corresponding constructions for proving equivalence of deterministic linear tree-to-word transducers."
],
[
"Introduction",
"In 2009, Staworko and Niehren observed that equivalence for sequential tree-to-word transducers [12] can be reduced to the morphism equivalence problem for context-free languages.",
"Since the latter problem is decidable in polynomial time [9], they thus proved that equivalence of sequential tree-to-word transducers is decidable in polynomial time.",
"This decision procedure was later accompanied by a canonical normal form which can be applied to learning [3], [4].",
"Sequentiality of transducers means that subtrees must always be processed from left to right.",
"This restriction was lifted by Boiret who provided a canonical normal form for unrestricted linear tree-to-word transducers [1].",
"Construction of that normal form, however, may require exponential time implying that the corresponding decision procedure requires exponential time as well.",
"In order to improve on that, Palenta and Boiret provided a polynomial time procedure which just normalizes the order in which an unrestricted linear tree-to-word transducer processes the subtrees of its input [2].",
"They proved that after that normalization, equivalent transducers are necessarily same-ordered.",
"As a consequence, equivalence of linear tree-to-word transducers can thus also be reduced to the morphism equivalence problem for context-free languages and thus can be decided in polynomial time.",
"Independently of that, Seidl, Maneth and Kemper showed by algebraic means, that equivalence of general (possibly non-linear) tree-to-word transducers is decidable [10].",
"Their techniques are also applicable if the outputs of transducers are not just in a free monoid of words, but also if outputs are in a free group.",
"The latter means that output words are considered as equivalent not just when they are literally equal, but also when they become equal after cancellation of matching positive and negative occurrences of letters.",
"For the special case of linear tree transducers with outputs either in a free monoid or a free group, Seidl et al.",
"provided a randomized polynomial time procedure for in-equivalence.",
"The question remained open whether for outputs in a free group, randomization can be omitted.",
"Here, we answer this question to the affirmative.",
"In fact, we follow the approach of [2] to normalize the order in which tree transducers produce their outputs.",
"For that normalization, we heavily rely on commutation laws as provided for the free group.",
"Due to these laws, the construction as well as the arguments for its correctness, are not only more general but also much cleaner than in the case of outputs in a free monoid only.",
"The observation that reasoning over the free group may simplify arguments has also been made, e.g., by Tomita and Seino and later by Senizergues when dealing with the equivalence problem for deterministic pushdown transducers [13], [11].",
"As morphism equivalence on context-free languages is decidable in polynomial time — even if the morphism outputs are in a free group [9], we obtain a polynomial time algorithm for equivalence of tree transducers with output in the free group."
],
[
"Preliminaries",
"We use $\\Sigma $ to denote a finite ranked alphabet, while ${\\textsf {A}}$ is used for an unranked alphabet.",
"$\\Sigma $ denotes the set of all trees (or terms) over $\\Sigma $ .",
"The depth $\\textsf {depth}(t)$ of a tree $t\\in \\Sigma $ equals 0, if $t= f()$ for some $f\\in \\Sigma $ of rank 0, and otherwise, $\\textsf {depth}(t) = 1+\\max \\lbrace \\textsf {depth}(t_i)\\mid i=1,\\ldots ,m\\rbrace $ for $t=f(t_1,\\ldots ,t_m)$ .",
"We denote by $\\mathcal {F}_{{\\textsf {A}}}$ the representation of the free group generated by ${\\textsf {A}}$ where the carrier is the set of reduced words instead of the usual quotient construction: For each $a\\in {\\textsf {A}}$ , we introduce its inverse $a^-$ .",
"The set of elements of $\\mathcal {F}_{\\textsf {A}}$ then consists of all words over the alphabet $\\lbrace a,a^-\\mid a\\in {\\textsf {A}}\\rbrace $ which do not contain $a\\,a^-$ or $a^-a$ as factors.",
"These words are also called reduced.",
"In particular, ${\\textsf {A}}^*\\subseteq \\mathcal {F}_{\\textsf {A}}$ .",
"The group operation “$\\cdot $ ” of $\\mathcal {F}_{\\textsf {A}}$ is concatenation, followed by reduction, i.e., repeated cancellation of subwords $a\\, a^-$ or $a^-a$ .",
"Thus, $a\\,b\\,c^-\\cdot c\\,b^-\\,a =_{\\mathcal {F}_{\\textsf {A}}}a\\,a$ .",
"The neutral element w.r.t.",
"this operation is the empty word $\\varepsilon $ , while the inverse $w^-$ of some element $w\\in \\mathcal {F}_{\\textsf {A}}$ is obtained by reverting the order of the letters in $w$ while replacing each letter $a$ with $a^-$ and each $a^-$ with $a$ .",
"Thus, e.g., $(a\\,b\\,c^-)^-= c\\,b^-a^-$ .",
"In light of the inverse operation $(\\,.\\,)^-$ , we have that $v\\cdot w =_{\\mathcal {F}_{\\textsf {A}}}v^{\\prime }w^{\\prime }$ where $v = v^{\\prime }u$ (as words) for a maximal suffix $u$ so that $u^-$ is a prefix of $w$ with $w= u^-w^{\\prime }$ .",
"For an element $w\\in \\mathcal {F}_{\\textsf {A}}$ , $\\langle w\\rangle = \\lbrace w^l\\mid l\\in \\mathbb {Z}\\rbrace $ denotes the cyclic subgroup of $\\mathcal {F}_{\\textsf {A}}$ generated from $w$ .",
"As usual, we use the convention that $w^0 = \\varepsilon $ , and $w^{-l} = (w^-)^l$ for $l>0$ .",
"An element $p\\in \\mathcal {F}_{\\textsf {A}}$ different from $\\varepsilon $ , is called primitive if $w^l=_{\\mathcal {F}_{\\textsf {A}}}p$ for some $w\\in \\mathcal {F}_{\\textsf {A}}$ and $l\\in \\mathbb {Z}$ implies that $w=_{\\mathcal {F}_{\\textsf {A}}}p$ or $w=_{\\mathcal {F}_{\\textsf {A}}}p^-$ , i.e., $p$ and $p^-$ are the only (trivial) roots of $p$ .",
"Thus, primitive elements generate maximal cyclic subgroups of $\\mathcal {F}_{\\textsf {A}}$ .",
"We state two crucial technical lemmas.",
"Assume that $y^m =_{\\mathcal {F}_{\\textsf {A}}}\\beta \\cdot y^n\\cdot \\beta ^-$ with $y\\in \\mathcal {F}_{\\textsf {A}}$ primitive.",
"Then $m=n$ , and $\\beta =_{\\mathcal {F}_{\\textsf {A}}}y^k$ for some $k\\in \\mathbb {Z}$ .",
"Since $\\beta \\cdot y^n\\cdot \\beta ^-=_{\\mathcal {F}_{\\textsf {A}}}(\\beta \\cdot y\\cdot \\beta ^-)^n$ , we find by [7] a primitive element $c$ such that $y$ and $\\beta \\cdot y\\cdot \\beta ^-$ are powers of $c$ .",
"As $y$ is primitive, $c$ can be chosen as $y$ .",
"Accordingly, ${\\small y^j=_{\\mathcal {F}_{\\textsf {A}}}\\beta \\cdot y\\cdot \\beta ^-}$ holds for some $j$ .",
"If $\\beta $ is a power of $y$ , then $\\beta \\cdot y\\cdot \\beta ^-=_{\\mathcal {F}_{\\textsf {A}}}y$ , and the assertion of the lemma holds.",
"Likewise if $j=1$ , then $\\beta $ and $y$ commute.",
"Since $y$ is primitive, then $\\beta $ necessarily must be a power of $y$ .",
"For a contradiction, therefore now assume that $\\beta $ is not a power of $y$ and $j\\ne 1$ .",
"W.l.o.g., we can assume that $j>1$ .",
"First, assume now that $y$ is cyclically reduced, i.e., the first and last letters, $a$ and $b$ , respectively, of $y$ are not mutually inverse.",
"Then for each $n>0$ , $y^n$ is obtained from $y$ by $n$ -concatenation of $y$ as a word (no reduction taking place).",
"Likewise, either the last letter of $\\beta $ is different $a^-$ or the first letter of $\\beta ^-$ is different from $b^-$ because these two letters are mutually inverse.",
"Assume that the former is the case.",
"Then $\\beta \\cdot y$ is obtained by concatenation of $\\beta $ and $y$ as words (no reduction taking place).",
"By (REF ), $\\beta \\cdot y^n=_{\\mathcal {F}_{\\textsf {A}}}y^{j\\cdot n}\\cdot \\beta $ .",
"for every $n\\ge 1$ .",
"Let $m>0$ denote the length of $\\beta $ as a word.",
"Since $\\beta $ can cancel only a suffix of $y^{j\\cdot n}$ of length at most $m$ , it follows, that the word $\\beta \\,y$ must a prefix of the word $y^{m+1}$ .",
"Since $\\beta $ is not a power of $y$ , the word $y$ can be factored into $y=y^{\\prime }c$ for some non-empty suffix $c$ such that $\\beta =y^{j^{\\prime }}y^{\\prime }$ , implying that $yc = cy$ holds.",
"As a consequence, $y=c^{l}$ for some $l>1$ — in contradiction to the irreducibility of $y$ .",
"If on the other hand, the first letter of $\\beta ^-$ is not the inverse of the last letter of $y$ , then $y\\cdot \\beta ^-$ is obtained as the concatenation of $y$ and $\\beta ^-$ as words.",
"As a consequence, $y\\beta ^-$ is a suffix of $y^{m+1}$ , and we arrive at a contradiction.",
"We conclude that the statement of the lemma holds whenever $y$ is cyclically reduced.",
"Now assume that $y$ is not yet cyclically reduced.",
"Then we can find a maximal suffix $r$ of $y$ (considered as a word) such that $y = r^-sr$ holds and $s$ is cyclically reduced.",
"Then $s$ is also necessarily primitive.",
"(If $s =_{\\mathcal {F}_{\\textsf {A}}}c^n$ , then $y=_{\\mathcal {F}_{\\textsf {A}}}(r^-cr)^n$ ).",
"Then assertion (REF ) can be equivalently formulated as ${\\small s^j=_{\\mathcal {F}_{\\textsf {A}}}(r\\cdot \\beta \\cdot r^-)\\cdot y\\cdot (r\\cdot \\beta \\cdot r^-)^-}$ We conclude that $r\\cdot \\beta \\cdot r^-=_{\\mathcal {F}_{\\textsf {A}}}s^l$ for some $l\\in \\mathbb {Z}$ .",
"But then $\\beta =_{\\mathcal {F}_{\\textsf {A}}}(r^-\\cdot s\\cdot r)^l =_{\\mathcal {F}_{\\textsf {A}}}y^l$ , and the claim of the lemma follows.",
"Assume that $x_1,x_2$ and $y_1,y_2$ are distinct elements in $\\mathcal {F}_{\\textsf {A}}$ and that ${\\small x_i\\cdot \\alpha \\cdot y_j\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\gamma \\cdot y^{\\prime }_j\\cdot \\alpha ^{\\prime }\\cdot x^{\\prime }_i\\cdot \\beta ^{\\prime }}$ holds for $i=1,2$ and $j=1,2$ .",
"Then there is some primitive element $p$ and exponents $r,s \\in \\mathbb {Z}$ such that $x_1\\cdot \\alpha =_{\\mathcal {F}_{\\textsf {A}}}x_2\\cdot \\alpha \\cdot p^r$ and $y_1 =_{\\mathcal {F}_{\\textsf {A}}}p^s\\cdot y_2$ .",
"By the assumption (REF ), ${\\small \\begin{array}{lll}\\gamma &=_{\\mathcal {F}_{\\textsf {A}}}&(x_1\\cdot \\alpha \\cdot y_j\\cdot \\beta )\\cdot (y^{\\prime }_j\\cdot \\alpha ^{\\prime }\\cdot x^{\\prime }_1\\cdot \\beta ^{\\prime })^-\\\\&=_{\\mathcal {F}_{\\textsf {A}}}&(x_2\\cdot \\alpha \\cdot y_j\\cdot \\beta )\\cdot (y^{\\prime }_j\\cdot \\alpha ^{\\prime }\\cdot x^{\\prime }_2\\cdot \\beta ^{\\prime })^-\\\\\\end{array}}$ for all $j=1,2$ .",
"Thus, ${\\small \\begin{array}{lll@{\\quad }l}x_1\\cdot \\alpha \\cdot y_j\\cdot \\beta {\\beta ^{\\prime }}^- {x^{\\prime }}^-_1{\\alpha ^{\\prime }}^-{y^{\\prime }}^-_j &=_{\\mathcal {F}_{\\textsf {A}}}&x_2\\cdot \\alpha \\cdot y_j\\cdot \\beta \\cdot {\\beta ^{\\prime }}^-\\cdot {x^{\\prime }}^-_2\\cdot {\\alpha ^{\\prime }}^-\\cdot {y^{\\prime }}^-_j&\\text{\\normalsize implying} \\\\y_j^-\\cdot \\alpha ^-\\cdot x_2^-\\cdot x_1\\cdot \\alpha \\cdot y_j &=_{\\mathcal {F}_{\\textsf {A}}}&\\beta \\cdot {\\beta ^{\\prime }}^-\\cdot {x^{\\prime }}^-_2\\cdot x^{\\prime }_1\\cdot \\beta ^{\\prime }\\cdot \\beta ^-\\end{array}}$ for $j=1,2$ .",
"Hence, ${\\small \\begin{array}{lll@{\\quad }l}y_1^-\\cdot \\alpha ^-\\cdot x_2^-\\cdot x_1\\cdot \\alpha \\cdot y_1 &=_{\\mathcal {F}_{\\textsf {A}}}&y_2^-\\cdot \\alpha ^-\\cdot x_2^-\\cdot x_1\\cdot \\alpha \\cdot y_2&\\text{\\normalsize implying}\\\\(x_2\\cdot \\alpha )^-x_1\\cdot \\alpha &=_{\\mathcal {F}_{\\textsf {A}}}& (y_1\\cdot y_2^-)\\cdot ((x_2\\cdot \\alpha )^-\\cdot x_1\\cdot \\alpha )\\cdot (y_1\\cdot y_2^-)^-\\end{array}}$ Since $x_1$ is different from $x_2$ , also $x_1\\cdot \\alpha $ is different from $x_2\\cdot \\alpha $ .",
"Let $p$ denote a primitive root of $(x_2\\cdot \\alpha )^-\\cdot x_1\\cdot \\alpha $ .",
"Then by Lemma , ${\\small \\begin{array}{lll}x_1\\cdot \\alpha &=_{\\mathcal {F}_{\\textsf {A}}}& x_2\\cdot \\alpha \\cdot p^r \\\\y_1 &=_{\\mathcal {F}_{\\textsf {A}}}& p^s\\cdot y_2\\end{array}}$ for suitable exponents $r,s\\in \\mathbb {Z}$ .",
"As the elements of $\\mathcal {F}_{\\textsf {A}}$ are words, they can be represented by straight-line programs (SLPs).",
"An SLP is a context-free grammar where each non-terminal occurs as the left-hand side of exactly one rule.",
"We briefly recall basic complexity results for operations on elements of $\\mathcal {F}_{\\textsf {A}}$ when represented as SLPs [6].",
"Let $U, V$ be SLPs representing words $w_1, w_2 \\in \\lbrace a, a^- \\mid a \\in {\\textsf {A}}\\rbrace $ , respectively.",
"Then the following computations/decision problems can be realized in polynomial time compute an SLP for $w_1^{-}$ ; compute the primitive root of $w_1$ if $w_1\\ne \\varepsilon $ ; compute an SLP for $w =_{\\mathcal {F}_{\\textsf {A}}}w_1$ with $w$ reduced; decide whether $w_1 =_{\\mathcal {F}_{\\textsf {A}}}w_2$ ; decide whether it exists $g \\in \\mathcal {F}_{\\textsf {A}}$ , such that $w_1 \\in g\\cdot \\langle w_2\\rangle $ and compute an SLP for such $g$ .",
"In the following, we introduce deterministic linear tree transducers which produce outputs in the free group $\\mathcal {F}_{\\textsf {A}}$ .",
"For convenience, we follow the approach in [10] where only total deterministic transducers are considered — but equivalence is relative w.r.t.",
"some top-down deterministic domain automaton $B$ .",
"A top-down deterministic automaton (DTA) $B$ is a tuple $(H,\\Sigma ,\\delta _B,h_0)$ where $H$ is a finite set of states, $\\Sigma $ is a finite ranked alphabet, $\\delta _B:H\\times \\Sigma \\rightarrow H^*$ is a partial function where $\\delta _B(h,f)\\in H^k$ if the the rank of $f$ equals $k$ , and $h_0$ is the start state of $B$ .",
"For every $h\\in H$ , we define the set $\\textsf {dom}(h)\\subseteq \\Sigma $ by $f(t_1,\\ldots ,t_m)\\in \\textsf {dom}(h)$ iff $\\delta _B(h,f) = h_1\\ldots h_m$ and $t_i\\in \\textsf {dom}(h_i)$ for all $i=1,\\ldots ,k$ .",
"$B$ is called reduced if $\\textsf {dom}(h)\\ne \\emptyset $ for all $h\\in H$ .",
"The language $\\mathcal {L}(B)$ accepted by $B$ is the set $\\textsf {dom}(h_0)$ .",
"We remark that for every DTA $B$ with $\\mathcal {L}(B)\\ne \\emptyset $ , a reduced DTA $B^{\\prime }$ can be constructed in polynomial time with $\\mathcal {L}(B) = \\mathcal {L}(B^{\\prime })$ .",
"Therefore, we subsequently assume w.l.o.g.",
"that each DTA $B$ is reduced.",
"A (total deterministic) linear tree transducer with output in $\\mathcal {F}_{\\textsf {A}}$ (${\\sf LT}_{\\textsf {A}}$ for short) is a tuple $M = (\\Sigma , {\\textsf {A}}, Q, S, R)$ where $\\Sigma $ is the ranked alphabet for the input trees, ${\\textsf {A}}$ is the finite (unranked) output alphabet, $Q$ is the set of states, $S$ is the axiom of the form $u_0$ or $u_0\\cdot q_0(x_0)\\cdot u_1$ with $u_0, u_1 \\in \\mathcal {F}_{\\textsf {A}}$ and $q_0\\in Q$ , and $R$ is the set of rules which contains for each state $q\\in Q$ and each input symbol $f\\in \\Sigma $ , one rule of the form ${\\small q(f(x_1, \\ldots , x_m)) \\rightarrow u_0\\cdot q_1(x_{\\sigma (1)})\\cdot \\ldots \\cdot u_{n-1}\\cdot q_n(x_{\\sigma (n)})\\cdot u_n}$ Here, $m$ is the rank of $f$ , $n \\le m$ , $u_0,\\ldots ,u_n\\in \\mathcal {F}_{\\textsf {A}}$ and $\\sigma $ is an injective mapping from $\\lbrace 1,\\ldots , n\\rbrace $ to $\\lbrace 1, \\ldots , m\\rbrace $ .",
"The semantics of a state $q$ is the function $\\llbracket q\\rrbracket :\\Sigma \\rightarrow \\mathcal {F}_{\\textsf {A}}$ defined by $\\llbracket q\\rrbracket (f(t_1,\\ldots ,t_m)) =_{\\mathcal {F}_{\\textsf {A}}}u_0 \\cdot \\llbracket q_1\\rrbracket (t_{\\sigma (1)}) \\cdot \\ldots \\cdot u_{n-1}\\cdot \\llbracket q_n\\rrbracket (t_{\\sigma (n)})\\cdot u_n$ if there is a rule of the form (REF ) in $R$ .",
"Then the translation of $M$ is the function $\\llbracket M\\rrbracket :\\Sigma \\rightarrow \\mathcal {F}_{\\textsf {A}}$ defined by $\\llbracket M\\rrbracket (t) =_{\\mathcal {F}_{\\textsf {A}}}u_0$ if the axiom of $M$ equals $u_0$ , and $\\llbracket M\\rrbracket (t) =_{\\mathcal {F}_{\\textsf {A}}}u_0\\cdot \\llbracket q\\rrbracket (t)\\cdot u_1$ if the axiom of $M$ is given by $u_0\\cdot q(x_0)\\cdot u_1$ .",
"Let ${\\textsf {A}}= \\lbrace a,b\\rbrace $ .",
"As a running example we consider the ${\\sf LT}_{\\textsf {A}}$ $M$ with input alphabet $\\Sigma =\\lbrace f^2, g^1, k^0\\rbrace $ where the superscripts indicate the rank of the input symbols.",
"$M$ has axiom $q_0(x_0)$ and the following rules ${\\small \\begin{array}{lll}q_0(f(x_1, x_2)) \\rightarrow q_1(x_2) b q_2(x_1) &q_0(g(x_1)) \\rightarrow q_0(x_1) &q_0(k) \\rightarrow \\varepsilon \\\\q_1(f(x_1, x_2)) \\rightarrow q_0(x_1) q_0(x_2) &q_1(g(x_1)) \\rightarrow ab q_1(x_1) &q_1(k) \\rightarrow a \\\\q_2(f(x_1, x_2)) \\rightarrow q_0(x_1) q_0(x_2) &q_2(g(x_1)) \\rightarrow ab q_2(x_1) &q_2(k) \\rightarrow ab\\end{array}}$ Two ${\\sf LT}_{\\textsf {A}}$ s $M$ , $M^{\\prime }$ are equivalent relative to the DTA $B$ iff their translations coincide on all input trees accepted by $B$ , i.e., $\\llbracket M\\rrbracket (t)=_{\\mathcal {F}_{\\textsf {A}}}\\llbracket M^{\\prime }\\rrbracket (t)$ for all $t\\in \\mathcal {L}(B)$ .",
"To relate the computations of the ${\\sf LT}_{\\textsf {A}}$ $M$ and the domain automaton $B$ , we introduce the following notion.",
"A mapping $\\iota :Q\\rightarrow H$ from the set of states of $M$ to the set of states of $B$ is called compatible if either the set of states of $M$ is empty (and thus the axiom of $M$ consists of an element of $\\mathcal {F}_{\\textsf {A}}$ only), or the following holds: $\\iota (q_0) = h_0$ ; If $\\iota (q) = h$ , $\\delta _B(h,f) = h_1\\ldots h_m$ , and there is a rule in $M$ of the form (REF ) then $\\iota (q_i) = h_{\\sigma (i)}$ for all $i=1,\\ldots ,n$ ; If $\\iota (q) = h$ and $\\delta _B(h,f)$ is undefined for some $f\\in \\Sigma $ of rank $m\\ge 0$ , then $M$ has the rule $q(f(x_1,\\ldots ,x_m))\\rightarrow \\bot $ for some dedicated symbol $\\bot $ which does not belong to ${\\textsf {A}}$ .",
"For an ${\\sf LT}_{\\textsf {A}}$ $M$ and a DTA $B=(H,\\Sigma ,\\delta _B,h_0)$ , an ${\\sf LT}_{\\textsf {A}}$ $M^{\\prime }$ with a set of states $Q^{\\prime }$ together with a mapping $\\iota :Q^{\\prime }\\rightarrow H$ can be constructed in polynomial time such that the following holds: $M$ and $M^{\\prime }$ are equivalent relative to $B$ ; $\\iota $ is compatible.",
"Let ${\\sf LT}_{\\textsf {A}}$ $M$ be defined as in Example REF .",
"Consider DTA $B$ with start state $h_0$ and the transition function $\\delta _B=\\lbrace (h_0,f) \\mapsto h_1 h_1, (h_1,g) \\mapsto h_1, (h_1,h) \\rightarrow \\varepsilon \\rbrace $ .",
"According to Lemma REF , ${\\sf LT}_{\\textsf {A}}$ $M^{\\prime }$ for $M$ then is defined as follows.",
"$M^{\\prime }$ has axiom $\\langle q_0, h_0\\rangle (x_0)$ and the rules ${\\small \\begin{array}{l@{\\qquad }l}\\multicolumn{2}{l}{\\langle q_0, h_0\\rangle (f(x_1, x_2)) \\rightarrow \\langle q_1, h_1\\rangle (x_2)\\, b\\, \\langle q_2, h_1\\rangle (x_1)} \\\\\\langle q_1, h_1\\rangle (g(x_1)) \\rightarrow ab\\, \\langle q_1, h_1\\rangle (x_1) & \\langle q_1, h_1\\rangle (k) \\rightarrow a \\\\\\langle q_2, h_1\\rangle (g(x_1)) \\rightarrow ab\\, \\langle q_2, h_1\\rangle (x_1) & \\langle q_2, h_1\\rangle (k) \\rightarrow ab \\\\\\end{array} }$ where the rules with left-hand sides $\\langle q_0, h_0\\rangle (g(x_1))$ , $\\langle q_0, h_0\\rangle (h)$ , $\\langle q_1, h_1\\rangle (f(x_1, x_2))$ , $\\langle q_2, h_1\\rangle (f(x_1, x_2))$, all have right-hand-sides $\\bot $ .",
"The compatible map $\\iota $ is then given by $\\iota = \\lbrace \\langle q_0,h_0\\rangle \\mapsto h_0, \\langle q_1,h_1\\rangle \\mapsto h_1,\\langle q_2,h_1\\rangle \\mapsto h_1\\rbrace $.",
"For convenience, we again denote the pairs $\\langle q_0,h_0\\rangle ,\\langle q_1,h_1\\rangle ,\\langle q_2,h_1\\rangle $ with $q_0,q_1,q_2$ , respectively.",
"Subsequently, we w.l.o.g.",
"assume that each ${\\sf LT}_{\\textsf {A}}$ $M$ with corresponding DTA $B$ for its domain, comes with a compatible map $\\iota $ .",
"Moreover, we define for each state $q$ of $M$ , the set $\\mathcal {L}(q) = \\lbrace \\llbracket q\\rrbracket (t)\\mid t\\in \\textsf {dom}(\\iota (q))\\rbrace $ of all outputs produced by state $q$ (on inputs in $\\textsf {dom}(\\iota (q))$ ), and $\\mathcal {L}^{(i)}(q)=\\lbrace \\llbracket q\\rrbracket (t)\\mid t\\in \\textsf {dom}(\\iota (q)),\\textsf {depth}(t)<i\\rbrace $ for $i\\ge 0$ .",
"Beyond the availability of a compatible map, we also require that all states of $M$ are non-trivial (relative to $B$ ).",
"Here, a state $q$ of $M$ is called trivial if $\\mathcal {L}(q)$ contains a single element only.",
"Otherwise, it is called non-trivial.",
"This property will be established in Theorem .",
"Deciding Equivalence In the first step, we show that equivalence relative to the DTA $B$ of same-ordered ${\\sf LT}_{\\textsf {A}}$ s is decidable.",
"For a DTA $B$ , consider the ${\\sf LT}_{\\textsf {A}}$ s $M$ and $M^{\\prime }$ with compatible mappings $\\iota $ and $\\iota ^{\\prime }$ , respectively.",
"$M$ and $M^{\\prime }$ are same-ordered relative to $B$ if they process their input trees in the same order.",
"We define set of pairs $\\langle q,q^{\\prime }\\rangle $ of co-reachable states of $M$ and $M^{\\prime }$ .",
"Let $u_0\\cdot q_0(x_1)\\cdot u_1$ and $u^{\\prime }_0\\cdot q^{\\prime }_0(x_1)\\cdot u^{\\prime }_1$ be the axioms of $M$ and $M^{\\prime }$ , respectively, where $\\iota (q_0) = \\iota ^{\\prime }(q^{\\prime }_0)$ is the start state of $B$ .",
"Then the pair $\\langle q_0,q^{\\prime }_0\\rangle $ is co-reachable.",
"Let $\\langle q,q^{\\prime }\\rangle $ be a pair of co-reachable states.",
"Then $\\iota (q)=\\iota ^{\\prime }(q^{\\prime })$ should hold.",
"For $f \\in \\Sigma $ , assume that $\\delta _B(\\iota (q),f)$ is defined.",
"Let ${\\small \\begin{array}{lll}q(f(x_1, \\ldots , x_m)) &\\rightarrow & u_0 q_1(x_{\\sigma (1)}) u_1 \\ldots u_{n-1} q_n(x_{\\sigma (n)}) u_n\\\\q^{\\prime }(f(x_1, \\ldots , x_m)) &\\rightarrow & u^{\\prime }_0 q^{\\prime }_1(x_{\\sigma ^{\\prime }(1)}) u^{\\prime }_1 \\ldots u^{\\prime }_{n-1} q^{\\prime }_n(x_{\\sigma ^{\\prime }(n^{\\prime })}) u^{\\prime }_{n^{\\prime }}\\end{array}} $ be the rules of $q, q^{\\prime }$ for $f$ , respectively.",
"Then $\\langle q_{j},q^{\\prime }_{j^{\\prime }}\\rangle $ is co-reachable whenever $\\sigma (j)=\\sigma ^{\\prime }(j^{\\prime })$ holds.",
"In particular, we then have $\\iota (q_j) = \\iota ^{\\prime }(q^{\\prime }_{j^{\\prime }})$ .",
"The pair $\\langle q,q^{\\prime }\\rangle $ of co-reachable states is called same-ordered, if for each corresponding pair of rules (REF ), $n = n^{\\prime }$ and $\\sigma = \\sigma ^{\\prime }$ .",
"Finally, $M$ and $M^{\\prime }$ are same-ordered if for every co-reachable pair $\\langle q,q^{\\prime }\\rangle $ of states of $M, M^{\\prime }$ , and every $f\\in \\Sigma $ , each pair of rules (REF ) is same-ordered whenever $\\delta _B(\\iota (q),f)$ is defined.",
"Given that the ${\\sf LT}_{\\textsf {A}}$ s $M$ and $M^{\\prime }$ are same-ordered relative to $B$ , we can represent the set of pairs of runs of $M$ and $M^{\\prime }$ on input trees by means of a single context-free grammar $G$ .",
"The set of nonterminals of $G$ consists of a distinct start nonterminal $S$ together with all co-reachable pairs $\\langle q,q^{\\prime }\\rangle $ of states $q,q^{\\prime }$ of $M,M^{\\prime }$ , respectively.",
"The set of terminal symbols $T$ of $G$ is given by $\\lbrace a,a^-,\\bar{a},\\bar{a}^-\\mid a\\in {\\textsf {A}}\\rbrace $ for fresh distinct symbols $\\bar{a},\\bar{a}^-,a\\in {\\textsf {A}}$ .",
"Let $\\langle q,q^{\\prime }\\rangle $ be a co-reachable pair of states of $M, M^{\\prime }$ , and $f \\in \\Sigma $ such that $\\delta _B(\\iota (q),f)$ is defined.",
"For each corresponding pair of rules (REF ), $G$ receives the rule $\\langle q,q^{\\prime }\\rangle \\rightarrow u_0 \\bar{u}^{\\prime }_0 \\langle q_1, q^{\\prime }_1\\rangle u_1 \\bar{u}^{\\prime }_1 \\ldots u_{n-1} \\bar{u}^{\\prime }_{n-1} \\langle q_n,q^{\\prime }_n\\rangle u_n \\bar{u}^{\\prime }_n$ where $\\bar{u}^{\\prime }_i$ is obtained from $u^{\\prime }_i$ by replacing each output symbol $a\\in {\\textsf {A}}$ with its barred copy $\\bar{a}$ as well as each inverse $a^-$ with its barred copy $\\bar{a}^-$ .",
"For the axioms $u_0 q(x_1) u_1$ and $u^{\\prime }_0 q^{\\prime }(x_1) u^{\\prime }_1$ of $M,M^{\\prime }$ , respectively, we introduce the rule $S \\rightarrow u_0 \\bar{u}^{\\prime }_0 \\langle q,q^{\\prime }\\rangle u_1 \\bar{u}^{\\prime }_1$ where again $\\bar{u}^{\\prime }_i$ are the barred copies of $u^{\\prime }_i$ .",
"We define morphisms $f,g:T^*\\rightarrow \\mathcal {F}_{\\textsf {A}}$ by ${\\small \\begin{array}{lll}f(a) = a & f(a^-) = a^-& f(\\bar{a})=f(\\bar{a}^-)=\\varepsilon \\\\g(\\bar{a}) = a & g(\\bar{a}^-) = a^-& g(a)=g(a^-)=\\varepsilon \\\\\\end{array}}$ for $a\\in {\\textsf {A}}$ .",
"Then $M$ and $M^{\\prime }$ are equivalent relative to $B$ iff $g(w) =_{\\mathcal {F}_{\\textsf {A}}}f(w)$ for all $w \\in \\mathcal {L}(G)$ .",
"Combining Plandowski's polynomial construction of a test set for a context-free language to check morphism equivalence over finitely generated free groups [9], with Lohrey's polynomial algorithm for checking equivalence of SLPs over the free group [5], we deduce that the equivalence of the morphisms $f$ and $g$ on all words generated by the context-free grammar $G$ , is decidable in polynomial time.",
"Consequently, we obtain: Equivalence of same-ordered ${\\sf LT}_{\\textsf {A}}$ s relative to a DTA $B$ is decidable in polynomial time.",
"$\\Box $ Next, we observe that for every ${\\sf LT}_{\\textsf {A}}$ $M$ with compatible map $\\iota $ and non-trivial states only, a canonical ordering can be established.",
"We call $M$ ordered (relative to $B$ ) if for all rules of the form (REF ), with $\\mathcal {L}(q_i)\\cdot u_i\\cdot \\ldots \\cdot u_{j-1}\\cdot \\mathcal {L}(q_j) \\subseteq v\\cdot \\langle p\\rangle $ , $p \\in \\mathcal {F}_{\\textsf {A}}$ the ordering $\\sigma (i) < \\ldots < \\sigma (j)$ holds.",
"Here we have naturally extended the operation “$\\cdot $ ” to sets of elements.",
"We show that two ordered ${\\sf LT}_{\\textsf {A}}$ s, when they are equivalent, are necessarily same-ordered.",
"The proof of this claim is split in two parts.",
"First, we prove that the set of indices of subtrees processed by equivalent co-reachable states are identical and second, that the order is the same.",
"Let $M, M^{\\prime }$ be ${\\sf LT}_{\\textsf {A}}$ s with compatible maps $\\iota $ and $\\iota ^{\\prime }$ , respectively, and non-trivial states only so that $M$ and $M^{\\prime }$ are equivalent relative to the DTA $B$ .",
"Let $\\langle q,q^{\\prime }\\rangle $ be a pair of co-reachable states of $M$ and $M^{\\prime }$ .",
"Assume that $\\delta _B(\\iota (q),f)$ is defined for some $f\\in \\Sigma $ and consider the corresponding pair of rules (REF ).",
"Then the following holds: $\\lbrace \\sigma (1),\\ldots , \\sigma (n)\\rbrace = \\lbrace \\sigma ^{\\prime }(1),\\ldots , \\sigma ^{\\prime }(n^{\\prime })\\rbrace $ ; $\\sigma = \\sigma ^{\\prime }$ .",
"Since $\\langle q,q^{\\prime }\\rangle $ is a co-reachable pair of states, there are elements $\\alpha , \\alpha ^{\\prime }, \\beta , \\beta ^{\\prime } \\in \\mathcal {F}_{\\textsf {A}}$ such that $\\alpha \\cdot \\llbracket q\\rrbracket (t)\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\alpha ^{\\prime }\\cdot \\llbracket q^{\\prime }\\rrbracket (t)\\cdot \\beta ^{\\prime }$ holds for all $t \\in \\textsf {dom}(\\iota (q))$ .",
"Consider the first statement.",
"Assume for a contradiction that $q_k(x_{j})$ occurs on the right-hand side of the rule for $q$ but $x_{j}$ does not occur on the right-hand side of the rule for $q^{\\prime }$ .",
"Then, there are input trees $t = f(t_1,\\ldots , t_m)$ and $t^{\\prime } = f(t^{\\prime }_1,\\ldots , t^{\\prime }_m)$ , both in $\\textsf {dom}(\\iota (q))$ , such that $\\llbracket q_k\\rrbracket (t_j) \\ne _{\\mathcal {F}_{\\textsf {A}}}\\llbracket q_k\\rrbracket (t^{\\prime }_j)$ and $t_i = t^{\\prime }_i$ for all $i \\ne j$ .",
"Moreover, there are $\\mu _1, \\mu _2\\in \\mathcal {F}_{\\textsf {A}}$ s.t.",
"${\\small \\alpha \\cdot \\llbracket q\\rrbracket (t)\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\alpha \\cdot \\mu _1\\cdot \\llbracket q_k\\rrbracket (t_{j})\\cdot \\mu _2\\cdot \\beta \\ne _{\\mathcal {F}_{\\textsf {A}}}\\alpha \\cdot \\mu _1\\cdot \\llbracket q_k\\rrbracket (t^{\\prime }_{j})\\cdot \\mu _2\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\alpha \\cdot \\llbracket q\\rrbracket (t^{\\prime })\\cdot \\beta }$ But then, ${\\small \\alpha \\cdot \\llbracket q\\rrbracket (t)\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\alpha ^{\\prime }\\cdot \\llbracket q^{\\prime }\\rrbracket (t)\\cdot \\beta ^{\\prime }=_{\\mathcal {F}_{\\textsf {A}}}\\alpha ^{\\prime }\\cdot \\llbracket q^{\\prime }\\rrbracket (t^{\\prime })\\cdot \\beta ^{\\prime }=_{\\mathcal {F}_{\\textsf {A}}}\\alpha \\cdot \\llbracket q\\rrbracket (t^{\\prime })\\cdot \\beta }$ — a contradiction.",
"By an analogous argument for some $x_j$ only occurring in the right-hand side of the rule for $q^{\\prime }$ the first statement follows.",
"Assume for contradiction that the mappings $\\sigma $ and $\\sigma ^{\\prime }$ in the corresponding rules (REF ) differ.",
"Let $k$ denote the minimal index so that $\\sigma (k) \\ne \\sigma ^{\\prime }(k)$ .",
"W.l.o.g., we assume that $\\sigma ^{\\prime }(k) < \\sigma (k)$ .",
"By the first statement, $n=n^{\\prime }$ and $\\lbrace \\sigma (1),\\ldots , \\sigma (n)\\rbrace = \\lbrace \\sigma ^{\\prime }(1),\\ldots , \\sigma ^{\\prime }(n)\\rbrace $ .",
"Then there are $\\ell , \\ell ^{\\prime } > k$ such that $\\sigma ^{\\prime }(k)=\\sigma (\\ell ) < \\sigma (k)=\\sigma ^{\\prime }(\\ell ^{\\prime })$ Let $t = f(t_1,\\ldots , t_n) \\in \\textsf {dom}(\\iota (q))$ be an input tree.",
"For that we obtain ${\\small \\begin{array}[t]{l}\\mu _0 := u_0\\cdot \\llbracket q_1\\rrbracket (t_{\\sigma (1)})\\cdot \\ldots \\cdot u_{k-1} \\\\\\mu _1 := u_k\\cdot \\llbracket q_{k+1}\\rrbracket (t_{\\sigma (k+1)})\\cdot \\ldots \\cdot u_{\\ell -1} \\\\\\mu _2 := u_\\ell \\cdot \\llbracket q_\\ell \\rrbracket (t_{\\sigma (\\ell )})\\cdot \\ldots \\cdot u_n\\end{array}\\quad \\begin{array}[t]{l}\\mu ^{\\prime }_0 := u^{\\prime }_0\\cdot \\llbracket q^{\\prime }_1\\rrbracket (t_{\\sigma ^{\\prime }(1)})\\cdot \\ldots \\cdot u^{\\prime }_{k-1} \\\\\\mu ^{\\prime }_1 := u^{\\prime }_k\\cdot \\llbracket q^{\\prime }_{k+1}\\rrbracket (t_{\\sigma ^{\\prime }(k+1)})\\cdot \\ldots \\cdot u^{\\prime }_{\\ell ^{\\prime }-1} \\\\\\mu ^{\\prime }_2 := u^{\\prime }_{\\ell ^{\\prime }}\\cdot \\llbracket q^{\\prime }_{\\ell ^{\\prime }}\\rrbracket (t_{\\sigma ^{\\prime }(\\ell ^{\\prime })})\\cdot \\ldots \\cdot u^{\\prime }_n\\end{array}}$ Then for all input trees $t^{\\prime } \\in \\textsf {dom}(\\iota (q_k))$ , $t^{\\prime \\prime } \\in \\mathcal {L}(\\textsf {dom}(\\iota (q^{\\prime }_k))$ , $\\alpha \\cdot \\mu _0\\cdot \\llbracket q_{k}\\rrbracket (t^{\\prime })\\cdot \\mu _1\\cdot \\llbracket q_{\\ell }\\rrbracket (t^{\\prime \\prime })\\cdot \\mu _2\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\alpha ^{\\prime }\\cdot \\mu ^{\\prime }_0\\cdot \\llbracket q^{\\prime }_{k}\\rrbracket (t^{\\prime \\prime })\\cdot \\mu ^{\\prime }_1\\cdot \\llbracket q^{\\prime }_{\\ell ^{\\prime }}\\rrbracket (t^{\\prime })\\cdot \\mu ^{\\prime }_2\\cdot \\beta ^{\\prime }$ Let $\\gamma ^{\\prime } = \\mu _0^-\\alpha ^-\\alpha ^{\\prime }\\mu ^{\\prime }_0$ .",
"Then $\\llbracket q_k\\rrbracket (t^{\\prime })\\cdot \\mu _1\\cdot \\llbracket q_\\ell \\rrbracket (t^{\\prime \\prime })\\cdot \\mu _2\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\gamma ^{\\prime }\\cdot \\llbracket q^{\\prime }_k\\rrbracket (t^{\\prime \\prime })\\cdot \\mu ^{\\prime }_1\\cdot \\llbracket q^{\\prime }_{\\ell ^{\\prime }}\\rrbracket (t^{\\prime })\\cdot \\mu ^{\\prime }_2\\cdot \\beta ^{\\prime }$ By Lemma , we obtain that for all $w_1, w_2 \\in \\mathcal {L}(q_k)$ and $v_1, v_2 \\in \\mathcal {L}(q_\\ell )$ , $w_2^-\\cdot w_1 \\in \\mu _1\\cdot \\langle p\\rangle \\cdot \\mu _1^-$ and $v_1\\cdot v_2^-\\in \\langle p\\rangle $ for some primitive $p$ .",
"If $\\ell = k+1$ , i.e., there is no further state between $q_k(x_{\\sigma (k)})$ and $q_\\ell (x_{\\sigma (\\ell )})$ , then $\\mu _1 =_{\\mathcal {F}_{\\textsf {A}}}u_k$ , $\\mathcal {L}(q_k) \\subseteq w\\cdot u_k\\cdot \\langle p\\rangle \\cdot u_k^-$ and $\\mathcal {L}(q_\\ell ) \\subseteq \\langle p\\rangle \\cdot v$ for some fixed $w \\in \\mathcal {L}(q_k)$ and $v \\in \\mathcal {L}(q_\\ell )$ .",
"As $\\sigma (k) > \\sigma ^{\\prime }(k) = \\sigma (\\ell )$ , this contradicts $M$ being ordered.",
"For the case that there is at least one occurrence of a state between $q_k(x_{\\sigma (k)})$ and $q_\\ell (x_{\\sigma (\\ell )})$ , we show that for all $\\alpha _1, \\alpha _2 \\in u_k\\cdot \\mathcal {L}(q_{k+1})\\cdot \\ldots \\cdot u_{\\ell -1} =_{\\mathcal {F}_{\\textsf {A}}}: \\hat{L}$ , $\\alpha _1^-\\alpha _2 \\in \\langle p\\rangle $ holds.",
"We fix $w_1, w_2 \\in \\mathcal {L}(q_k)$ and $v_1,v_2\\in \\mathcal {L}(q_\\ell )$ with $w_1 \\ne w_2$ and $v_1\\ne v_2$ .",
"For every $\\alpha \\in \\hat{L}$ , we find by Lemma , primitive $p_\\alpha $ and exponent $r_\\alpha \\in \\mathbb {Z}$ such that $v_1\\cdot v_2^-=_{\\mathcal {F}_{\\textsf {A}}}p_\\alpha ^{r_\\alpha }$ holds.",
"Since $p_\\alpha $ is primitive, this means that $p_\\alpha =_{\\mathcal {F}_{\\textsf {A}}}p$ or $p_\\alpha =_{\\mathcal {F}_{\\textsf {A}}}p^-$ .",
"Furthermore, there must be some exponent $r^{\\prime }_\\alpha $ such that $w_1^-\\cdot w_2 =_{\\mathcal {F}_{\\textsf {A}}}\\alpha \\cdot p^{r^{\\prime }_\\alpha }\\cdot \\alpha ^-$ .",
"For $\\alpha _1,\\alpha _2 \\in \\hat{L}$ , we therefore have that ${\\small p^{r^{\\prime }_{\\alpha _1}} =_{\\mathcal {F}_{\\textsf {A}}}(\\alpha _1^-\\cdot \\alpha _2)\\cdot p^{r^{\\prime }_{\\alpha _2}}\\cdot (\\alpha _1^-\\cdot \\alpha _2)^-}$ Therefore by Lemma , $\\alpha _1^-\\cdot \\alpha _2 \\in \\langle p\\rangle $ .",
"Let us fix some $w_k \\in \\mathcal {L}(q_k)$ , $\\alpha \\in \\hat{L} =_{\\mathcal {F}_{\\textsf {A}}}u_k\\cdot \\mathcal {L}(q_{k+1})\\cdot \\ldots \\cdot u_{\\ell -1}$ , and $w_l \\in \\mathcal {L}(q_l)$ .",
"Then $\\mathcal {L}(q_k) \\subseteq w_k\\cdot \\alpha \\cdot \\langle p\\rangle \\cdot \\alpha ^-$ , $\\hat{L} \\subseteq \\alpha \\cdot \\langle p\\rangle $ and $\\mathcal {L}(q_l) \\subseteq \\langle p\\rangle \\cdot w_l$ .",
"Therefore, ${\\small \\mathcal {L}(q_k)\\cdot u_k\\cdot \\ldots \\cdot \\mathcal {L}(q_\\ell ) \\subseteq w_k\\cdot \\alpha \\cdot \\langle p\\rangle \\cdot \\alpha ^-\\cdot \\alpha \\cdot \\langle p\\rangle \\cdot \\langle p\\rangle \\cdot w_l=_{\\mathcal {F}_{\\textsf {A}}}w_k\\cdot \\alpha \\cdot \\langle p\\rangle \\cdot w_l}$ As $\\sigma (k) > \\sigma ^{\\prime }(k) = \\sigma (\\ell )$ , this again contradicts $M$ being ordered.",
"It remains to show that every ${\\sf LT}_{\\textsf {A}}$ can be ordered in polynomial time.",
"For that, we rely on the following characterization.",
"Assume that $L_1,\\ldots ,L_n$ are neither empty nor singleton subsets of $\\mathcal {F}_{\\textsf {A}}$ and $u_1, \\ldots , u_{n-1} \\in \\mathcal {F}_{\\textsf {A}}$ .",
"Then there are $v_1,\\ldots ,v_n\\in \\mathcal {F}_{\\textsf {A}}$ such that ${\\small L_1\\cdot u_1\\cdot \\ldots \\cdot L_{n-1}\\cdot u_{n-1}\\cdot L_n \\subseteq v\\cdot \\langle p\\rangle }$ holds if and only if for $i=1,\\ldots ,n$ , $L_i \\subseteq v_i\\cdot \\langle p_i\\rangle $ with ${\\small \\begin{array}{lll}p_n &=_{\\mathcal {F}_{\\textsf {A}}}& p\\\\p_i &=_{\\mathcal {F}_{\\textsf {A}}}& (u_i\\cdot v_{i+1})\\cdot p_{i+1}\\cdot (u_i\\cdot v_{i+1})^-\\quad \\text{\\normalsize for }i<n\\end{array}}$ and ${\\small v^-\\cdot v_1\\cdot u_1\\cdot \\ldots \\cdot v_{n-1}\\cdot u_{n-1}\\cdot v_n\\in \\langle p\\rangle }$ Let $s_1 = \\varepsilon $ .",
"For $i = 2,\\ldots , n$ we fix some word $s_i \\in L_1\\cdot u_1\\cdot L_2\\cdot \\ldots \\cdot L_{i-1}\\cdot u_{i-1}$ .",
"Likewise, let $t_n = \\varepsilon $ and for $i = 1,\\ldots , n-1$ fix some word $t_i \\in u_i\\cdot L_{i+1}\\cdot \\ldots \\cdot L_{n}$ , and define $v_i =_{\\mathcal {F}_{\\textsf {A}}}s_i^-\\cdot v\\cdot t_i^-$ .",
"First assume that the inclusion (REF ) holds.",
"Let $p^{\\prime }_i=_{\\mathcal {F}_{\\textsf {A}}}t_i\\cdot p\\cdot t_i^-$ .",
"Then for all $i$ , $s_i\\cdot L_i\\cdot t_i \\subseteq v\\cdot \\langle p\\rangle $ , and therefore $L_i \\subseteq s_i^-\\cdot v\\cdot \\langle p\\rangle \\cdot t_i^-=_{\\mathcal {F}_{\\textsf {A}}}s_i^-\\cdot v\\cdot t_i^-\\cdot t_i\\cdot \\langle p\\rangle \\cdot t_i^-=_{\\mathcal {F}_{\\textsf {A}}}v_i \\langle p^{\\prime }_i\\rangle $ We claim that $p^{\\prime }_i = p_i$ for all $i=1,\\ldots ,n$ .",
"We proceed by induction on $n-i$ .",
"As $t_n = \\varepsilon $ , we have that $p^{\\prime }_n = p = p_n$ .",
"For $i<n$ , we can rewrite $t_i =_{\\mathcal {F}_{\\textsf {A}}}u_i\\cdot w_{i+1}\\cdot t_{i+1}$ where $w_{i+1}\\in L_{i+1}$ and thus is of the form $v_{i+1}\\cdot p_{i+1}^{k_{i+1}}$ for some exponent $k_{i+1}$ .",
"${\\small \\begin{array}{lll}p^{\\prime }_i &=_{\\mathcal {F}_{\\textsf {A}}}& t_i\\cdot p\\cdot t_i^-\\\\&=_{\\mathcal {F}_{\\textsf {A}}}& u_i\\cdot w_{i+1}\\cdot t_{i+1}\\cdot p\\cdot t_{i+1}^-\\cdot w_{i+1}^-\\cdot u_i^-\\\\&=_{\\mathcal {F}_{\\textsf {A}}}& u_i\\cdot w_{i+1}\\cdot p_{i+1} \\cdot w_{i+1}^-\\cdot u_i^-\\qquad \\text{\\normalsize by I.H.",
"}\\\\&=_{\\mathcal {F}_{\\textsf {A}}}& u_i\\cdot v_{i+1}\\cdot p_{i+1} \\cdot v_{i+1}^-\\cdot u_i^-\\\\&=_{\\mathcal {F}_{\\textsf {A}}}& p_i\\end{array}}$ It remains to prove the inclusion (REF ).",
"Since $w_i\\in L_i$ , we have by (REF ) that $v^-w_1\\cdot u_1\\cdot \\ldots w_n\\cdot u_n\\in \\langle p\\rangle $ holds.",
"Now we calculate: ${\\small \\begin{array}{lll}v^-\\cdot w_1\\cdot u_1\\cdot \\ldots u_{n-1}\\cdot w_n&=_{\\mathcal {F}_{\\textsf {A}}}& v^-\\cdot v_1\\cdot p_1^{k_1}\\cdot u_1\\cdot \\ldots \\cdot u_{n-1} \\cdot v_n\\cdot p_n^{k_n} \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v^-\\cdot v_1\\cdot u_1\\cdot v_2\\cdot p_2^{k_1+k_2}\\cdot u_2\\cdot \\ldots \\cdot u_{n-1} \\cdot v_n\\cdot p_n^{k_n} \\\\&\\ldots & \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v^-\\cdot v_1\\cdot u_1\\cdot \\ldots v_{n-1}\\cdot u_{n-1}\\cdot v_n\\cdot p_n^k\\end{array}}$ where $k= k_1+\\ldots +k_n$ .",
"Since $p_n=p$ , the claim follows.",
"The other direction of the claim of the lemma follows directly: ${\\small \\begin{array}{lll}L_1 u_1 \\ldots L_{n-1} u_{n-1} L_n&\\subseteq & v_1\\cdot \\langle p_1\\rangle \\cdot u_1\\cdot \\ldots \\cdot v_{n-1}\\cdot \\langle p_{n-1}\\rangle \\cdot u_{n-1}\\cdot v_n\\cdot \\langle p_n\\rangle \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v_1\\cdot u_1\\cdot v_2\\cdot \\langle p_2\\rangle \\cdot \\langle p_2\\rangle \\cdot u_2\\cdot \\ldots \\cdot v_{n-1}\\cdot \\langle p_{n-1}\\rangle \\cdot u_{n-1}\\cdot v_n\\cdot \\langle p_n\\rangle \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v_1\\cdot u_1\\cdot v_2\\cdot \\langle p_2\\rangle \\cdot u_2\\cdot \\ldots \\cdot v_{n-1}\\cdot \\langle p_{n-1}\\rangle \\cdot u_{n-1}\\cdot v_n\\cdot \\langle p_n\\rangle \\\\&\\cdots & \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v_1\\cdot u_1\\cdot v_2\\cdot \\ldots \\cdot u_{n-1}\\cdot v_n\\cdot \\langle p_n\\rangle \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v_1\\cdot u_1\\cdot v_2\\cdot \\ldots \\cdot u_{n-1}\\cdot v_n\\cdot \\langle p\\rangle \\\\&\\subseteq &v\\cdot \\langle p\\rangle \\end{array}}$ where the last inclusion follows from (REF ).",
"Let us call a non-empty, non-singleton language $L\\subseteq \\mathcal {F}_{\\textsf {A}}$ periodic, if $L\\subseteq v\\cdot \\langle p\\rangle $ for some $v,p\\in \\mathcal {F}_{\\textsf {A}}$ .",
"Lemma then implies that if a concatenation of languages and elements from $\\mathcal {F}_{\\textsf {A}}$ is periodic, then so must be all non-singleton component languages.",
"In fact, the languages in the composition can then be arbitrarily permuted.",
"Assume for non-empty, nonsingleton languages $L_1,\\ldots ,L_n\\subseteq \\mathcal {F}_{\\textsf {A}}$ and $u_1, \\ldots , u_{n-1} \\in \\mathcal {F}_{\\textsf {A}}$ that property (REF ) holds.",
"Then for every permutation $\\pi $ , there are elements $u_{\\pi ,0},\\ldots ,u_{\\pi ,n}\\in \\mathcal {F}_{\\textsf {A}}$ such that $L_1\\cdot u_1\\cdot \\ldots \\cdot L_{n-1}\\cdot u_{n-1}\\cdot L_n =u_{\\pi ,0}\\cdot L_{\\pi (1)}\\cdot u_{\\pi ,1}\\cdot \\ldots \\cdot u_{\\pi _n-1}\\cdot L_{\\pi (n)}\\cdot u_{\\pi ,n}$ We reconsider ${\\sf LT}_{\\textsf {A}}$ $M^{\\prime }$ and DTA $B$ from Example REF .",
"We observe that $\\mathcal {L}(q_1) \\subseteq a\\cdot \\langle ba\\rangle $ , $\\mathcal {L}(q_2) \\subseteq \\langle ab\\rangle $ , and thus $\\mathcal {L}(q_0) = \\mathcal {L}(q_1)\\cdot b\\cdot \\mathcal {L}(q_2) \\subseteq \\langle ab\\rangle $ .",
"Accordingly, the rule for state $q_0$ and input symbol $f$ is not ordered.",
"Following the notation of Corollary , we find $v_1 = a$ , $u_1 = b$ and $v_2 = \\varepsilon $ , and the rule for $q_0$ and $f$ can be reordered to ${\\small q_0(f(x_1, x_2)) \\rightarrow ab\\cdot q_2(x_1)\\cdot a^-\\cdot q_1(x_2)}$ This example shows major improvements compared to the construction in [2].",
"Since we have inverses at hand, only local changes must be applied to the sub-sequence $q_1(x_2)\\cdot b\\cdot q_2(x_1)$ .",
"In contrast to the construction in [2], neither auxiliary states nor further changes to the rules of $q_1$ and $q_2$ are required.",
"By Corollary , the order of occurrences of terms $q_k(x_{\\sigma (k)})$ can be permuted in every sub-sequence $q_i(x_{\\sigma (i)})\\cdot u_i\\cdot \\ldots \\cdot u_{j-1} q_j(x_{\\sigma (j)})$ where $\\mathcal {L}(q_i)\\cdot u_i\\cdot \\ldots \\cdot u_{j-1}\\cdot \\mathcal {L}(q_j) \\in u\\cdot \\langle p\\rangle $ is periodic, to satisfy the requirements of an ordered ${\\sf LT}_{\\textsf {A}}$ .",
"A sufficient condition for that is, according to Lemma , that $\\mathcal {L}(q_k)$ is periodic for each $q_k$ occurring in that sub-sequence.",
"Therefore we will determine the subset of all states $q$ where $\\mathcal {L}(q)$ is periodic, and if so elements $v_q,p_q$ such that $\\mathcal {L}(q)\\subseteq v_q\\cdot \\langle p_q\\rangle $ .",
"In order to do so we compute an abstraction of the sets $\\mathcal {L}(q)$ by means of a complete lattice which both reports constant values and also captures periodicity.",
"Let $ 2^{\\mathcal {F}_{\\textsf {A}}}$ denote the complete lattice of subsets of the free group $\\mathcal {F}_{\\textsf {A}}$ .",
"We define a projection $\\alpha : by$ ()=$, $ ({g}) = {g}$, and for languages $ L$ withat least two elements,$${\\small \\alpha (L) = {\\left\\lbrace \\begin{array}{ll}g\\langle p\\rangle & \\text{if } L \\subseteq g\\langle p\\rangle \\text{ and $p$ is primitive} \\\\\\mathcal {F}_{\\textsf {A}}& \\text{otherwise}\\end{array}\\right.",
"}}$$The projection $$ is a \\emph {closure} operator, i.e., is a monotonic function with$ L(L)$, and $ ((L)) = (L)$.The image of $$ can be considered as an \\emph {abstract} complete lattice $ ♯$,partially ordered by subset inclusion.Thereby, the abstraction $$ commutes with least upper bounds as well as withthe group operation.",
"For that, we define \\emph {abstract} versions$ ,:(♯)2♯$of set union and the group operation by$$A_1\\sqcup A_2 = \\alpha (A_1\\cup A_2)\\qquad A_1\\star A_2 = \\alpha (A_1\\cdot A_2)$$In fact, ``$$^{\\prime \\prime } is the least upper bound operation for $ ♯$.The two abstract operators can also be more explicitly defined by:$${\\small \\begin{array}{lllll}\\emptyset \\sqcup L &=& L\\sqcup \\emptyset &=& L \\\\\\mathcal {F}_{\\textsf {A}}\\sqcup L &=& L\\sqcup \\mathcal {F}_{\\textsf {A}}&=& \\mathcal {F}_{\\textsf {A}}\\\\\\lbrace g_1\\rbrace \\sqcup \\lbrace g_2\\rbrace &=&\\multicolumn{3}{l}{{\\left\\lbrace \\begin{array}{ll}\\lbrace g_1\\rbrace & \\text{if } g_1 = g_2 \\\\g_1\\cdot \\langle p\\rangle & \\text{if } g_1\\ne g_2, p \\text{ primitive root of } g_1^-\\cdot g_2\\end{array}\\right.}}",
"\\\\\\lbrace g_1\\rbrace \\sqcup g_2\\cdot \\langle p\\rangle &=&g_2\\cdot \\langle p\\rangle \\sqcup \\lbrace g_1\\rbrace &=&{\\left\\lbrace \\begin{array}{ll}g_2\\cdot \\langle p\\rangle & \\text{if } g_1\\in g_2\\cdot \\langle p\\rangle \\\\\\mathcal {F}_{\\textsf {A}}& \\text{otherwise}\\end{array}\\right.}",
"\\\\g_1\\cdot \\langle p_1\\rangle \\sqcup g_2\\cdot \\langle p_2\\rangle &=&\\multicolumn{3}{l}{{\\left\\lbrace \\begin{array}{ll}g_1\\cdot \\langle p_1\\rangle & \\text{if } p_2\\in \\langle p_1\\rangle \\text{ and }g_2^-\\cdot g_1\\in \\langle p_1\\rangle \\\\\\mathcal {F}_{\\textsf {A}}& \\text{otherwise}\\end{array}\\right.}}",
"\\\\\\end{array}}$$$ ${\\small \\begin{array}{lllll}\\emptyset \\star L &=& L\\star \\emptyset &=& \\emptyset \\\\\\mathcal {F}_{\\textsf {A}}\\star L &=& L\\star \\mathcal {F}_{\\textsf {A}}&=& F_{\\textsf {A}}\\qquad \\text{for } L\\ne \\emptyset \\\\\\lbrace g_1\\rbrace \\star \\lbrace g_2\\rbrace &=& \\multicolumn{3}{l}{\\lbrace g_1\\cdot g_2\\rbrace } \\\\\\lbrace g_1\\rbrace \\star g_2\\cdot \\langle p\\rangle &=& \\multicolumn{3}{l}{(g_1\\cdot g_2)\\cdot \\langle p\\rangle } \\\\g_1\\cdot \\langle p\\rangle \\star \\lbrace g_2\\rbrace &=& \\multicolumn{3}{l}{(g_1\\cdot g_2)\\cdot \\langle g_2^-\\cdot p\\cdot g_2\\rangle } \\\\g_1\\cdot \\langle p_1\\rangle \\star g_2\\cdot \\langle p_2\\rangle &=&\\multicolumn{3}{l}{{\\left\\lbrace \\begin{array}{ll}(g_1\\cdot g_2)\\cdot \\langle p_2\\rangle &\\text{if } g_2^-\\cdot p_1\\cdot g_2\\in \\langle p_2\\rangle \\\\\\mathcal {F}_{\\textsf {A}}& \\text{otherwise}\\end{array}\\right.",
"}}\\end{array}}$ For all subsets $L_1,L_2\\subseteq \\mathcal {F}_{\\textsf {A}}$ , $\\alpha (L_1\\cup L_2) = \\alpha (L_1)\\sqcup \\alpha (L_2)$ and $\\alpha (L_1\\cdot L_2) = \\alpha (L_1)\\star \\alpha (L_2)$ .",
"We conclude that $\\alpha $ in fact represents a precise abstract interpretation in the sense of [8].",
"Accordingly, we obtain: For every ${\\sf LT}_{\\textsf {A}}$ $M$ and DTA $B$ with compatible map $\\iota $ , the sets $\\alpha (\\mathcal {L}(q))$ , $q$ state of $M$ , can be computed in polynomial time.",
"We introduce one unknown $X_q$ for every state $q$ of $M$ , and one constraint for each rule of $M$ of the form (REF ) where $\\delta (\\iota (q),f)$ is defined in $B$ .",
"This constraint is given by: ${\\small X_q\\sqsupseteq u_0\\star X_{q_1}\\star \\ldots \\star u_{n-1}\\star X_{q_n}\\star u_n} $ As the right-hand sides of the constraints (REF ) all represent monotonic functions, the given system of constraints has a least solution.",
"In order to obtain this solution, we consider for each state $q$ of $M$ , the sequence $X_{q}^{(i)},i\\ge 0$ of values in $\\sharp $ where $X_q^{(0)} = \\emptyset $ , and for $i>0$ , we set $X_q^{(i)}$ as the least upper bound of the values obtained from the constraints with left-hand side $X_q$ of the form (REF ) by replacing the unknowns $X_{q_j}$ on the right-hand side with the values $X_{q_j}^{(i-1)}$ .",
"By induction on $i\\ge 0$ , we verify that for all states $q$ of $M$ , ${\\small X_q^{(i)} = \\alpha (\\mathcal {L}^{(i)}(q)})$ holds.",
"Note that the induction step thereby, relies on Lemma .",
"As each strictly increasing chain of elements in $\\sharp $ consists of at most four elements, we have that the least solution of the constraint system is attained after at most $3\\cdot N$ iterations, if $N$ is the number of states of $M$ , i.e., for each state $q$ of $M$ , $X_q^{(3N)} = X_q^{(i)}$ for all $i\\ge 3N$ .",
"The elements of $\\sharp $ can be represented by SLPs where the operations $\\star $ and $\\sqcup $ run in polynomial time, cf.",
"Lemma .",
"Since each iteration requires only a polynomial number of operations $\\star $ and $\\sqcup $ , the statement of the lemma follows.",
"We now exploit the information provided by the $\\alpha (\\mathcal {L}(q))$ to remove trivial states as well as order subsequences of right-hand sides which are periodic.",
"Let $B$ be a DTA such that $\\mathcal {L}(B)\\ne \\emptyset $ .",
"For every ${\\sf LT}_{\\textsf {A}}$ $M$ with compatible map $\\iota $ , an ${\\sf LT}_{\\textsf {A}}$ $M^{\\prime }$ with compatible map $\\iota ^{\\prime }$ can be constructed in polynomial time such that $M$ and $M^{\\prime }$ are equivalent relative to $B$ ; $M^{\\prime }$ has no trivial states; $M^{\\prime }$ is ordered.",
"By Lemma , we can, in polynomial time, determine for every state $q$ of $M$ , the value $\\alpha (\\mathcal {L}(q))$ .",
"We use this information to remove from $M$ all trivial states.",
"W.l.o.g., assume that the axiom of $M$ is given by $u_0\\cdot q_0(x_0)\\cdot u_1$ .",
"If the state $q_0$ occurring in the axiom of $M$ is trivial with $\\mathcal {L}(q_0) = \\lbrace v\\rbrace $ , then $M_1$ has no states or rules, but the axiom $u_0\\cdot v\\cdot u_1$ .",
"Therefore now assume that $q_0$ is non-trivial.",
"We then construct an ${\\sf LT}_{\\textsf {A}}$ $M_1$ whose set of states $Q_1$ consists of all non-trivial states $q$ of $M$ where the compatible map $\\iota _1$ of $M_1$ is obtained from $\\iota $ by restriction to $Q_1$ .",
"Since $\\mathcal {L}(M)\\ne \\emptyset $ , the state of $M$ occurring in the axiom is non-trivial.",
"Accordingly, the axiom of $M$ is also used as axiom for $M_1$ .",
"Consider a non-trivial state $q$ of $M$ and $f\\in \\Sigma $ .",
"If $\\delta (\\iota (q),f)$ is not defined $M_1$ has the rule $q(f(x_1,\\ldots ,x_m)\\rightarrow \\bot $ .",
"Assume that $\\delta (\\iota (q),f)$ is defined and $M$ has a rule of the form (REF ).",
"Then $M_1$ has the rule ${\\small q(f(x_1,\\ldots ,x_m))\\rightarrow u_0\\cdot g_1\\cdot \\ldots \\cdot u_{n-1}\\cdot g_n\\cdot u_n}$ where for $i=1,\\ldots ,n$ , $g_i$ equals $q_i(x_{\\sigma (i)})$ if $q_i$ is non-trivial, and equals the single word in $\\mathcal {L}(q_i)$ otherwise.",
"Obviously, $M$ and $M_1$ are equivalent relative to $B$ where $M_1$ now has no trivial states, while for every non-trivial state $q$ , the semantics of $q$ in $M$ and $M_1$ are the same relative to $B$ .",
"Our goal now is to equivalently rewrite the right-hand side of each rule of $M_1$ so that the result is ordered.",
"For each state $q$ of the ${\\sf LT}_{\\textsf {A}}$ we determine whether there are $v, p \\in \\textsf {B}^*$ such that $\\mathcal {L}(q) \\subseteq v\\langle p\\rangle $ , cf.",
"Lemma .",
"So consider a rule of $M_1$ of the form (REF ).",
"By means of the values $\\alpha (\\mathcal {L}(q_i))$ , $i=1,\\ldots ,n$ , together with the abstract operation “$\\star $ ”, we can determine maximal intervals $[i,j]$ such that $\\mathcal {L}(q_i)\\cdot u_i\\cdot \\ldots \\cdot u_{j-1}\\cdot \\mathcal {L}(q_j)$ is periodic, i.e., $\\alpha (\\mathcal {L}(q_i))\\star u_i\\cdot \\ldots \\star u_{j-1}\\star \\alpha (\\mathcal {L}(q_j)) \\subseteq v\\cdot \\langle p\\rangle $ for some $v,p\\in \\mathcal {F}_{\\textsf {A}}$ .",
"We remark that these maximal intervals are necessarily disjoint.",
"By Corollary , for every permutation $\\pi :[i,j]\\rightarrow [i,j]$ , elements $u^{\\prime },u^{\\prime }_i,\\ldots ,u^{\\prime }_j,u^{\\prime \\prime }\\in \\mathcal {F}_{\\textsf {A}}$ can be found so that $q_i(x_{\\sigma (i)})\\cdot u_i\\cdot \\ldots \\cdot u_{j-1}\\cdot q_j(x_{\\sigma (j)})$ is equivalent to $u^{\\prime }\\cdot q_{\\pi (i)}(x_{\\sigma (\\pi (i))})\\cdot u^{\\prime }_i\\cdot \\ldots \\cdot u^{\\prime }_{j-1}\\cdot q_{\\pi (j)}(x_{\\sigma (\\pi (j))})\\cdot u^{\\prime \\prime }$ .",
"In particular, this is true for the permutation $\\pi $ with $\\sigma (\\pi (i)) < \\ldots <\\sigma (\\pi (j))$ .",
"Assuming that all group elements are represented as SLPs, the overall construction runs in polynomial time.",
"In summary, we arrive at the main theorem of this paper.",
"The equivalence of ${\\sf LT}_{\\textsf {A}}$ s relative to some DTA $B$ can be decided in polynomial time.",
"Assume we are given ${\\sf LT}_{\\textsf {A}}$ s $M,M^{\\prime }$ with compatible maps (relative to $B$ ).",
"By Theorem , we may w.l.o.g.",
"assume that $M$ and $M^{\\prime }$ both have no trivial states and are ordered.",
"It can be checked in polynomial time whether or not $M$ and $M^{\\prime }$ are same-ordered.",
"If they are not, then by Lemma , they cannot be equivalent relative to $B$ .",
"Therefore now assume that $M$ and $M^{\\prime }$ are same-ordered.",
"Then their equivalence relative to $B$ is decidable in polynomial time by Corollary .",
"Altogether we thus obtain a polynomial decision procedure for equivalence of ${\\sf LT}_{\\textsf {A}}$ s relative to some DTA $B$ .",
"Conclusion We have shown that equivalence of ${\\sf LT}_{\\textsf {A}}$ s relative to a given DTA $B$ can be decided in polynomial time.",
"For that, we considered total transducers only, but defined the domain of allowed input trees separately by means of the DTA.",
"This does not impose any restriction of generality, since any (possibly partial) linear deterministic top-down tree transducer can be translated in polynomial time to a corresponding total ${\\sf LT}_{\\textsf {A}}$ together with a corresponding DTA (see, e.g., [10]).",
"The required constructions for ${\\sf LT}_{\\textsf {A}}$ s which we have presented here, turn out to be more general than the constructions provided in [2] since they apply to transducers which may not only output symbols $a\\in {\\textsf {A}}$ , but also their inverses $a^-$ .",
"At the same time, they are simpler and easier to be proven correct due to the combinatorial and algebraic properties provided by the free group.",
"Proof of Lemma REF For an ${\\sf LT}_{\\textsf {A}}$ $M$ and a DTA $B=(H,\\Sigma ,\\delta _B,h_0)$ , an ${\\sf LT}_{\\textsf {A}}$ $M^{\\prime }$ with a set of states $Q^{\\prime }$ together with a mapping $\\iota :Q^{\\prime }\\rightarrow H$ can be constructed in polynomial time such that the following holds: $M$ and $M^{\\prime }$ are equivalent relative to $B$ ; $\\iota $ is compatible.",
"In case that the axiom of $M$ is in $\\mathcal {F}_{\\textsf {A}}$ , we obtain $M^{\\prime }$ from $M$ using the axiom of $M$ and using empty sets of states and rules, respectively.",
"Assume therefore that the axiom of $M$ is of the form $u_0\\cdot q_0(x_0)\\cdot u_1$ .",
"Then ${\\sf LT}_{\\textsf {A}}$ $M^{\\prime }$ is constructed as follows.",
"The set $Q^{\\prime }$ of states of $M^{\\prime }$ consists of pairs $\\langle q,h\\rangle $ , $q\\in Q, h\\in H$ where $\\iota (\\langle q,h\\rangle ) = h$ .",
"In particular, $\\langle q_0,h_0\\rangle \\in Q^{\\prime }$ .",
"As the axiom of $M^{\\prime }$ we then use $u_0\\cdot \\langle q_0,h_0\\rangle (x_0)\\cdot u_1$ .",
"For a state $\\langle q,h\\rangle \\in Q^{\\prime }$ , consider each input symbol $f\\in \\Sigma $ .",
"Let $m\\ge 0$ denote the rank of $f$ .",
"If $\\delta _B(h,f)$ is not defined, $M^{\\prime }$ has the rule $\\langle q,h\\rangle (f(x_1,\\ldots ,x_m))\\rightarrow \\bot $ Otherwise, let $\\delta _H(h,f) = h_1\\ldots h_m$ , and assume that $M$ has a rule of the form (REF ).",
"Then we add the states $\\langle q_i,h_{\\sigma (i)}\\rangle $ to $Q^{\\prime }$ together with the rule $\\langle q,h\\rangle (f(x_1,\\ldots ,x_m))\\rightarrow u_0\\cdot \\langle q_1,h_{\\sigma (1)}\\rangle (x_{\\sigma (1)})\\cdot \\ldots \\cdot u_{n-1}\\cdot \\langle q_n,h_{\\sigma (n)}\\rangle (x_{\\sigma (n)})\\cdot u_n$ By construction, the mapping $\\iota $ is compatible.",
"We verify for each state $\\langle q,h\\rangle $ of $M^{\\prime }$ and each input tree $t\\in \\textsf {dom}(h)$ that $\\llbracket q\\rrbracket (t) = \\llbracket \\langle q,h\\rangle \\rrbracket (t)$ holds.",
"This proof is by induction on the structure of $t$ .",
"From that, the equivalence of $M$ and $M^{\\prime }$ relative to $B$ follows.",
"Proof of Corollary Assume for non-empty, nonsingleton languages $L_1,\\ldots ,L_n\\subseteq \\mathcal {F}_{\\textsf {A}}$ and $u_1, \\ldots , u_{n-1} \\in \\mathcal {F}_{\\textsf {A}}$ that property (REF ) holds.",
"Then for every permutation $\\pi $ , there are elements $u_{\\pi ,0},\\ldots ,u_{\\pi ,n}\\in \\mathcal {F}_{\\textsf {A}}$ such that $L_1\\cdot u_1\\cdot \\ldots \\cdot L_{n-1}\\cdot u_{n-1}\\cdot L_n =u_{\\pi ,0}\\cdot L_{\\pi (1)}\\cdot u_{\\pi ,1}\\cdot \\ldots \\cdot u_{\\pi _n-1}\\cdot L_{\\pi (n)}\\cdot u_{\\pi ,n}$ For $i=1,\\ldots ,n$ , let $v_i$ and $p_i$ be defined as in Lemma .",
"Then for all $i$ , $L_i \\subseteq v_i \\langle p_i\\rangle $ .",
"Moreover, the languages $L^{\\prime }_i$ defined by $L^{\\prime }_n = v_n^-\\cdot L_n$ and for $i<n$ , ${\\small L^{\\prime }_i = (u_{i}\\cdot v_{i+1}\\cdot \\ldots \\cdot u_{n-1}\\cdot v_n)^-\\cdot (v_i^-\\cdot L_i)\\cdot (u_{i}\\cdot v_{i+1}\\cdot \\ldots \\cdot u_{n-1}\\cdot v_n)}$ all are subsets of $\\langle p\\rangle $ .",
"Therefore their compositions can arbitrarily be permuted.",
"At the same time, ${\\small L_1\\cdot u_1\\cdot \\ldots \\cdot L_{n-1}\\cdot u_{n-1}\\cdot L_n=_{\\mathcal {F}_{\\textsf {A}}}v_1\\cdot u_1\\cdot \\ldots \\cdot v_{n-1}\\cdot u_n\\cdot v_n\\cdot L^{\\prime }_1\\cdot \\ldots L^{\\prime }_n}$ From that, the corollary follows.",
"Proof of Lemma For all subsets $L_1,L_2\\subseteq \\mathcal {F}_{\\textsf {A}}$ , $\\alpha (L_1\\cup L_2) = \\alpha (L_1)\\sqcup \\alpha (L_2)$ and $\\alpha (L_1\\cdot L_2) = \\alpha (L_1)\\star \\alpha (L_2)$ .",
"As $\\emptyset \\cup L = L \\cup \\emptyset = \\emptyset $ , it follows that $\\alpha (\\emptyset \\cup L) = \\alpha (L \\cup \\emptyset ) = \\alpha (\\emptyset ) = \\emptyset = \\emptyset \\sqcup L^{\\prime } = L^{\\prime } \\sqcup \\emptyset = \\alpha (\\emptyset ) \\sqcup \\alpha (L) = \\alpha (L) \\sqcup \\alpha (\\emptyset )$ .",
"Assume that $\\alpha (L_1) = \\mathcal {F}_\\textsf {A}$ .",
"Let $L_2$ be some language, then $\\alpha (L_1 \\cup L_2) = \\alpha (L_2 \\cup L_1) = \\mathcal {F}_\\textsf {A}$ and $\\alpha (L_1) \\sqcup \\alpha (L_2) = \\mathcal {F}_\\textsf {A}\\sqcup \\alpha (L_2) = \\mathcal {F}_\\textsf {A}= \\alpha (L_2) \\sqcup \\mathcal {F}_\\textsf {A}= \\alpha (L_2) \\sqcup \\alpha (L_1)$ .",
"The case where $\\alpha (L_2)=\\mathcal {F}_\\textsf {A}$ is analogous.",
"For $\\alpha (L_1) = \\lbrace g_1\\rbrace $ , $\\alpha (L_2) = \\lbrace g_2\\rbrace $ , both languages are singleton, and we obtain that $\\lbrace g_1\\rbrace \\cup \\lbrace g_2\\rbrace = \\lbrace g_1\\rbrace $ if and only if $g_1 = g_2$ .",
"Accordingly, $\\alpha (L_1 \\cup L_2) = \\alpha (\\lbrace g_1\\rbrace ) = \\lbrace g_1\\rbrace =\\alpha (\\lbrace g_1\\rbrace ) \\sqcup \\alpha (\\lbrace g_2\\rbrace )$ .",
"If $g_1 \\ne g_2$ then $\\lbrace g_1\\rbrace \\cup \\lbrace g_2\\rbrace \\subseteq g_1\\langle g_1^-g_2\\rangle $ and $\\alpha (L_1 \\cup L_2) = g_1\\langle p\\rangle $ with $p$ the primitive root of $g_1^- g_2$ .",
"Therefore, $\\alpha (L_1 \\cup L_2) = g_1\\langle p\\rangle = \\alpha (\\lbrace g_1\\rbrace ) \\sqcup \\alpha (\\lbrace g_2\\rbrace )$ .",
"Assume that $\\alpha (L_1)=\\lbrace g_1\\rbrace $ and $\\alpha (L_2)=g_2\\langle p_2\\rangle $ for some primitive $p_2$ .",
"If $g_1\\in g_2\\langle p_2\\rangle $ , then $\\alpha (L_1\\cup L_2)= g_2\\langle p_2\\rangle = \\alpha (L_1)\\sqcup \\alpha (L_2)$ .",
"Otherwise, i.e., if $g_1 \\notin g_2\\langle p_2\\rangle $ , then $L_1\\cup L_2$ is not contained in $g\\langle p\\rangle $ for any $p$ (since $p_2$ was chosen primitive), and therefore, $\\alpha (L_1\\cup L_2) = \\mathcal {F}_\\textsf {A}= \\alpha (L_1)\\sqcup \\alpha (L_2)$ .",
"A similar argument applies if $\\alpha (L_2) = \\lbrace g_1\\rbrace $ , and $\\alpha (L_1) = g_2\\langle p_2\\rangle $ .",
"Assume that $\\alpha (L_1)=g_1\\langle p_1\\rangle $ and $\\alpha (L_2)=g_2\\langle p_2\\rangle $ for some primitive $p_1,p_2$ .",
"If $p_2\\in \\langle p_1\\rangle $ as well as $g_2^-g_1\\in \\langle p_1\\rangle $ , then $g_1\\langle p_1\\rangle = g_2\\langle p_2\\rangle $ (due to primitivity of $p_1,p_2$ ).",
"Moreover, $\\alpha (L_1\\cup L_2) = g_1\\langle p_1\\rangle = \\alpha (L_1)\\sqcup \\alpha (L_2)$ .",
"Otherwise, i.e., if $p_2\\notin \\langle p_1\\rangle $ or $g_2^- g_1\\notin \\langle p_1\\rangle $ , then $L_1\\cup L_2$ cannot be subset of $g\\langle p\\rangle $ for any $g,p\\in \\mathcal {F}_\\textsf {A}$ .",
"Therefore, $\\alpha (L_1\\cup L_2) = \\mathcal {F}_\\textsf {A}= \\alpha (L_1)\\sqcup \\alpha (L_2)$ .",
"For the concatenation with the empty set and the product operator we find $\\alpha (\\emptyset \\cdot L) = \\alpha (L \\cdot \\emptyset ) = \\alpha (\\emptyset ) = \\emptyset = \\alpha (\\emptyset ) \\star \\alpha (L) = \\alpha (L) \\star \\alpha (\\emptyset )$ .",
"Assume that $\\alpha (L_1) =\\mathcal {F}_\\textsf {A}$ .",
"Then $L_1 \\lnot \\subseteq g\\langle p\\rangle $ for any $g,p\\in \\mathcal {F}_\\textsf {A}$ .",
"Assume that $L_2\\subseteq \\mathcal {F}_\\textsf {A}$ is nonempty.",
"Then by Lemma , $L_1 \\cdot L_2$ and $L_2\\cdot L_1$ cannot be contained in $g^{\\prime }\\langle p^{\\prime }\\rangle $ for any $g^{\\prime },p^{\\prime }$ .",
"Therefore, $\\alpha (L_1 \\cdot L_2) = \\alpha (L_2 \\cdot L_1) = \\mathcal {F}_\\textsf {A}=\\alpha (L_1) \\star \\alpha (L_2) = \\alpha (L_2) \\star \\alpha (L_1)$ .",
"For $\\alpha (L_1)=\\lbrace g_1\\rbrace , \\alpha (L_2)=\\lbrace g_2\\rbrace $ , both languages are singletons, and we obtain $\\alpha (L_1\\cdot L_2) = \\lbrace g_1g_2\\rbrace = \\lbrace g_1\\rbrace \\star \\lbrace g_2\\rbrace = \\alpha (L_1) \\star \\alpha (L_2)$ .",
"Now assume that $\\alpha (L_1)=\\lbrace g_1\\rbrace $ and $\\alpha (L_2)=g_2\\langle p_2\\rangle $ .",
"Then $L_1=\\lbrace g_1\\rbrace $ , while $L_1\\cdot L_2$ is not a singleton language, but contained in $g_1g_2\\langle p_2\\rangle $ .",
"Therefore, $\\alpha (L_1\\cdot L_2) = g_1g_2 \\langle p_2\\rangle = \\alpha (L_1) \\star \\alpha (L_2)$ .",
"Likewise, if $\\alpha (L_1)=g_1\\langle p_1\\rangle $ and $\\alpha (L_2)=\\lbrace g_2\\rbrace $ , then $L_2=\\lbrace g_2\\rbrace $ , and $L_1\\cdot L_2$ is a non-singleton language contained in $g_1g_2\\langle g_2^-p_1g_2\\rangle $ .",
"Therefore, $\\alpha (L_1\\cdot L_2) = g_1g_2 \\langle g_2^-p_1g_2\\rangle = \\alpha (L_1) \\star \\alpha (L_2)$ .",
"Finally, let $\\alpha (L_1) = g_1\\langle p_1\\rangle $ and $\\alpha (L_2) = g_2\\langle p_2\\rangle $ be both ultimately periodic languages.",
"By Lemma , $L_1 \\cdot L_2$ is ultimately periodic if and only if $g_2^-p_1g_2 \\in \\langle p_2\\rangle $ .",
"Thus if $L_1\\cdot L_2$ is ultimately periodic, then $\\alpha (L_1 \\cdot L_2) = g_1 g_2 \\langle p_2\\rangle = \\alpha (L_1) \\star \\alpha (L_2)$ .",
"Otherwise, $L_1 \\cdot L_2 \\lnot \\subseteq g\\langle p\\rangle $ for any $g,p\\in \\mathcal {F}_\\textsf {A}$ , and therefore $\\alpha (L_1 \\cdot L_2) = \\mathcal {F}_\\textsf {A}= \\alpha (L_1) \\star \\alpha (L_2)$ ."
],
[
"Deciding Equivalence",
"In the first step, we show that equivalence relative to the DTA $B$ of same-ordered ${\\sf LT}_{\\textsf {A}}$ s is decidable.",
"For a DTA $B$ , consider the ${\\sf LT}_{\\textsf {A}}$ s $M$ and $M^{\\prime }$ with compatible mappings $\\iota $ and $\\iota ^{\\prime }$ , respectively.",
"$M$ and $M^{\\prime }$ are same-ordered relative to $B$ if they process their input trees in the same order.",
"We define set of pairs $\\langle q,q^{\\prime }\\rangle $ of co-reachable states of $M$ and $M^{\\prime }$ .",
"Let $u_0\\cdot q_0(x_1)\\cdot u_1$ and $u^{\\prime }_0\\cdot q^{\\prime }_0(x_1)\\cdot u^{\\prime }_1$ be the axioms of $M$ and $M^{\\prime }$ , respectively, where $\\iota (q_0) = \\iota ^{\\prime }(q^{\\prime }_0)$ is the start state of $B$ .",
"Then the pair $\\langle q_0,q^{\\prime }_0\\rangle $ is co-reachable.",
"Let $\\langle q,q^{\\prime }\\rangle $ be a pair of co-reachable states.",
"Then $\\iota (q)=\\iota ^{\\prime }(q^{\\prime })$ should hold.",
"For $f \\in \\Sigma $ , assume that $\\delta _B(\\iota (q),f)$ is defined.",
"Let ${\\small \\begin{array}{lll}q(f(x_1, \\ldots , x_m)) &\\rightarrow & u_0 q_1(x_{\\sigma (1)}) u_1 \\ldots u_{n-1} q_n(x_{\\sigma (n)}) u_n\\\\q^{\\prime }(f(x_1, \\ldots , x_m)) &\\rightarrow & u^{\\prime }_0 q^{\\prime }_1(x_{\\sigma ^{\\prime }(1)}) u^{\\prime }_1 \\ldots u^{\\prime }_{n-1} q^{\\prime }_n(x_{\\sigma ^{\\prime }(n^{\\prime })}) u^{\\prime }_{n^{\\prime }}\\end{array}} $ be the rules of $q, q^{\\prime }$ for $f$ , respectively.",
"Then $\\langle q_{j},q^{\\prime }_{j^{\\prime }}\\rangle $ is co-reachable whenever $\\sigma (j)=\\sigma ^{\\prime }(j^{\\prime })$ holds.",
"In particular, we then have $\\iota (q_j) = \\iota ^{\\prime }(q^{\\prime }_{j^{\\prime }})$ .",
"The pair $\\langle q,q^{\\prime }\\rangle $ of co-reachable states is called same-ordered, if for each corresponding pair of rules (REF ), $n = n^{\\prime }$ and $\\sigma = \\sigma ^{\\prime }$ .",
"Finally, $M$ and $M^{\\prime }$ are same-ordered if for every co-reachable pair $\\langle q,q^{\\prime }\\rangle $ of states of $M, M^{\\prime }$ , and every $f\\in \\Sigma $ , each pair of rules (REF ) is same-ordered whenever $\\delta _B(\\iota (q),f)$ is defined.",
"Given that the ${\\sf LT}_{\\textsf {A}}$ s $M$ and $M^{\\prime }$ are same-ordered relative to $B$ , we can represent the set of pairs of runs of $M$ and $M^{\\prime }$ on input trees by means of a single context-free grammar $G$ .",
"The set of nonterminals of $G$ consists of a distinct start nonterminal $S$ together with all co-reachable pairs $\\langle q,q^{\\prime }\\rangle $ of states $q,q^{\\prime }$ of $M,M^{\\prime }$ , respectively.",
"The set of terminal symbols $T$ of $G$ is given by $\\lbrace a,a^-,\\bar{a},\\bar{a}^-\\mid a\\in {\\textsf {A}}\\rbrace $ for fresh distinct symbols $\\bar{a},\\bar{a}^-,a\\in {\\textsf {A}}$ .",
"Let $\\langle q,q^{\\prime }\\rangle $ be a co-reachable pair of states of $M, M^{\\prime }$ , and $f \\in \\Sigma $ such that $\\delta _B(\\iota (q),f)$ is defined.",
"For each corresponding pair of rules (REF ), $G$ receives the rule $\\langle q,q^{\\prime }\\rangle \\rightarrow u_0 \\bar{u}^{\\prime }_0 \\langle q_1, q^{\\prime }_1\\rangle u_1 \\bar{u}^{\\prime }_1 \\ldots u_{n-1} \\bar{u}^{\\prime }_{n-1} \\langle q_n,q^{\\prime }_n\\rangle u_n \\bar{u}^{\\prime }_n$ where $\\bar{u}^{\\prime }_i$ is obtained from $u^{\\prime }_i$ by replacing each output symbol $a\\in {\\textsf {A}}$ with its barred copy $\\bar{a}$ as well as each inverse $a^-$ with its barred copy $\\bar{a}^-$ .",
"For the axioms $u_0 q(x_1) u_1$ and $u^{\\prime }_0 q^{\\prime }(x_1) u^{\\prime }_1$ of $M,M^{\\prime }$ , respectively, we introduce the rule $S \\rightarrow u_0 \\bar{u}^{\\prime }_0 \\langle q,q^{\\prime }\\rangle u_1 \\bar{u}^{\\prime }_1$ where again $\\bar{u}^{\\prime }_i$ are the barred copies of $u^{\\prime }_i$ .",
"We define morphisms $f,g:T^*\\rightarrow \\mathcal {F}_{\\textsf {A}}$ by ${\\small \\begin{array}{lll}f(a) = a & f(a^-) = a^-& f(\\bar{a})=f(\\bar{a}^-)=\\varepsilon \\\\g(\\bar{a}) = a & g(\\bar{a}^-) = a^-& g(a)=g(a^-)=\\varepsilon \\\\\\end{array}}$ for $a\\in {\\textsf {A}}$ .",
"Then $M$ and $M^{\\prime }$ are equivalent relative to $B$ iff $g(w) =_{\\mathcal {F}_{\\textsf {A}}}f(w)$ for all $w \\in \\mathcal {L}(G)$ .",
"Combining Plandowski's polynomial construction of a test set for a context-free language to check morphism equivalence over finitely generated free groups [9], with Lohrey's polynomial algorithm for checking equivalence of SLPs over the free group [5], we deduce that the equivalence of the morphisms $f$ and $g$ on all words generated by the context-free grammar $G$ , is decidable in polynomial time.",
"Consequently, we obtain: Equivalence of same-ordered ${\\sf LT}_{\\textsf {A}}$ s relative to a DTA $B$ is decidable in polynomial time.",
"$\\Box $ Next, we observe that for every ${\\sf LT}_{\\textsf {A}}$ $M$ with compatible map $\\iota $ and non-trivial states only, a canonical ordering can be established.",
"We call $M$ ordered (relative to $B$ ) if for all rules of the form (REF ), with $\\mathcal {L}(q_i)\\cdot u_i\\cdot \\ldots \\cdot u_{j-1}\\cdot \\mathcal {L}(q_j) \\subseteq v\\cdot \\langle p\\rangle $ , $p \\in \\mathcal {F}_{\\textsf {A}}$ the ordering $\\sigma (i) < \\ldots < \\sigma (j)$ holds.",
"Here we have naturally extended the operation “$\\cdot $ ” to sets of elements.",
"We show that two ordered ${\\sf LT}_{\\textsf {A}}$ s, when they are equivalent, are necessarily same-ordered.",
"The proof of this claim is split in two parts.",
"First, we prove that the set of indices of subtrees processed by equivalent co-reachable states are identical and second, that the order is the same.",
"Let $M, M^{\\prime }$ be ${\\sf LT}_{\\textsf {A}}$ s with compatible maps $\\iota $ and $\\iota ^{\\prime }$ , respectively, and non-trivial states only so that $M$ and $M^{\\prime }$ are equivalent relative to the DTA $B$ .",
"Let $\\langle q,q^{\\prime }\\rangle $ be a pair of co-reachable states of $M$ and $M^{\\prime }$ .",
"Assume that $\\delta _B(\\iota (q),f)$ is defined for some $f\\in \\Sigma $ and consider the corresponding pair of rules (REF ).",
"Then the following holds: $\\lbrace \\sigma (1),\\ldots , \\sigma (n)\\rbrace = \\lbrace \\sigma ^{\\prime }(1),\\ldots , \\sigma ^{\\prime }(n^{\\prime })\\rbrace $ ; $\\sigma = \\sigma ^{\\prime }$ .",
"Since $\\langle q,q^{\\prime }\\rangle $ is a co-reachable pair of states, there are elements $\\alpha , \\alpha ^{\\prime }, \\beta , \\beta ^{\\prime } \\in \\mathcal {F}_{\\textsf {A}}$ such that $\\alpha \\cdot \\llbracket q\\rrbracket (t)\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\alpha ^{\\prime }\\cdot \\llbracket q^{\\prime }\\rrbracket (t)\\cdot \\beta ^{\\prime }$ holds for all $t \\in \\textsf {dom}(\\iota (q))$ .",
"Consider the first statement.",
"Assume for a contradiction that $q_k(x_{j})$ occurs on the right-hand side of the rule for $q$ but $x_{j}$ does not occur on the right-hand side of the rule for $q^{\\prime }$ .",
"Then, there are input trees $t = f(t_1,\\ldots , t_m)$ and $t^{\\prime } = f(t^{\\prime }_1,\\ldots , t^{\\prime }_m)$ , both in $\\textsf {dom}(\\iota (q))$ , such that $\\llbracket q_k\\rrbracket (t_j) \\ne _{\\mathcal {F}_{\\textsf {A}}}\\llbracket q_k\\rrbracket (t^{\\prime }_j)$ and $t_i = t^{\\prime }_i$ for all $i \\ne j$ .",
"Moreover, there are $\\mu _1, \\mu _2\\in \\mathcal {F}_{\\textsf {A}}$ s.t.",
"${\\small \\alpha \\cdot \\llbracket q\\rrbracket (t)\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\alpha \\cdot \\mu _1\\cdot \\llbracket q_k\\rrbracket (t_{j})\\cdot \\mu _2\\cdot \\beta \\ne _{\\mathcal {F}_{\\textsf {A}}}\\alpha \\cdot \\mu _1\\cdot \\llbracket q_k\\rrbracket (t^{\\prime }_{j})\\cdot \\mu _2\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\alpha \\cdot \\llbracket q\\rrbracket (t^{\\prime })\\cdot \\beta }$ But then, ${\\small \\alpha \\cdot \\llbracket q\\rrbracket (t)\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\alpha ^{\\prime }\\cdot \\llbracket q^{\\prime }\\rrbracket (t)\\cdot \\beta ^{\\prime }=_{\\mathcal {F}_{\\textsf {A}}}\\alpha ^{\\prime }\\cdot \\llbracket q^{\\prime }\\rrbracket (t^{\\prime })\\cdot \\beta ^{\\prime }=_{\\mathcal {F}_{\\textsf {A}}}\\alpha \\cdot \\llbracket q\\rrbracket (t^{\\prime })\\cdot \\beta }$ — a contradiction.",
"By an analogous argument for some $x_j$ only occurring in the right-hand side of the rule for $q^{\\prime }$ the first statement follows.",
"Assume for contradiction that the mappings $\\sigma $ and $\\sigma ^{\\prime }$ in the corresponding rules (REF ) differ.",
"Let $k$ denote the minimal index so that $\\sigma (k) \\ne \\sigma ^{\\prime }(k)$ .",
"W.l.o.g., we assume that $\\sigma ^{\\prime }(k) < \\sigma (k)$ .",
"By the first statement, $n=n^{\\prime }$ and $\\lbrace \\sigma (1),\\ldots , \\sigma (n)\\rbrace = \\lbrace \\sigma ^{\\prime }(1),\\ldots , \\sigma ^{\\prime }(n)\\rbrace $ .",
"Then there are $\\ell , \\ell ^{\\prime } > k$ such that $\\sigma ^{\\prime }(k)=\\sigma (\\ell ) < \\sigma (k)=\\sigma ^{\\prime }(\\ell ^{\\prime })$ Let $t = f(t_1,\\ldots , t_n) \\in \\textsf {dom}(\\iota (q))$ be an input tree.",
"For that we obtain ${\\small \\begin{array}[t]{l}\\mu _0 := u_0\\cdot \\llbracket q_1\\rrbracket (t_{\\sigma (1)})\\cdot \\ldots \\cdot u_{k-1} \\\\\\mu _1 := u_k\\cdot \\llbracket q_{k+1}\\rrbracket (t_{\\sigma (k+1)})\\cdot \\ldots \\cdot u_{\\ell -1} \\\\\\mu _2 := u_\\ell \\cdot \\llbracket q_\\ell \\rrbracket (t_{\\sigma (\\ell )})\\cdot \\ldots \\cdot u_n\\end{array}\\quad \\begin{array}[t]{l}\\mu ^{\\prime }_0 := u^{\\prime }_0\\cdot \\llbracket q^{\\prime }_1\\rrbracket (t_{\\sigma ^{\\prime }(1)})\\cdot \\ldots \\cdot u^{\\prime }_{k-1} \\\\\\mu ^{\\prime }_1 := u^{\\prime }_k\\cdot \\llbracket q^{\\prime }_{k+1}\\rrbracket (t_{\\sigma ^{\\prime }(k+1)})\\cdot \\ldots \\cdot u^{\\prime }_{\\ell ^{\\prime }-1} \\\\\\mu ^{\\prime }_2 := u^{\\prime }_{\\ell ^{\\prime }}\\cdot \\llbracket q^{\\prime }_{\\ell ^{\\prime }}\\rrbracket (t_{\\sigma ^{\\prime }(\\ell ^{\\prime })})\\cdot \\ldots \\cdot u^{\\prime }_n\\end{array}}$ Then for all input trees $t^{\\prime } \\in \\textsf {dom}(\\iota (q_k))$ , $t^{\\prime \\prime } \\in \\mathcal {L}(\\textsf {dom}(\\iota (q^{\\prime }_k))$ , $\\alpha \\cdot \\mu _0\\cdot \\llbracket q_{k}\\rrbracket (t^{\\prime })\\cdot \\mu _1\\cdot \\llbracket q_{\\ell }\\rrbracket (t^{\\prime \\prime })\\cdot \\mu _2\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\alpha ^{\\prime }\\cdot \\mu ^{\\prime }_0\\cdot \\llbracket q^{\\prime }_{k}\\rrbracket (t^{\\prime \\prime })\\cdot \\mu ^{\\prime }_1\\cdot \\llbracket q^{\\prime }_{\\ell ^{\\prime }}\\rrbracket (t^{\\prime })\\cdot \\mu ^{\\prime }_2\\cdot \\beta ^{\\prime }$ Let $\\gamma ^{\\prime } = \\mu _0^-\\alpha ^-\\alpha ^{\\prime }\\mu ^{\\prime }_0$ .",
"Then $\\llbracket q_k\\rrbracket (t^{\\prime })\\cdot \\mu _1\\cdot \\llbracket q_\\ell \\rrbracket (t^{\\prime \\prime })\\cdot \\mu _2\\cdot \\beta =_{\\mathcal {F}_{\\textsf {A}}}\\gamma ^{\\prime }\\cdot \\llbracket q^{\\prime }_k\\rrbracket (t^{\\prime \\prime })\\cdot \\mu ^{\\prime }_1\\cdot \\llbracket q^{\\prime }_{\\ell ^{\\prime }}\\rrbracket (t^{\\prime })\\cdot \\mu ^{\\prime }_2\\cdot \\beta ^{\\prime }$ By Lemma , we obtain that for all $w_1, w_2 \\in \\mathcal {L}(q_k)$ and $v_1, v_2 \\in \\mathcal {L}(q_\\ell )$ , $w_2^-\\cdot w_1 \\in \\mu _1\\cdot \\langle p\\rangle \\cdot \\mu _1^-$ and $v_1\\cdot v_2^-\\in \\langle p\\rangle $ for some primitive $p$ .",
"If $\\ell = k+1$ , i.e., there is no further state between $q_k(x_{\\sigma (k)})$ and $q_\\ell (x_{\\sigma (\\ell )})$ , then $\\mu _1 =_{\\mathcal {F}_{\\textsf {A}}}u_k$ , $\\mathcal {L}(q_k) \\subseteq w\\cdot u_k\\cdot \\langle p\\rangle \\cdot u_k^-$ and $\\mathcal {L}(q_\\ell ) \\subseteq \\langle p\\rangle \\cdot v$ for some fixed $w \\in \\mathcal {L}(q_k)$ and $v \\in \\mathcal {L}(q_\\ell )$ .",
"As $\\sigma (k) > \\sigma ^{\\prime }(k) = \\sigma (\\ell )$ , this contradicts $M$ being ordered.",
"For the case that there is at least one occurrence of a state between $q_k(x_{\\sigma (k)})$ and $q_\\ell (x_{\\sigma (\\ell )})$ , we show that for all $\\alpha _1, \\alpha _2 \\in u_k\\cdot \\mathcal {L}(q_{k+1})\\cdot \\ldots \\cdot u_{\\ell -1} =_{\\mathcal {F}_{\\textsf {A}}}: \\hat{L}$ , $\\alpha _1^-\\alpha _2 \\in \\langle p\\rangle $ holds.",
"We fix $w_1, w_2 \\in \\mathcal {L}(q_k)$ and $v_1,v_2\\in \\mathcal {L}(q_\\ell )$ with $w_1 \\ne w_2$ and $v_1\\ne v_2$ .",
"For every $\\alpha \\in \\hat{L}$ , we find by Lemma , primitive $p_\\alpha $ and exponent $r_\\alpha \\in \\mathbb {Z}$ such that $v_1\\cdot v_2^-=_{\\mathcal {F}_{\\textsf {A}}}p_\\alpha ^{r_\\alpha }$ holds.",
"Since $p_\\alpha $ is primitive, this means that $p_\\alpha =_{\\mathcal {F}_{\\textsf {A}}}p$ or $p_\\alpha =_{\\mathcal {F}_{\\textsf {A}}}p^-$ .",
"Furthermore, there must be some exponent $r^{\\prime }_\\alpha $ such that $w_1^-\\cdot w_2 =_{\\mathcal {F}_{\\textsf {A}}}\\alpha \\cdot p^{r^{\\prime }_\\alpha }\\cdot \\alpha ^-$ .",
"For $\\alpha _1,\\alpha _2 \\in \\hat{L}$ , we therefore have that ${\\small p^{r^{\\prime }_{\\alpha _1}} =_{\\mathcal {F}_{\\textsf {A}}}(\\alpha _1^-\\cdot \\alpha _2)\\cdot p^{r^{\\prime }_{\\alpha _2}}\\cdot (\\alpha _1^-\\cdot \\alpha _2)^-}$ Therefore by Lemma , $\\alpha _1^-\\cdot \\alpha _2 \\in \\langle p\\rangle $ .",
"Let us fix some $w_k \\in \\mathcal {L}(q_k)$ , $\\alpha \\in \\hat{L} =_{\\mathcal {F}_{\\textsf {A}}}u_k\\cdot \\mathcal {L}(q_{k+1})\\cdot \\ldots \\cdot u_{\\ell -1}$ , and $w_l \\in \\mathcal {L}(q_l)$ .",
"Then $\\mathcal {L}(q_k) \\subseteq w_k\\cdot \\alpha \\cdot \\langle p\\rangle \\cdot \\alpha ^-$ , $\\hat{L} \\subseteq \\alpha \\cdot \\langle p\\rangle $ and $\\mathcal {L}(q_l) \\subseteq \\langle p\\rangle \\cdot w_l$ .",
"Therefore, ${\\small \\mathcal {L}(q_k)\\cdot u_k\\cdot \\ldots \\cdot \\mathcal {L}(q_\\ell ) \\subseteq w_k\\cdot \\alpha \\cdot \\langle p\\rangle \\cdot \\alpha ^-\\cdot \\alpha \\cdot \\langle p\\rangle \\cdot \\langle p\\rangle \\cdot w_l=_{\\mathcal {F}_{\\textsf {A}}}w_k\\cdot \\alpha \\cdot \\langle p\\rangle \\cdot w_l}$ As $\\sigma (k) > \\sigma ^{\\prime }(k) = \\sigma (\\ell )$ , this again contradicts $M$ being ordered.",
"It remains to show that every ${\\sf LT}_{\\textsf {A}}$ can be ordered in polynomial time.",
"For that, we rely on the following characterization.",
"Assume that $L_1,\\ldots ,L_n$ are neither empty nor singleton subsets of $\\mathcal {F}_{\\textsf {A}}$ and $u_1, \\ldots , u_{n-1} \\in \\mathcal {F}_{\\textsf {A}}$ .",
"Then there are $v_1,\\ldots ,v_n\\in \\mathcal {F}_{\\textsf {A}}$ such that ${\\small L_1\\cdot u_1\\cdot \\ldots \\cdot L_{n-1}\\cdot u_{n-1}\\cdot L_n \\subseteq v\\cdot \\langle p\\rangle }$ holds if and only if for $i=1,\\ldots ,n$ , $L_i \\subseteq v_i\\cdot \\langle p_i\\rangle $ with ${\\small \\begin{array}{lll}p_n &=_{\\mathcal {F}_{\\textsf {A}}}& p\\\\p_i &=_{\\mathcal {F}_{\\textsf {A}}}& (u_i\\cdot v_{i+1})\\cdot p_{i+1}\\cdot (u_i\\cdot v_{i+1})^-\\quad \\text{\\normalsize for }i<n\\end{array}}$ and ${\\small v^-\\cdot v_1\\cdot u_1\\cdot \\ldots \\cdot v_{n-1}\\cdot u_{n-1}\\cdot v_n\\in \\langle p\\rangle }$ Let $s_1 = \\varepsilon $ .",
"For $i = 2,\\ldots , n$ we fix some word $s_i \\in L_1\\cdot u_1\\cdot L_2\\cdot \\ldots \\cdot L_{i-1}\\cdot u_{i-1}$ .",
"Likewise, let $t_n = \\varepsilon $ and for $i = 1,\\ldots , n-1$ fix some word $t_i \\in u_i\\cdot L_{i+1}\\cdot \\ldots \\cdot L_{n}$ , and define $v_i =_{\\mathcal {F}_{\\textsf {A}}}s_i^-\\cdot v\\cdot t_i^-$ .",
"First assume that the inclusion (REF ) holds.",
"Let $p^{\\prime }_i=_{\\mathcal {F}_{\\textsf {A}}}t_i\\cdot p\\cdot t_i^-$ .",
"Then for all $i$ , $s_i\\cdot L_i\\cdot t_i \\subseteq v\\cdot \\langle p\\rangle $ , and therefore $L_i \\subseteq s_i^-\\cdot v\\cdot \\langle p\\rangle \\cdot t_i^-=_{\\mathcal {F}_{\\textsf {A}}}s_i^-\\cdot v\\cdot t_i^-\\cdot t_i\\cdot \\langle p\\rangle \\cdot t_i^-=_{\\mathcal {F}_{\\textsf {A}}}v_i \\langle p^{\\prime }_i\\rangle $ We claim that $p^{\\prime }_i = p_i$ for all $i=1,\\ldots ,n$ .",
"We proceed by induction on $n-i$ .",
"As $t_n = \\varepsilon $ , we have that $p^{\\prime }_n = p = p_n$ .",
"For $i<n$ , we can rewrite $t_i =_{\\mathcal {F}_{\\textsf {A}}}u_i\\cdot w_{i+1}\\cdot t_{i+1}$ where $w_{i+1}\\in L_{i+1}$ and thus is of the form $v_{i+1}\\cdot p_{i+1}^{k_{i+1}}$ for some exponent $k_{i+1}$ .",
"${\\small \\begin{array}{lll}p^{\\prime }_i &=_{\\mathcal {F}_{\\textsf {A}}}& t_i\\cdot p\\cdot t_i^-\\\\&=_{\\mathcal {F}_{\\textsf {A}}}& u_i\\cdot w_{i+1}\\cdot t_{i+1}\\cdot p\\cdot t_{i+1}^-\\cdot w_{i+1}^-\\cdot u_i^-\\\\&=_{\\mathcal {F}_{\\textsf {A}}}& u_i\\cdot w_{i+1}\\cdot p_{i+1} \\cdot w_{i+1}^-\\cdot u_i^-\\qquad \\text{\\normalsize by I.H.",
"}\\\\&=_{\\mathcal {F}_{\\textsf {A}}}& u_i\\cdot v_{i+1}\\cdot p_{i+1} \\cdot v_{i+1}^-\\cdot u_i^-\\\\&=_{\\mathcal {F}_{\\textsf {A}}}& p_i\\end{array}}$ It remains to prove the inclusion (REF ).",
"Since $w_i\\in L_i$ , we have by (REF ) that $v^-w_1\\cdot u_1\\cdot \\ldots w_n\\cdot u_n\\in \\langle p\\rangle $ holds.",
"Now we calculate: ${\\small \\begin{array}{lll}v^-\\cdot w_1\\cdot u_1\\cdot \\ldots u_{n-1}\\cdot w_n&=_{\\mathcal {F}_{\\textsf {A}}}& v^-\\cdot v_1\\cdot p_1^{k_1}\\cdot u_1\\cdot \\ldots \\cdot u_{n-1} \\cdot v_n\\cdot p_n^{k_n} \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v^-\\cdot v_1\\cdot u_1\\cdot v_2\\cdot p_2^{k_1+k_2}\\cdot u_2\\cdot \\ldots \\cdot u_{n-1} \\cdot v_n\\cdot p_n^{k_n} \\\\&\\ldots & \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v^-\\cdot v_1\\cdot u_1\\cdot \\ldots v_{n-1}\\cdot u_{n-1}\\cdot v_n\\cdot p_n^k\\end{array}}$ where $k= k_1+\\ldots +k_n$ .",
"Since $p_n=p$ , the claim follows.",
"The other direction of the claim of the lemma follows directly: ${\\small \\begin{array}{lll}L_1 u_1 \\ldots L_{n-1} u_{n-1} L_n&\\subseteq & v_1\\cdot \\langle p_1\\rangle \\cdot u_1\\cdot \\ldots \\cdot v_{n-1}\\cdot \\langle p_{n-1}\\rangle \\cdot u_{n-1}\\cdot v_n\\cdot \\langle p_n\\rangle \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v_1\\cdot u_1\\cdot v_2\\cdot \\langle p_2\\rangle \\cdot \\langle p_2\\rangle \\cdot u_2\\cdot \\ldots \\cdot v_{n-1}\\cdot \\langle p_{n-1}\\rangle \\cdot u_{n-1}\\cdot v_n\\cdot \\langle p_n\\rangle \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v_1\\cdot u_1\\cdot v_2\\cdot \\langle p_2\\rangle \\cdot u_2\\cdot \\ldots \\cdot v_{n-1}\\cdot \\langle p_{n-1}\\rangle \\cdot u_{n-1}\\cdot v_n\\cdot \\langle p_n\\rangle \\\\&\\cdots & \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v_1\\cdot u_1\\cdot v_2\\cdot \\ldots \\cdot u_{n-1}\\cdot v_n\\cdot \\langle p_n\\rangle \\\\&=_{\\mathcal {F}_{\\textsf {A}}}& v_1\\cdot u_1\\cdot v_2\\cdot \\ldots \\cdot u_{n-1}\\cdot v_n\\cdot \\langle p\\rangle \\\\&\\subseteq &v\\cdot \\langle p\\rangle \\end{array}}$ where the last inclusion follows from (REF ).",
"Let us call a non-empty, non-singleton language $L\\subseteq \\mathcal {F}_{\\textsf {A}}$ periodic, if $L\\subseteq v\\cdot \\langle p\\rangle $ for some $v,p\\in \\mathcal {F}_{\\textsf {A}}$ .",
"Lemma then implies that if a concatenation of languages and elements from $\\mathcal {F}_{\\textsf {A}}$ is periodic, then so must be all non-singleton component languages.",
"In fact, the languages in the composition can then be arbitrarily permuted.",
"Assume for non-empty, nonsingleton languages $L_1,\\ldots ,L_n\\subseteq \\mathcal {F}_{\\textsf {A}}$ and $u_1, \\ldots , u_{n-1} \\in \\mathcal {F}_{\\textsf {A}}$ that property (REF ) holds.",
"Then for every permutation $\\pi $ , there are elements $u_{\\pi ,0},\\ldots ,u_{\\pi ,n}\\in \\mathcal {F}_{\\textsf {A}}$ such that $L_1\\cdot u_1\\cdot \\ldots \\cdot L_{n-1}\\cdot u_{n-1}\\cdot L_n =u_{\\pi ,0}\\cdot L_{\\pi (1)}\\cdot u_{\\pi ,1}\\cdot \\ldots \\cdot u_{\\pi _n-1}\\cdot L_{\\pi (n)}\\cdot u_{\\pi ,n}$ We reconsider ${\\sf LT}_{\\textsf {A}}$ $M^{\\prime }$ and DTA $B$ from Example REF .",
"We observe that $\\mathcal {L}(q_1) \\subseteq a\\cdot \\langle ba\\rangle $ , $\\mathcal {L}(q_2) \\subseteq \\langle ab\\rangle $ , and thus $\\mathcal {L}(q_0) = \\mathcal {L}(q_1)\\cdot b\\cdot \\mathcal {L}(q_2) \\subseteq \\langle ab\\rangle $ .",
"Accordingly, the rule for state $q_0$ and input symbol $f$ is not ordered.",
"Following the notation of Corollary , we find $v_1 = a$ , $u_1 = b$ and $v_2 = \\varepsilon $ , and the rule for $q_0$ and $f$ can be reordered to ${\\small q_0(f(x_1, x_2)) \\rightarrow ab\\cdot q_2(x_1)\\cdot a^-\\cdot q_1(x_2)}$ This example shows major improvements compared to the construction in [2].",
"Since we have inverses at hand, only local changes must be applied to the sub-sequence $q_1(x_2)\\cdot b\\cdot q_2(x_1)$ .",
"In contrast to the construction in [2], neither auxiliary states nor further changes to the rules of $q_1$ and $q_2$ are required.",
"By Corollary , the order of occurrences of terms $q_k(x_{\\sigma (k)})$ can be permuted in every sub-sequence $q_i(x_{\\sigma (i)})\\cdot u_i\\cdot \\ldots \\cdot u_{j-1} q_j(x_{\\sigma (j)})$ where $\\mathcal {L}(q_i)\\cdot u_i\\cdot \\ldots \\cdot u_{j-1}\\cdot \\mathcal {L}(q_j) \\in u\\cdot \\langle p\\rangle $ is periodic, to satisfy the requirements of an ordered ${\\sf LT}_{\\textsf {A}}$ .",
"A sufficient condition for that is, according to Lemma , that $\\mathcal {L}(q_k)$ is periodic for each $q_k$ occurring in that sub-sequence.",
"Therefore we will determine the subset of all states $q$ where $\\mathcal {L}(q)$ is periodic, and if so elements $v_q,p_q$ such that $\\mathcal {L}(q)\\subseteq v_q\\cdot \\langle p_q\\rangle $ .",
"In order to do so we compute an abstraction of the sets $\\mathcal {L}(q)$ by means of a complete lattice which both reports constant values and also captures periodicity.",
"Let $ 2^{\\mathcal {F}_{\\textsf {A}}}$ denote the complete lattice of subsets of the free group $\\mathcal {F}_{\\textsf {A}}$ .",
"We define a projection $\\alpha : by$ ()=$, $ ({g}) = {g}$, and for languages $ L$ withat least two elements,$${\\small \\alpha (L) = {\\left\\lbrace \\begin{array}{ll}g\\langle p\\rangle & \\text{if } L \\subseteq g\\langle p\\rangle \\text{ and $p$ is primitive} \\\\\\mathcal {F}_{\\textsf {A}}& \\text{otherwise}\\end{array}\\right.",
"}}$$The projection $$ is a \\emph {closure} operator, i.e., is a monotonic function with$ L(L)$, and $ ((L)) = (L)$.The image of $$ can be considered as an \\emph {abstract} complete lattice $ ♯$,partially ordered by subset inclusion.Thereby, the abstraction $$ commutes with least upper bounds as well as withthe group operation.",
"For that, we define \\emph {abstract} versions$ ,:(♯)2♯$of set union and the group operation by$$A_1\\sqcup A_2 = \\alpha (A_1\\cup A_2)\\qquad A_1\\star A_2 = \\alpha (A_1\\cdot A_2)$$In fact, ``$$^{\\prime \\prime } is the least upper bound operation for $ ♯$.The two abstract operators can also be more explicitly defined by:$${\\small \\begin{array}{lllll}\\emptyset \\sqcup L &=& L\\sqcup \\emptyset &=& L \\\\\\mathcal {F}_{\\textsf {A}}\\sqcup L &=& L\\sqcup \\mathcal {F}_{\\textsf {A}}&=& \\mathcal {F}_{\\textsf {A}}\\\\\\lbrace g_1\\rbrace \\sqcup \\lbrace g_2\\rbrace &=&\\multicolumn{3}{l}{{\\left\\lbrace \\begin{array}{ll}\\lbrace g_1\\rbrace & \\text{if } g_1 = g_2 \\\\g_1\\cdot \\langle p\\rangle & \\text{if } g_1\\ne g_2, p \\text{ primitive root of } g_1^-\\cdot g_2\\end{array}\\right.}}",
"\\\\\\lbrace g_1\\rbrace \\sqcup g_2\\cdot \\langle p\\rangle &=&g_2\\cdot \\langle p\\rangle \\sqcup \\lbrace g_1\\rbrace &=&{\\left\\lbrace \\begin{array}{ll}g_2\\cdot \\langle p\\rangle & \\text{if } g_1\\in g_2\\cdot \\langle p\\rangle \\\\\\mathcal {F}_{\\textsf {A}}& \\text{otherwise}\\end{array}\\right.}",
"\\\\g_1\\cdot \\langle p_1\\rangle \\sqcup g_2\\cdot \\langle p_2\\rangle &=&\\multicolumn{3}{l}{{\\left\\lbrace \\begin{array}{ll}g_1\\cdot \\langle p_1\\rangle & \\text{if } p_2\\in \\langle p_1\\rangle \\text{ and }g_2^-\\cdot g_1\\in \\langle p_1\\rangle \\\\\\mathcal {F}_{\\textsf {A}}& \\text{otherwise}\\end{array}\\right.}}",
"\\\\\\end{array}}$$$ ${\\small \\begin{array}{lllll}\\emptyset \\star L &=& L\\star \\emptyset &=& \\emptyset \\\\\\mathcal {F}_{\\textsf {A}}\\star L &=& L\\star \\mathcal {F}_{\\textsf {A}}&=& F_{\\textsf {A}}\\qquad \\text{for } L\\ne \\emptyset \\\\\\lbrace g_1\\rbrace \\star \\lbrace g_2\\rbrace &=& \\multicolumn{3}{l}{\\lbrace g_1\\cdot g_2\\rbrace } \\\\\\lbrace g_1\\rbrace \\star g_2\\cdot \\langle p\\rangle &=& \\multicolumn{3}{l}{(g_1\\cdot g_2)\\cdot \\langle p\\rangle } \\\\g_1\\cdot \\langle p\\rangle \\star \\lbrace g_2\\rbrace &=& \\multicolumn{3}{l}{(g_1\\cdot g_2)\\cdot \\langle g_2^-\\cdot p\\cdot g_2\\rangle } \\\\g_1\\cdot \\langle p_1\\rangle \\star g_2\\cdot \\langle p_2\\rangle &=&\\multicolumn{3}{l}{{\\left\\lbrace \\begin{array}{ll}(g_1\\cdot g_2)\\cdot \\langle p_2\\rangle &\\text{if } g_2^-\\cdot p_1\\cdot g_2\\in \\langle p_2\\rangle \\\\\\mathcal {F}_{\\textsf {A}}& \\text{otherwise}\\end{array}\\right.",
"}}\\end{array}}$ For all subsets $L_1,L_2\\subseteq \\mathcal {F}_{\\textsf {A}}$ , $\\alpha (L_1\\cup L_2) = \\alpha (L_1)\\sqcup \\alpha (L_2)$ and $\\alpha (L_1\\cdot L_2) = \\alpha (L_1)\\star \\alpha (L_2)$ .",
"We conclude that $\\alpha $ in fact represents a precise abstract interpretation in the sense of [8].",
"Accordingly, we obtain: For every ${\\sf LT}_{\\textsf {A}}$ $M$ and DTA $B$ with compatible map $\\iota $ , the sets $\\alpha (\\mathcal {L}(q))$ , $q$ state of $M$ , can be computed in polynomial time.",
"We introduce one unknown $X_q$ for every state $q$ of $M$ , and one constraint for each rule of $M$ of the form (REF ) where $\\delta (\\iota (q),f)$ is defined in $B$ .",
"This constraint is given by: ${\\small X_q\\sqsupseteq u_0\\star X_{q_1}\\star \\ldots \\star u_{n-1}\\star X_{q_n}\\star u_n} $ As the right-hand sides of the constraints (REF ) all represent monotonic functions, the given system of constraints has a least solution.",
"In order to obtain this solution, we consider for each state $q$ of $M$ , the sequence $X_{q}^{(i)},i\\ge 0$ of values in $\\sharp $ where $X_q^{(0)} = \\emptyset $ , and for $i>0$ , we set $X_q^{(i)}$ as the least upper bound of the values obtained from the constraints with left-hand side $X_q$ of the form (REF ) by replacing the unknowns $X_{q_j}$ on the right-hand side with the values $X_{q_j}^{(i-1)}$ .",
"By induction on $i\\ge 0$ , we verify that for all states $q$ of $M$ , ${\\small X_q^{(i)} = \\alpha (\\mathcal {L}^{(i)}(q)})$ holds.",
"Note that the induction step thereby, relies on Lemma .",
"As each strictly increasing chain of elements in $\\sharp $ consists of at most four elements, we have that the least solution of the constraint system is attained after at most $3\\cdot N$ iterations, if $N$ is the number of states of $M$ , i.e., for each state $q$ of $M$ , $X_q^{(3N)} = X_q^{(i)}$ for all $i\\ge 3N$ .",
"The elements of $\\sharp $ can be represented by SLPs where the operations $\\star $ and $\\sqcup $ run in polynomial time, cf.",
"Lemma .",
"Since each iteration requires only a polynomial number of operations $\\star $ and $\\sqcup $ , the statement of the lemma follows.",
"We now exploit the information provided by the $\\alpha (\\mathcal {L}(q))$ to remove trivial states as well as order subsequences of right-hand sides which are periodic.",
"Let $B$ be a DTA such that $\\mathcal {L}(B)\\ne \\emptyset $ .",
"For every ${\\sf LT}_{\\textsf {A}}$ $M$ with compatible map $\\iota $ , an ${\\sf LT}_{\\textsf {A}}$ $M^{\\prime }$ with compatible map $\\iota ^{\\prime }$ can be constructed in polynomial time such that $M$ and $M^{\\prime }$ are equivalent relative to $B$ ; $M^{\\prime }$ has no trivial states; $M^{\\prime }$ is ordered.",
"By Lemma , we can, in polynomial time, determine for every state $q$ of $M$ , the value $\\alpha (\\mathcal {L}(q))$ .",
"We use this information to remove from $M$ all trivial states.",
"W.l.o.g., assume that the axiom of $M$ is given by $u_0\\cdot q_0(x_0)\\cdot u_1$ .",
"If the state $q_0$ occurring in the axiom of $M$ is trivial with $\\mathcal {L}(q_0) = \\lbrace v\\rbrace $ , then $M_1$ has no states or rules, but the axiom $u_0\\cdot v\\cdot u_1$ .",
"Therefore now assume that $q_0$ is non-trivial.",
"We then construct an ${\\sf LT}_{\\textsf {A}}$ $M_1$ whose set of states $Q_1$ consists of all non-trivial states $q$ of $M$ where the compatible map $\\iota _1$ of $M_1$ is obtained from $\\iota $ by restriction to $Q_1$ .",
"Since $\\mathcal {L}(M)\\ne \\emptyset $ , the state of $M$ occurring in the axiom is non-trivial.",
"Accordingly, the axiom of $M$ is also used as axiom for $M_1$ .",
"Consider a non-trivial state $q$ of $M$ and $f\\in \\Sigma $ .",
"If $\\delta (\\iota (q),f)$ is not defined $M_1$ has the rule $q(f(x_1,\\ldots ,x_m)\\rightarrow \\bot $ .",
"Assume that $\\delta (\\iota (q),f)$ is defined and $M$ has a rule of the form (REF ).",
"Then $M_1$ has the rule ${\\small q(f(x_1,\\ldots ,x_m))\\rightarrow u_0\\cdot g_1\\cdot \\ldots \\cdot u_{n-1}\\cdot g_n\\cdot u_n}$ where for $i=1,\\ldots ,n$ , $g_i$ equals $q_i(x_{\\sigma (i)})$ if $q_i$ is non-trivial, and equals the single word in $\\mathcal {L}(q_i)$ otherwise.",
"Obviously, $M$ and $M_1$ are equivalent relative to $B$ where $M_1$ now has no trivial states, while for every non-trivial state $q$ , the semantics of $q$ in $M$ and $M_1$ are the same relative to $B$ .",
"Our goal now is to equivalently rewrite the right-hand side of each rule of $M_1$ so that the result is ordered.",
"For each state $q$ of the ${\\sf LT}_{\\textsf {A}}$ we determine whether there are $v, p \\in \\textsf {B}^*$ such that $\\mathcal {L}(q) \\subseteq v\\langle p\\rangle $ , cf.",
"Lemma .",
"So consider a rule of $M_1$ of the form (REF ).",
"By means of the values $\\alpha (\\mathcal {L}(q_i))$ , $i=1,\\ldots ,n$ , together with the abstract operation “$\\star $ ”, we can determine maximal intervals $[i,j]$ such that $\\mathcal {L}(q_i)\\cdot u_i\\cdot \\ldots \\cdot u_{j-1}\\cdot \\mathcal {L}(q_j)$ is periodic, i.e., $\\alpha (\\mathcal {L}(q_i))\\star u_i\\cdot \\ldots \\star u_{j-1}\\star \\alpha (\\mathcal {L}(q_j)) \\subseteq v\\cdot \\langle p\\rangle $ for some $v,p\\in \\mathcal {F}_{\\textsf {A}}$ .",
"We remark that these maximal intervals are necessarily disjoint.",
"By Corollary , for every permutation $\\pi :[i,j]\\rightarrow [i,j]$ , elements $u^{\\prime },u^{\\prime }_i,\\ldots ,u^{\\prime }_j,u^{\\prime \\prime }\\in \\mathcal {F}_{\\textsf {A}}$ can be found so that $q_i(x_{\\sigma (i)})\\cdot u_i\\cdot \\ldots \\cdot u_{j-1}\\cdot q_j(x_{\\sigma (j)})$ is equivalent to $u^{\\prime }\\cdot q_{\\pi (i)}(x_{\\sigma (\\pi (i))})\\cdot u^{\\prime }_i\\cdot \\ldots \\cdot u^{\\prime }_{j-1}\\cdot q_{\\pi (j)}(x_{\\sigma (\\pi (j))})\\cdot u^{\\prime \\prime }$ .",
"In particular, this is true for the permutation $\\pi $ with $\\sigma (\\pi (i)) < \\ldots <\\sigma (\\pi (j))$ .",
"Assuming that all group elements are represented as SLPs, the overall construction runs in polynomial time.",
"In summary, we arrive at the main theorem of this paper.",
"The equivalence of ${\\sf LT}_{\\textsf {A}}$ s relative to some DTA $B$ can be decided in polynomial time.",
"Assume we are given ${\\sf LT}_{\\textsf {A}}$ s $M,M^{\\prime }$ with compatible maps (relative to $B$ ).",
"By Theorem , we may w.l.o.g.",
"assume that $M$ and $M^{\\prime }$ both have no trivial states and are ordered.",
"It can be checked in polynomial time whether or not $M$ and $M^{\\prime }$ are same-ordered.",
"If they are not, then by Lemma , they cannot be equivalent relative to $B$ .",
"Therefore now assume that $M$ and $M^{\\prime }$ are same-ordered.",
"Then their equivalence relative to $B$ is decidable in polynomial time by Corollary .",
"Altogether we thus obtain a polynomial decision procedure for equivalence of ${\\sf LT}_{\\textsf {A}}$ s relative to some DTA $B$ ."
],
[
"Conclusion",
"We have shown that equivalence of ${\\sf LT}_{\\textsf {A}}$ s relative to a given DTA $B$ can be decided in polynomial time.",
"For that, we considered total transducers only, but defined the domain of allowed input trees separately by means of the DTA.",
"This does not impose any restriction of generality, since any (possibly partial) linear deterministic top-down tree transducer can be translated in polynomial time to a corresponding total ${\\sf LT}_{\\textsf {A}}$ together with a corresponding DTA (see, e.g., [10]).",
"The required constructions for ${\\sf LT}_{\\textsf {A}}$ s which we have presented here, turn out to be more general than the constructions provided in [2] since they apply to transducers which may not only output symbols $a\\in {\\textsf {A}}$ , but also their inverses $a^-$ .",
"At the same time, they are simpler and easier to be proven correct due to the combinatorial and algebraic properties provided by the free group."
],
[
"Proof of Lemma ",
"For an ${\\sf LT}_{\\textsf {A}}$ $M$ and a DTA $B=(H,\\Sigma ,\\delta _B,h_0)$ , an ${\\sf LT}_{\\textsf {A}}$ $M^{\\prime }$ with a set of states $Q^{\\prime }$ together with a mapping $\\iota :Q^{\\prime }\\rightarrow H$ can be constructed in polynomial time such that the following holds: $M$ and $M^{\\prime }$ are equivalent relative to $B$ ; $\\iota $ is compatible.",
"In case that the axiom of $M$ is in $\\mathcal {F}_{\\textsf {A}}$ , we obtain $M^{\\prime }$ from $M$ using the axiom of $M$ and using empty sets of states and rules, respectively.",
"Assume therefore that the axiom of $M$ is of the form $u_0\\cdot q_0(x_0)\\cdot u_1$ .",
"Then ${\\sf LT}_{\\textsf {A}}$ $M^{\\prime }$ is constructed as follows.",
"The set $Q^{\\prime }$ of states of $M^{\\prime }$ consists of pairs $\\langle q,h\\rangle $ , $q\\in Q, h\\in H$ where $\\iota (\\langle q,h\\rangle ) = h$ .",
"In particular, $\\langle q_0,h_0\\rangle \\in Q^{\\prime }$ .",
"As the axiom of $M^{\\prime }$ we then use $u_0\\cdot \\langle q_0,h_0\\rangle (x_0)\\cdot u_1$ .",
"For a state $\\langle q,h\\rangle \\in Q^{\\prime }$ , consider each input symbol $f\\in \\Sigma $ .",
"Let $m\\ge 0$ denote the rank of $f$ .",
"If $\\delta _B(h,f)$ is not defined, $M^{\\prime }$ has the rule $\\langle q,h\\rangle (f(x_1,\\ldots ,x_m))\\rightarrow \\bot $ Otherwise, let $\\delta _H(h,f) = h_1\\ldots h_m$ , and assume that $M$ has a rule of the form (REF ).",
"Then we add the states $\\langle q_i,h_{\\sigma (i)}\\rangle $ to $Q^{\\prime }$ together with the rule $\\langle q,h\\rangle (f(x_1,\\ldots ,x_m))\\rightarrow u_0\\cdot \\langle q_1,h_{\\sigma (1)}\\rangle (x_{\\sigma (1)})\\cdot \\ldots \\cdot u_{n-1}\\cdot \\langle q_n,h_{\\sigma (n)}\\rangle (x_{\\sigma (n)})\\cdot u_n$ By construction, the mapping $\\iota $ is compatible.",
"We verify for each state $\\langle q,h\\rangle $ of $M^{\\prime }$ and each input tree $t\\in \\textsf {dom}(h)$ that $\\llbracket q\\rrbracket (t) = \\llbracket \\langle q,h\\rangle \\rrbracket (t)$ holds.",
"This proof is by induction on the structure of $t$ .",
"From that, the equivalence of $M$ and $M^{\\prime }$ relative to $B$ follows."
],
[
"Proof of Corollary ",
"Assume for non-empty, nonsingleton languages $L_1,\\ldots ,L_n\\subseteq \\mathcal {F}_{\\textsf {A}}$ and $u_1, \\ldots , u_{n-1} \\in \\mathcal {F}_{\\textsf {A}}$ that property (REF ) holds.",
"Then for every permutation $\\pi $ , there are elements $u_{\\pi ,0},\\ldots ,u_{\\pi ,n}\\in \\mathcal {F}_{\\textsf {A}}$ such that $L_1\\cdot u_1\\cdot \\ldots \\cdot L_{n-1}\\cdot u_{n-1}\\cdot L_n =u_{\\pi ,0}\\cdot L_{\\pi (1)}\\cdot u_{\\pi ,1}\\cdot \\ldots \\cdot u_{\\pi _n-1}\\cdot L_{\\pi (n)}\\cdot u_{\\pi ,n}$ For $i=1,\\ldots ,n$ , let $v_i$ and $p_i$ be defined as in Lemma .",
"Then for all $i$ , $L_i \\subseteq v_i \\langle p_i\\rangle $ .",
"Moreover, the languages $L^{\\prime }_i$ defined by $L^{\\prime }_n = v_n^-\\cdot L_n$ and for $i<n$ , ${\\small L^{\\prime }_i = (u_{i}\\cdot v_{i+1}\\cdot \\ldots \\cdot u_{n-1}\\cdot v_n)^-\\cdot (v_i^-\\cdot L_i)\\cdot (u_{i}\\cdot v_{i+1}\\cdot \\ldots \\cdot u_{n-1}\\cdot v_n)}$ all are subsets of $\\langle p\\rangle $ .",
"Therefore their compositions can arbitrarily be permuted.",
"At the same time, ${\\small L_1\\cdot u_1\\cdot \\ldots \\cdot L_{n-1}\\cdot u_{n-1}\\cdot L_n=_{\\mathcal {F}_{\\textsf {A}}}v_1\\cdot u_1\\cdot \\ldots \\cdot v_{n-1}\\cdot u_n\\cdot v_n\\cdot L^{\\prime }_1\\cdot \\ldots L^{\\prime }_n}$ From that, the corollary follows."
],
[
"Proof of Lemma ",
"For all subsets $L_1,L_2\\subseteq \\mathcal {F}_{\\textsf {A}}$ , $\\alpha (L_1\\cup L_2) = \\alpha (L_1)\\sqcup \\alpha (L_2)$ and $\\alpha (L_1\\cdot L_2) = \\alpha (L_1)\\star \\alpha (L_2)$ .",
"As $\\emptyset \\cup L = L \\cup \\emptyset = \\emptyset $ , it follows that $\\alpha (\\emptyset \\cup L) = \\alpha (L \\cup \\emptyset ) = \\alpha (\\emptyset ) = \\emptyset = \\emptyset \\sqcup L^{\\prime } = L^{\\prime } \\sqcup \\emptyset = \\alpha (\\emptyset ) \\sqcup \\alpha (L) = \\alpha (L) \\sqcup \\alpha (\\emptyset )$ .",
"Assume that $\\alpha (L_1) = \\mathcal {F}_\\textsf {A}$ .",
"Let $L_2$ be some language, then $\\alpha (L_1 \\cup L_2) = \\alpha (L_2 \\cup L_1) = \\mathcal {F}_\\textsf {A}$ and $\\alpha (L_1) \\sqcup \\alpha (L_2) = \\mathcal {F}_\\textsf {A}\\sqcup \\alpha (L_2) = \\mathcal {F}_\\textsf {A}= \\alpha (L_2) \\sqcup \\mathcal {F}_\\textsf {A}= \\alpha (L_2) \\sqcup \\alpha (L_1)$ .",
"The case where $\\alpha (L_2)=\\mathcal {F}_\\textsf {A}$ is analogous.",
"For $\\alpha (L_1) = \\lbrace g_1\\rbrace $ , $\\alpha (L_2) = \\lbrace g_2\\rbrace $ , both languages are singleton, and we obtain that $\\lbrace g_1\\rbrace \\cup \\lbrace g_2\\rbrace = \\lbrace g_1\\rbrace $ if and only if $g_1 = g_2$ .",
"Accordingly, $\\alpha (L_1 \\cup L_2) = \\alpha (\\lbrace g_1\\rbrace ) = \\lbrace g_1\\rbrace =\\alpha (\\lbrace g_1\\rbrace ) \\sqcup \\alpha (\\lbrace g_2\\rbrace )$ .",
"If $g_1 \\ne g_2$ then $\\lbrace g_1\\rbrace \\cup \\lbrace g_2\\rbrace \\subseteq g_1\\langle g_1^-g_2\\rangle $ and $\\alpha (L_1 \\cup L_2) = g_1\\langle p\\rangle $ with $p$ the primitive root of $g_1^- g_2$ .",
"Therefore, $\\alpha (L_1 \\cup L_2) = g_1\\langle p\\rangle = \\alpha (\\lbrace g_1\\rbrace ) \\sqcup \\alpha (\\lbrace g_2\\rbrace )$ .",
"Assume that $\\alpha (L_1)=\\lbrace g_1\\rbrace $ and $\\alpha (L_2)=g_2\\langle p_2\\rangle $ for some primitive $p_2$ .",
"If $g_1\\in g_2\\langle p_2\\rangle $ , then $\\alpha (L_1\\cup L_2)= g_2\\langle p_2\\rangle = \\alpha (L_1)\\sqcup \\alpha (L_2)$ .",
"Otherwise, i.e., if $g_1 \\notin g_2\\langle p_2\\rangle $ , then $L_1\\cup L_2$ is not contained in $g\\langle p\\rangle $ for any $p$ (since $p_2$ was chosen primitive), and therefore, $\\alpha (L_1\\cup L_2) = \\mathcal {F}_\\textsf {A}= \\alpha (L_1)\\sqcup \\alpha (L_2)$ .",
"A similar argument applies if $\\alpha (L_2) = \\lbrace g_1\\rbrace $ , and $\\alpha (L_1) = g_2\\langle p_2\\rangle $ .",
"Assume that $\\alpha (L_1)=g_1\\langle p_1\\rangle $ and $\\alpha (L_2)=g_2\\langle p_2\\rangle $ for some primitive $p_1,p_2$ .",
"If $p_2\\in \\langle p_1\\rangle $ as well as $g_2^-g_1\\in \\langle p_1\\rangle $ , then $g_1\\langle p_1\\rangle = g_2\\langle p_2\\rangle $ (due to primitivity of $p_1,p_2$ ).",
"Moreover, $\\alpha (L_1\\cup L_2) = g_1\\langle p_1\\rangle = \\alpha (L_1)\\sqcup \\alpha (L_2)$ .",
"Otherwise, i.e., if $p_2\\notin \\langle p_1\\rangle $ or $g_2^- g_1\\notin \\langle p_1\\rangle $ , then $L_1\\cup L_2$ cannot be subset of $g\\langle p\\rangle $ for any $g,p\\in \\mathcal {F}_\\textsf {A}$ .",
"Therefore, $\\alpha (L_1\\cup L_2) = \\mathcal {F}_\\textsf {A}= \\alpha (L_1)\\sqcup \\alpha (L_2)$ .",
"For the concatenation with the empty set and the product operator we find $\\alpha (\\emptyset \\cdot L) = \\alpha (L \\cdot \\emptyset ) = \\alpha (\\emptyset ) = \\emptyset = \\alpha (\\emptyset ) \\star \\alpha (L) = \\alpha (L) \\star \\alpha (\\emptyset )$ .",
"Assume that $\\alpha (L_1) =\\mathcal {F}_\\textsf {A}$ .",
"Then $L_1 \\lnot \\subseteq g\\langle p\\rangle $ for any $g,p\\in \\mathcal {F}_\\textsf {A}$ .",
"Assume that $L_2\\subseteq \\mathcal {F}_\\textsf {A}$ is nonempty.",
"Then by Lemma , $L_1 \\cdot L_2$ and $L_2\\cdot L_1$ cannot be contained in $g^{\\prime }\\langle p^{\\prime }\\rangle $ for any $g^{\\prime },p^{\\prime }$ .",
"Therefore, $\\alpha (L_1 \\cdot L_2) = \\alpha (L_2 \\cdot L_1) = \\mathcal {F}_\\textsf {A}=\\alpha (L_1) \\star \\alpha (L_2) = \\alpha (L_2) \\star \\alpha (L_1)$ .",
"For $\\alpha (L_1)=\\lbrace g_1\\rbrace , \\alpha (L_2)=\\lbrace g_2\\rbrace $ , both languages are singletons, and we obtain $\\alpha (L_1\\cdot L_2) = \\lbrace g_1g_2\\rbrace = \\lbrace g_1\\rbrace \\star \\lbrace g_2\\rbrace = \\alpha (L_1) \\star \\alpha (L_2)$ .",
"Now assume that $\\alpha (L_1)=\\lbrace g_1\\rbrace $ and $\\alpha (L_2)=g_2\\langle p_2\\rangle $ .",
"Then $L_1=\\lbrace g_1\\rbrace $ , while $L_1\\cdot L_2$ is not a singleton language, but contained in $g_1g_2\\langle p_2\\rangle $ .",
"Therefore, $\\alpha (L_1\\cdot L_2) = g_1g_2 \\langle p_2\\rangle = \\alpha (L_1) \\star \\alpha (L_2)$ .",
"Likewise, if $\\alpha (L_1)=g_1\\langle p_1\\rangle $ and $\\alpha (L_2)=\\lbrace g_2\\rbrace $ , then $L_2=\\lbrace g_2\\rbrace $ , and $L_1\\cdot L_2$ is a non-singleton language contained in $g_1g_2\\langle g_2^-p_1g_2\\rangle $ .",
"Therefore, $\\alpha (L_1\\cdot L_2) = g_1g_2 \\langle g_2^-p_1g_2\\rangle = \\alpha (L_1) \\star \\alpha (L_2)$ .",
"Finally, let $\\alpha (L_1) = g_1\\langle p_1\\rangle $ and $\\alpha (L_2) = g_2\\langle p_2\\rangle $ be both ultimately periodic languages.",
"By Lemma , $L_1 \\cdot L_2$ is ultimately periodic if and only if $g_2^-p_1g_2 \\in \\langle p_2\\rangle $ .",
"Thus if $L_1\\cdot L_2$ is ultimately periodic, then $\\alpha (L_1 \\cdot L_2) = g_1 g_2 \\langle p_2\\rangle = \\alpha (L_1) \\star \\alpha (L_2)$ .",
"Otherwise, $L_1 \\cdot L_2 \\lnot \\subseteq g\\langle p\\rangle $ for any $g,p\\in \\mathcal {F}_\\textsf {A}$ , and therefore $\\alpha (L_1 \\cdot L_2) = \\mathcal {F}_\\textsf {A}= \\alpha (L_1) \\star \\alpha (L_2)$ ."
]
] | 2001.03480 |
[
[
"Unitarily invariant valuations and Tutte's sequence"
],
[
"Abstract We prove Fu's power series conjecture which relates the algebra of isometry invariant valuations on complex space forms to a formal power series from combinatorics which was introduced by Tutte.",
"The $n$-th coefficient of this series is the number of triangulations of a triangle with $3n$ internal edges; or the number of intervals in Tamari's lattice $Y_n$."
],
[
"Statement of the result and background",
"Thanks to the groundbreaking work of Alesker [2], [4], [5], the space of valuations on manifolds (i.e.",
"finitely additive functionals on some class of sufficiently regular sets) was endowed with a product structure which satisfies a version of Poincaré duality.",
"The product of invariant valuations on isotropic manifolds (i.e.",
"Riemannian manifolds such that the isometry group acts transitively on the sphere bundle) encodes the kinematic formulas on such spaces [9], [11].",
"Using this powerful new branch of integral geometry, which is called algebraic integral geometry, it was possible to write down in explicit form the kinematic formulas on all complex space forms [10], [11].",
"We refer to [8], [16] for a survey on these developments.",
"The starting point is a theorem by J. Fu which describes the algebra of invariant valuations on hermitian space $(n,\\mathbb {U}(n))$ .",
"Theorem 1.1 The algebra $\\operatorname{Val}^{\\mathrm {U}(n)}$ of translation-invariant, continuous and $\\mathrm {U}(n)$ -invariant valuations on $n$ is isomorphic to the polynomial algebra $\\mathbb {R}[t,s]/(f_{n+1},f_{n+2}),$ where $\\deg t=1, \\deg s=2$ , and $\\log (1+t+s)=\\sum _{i \\ge 1} f_i(t,s)$ is the decomposition into homogeneous components.",
"The valuation $t \\in \\operatorname{Val}^{\\mathrm {U}(n)}$ equals, up to normalization, the first intrinsic volume.",
"The valuation $s$ equals on a compact convex subset $K \\subset n$ the measure of complex hyperplanes intersecting $K \\subset n$ (again up to normalization).",
"This theorem was the main entry into a deep study of the integral geometry of $(n,\\mathrm {U}(n))$ .",
"In [10], several geometrically interesting bases of $\\operatorname{Val}^{\\mathrm {U}(n)}$ were introduced, their mutual relations were described and the kinematic formulas were explicitly written down.",
"After these results in the flat case, the next challenge was to find a similar approach in the curved case.",
"Given a real number $\\lambda $ , we denote by $\\mathbb {CP}^n_\\lambda $ the complex space form of constant holomorphic curvature $4\\lambda $ , endowed with the group $G_\\lambda $ of holomorphic isometries.",
"If $\\lambda >0$ , then $\\mathbb {CP}^n_\\lambda $ is a rescaling of complex projective space.",
"If $\\lambda <0$ , then $\\mathbb {CP}^n_\\lambda $ is a rescaling of complex hyperbolic space, while $\\mathbb {CP}^n_0$ is the flat space $n$ .",
"A natural guess is that the algebra structure of $\\mathcal {V}(\\mathbb {CP}^n_\\lambda )^{G_\\lambda }$ can be described in a way analogous to Theorem REF , with some modification of the $f_i$ depending on $\\lambda $ .",
"Based on numerical evidence, J. Fu stated in several talks around 2008 and in written form in [16] the following conjecture.",
"Conjecture 1.2 (Fu's power series conjecture) Define Tutte's series as $\\tau (\\lambda ):=\\sum _{i=1}^\\infty \\frac{2(4i+1)!}{(i+1)!(3i+2)!",
"}\\lambda ^i=\\lambda +3\\lambda ^2+13\\lambda ^3+68\\lambda ^4+399\\lambda ^5+\\ldots $ Then the algebra of invariant valuations on $\\mathbb {CP}_\\lambda ^n$ is isomorphic to $\\mathbb {R}[[t,s]]/(\\bar{f}_{n+1}^\\lambda ,\\bar{f}_{n+2}^\\lambda ),$ where the formal power series $\\bar{f}_k^\\lambda (t,s) \\in \\mathbb {R}[[t,s]]$ is defined as the degree $k$ -part in the expansion of $\\log \\left(1+t+s+\\tau (\\lambda )\\right).$ Here $t$ is of degree 1, $s$ is of degree 2, and $\\lambda $ is of degree $(-2)$ .",
"In this conjecture, $t$ denotes a certain multiple of the first intrinsic volume (which may be defined canonically on any Riemannian manifold), and $s$ is the average Euler characteristic of the intersection with a totally geodesic complex hyperplane in $\\mathbb {CP}^n_\\lambda $ .",
"If we prefer an ideal in $\\mathbb {R}[t,s]$ instead of $\\mathbb {R}[[t,s]]$ , we may truncate $\\bar{f}_{n+1}^\\lambda ,\\bar{f}_{n+2}^\\lambda $ and add the monomials $t^{2n+1},\\ldots ,s^nt$ .",
"This is the original form of the conjecture, see [11].",
"It will also come out of the proof that all $\\bar{f}_i^\\lambda , i>n$ , vanish on $\\mathbb {CP}_\\lambda ^n$ , as conjectured in [16] The first few terms of the conjecture can be confirmed using the template method and some formulas involving binomial coefficients.",
"These formulas can be shown using Zeilberger's algorithm, but get more and more involved for higher orders of $\\lambda $ , compare the discussion in [16].",
"Interestingly, the sequence of coefficients $1,3,13,68,399,2530,\\ldots $ has various combinatorial interpretations.",
"Tutte [19] has shown that the coefficient of $\\lambda ^i,i>0$ in $\\tau (\\lambda )$ is the number of non-isomorphic planar triangulations of a triangle with $3i$ internal edges.",
"In [14], the same coefficient appears as the number of description trees of type $(1,1)$ and size $i$ .",
"In [13] it was shown that this coefficient also equals the number of intervals in Tamari's lattice $Y_i$ .",
"In [7] some explicit bijections explainig these numerical coincidences are constructed.",
"See [18] for many other appearances of this sequence.",
"Some years after the statement of the conjecture, and without using it, the integral geometry of complex space forms was worked out in [11].",
"A surprising result, which is based on some computations and not yet fully understood, is the following.",
"Theorem 1.3 The algebras $\\operatorname{Val}^{\\mathrm {U}(n)}$ and $\\mathcal {V}(\\mathbb {CP}^n_\\lambda )^{G_\\lambda }$ are isomorphic for each $\\lambda $ .",
"There are many different isomorphisms, and one of them, given in Proposition REF below, will play a central role in this paper.",
"Despite the progress in integral geometry of complex space forms (see also [1], [3], [12], [20], [21]), the original conjecture remained previously open.",
"In this paper we are going to prove it.",
"Theorem 1 Fu's power series conjecture is true.",
"Our proof is a mixture of the template method and some algebraic manipulations of generating functions.",
"One of the main ingredients is the fact that $\\tau (\\lambda )$ satisfies an algebraic equation over $\\mathbb {R}(\\lambda )$ , which implies that some auxiliary power series in $s$ and $\\lambda $ , which comes out of the template method, is of non-positive degree (where $\\deg s=2, \\deg \\lambda =-2$ ).",
"This part of the proof uses the theory of holonomic functions.",
"We refer to the very recent lecture notes [17] for more information on this subject.",
"We do not know if there is a more direct link between integral geometry and the combinatorics of triangulations, Tamari's lattice or description trees.",
"It is a priori not even clear that the coefficients in the power series in Fu's conjecture have to be integers and not just any real numbers.",
"Also the way they appear in the conjecture (via the relations satisfied in some algebra) does not seem to be related to any counting of objects."
],
[
"Thanks",
"I would like to thank Joseph Fu for many interesting discussions about hermitian integral geometry and for the (still ongoing) collaboration on this subject.",
"I thank Anna-Laura Sattelberger for her helpful explanations about D-modules and holonomic functions.",
"I thank the anonymous referee for useful suggestions for improving the presentation."
],
[
"An isomorphism",
"Proposition 2.1 The map given by $t \\mapsto t, s \\mapsto \\frac{s}{1-\\lambda s}$ induces an algebra isomorphism between $\\operatorname{Val}^{\\mathrm {U}(n)}$ and $\\mathcal {V}(\\mathbb {CP}^n_\\lambda )^{G_\\lambda }$ .",
"For each $\\lambda \\in \\mathbb {R}$ , the map $\\mathbb {R}[[t,s]] \\rightarrow \\mathbb {R}[[t,s]], t \\mapsto \\frac{t}{\\sqrt{1-\\lambda s}}, s \\mapsto \\frac{s}{1-\\lambda s}$ covers an isomorphism $\\operatorname{Val}^{\\mathrm {U}(n)} \\rightarrow \\operatorname{Val}^{\\mathrm {U}(n)}$ , since the ideal defining $\\operatorname{Val}^{\\mathrm {U}(n)}$ is homogeneous (see Theorem REF ).",
"By [11], the map $t \\mapsto t \\sqrt{1-\\lambda s}, s \\mapsto s$ covers an isomorphism from $\\operatorname{Val}^{\\mathrm {U}(n)}$ to $\\mathcal {V}(\\mathbb {CP}_\\lambda ^n)^{G_\\lambda }$ .",
"The displayed map is the composition of these two isomorphisms, and hence an isomorphism as well.",
"The proposition implies that the map $t \\mapsto t, s \\mapsto \\frac{s}{1+\\lambda s}$ is an isomorphism from $\\mathcal {V}(\\mathbb {CP}_\\lambda ^n)^{G_\\lambda }$ to $\\operatorname{Val}^{\\mathrm {U}(n)}$ .",
"Therefore, Theorem REF is equivalent to saying that $\\operatorname{Val}^{\\mathrm {U}(n)}$ is isomorphic to $\\mathbb {\\mathbb {R}}[[t,s]]/(f_{n+1}^\\lambda ,f_{n+2}^\\lambda )$ , where $f_k^\\lambda (t,s)$ is the degree $k$ -component in $\\log \\left(1+t+\\frac{s}{1+\\lambda s}+\\tau (\\lambda )\\right).$ We will prove it in this form."
],
[
"The template method",
"We first need some preparations.",
"All computations will be done with formal power series, so no convergence is required.",
"Let us recall the formal power series $\\log (1+x) & = \\sum _{k=1}^\\infty \\frac{(-1)^k}{k}x^k,\\\\(1+x)^\\alpha & = \\sum _{k=0}^\\infty \\binom{\\alpha }{k} x^k.$ Taking $\\alpha :=-\\frac{1}{2}$ and rewriting the binomial coefficient we get $ \\frac{1}{\\sqrt{1-4x}}=\\sum _{l=0}^\\infty \\binom{2l}{l} x^l.$ Lemma 3.1 For all $m \\ge 0$ we have $ \\sum _{\\begin{array}{c}k \\equiv m \\mod {2}\\\\k>0\\end{array}} \\binom{k+m}{\\frac{k+m}{2}} \\frac{x^k}{k} & =Q_1^m\\left(\\frac{1}{x}\\right)+Q_2^m \\left(\\frac{1}{x}\\right)\\sqrt{1-4x^2} \\nonumber \\\\& \\quad \\underbrace{-\\binom{m}{\\frac{m}{2}} \\log (1+\\sqrt{1-4x^2})}_{\\text{ if } m \\text{ is even}},$ where $Q_1^m,Q_2^m$ are polynomials of degree $m$ .",
"If $m$ is even, then $Q_1^m,Q_2^m$ are even polynomials, and $Q_2^m$ does not contain an absolute term.",
"If $m$ is odd, then $Q_1^m,Q_2^m$ are odd polynomials, and the logarithmic term does not appear.",
"Set $Q_1^m(y) := \\sum _{\\begin{array}{c}1 \\le i \\le m\\\\i \\equiv m \\mod {2}\\end{array}} \\binom{m-i}{\\frac{m-i}{2}} \\frac{y^i}{i}$ and $F(x):=\\sum _{\\begin{array}{c}k \\equiv m \\mod {2}\\\\ k>0\\end{array}} \\binom{k+m}{\\frac{k+m}{2}} \\frac{x^k}{k}-Q_1^m\\left(\\frac{1}{x}\\right)+ \\underbrace{\\binom{m}{\\frac{m}{2}} \\log (1+\\sqrt{1-4x^2})}_{\\text{ if } m \\text{ is even}}.$ Using (REF ) we compute $F^{\\prime }(x) & =\\sum _{\\begin{array}{c}k \\equiv m \\mod {2}\\\\ k>0\\end{array}} \\binom{k+m}{\\frac{k+m}{2}} x^{k-1}+\\sum _{\\begin{array}{c}1 \\le i \\le m\\\\i \\equiv m \\mod {2}\\end{array}} \\binom{m-i}{\\frac{m-i}{2}} \\frac{1}{x^{i+1}}-\\underbrace{\\binom{m}{\\frac{m}{2}} \\frac{1-\\sqrt{1-4x^2}}{x \\sqrt{1-4x^2}}}_{\\text{ if } m \\text{ is even}}\\\\& = \\sum _{j>\\frac{m}{2}} \\binom{2j}{j} \\frac{x^{2j}}{x^{m+1}}+\\sum _{0 \\le j<\\frac{m}{2}} \\binom{2j}{j} \\frac{x^{2j}}{x^{m+1}}+\\underbrace{\\binom{m}{\\frac{m}{2}} \\frac{x^m}{x^{m+1}}-\\binom{m}{\\frac{m}{2}} \\frac{1}{x \\sqrt{1-4x^2}}}_{\\text{ if } m \\text{ is even}}\\\\& = \\frac{1}{x^{m+1}\\sqrt{1-4x^2}}- \\underbrace{\\binom{m}{\\frac{m}{2}} \\frac{1}{x\\sqrt{1-4x^2}}}_{\\text{ if } m \\text{ is even}}.$ We now look for some polynomial $Q_2^m$ such that $\\left[ Q_2^m \\left(\\frac{1}{x}\\right)\\sqrt{1-4x^2} \\right]^{\\prime }=F^{\\prime }(x)=\\frac{1}{x^{m+1}\\sqrt{1-4x^2}}- \\underbrace{\\binom{m}{\\frac{m}{2}} \\frac{1}{x\\sqrt{1-4x^2}}}_{\\text{ if } m \\text{ is even}}.$ This is equivalent to $(4-y^2) \\frac{d}{dy} Q_2^{m}(y)-\\frac{4}{y} Q_2^m(y)=y^{m+1}-\\underbrace{\\binom{m}{\\frac{m}{2}} y}_{\\text{ if } m \\text{ is even}}$ and admits the solution $Q_2^m(y):=- \\sum _{\\begin{array}{c}1 \\le i \\le m\\\\i \\equiv m \\mod {2}\\end{array}} \\frac{2^{m-i}i!!(m-1)!!",
"}{i (i-1)!!m!!}",
"y^i,$ where we use the standard notation $i!",
"!=i \\cdot (i-2) \\cdot (i-4) \\cdot \\ldots $ , with the convention $(-1)!",
"!=0, 0!",
"!=1$ .",
"It follows that with these choices of $Q_1^m,Q_2^m$ , the derivative of (REF ) with respect to $x$ vanishes, hence the equation is correct up to a constant.",
"If $m$ is odd, then both sides of the equation are odd, hence the constant must be 0.",
"If $m$ is even, we may add the constant to $Q_1^m$ .",
"Definition 3.2 A formal power series $b=b(s,\\lambda )$ is called of degree $\\le 2m$ if each monomial $s^i \\lambda ^j$ appearing in $b$ has $2i-2j \\le 2m$ .",
"Equivalently, $b^{(k)}(0):= \\left(\\left.\\frac{\\partial }{\\partial \\lambda }\\right|_{\\lambda =0}\\right)^k b(s,\\lambda )$ is a polynomial in $s$ of degree $\\le 2(m+k)$ for each $k \\ge 0$ .",
"It is clear that the sum of two series of degrees $\\le 2m$ is again of degree $\\le 2m$ and that the product of a series of degree $\\le 2m_1$ and a series of degree $\\le 2m_2$ is a series of degree $\\le 2(m_1+m_2)$ .",
"If $c>0$ and $b$ has no constant term, then $\\log (c+b)=\\log (c)+\\log \\left(1+\\frac{b}{c}\\right)=\\log (c)+\\sum _{n=1}^\\infty \\frac{(-1)^{n+1}}{n} \\frac{b^n}{c^n}$ is a well-defined formal power series.",
"If moreover $b$ is of degree $\\le 0$ , then $\\log (c+b)$ is also of degree $\\le 0$ .",
"Proposition 3.3 Let $r:=\\frac{s}{1+\\lambda s}+\\tau (\\lambda )$ , where $\\tau (\\lambda )$ is Tutte's sequence.",
"Then the formal power series $b(s,\\lambda ):= s+(1+\\lambda s) \\sqrt{(1+r)^2-4s}$ is of degree $\\le 0$ .",
"We postpone the technical proof to the next section.",
"Given a formal power series $p$ in $t,s$ , let us write $\\int _{n} p:=p_{2n}(D_n^,$ where $p_{2n}$ denotes the $(2n)$ -homogeneous component of $p$ , considered as an element of $\\operatorname{Val}_{2n}^{\\mathrm {U}(n)}$ under the isomorphism from Theorem REF , and $D_n^n$ is the unit ball.",
"By [16] we have $ \\int _{n} s^k t^{2n-k}=\\binom{2n-2k}{n-k}.$ This implies that for every $p$ and $n \\ge 0$ we have $ \\int _{{n}} sp=\\int _{{n-1}} p,$ where we formally set $\\int _{{n-1}}p:=0$ if $n=0$ .",
"Let us write $p \\equiv q$ if $p,q$ are formal power series in $t,s$ such that $\\int _{n} p=\\int _{n} q$ for all $n \\ge 0$ .",
"From (REF ) it follows that $t^k \\equiv {\\left\\lbrace \\begin{array}{ll} \\binom{k}{\\frac{k}{2}} s^{\\frac{k}{2}} & \\text{if } k \\text{ is even,}\\\\0 & \\text{if } k \\text{ is odd}.\\end{array}\\right.",
"}$ If $p \\equiv q$ then (REF ) implies that $sp \\equiv sq$ .",
"Lemma 3.4 Let $r=\\frac{s}{1+\\lambda s}+\\tau (\\lambda )$ as above.",
"There exists a formal power series $h_m(s,\\lambda ) \\in \\mathbb {R}[[s,\\lambda ]]$ of degree $\\le 2m$ such that $ t^{m}\\log \\left(1+t+r\\right) \\equiv h_m(s,\\lambda ).$ We compute $t^{m} & \\log (1+t+r) = \\sum _{n=1}^\\infty t^{m} \\frac{(-1)^{n+1}}{n}(t+r)^n\\\\& = \\sum _{n=1}^\\infty \\frac{(-1)^{n+1}}{n} \\sum _{k=0}^n \\binom{n}{k} t^{k+m} r^{n-k}\\\\& \\equiv \\sum _{n=1}^\\infty \\frac{(-1)^{n+1}}{n} \\sum _{k \\equiv m \\mod {2}} \\binom{n}{k} \\binom{k+m}{\\frac{k+m}{2}} s^{\\frac{k+m}{2}} r^{n-k}\\\\& =\\underbrace{ \\binom{m}{\\frac{m}{2}}s^{\\frac{m}{2}} \\log (1+r)}_{\\text{ if } m \\text{ even}}+\\sum _{k \\equiv m \\mod {2}, k>0} \\binom{k+m}{\\frac{k+m}{2}} s^{\\frac{k+m}{2}} \\sum _{n=k}^\\infty \\frac{(-1)^{n+1}}{n} \\binom{n}{k} r^{n-k}\\\\& = \\underbrace{ \\binom{m}{\\frac{m}{2}}s^{\\frac{m}{2}} \\log (1+r)}_{\\text{ if } m \\text{ even}}+(-1)^{m+1} \\sum _{k \\equiv m \\mod {2}, k>0} \\binom{k+m}{\\frac{k+m}{2}} \\frac{s^{\\frac{k+m}{2}}}{k(1+r)^{k}}\\\\& = h_m(s,\\lambda ),$ where $h_{m}(s,\\lambda ) & := \\binom{m}{\\frac{m}{2}}s^{\\frac{m}{2}} \\left(\\log (1+r)+\\log \\left(1+\\sqrt{1-\\frac{4s}{(1+r)^2}}\\right)\\right)\\\\& \\quad +(-1)^{m+1} s^{\\frac{m}{2}} \\left(Q_1^{m}\\left(\\frac{1+r}{\\sqrt{s}}\\right)+Q_2^m\\left(\\frac{1+r}{\\sqrt{s}}\\right)\\sqrt{1-\\frac{4s}{(1+r)^2}}\\right),$ and the logarithmic terms only appear if $m$ is even.",
"It remains to show that $h_m$ is of degree $\\le 2m$ .",
"Consider a monomial in the $Q_1$ -term.",
"Let $0 \\le i \\le m, i \\equiv m \\mod {2}$ .",
"Then $s^{\\frac{m}{2}} \\left(\\frac{1+r}{\\sqrt{s}}\\right)^{i} & = s^{\\frac{m-i}{2}} \\left(1+\\frac{s}{1+\\lambda s}+\\tau (\\lambda )\\right)^{i}\\\\& = s^{\\frac{m-i}{2}} \\sum _{a=0}^{i} \\binom{i}{a}\\left(\\frac{s}{1+\\lambda s}\\right)^a (1+\\tau (\\lambda ))^{i-a} \\\\& = \\sum _{a=0}^{i} \\binom{i}{a} s^{\\frac{m-i}{2}+a} \\frac{(1+\\tau (\\lambda ))^{i-a}}{(1+\\lambda s)^a}.$ Since $a \\le i$ , each summand is of degree $\\le 2m$ .",
"Next, we consider a monomial in the $Q_2$ -term.",
"Let $1 \\le i \\le m, i \\equiv m \\mod {2}$ .",
"Then $s^\\frac{m}{2} \\left(\\frac{1+r}{\\sqrt{s}}\\right)^{i} \\sqrt{1-\\frac{4s}{(1+r)^2}} & = s^\\frac{m-1}{2} \\left(\\frac{1+r}{\\sqrt{s}}\\right)^{i-1} \\sqrt{(1+r)^2-4s}\\\\& = s^\\frac{m-1}{2} \\left(\\frac{1+r}{\\sqrt{s}}\\right)^{i-1} \\frac{1}{1+\\lambda s} (b(s,\\lambda )-s).$ By Proposition REF , $(b(s,\\lambda )-s)$ is of degree $\\le 2$ .",
"As we have seen above, $s^\\frac{m-1}{2} \\left(\\frac{1+r}{\\sqrt{s}}\\right)^{i-1}$ is of degree $\\le 2(m-1)$ .",
"Hence the whole term is of degree $\\le 2m$ .",
"Let us finally consider the logarithmic term, which only appears if $m$ is even.",
"We have $\\log (1+r)+& \\log \\left(1+\\sqrt{1-\\frac{4s}{(1+r)^2}}\\right) \\\\& = \\log \\left(1+r+\\sqrt{(1+r)^2-4s}\\right)\\\\& = \\log \\left(1+\\lambda s + (1+\\lambda s)\\tau (\\lambda )+b(s,\\lambda )\\right)-\\log (1+\\lambda s).$ By Proposition REF , this term is of degree $\\le 0$ .",
"Since we multiply it by $\\binom{m}{\\frac{m}{2}} s^{\\frac{m}{2}}$ , we get a term of degree $\\le m \\le 2m$ .",
"Let us write $f_n^\\lambda =\\sum _{i=0}^\\infty f_{n,i}(t,s)\\lambda ^i, \\quad \\deg f_{n,i}(t,s)=n+2i,$ and $h_m(s,\\lambda )=\\sum _{i \\ge 0} h_{m,i}(s) \\lambda ^i.$ Let us show that $\\int _{n} t^{m}s^l f_{n+1}^\\lambda =0$ for all $n,m,l$ or equivalently that $\\int _{n} t^{m}s^l f_{n+1,i}=0$ for all $n,i,m,l$ .",
"Alesker duality [4] then implies that $f_{n+1,i}=0$ for all $i$ and hence $f_{n+1}^\\lambda =0$ on $n$ .",
"By degree reasons, we have to show this only if $n=m+2l+2i+1$ .",
"Now $\\int _{{m+2l+2i+1}} t^{m}s^l f_{m+2i+2,i}(t,s) & =\\int _{{m+2l+2i+1}} t^{m}s^l \\sum _q f_{q,i}(t,s)\\\\& =\\int _{{m+2l+2i+1}} s^lh_{m,i}(s),$ where the last equation follows from comparing the coefficients of $\\lambda ^i$ in (REF ).",
"By Lemma REF , we have $\\deg s^lh_{m,i} \\le 2l+2(m+i)<2(m+2l+2i+1)$ , hence the integral on the right hand side vanishes.",
"This finishes the proof that $f_{n+1}^\\lambda =0$ on $n$ for all $n$ .",
"If $i>1$ , then $f_{n+i}^\\lambda =f_{(n+i-1)+1}^\\lambda =0$ on ${n+i-1}$ by what we have shown and hence $f_{n+i}^\\lambda =0$ on $n$ by restriction from ${n+i-1}$ to $n$ .",
"Let us write $\\Phi :\\mathbb {R}[[t,s]] \\rightarrow \\mathcal {V}(\\mathbb {CP}_\\lambda ^n)^{G_\\lambda }$ for the natural algebra morphism.",
"This map is surjective, and we have shown that $\\bar{f}_{n+1}^\\lambda ,\\bar{f}_{n+2}^\\lambda \\in \\ker \\Phi $ .",
"To finish the proof, we have to show that these two power series generate the kernel.",
"By [6] the algebra $\\mathcal {V}(\\mathbb {CP}_\\lambda ^n)$ has a natural filtration $\\mathcal {V}(\\mathbb {CP}_\\lambda ^n) = W_0(\\mathbb {CP}_\\lambda ^n) \\supset W_1(\\mathbb {CP}_\\lambda ^n) \\supset \\cdots \\supset W_{2n}(\\mathbb {CP}_\\lambda ^n),$ which is compatible with the product structure (where we put $W_k(\\mathbb {CP}_\\lambda ^n)=\\lbrace 0\\rbrace $ for $k>2n$ ).",
"Fix a point $p_0 \\in \\mathbb {CP}^n_\\lambda $ .",
"By [6] we have a linear map $\\Xi _k:W_k(\\mathbb {CP}_\\lambda ^n)^{G_\\lambda } \\rightarrow \\operatorname{Val}_k(T_{p_0}\\mathbb {CP}^n_\\lambda )^{\\mathrm {U}(n)} \\cong \\operatorname{Val}_k^{\\mathrm {U}(n)}.$ Since $t \\in W_1(\\mathbb {CP}_\\lambda ^n)^{G_\\lambda }, s \\in W_2(\\mathbb {CP}_\\lambda ^n)^{G_\\lambda }$ it follows that if the lowest degree of $h \\in \\mathbb {R}[[t,s]]$ is $\\ge k$ , then $\\Phi (h) \\in W_k(\\mathbb {CP}_\\lambda ^n)^{G_\\lambda }$ , and $\\Xi _k \\circ \\Phi (h)$ equals the $k$ -homogeneous part of $h$ , which we denote by $h_k$ .",
"Suppose now that $h \\in \\mathbb {R}[[t,s]]$ is in the kernel of $\\Phi $ and is of lowest degree $k$ .",
"Then $h_k=\\Xi _k \\circ \\Phi (f)=0$ in $\\operatorname{Val}^{\\mathrm {U}(n)}$ .",
"Theorem REF implies the existence of a polynomial $p_k \\in \\mathbb {R}[t,s]$ of degree $(k-n-1)$ and a polynomial $q_k \\in \\mathbb {R}[t,s]$ of degree $(k-n-2)$ such that $h_k=p_k f_{n+1}+q_k f_{n+2}$ .",
"The lowest degree of $h-p_k \\bar{f}_{n+1}^\\lambda -q_k \\bar{f}_{n+2}^\\lambda \\in \\mathbb {R}[[t,s]]$ is then at least $(k+1)$ .",
"Repeating the same argument, we find sequences of polynomials $p_i,q_i \\in \\mathbb {R}[t,s],i=k,k+1,\\ldots $ with $\\deg p_i=i-n-1, \\deg q_i=i-n-2$ such that $h-\\sum _{i=k}^l (p_i \\bar{f}_{n+1}^\\lambda -q_i \\bar{f}_{n+2}^\\lambda )$ is of lowest degree at least $(l+1)$ for each $l \\ge k$ .",
"Setting $p:=\\sum _{i=k}^\\infty p_i \\in \\mathbb {R}[[t,s]], q:=\\sum _{i=k}^\\infty q_i \\in \\mathbb {R}[[t,s]]$ , we find that $h=p \\bar{f}_{n+1}^\\lambda +q \\bar{f}_{n+2}^\\lambda $ ."
],
[
"Proof of Proposition ",
"The proof in this section is based on some terminology and results from D-modules and holonomic functions.",
"We refer to [15], [17], [22] for more information on this subject.",
"For the reader's convenience, we have spelled out the details so that it should be possible to understand the proof without any prior knowledge of holonomic functions.",
"Let us first sketch the idea.",
"Tutte has shown that $\\tau (\\lambda )$ is an algebraic function [19] (see also [14], [13]).",
"The formal power series $b(s,\\lambda )$ is therefore the composition of a holonomic and an algebraic function and is itself holonomic [17].",
"The holonomic rank turns out to be 4.",
"In particular, if we consider $b$ as a formal power series in $\\lambda $ with coefficients in the space of formal power series in $s$ , it satisfies some differential equation of degree 4 whose coefficients are polynomials in $s$ and $\\lambda $ .",
"The differential equation gives a recursive relation for the higher derivatives $\\left.\\frac{d^i}{d\\lambda ^i}\\right|_{\\lambda =0} b(s,\\lambda )$ which will be enough to prove the proposition.",
"Let us now work out the details.",
"Define an algebraic power series $\\phi (\\lambda )$ by the equation $P(\\lambda ,\\phi (\\lambda ))=0$ , where $P(\\lambda ,\\phi ):=\\phi -\\lambda (1+\\phi )^4 \\in \\mathbb {R}[\\lambda ,\\phi ]$ is irreducible.",
"Then, by [13], $\\tau (\\lambda )=\\phi (\\lambda )(1-\\phi (\\lambda )-\\phi (\\lambda )^2)$ .",
"Lemma 4.1 There exists some $P_0 \\in \\mathbb {R}(\\lambda )[\\phi ]$ of degree 3 such that $\\phi ^{\\prime }(\\lambda )=P_0(\\lambda ,\\phi (\\lambda )).$ Taking derivatives of the equation $P(\\lambda ,\\phi (\\lambda ))=0$ yields $\\phi ^{\\prime }(\\lambda )=- \\frac{\\frac{\\partial P}{\\partial \\lambda }(\\lambda ,\\phi (\\lambda ))}{\\frac{\\partial P}{\\partial \\phi }(\\lambda ,\\phi (\\lambda ))}=\\frac{(1+\\phi (\\lambda ))^4}{1-4\\lambda (1+\\phi (\\lambda ))^3}.$ Applying the extended Euclidean algorithm to $P$ and $\\frac{\\partial P}{\\partial \\phi }$ gives polynomials $U,V \\in \\mathbb {R}(\\lambda )[\\phi ]$ such that $U P +V \\frac{\\partial P}{\\partial \\phi }=1$ , and we deduce that $\\phi ^{\\prime }(\\lambda )=- \\frac{V \\frac{\\partial P}{\\partial \\lambda }(\\lambda ,\\phi (\\lambda ))}{V \\frac{\\partial P}{\\partial \\phi } (\\lambda ,\\phi (\\lambda ))}=-V \\frac{\\partial P}{\\partial \\phi }(\\lambda ,\\phi (\\lambda )).$ Let $P_0 \\in \\mathbb {R}(\\lambda )[\\phi ]$ be the remainder of $-V \\frac{\\partial P}{\\partial \\phi }$ by division by $P$ .",
"Then $\\phi ^{\\prime }(\\lambda )=P_0(\\lambda ,\\phi (\\lambda ))$ .",
"Doing the computations explicitly, we find that $P_0(\\lambda ,\\phi )=\\frac{12 \\lambda \\phi ^3+52 \\lambda \\phi ^2+4\\lambda \\phi -36\\lambda +9\\phi }{(256\\lambda -27)\\lambda }.$ Lemma 4.2 Set $F(s,\\lambda ,\\phi ):=\\sqrt{\\left[1+\\lambda s+s+(1+\\lambda s)\\phi (1-\\phi -\\phi ^2)\\right]^2-4s(1+\\lambda s)^2},$ so that $b(s,\\lambda )=s+F(s,\\lambda ,\\phi (\\lambda ))$ .",
"Then there exists $Q_1 \\in \\mathbb {R}(s,\\lambda )[\\phi ]$ of degree $\\le 3$ with $\\frac{d}{d\\lambda } F(s,\\lambda ,\\phi (\\lambda )) = Q_1(s,\\lambda ,\\phi (\\lambda )) F(s,\\lambda ,\\phi (\\lambda )).$ Noting that $F^2$ is a polynomial in $s,\\lambda ,\\phi $ , we see that $\\frac{\\partial F}{\\partial \\lambda } & = \\frac{1}{2F^2} \\frac{\\partial F^2}{\\partial \\lambda } F=\\tilde{P}_1 F,\\\\\\frac{\\partial F}{\\partial \\phi } & = \\frac{1}{2F^2} \\frac{\\partial F^2}{\\partial \\phi } F=\\tilde{P}_2 F,\\\\\\frac{\\partial F}{\\partial s} & = \\frac{1}{2F^2} \\frac{\\partial F^2}{\\partial s} F=\\tilde{P}_3 F,$ where $\\tilde{P}_1,\\tilde{P}_2, \\tilde{P}_3$ are rational functions of $s,\\lambda ,\\phi $ .",
"This means that the formal power series $F$ is holonomic of rank 1.",
"Arguing as in the previous lemma, we find $P_1,P_2 \\in \\mathbb {R}(s,\\lambda )[\\phi ]$ of degree $\\le 3$ such that $\\frac{\\partial F}{\\partial \\lambda }(s,\\lambda ,\\phi (\\lambda )) & = P_1(s,\\lambda ,\\phi (\\lambda )) F(s,\\lambda ,\\phi (\\lambda ))\\\\\\frac{\\partial F}{\\partial \\phi }(s,\\lambda ,\\phi (\\lambda )) & = P_2(s,\\lambda ,\\phi (\\lambda )) F(s,\\lambda ,\\phi (\\lambda )).$ By the chain rule we find that $\\frac{d}{d\\lambda } F(s,\\lambda ,\\phi (\\lambda )) & = \\frac{\\partial F}{\\partial \\lambda }(s,\\lambda ,\\phi (\\lambda )) + \\frac{\\partial F}{\\partial \\phi }(s,\\lambda ,\\phi (\\lambda )) \\phi ^{\\prime }(\\lambda )\\\\& = \\left[P_1(s,\\lambda ,\\phi (\\lambda ))+P_0(\\lambda ,\\phi (\\lambda )) P_2(s,\\lambda ,\\phi (\\lambda ))\\right]F(s,\\lambda ,\\phi (\\lambda )).$ Letting $Q_1$ be the reduction of $P_1+P_0P_2$ modulo $P$ finishes the proof.",
"Since $\\phi (\\lambda )$ is algebraic of rank 4 and $F$ is holonomic of degree 1, it follows that $F(s,\\lambda ,\\phi (\\lambda ))$ is holonomic of degree 4.",
"More precisely, we obtain the following lemma.",
"Lemma 4.3 There are $Q_i \\in \\mathbb {R}(s,\\lambda )[\\phi ]$ of degree $\\le 3$ such that $ \\frac{d^i F}{d \\lambda ^i}(s,\\lambda ,\\phi (\\lambda )) = Q_i(s,\\lambda ,\\phi (\\lambda )) F(s,\\lambda ,\\phi (\\lambda )).$ We construct $Q_i$ by induction on $i$ .",
"With $Q_0:=1$ and with $Q_1$ as in the previous lemma, the cases $i=0,1$ are already done.",
"Once $Q_i$ is defined, we take the derivative to find that $\\frac{d^{i+1} F}{d \\lambda ^{i+1}}(s,\\lambda ,\\phi (\\lambda )) = \\tilde{Q}_{i+1}(s,\\lambda ,\\phi (\\lambda )) F(s,\\lambda ,\\phi (\\lambda )),$ where $\\tilde{Q}_{i+1}(s,\\lambda ,\\phi ) & :=\\frac{\\partial Q_i}{\\partial \\lambda } (s,\\lambda ,\\phi )+\\frac{\\partial Q_i}{\\partial \\phi } (s,\\lambda ,\\phi ) P_0(\\lambda ,\\phi ) \\\\& \\quad + Q_i(s,\\lambda ,\\phi ) Q_1(s,\\lambda ,\\phi ) \\in \\mathbb {R}(s,\\lambda )[\\phi ].$ We let $Q_{i+1} \\in \\mathbb {R}(s,\\lambda )[\\phi ]$ be the remainder of $\\tilde{Q}_{i+1}$ by division by $P$ .",
"Then (REF ) is satisfied with $i$ replaced by $(i+1)$ .",
"Write $b(s,\\lambda )=\\sum _{l=0}^\\infty \\frac{b_l}{l!}",
"\\lambda ^l, \\quad b_l \\in \\mathbb {R}(s).$ The statement of the proposition is equivalent to $\\deg b_l \\le 2l$ (in particular $b_l \\in \\mathbb {R}[s]$ , which is a non trivial fact).",
"Lemma 4.4 Suppose that $b_m$ is a polynomial in $s$ for all $m \\le l$ .",
"Then $(s-1)b_{l+1}$ is a polynomial in $s$ .",
"The space of polynomials in $\\phi $ of degree $\\le 3$ with coefficients in $\\mathbb {R}(s,\\lambda )$ is a vector space of dimension 4 over the field $\\mathbb {R}(s,\\lambda )$ .",
"Hence there exist rational functions $\\tilde{R}_i \\in \\mathbb {R}(s,\\lambda )$ which are not all zero with $\\sum _{i=0}^4 \\tilde{R}_i(s,\\lambda ) \\frac{1}{i!}",
"Q_i (s,\\lambda ,\\phi )=0.$ Clearing denominators and multiplying by $F(s,\\lambda ,\\phi (\\lambda ))$ we find polynomials $R_i \\in \\mathbb {R}[s,\\lambda ]$ with $\\sum _{i=0}^4 R_i(s,\\lambda ) \\frac{1}{i!}",
"\\frac{d^i F}{d \\lambda ^i}(s,\\lambda ,\\phi (\\lambda )) =0.$ Write $R_i=\\sum _{k=0}^\\infty \\frac{1}{k!}",
"R_{ik}(s)\\lambda ^k$ with $R_{ik} \\in \\mathbb {R}[s]$ .",
"The explicit computation of these polynomials is tedious, but a computer algebra software can handle this very quickly.",
"We only need the following properties.",
"$R_{ik} & = 0 \\quad \\text{ if }k<i-1, \\\\R_{i,i-1} & = c_i(s-1), c_i>0, i=1,\\ldots ,4 .$ We have $0=R_0 (b(\\lambda )-s)+\\sum _{i=1}^4 R_i b^{(i)}(\\lambda )=R_0 (b(\\lambda )-s)+\\sum _{i=1}^4 \\sum _{j,k=0}^\\infty \\frac{R_{ik}}{k!}",
"\\frac{b_{j+i}}{j!}",
"\\lambda ^{j+k}.$ Comparing the coefficient of $\\lambda ^l, l \\ge 0$ and using (REF ) yields $\\sum _{i=1}^4 \\binom{l}{l+1-i} R_{i,i-1}b_{l+1}+ \\sum _{m=0}^{l} \\sum _{i=0}^4 \\binom{l}{m-i} R_{i,l+i-m}b_m=R_{0,l}s.$ By (), $\\sum _{i=1}^4 \\binom{l}{l+1-i} R_{i,i-1}$ is a non-zero scalar multiple of $(s-1)$ .",
"The second sum and the right hand side are polynomials in $s$ by assumption, hence $(s-1)b_{l+1}$ is a polynomial in $s$ .",
"Lemma 4.5 Suppose that $b_m$ is a polynomial in $s$ for all $m \\le l+1$ .",
"Then $(3s+1)b_{l+2}$ is a polynomial in $s$ .",
"We argue as in the previous lemma, but this time using that the polynomials $\\frac{1}{i!}",
"Q_i(s,\\lambda ,\\phi ),i=1,\\ldots ,5$ are linearly dependent.",
"We then find polynomials $\\hat{R}_i \\in \\mathbb {R}[s,\\lambda ],i=1,\\ldots ,5$ such that $\\sum _{i=1}^5 \\hat{R}_i(s,\\lambda ) \\frac{1}{i!}",
"\\frac{d^iF}{d\\lambda ^i}(s,\\lambda ,\\phi (\\lambda ))=0.$ We decompose $\\hat{R}_i=\\sum _{k}^\\infty \\frac{1}{k!}",
"\\hat{R}_{ik}(s)\\lambda ^k.$ We only need that $ \\deg \\hat{R}_i\\le 2(3-i), \\quad \\hat{R}_{ik}=0 \\text{ if } k<i-2,$ and $ \\hat{R}_{i,i-2}=\\hat{c}_i (3s+1), \\hat{c}_i>0, i=2,\\ldots ,5.$ Comparing coefficients as above gives us for every $l \\ge 0$ $ \\sum _{i=2}^5 \\binom{l}{l+2-i} \\hat{R}_{i,i-2} b_{l+2}+\\sum _{m=0}^{l+1} \\sum _{i=1}^5 \\binom{l}{m-i} \\hat{R}_{i,l+i-m} b_m=0.$ By (REF ), $\\sum _{i=2}^5 \\binom{l}{l+2-i} \\hat{R}_{i,i-2}$ is a non-zero multiple of $(3s+1)$ .",
"The second sum is a polynomial in $s$ by assumption and the statement follows.",
"Lemma 4.6 $b_{l}$ is a polynomial in $s$ of degree $\\le 2l$ for all $l$ .",
"We use induction on $l$ .",
"Clearly $b_0=1$ and $b_1=3s+1$ are polynomials.",
"If all $b_m,m \\le l+1$ are polynomials and $l \\ge 0$ , then $(s-1)b_{l+2}$ and $(3s+1)b_{l+2}$ are polynomials.",
"Since $(s-1)$ and $(3s+1)$ are relatively prime, this implies that $b_{l+2}$ is a polynomial in $s$ .",
"The induction hypothesis and (REF ) imply that for each $1 \\le i \\le 5, 0 \\le m \\le l+1$ we have $\\deg \\hat{R}_{i,l+i-m}b_m = \\deg \\hat{R}_{i,l+i-m}+\\deg b_m \\le 2(l-m+3)+2m = 2(l+3).$ From (REF ) and $\\deg (3s+1)=2$ we conclude that $\\deg b_{l+2} \\le 2(l+2)$ ."
]
] | 2001.03372 |
[
[
"Non-Newtonian effects on draw resonance in film casting"
],
[
"Abstract In this paper, the influence of non-Newtonian material properties on the draw resonance instability in film casting is investigated.",
"Viscoelastic models of infinite width film casting are derived systematically following an asymptotic expansion and using two well-known constitutive equations: the Giesekus model and the simplified Phan-Thien/Tanner~(PTT) model.",
"Based on a steady state analysis, a numerical boundary condition for the inlet stresses is formulated, which suppresses the unknown deformation history of the die flow.",
"The critical draw ratio in dependence of both the Deborah number and the nonlinear parameters is calculated by means of linear stability analysis.",
"For both models, the most unstable instability mode may switch under variation of the control parameters, leading to a non-continuous change in the oscillation frequency at criticality.",
"The effective elongational viscosity, which depends exclusively on the local Weissenberg number, is analyzed and identified as crucial quantity as long as the Deborah number is not too high.",
"This is demonstrated by using a generalized Newtonian fluid model to approximate the PTT model.",
"Based on such a generalized Newtonian fluid model, the effects of strain hardening and strain thinning are finally explored, revealing two opposing mechanisms underlying the non-Newtonian stability behavior."
],
[
"Introduction",
"In film casting processes, molten material is extruded through a slit die, here referred to as the inlet, and taken-up by a distant, rotating chill roll, here referred to as the outlet (see Fig.",
"REF (a)).",
"Usually, the rotation speed at the outlet is desired to be high in order to produce films of preferably small thickness.",
"The so-called draw resonance instability imposes an upper bound on the draw ratio, which is defined as the ratio of outlet to inlet velocity, and manifests itself in steady oscillations of flow velocity and both film thickness and width [1].",
"Many theoretical studies on this phenomenon were undertaken in the past decades, starting with a linear stability analysis of a purely viscous, one-dimensional model for infinite width film casting of a Newtonian fluid [2].",
"This model had been extended afterwards in several works to cover additional effects like inertia and gravity [3], [4], neck-in [5], [6], and cooling [7].",
"In many cases, viscoelastic materials like polymer melts are processed and the non-Newtonian flow properties need to be taken into account in the modeling.",
"First attempts were made employing a power-law equation in generalized Newtonian fluid (GNF) models [8], [9], [10].",
"A power-law index larger than unity, i.e., pure strain hardening, was found to increase the critical draw ratio, while an index below unity, i.e., pure strain thinning, decreases the stability as compared to the well-known Newtonian value of $20.218$ .",
"[5] investigated the linear stability analysis of an upper-convected Maxwell (UCM) fluid.",
"A strongly stabilizing effect of increasing Deborah number is predicted, including a second critical draw ratio beyond which the process becomes stable again.",
"Moreover, an unattainable region for high Deborah numbers and draw ratios is reported, where no steady state solutions exist.",
"Stability studies utilizing the Phan-Thien/Tanner (PTT) [11], [12] and the eXtended Pom-Pom model [13], [14] constitutive equations reveal in contrast to that a mostly destabilizing effect of increasing elastic properties of the material, which is in qualitative agreement with experimental findings [15], [16].",
"[17], [18] performed a steady state and a linear stability analysis of one-dimensional, infinite width film casting of a modified Giesekus fluid.",
"The results are correlated with the dependence of the steady shear and uniaxial elongational viscosities on the deformation rates and shear thinning is found to lead to destabilization, while strain hardening can lead to a stabilization depending on the range of relaxation time and strain rates occurring within the process.",
"While all of the above mentioned studies employ single mode models, there exist up to now also some works on draw resonance for a multi-relaxation-mode PTT model [19], [20], [21].",
"In two of these studies, the relaxation spectra are obtained from measurements of actual polymeric fluids [19], [21].",
"Even though already many investigations on draw resonance can be found, including viscoelastic effects, some fundamental questions remain still unanswered.",
"First of all, the influences of non-Newtonian, viscous effects and purely elastic effects on the critical draw ratio are not well separated yet.",
"This includes the question, whether a GNF model is sufficient for essentially predicting the critical draw ratio or not.",
"While [18] attempted to investigate the stability mechanism of a deformation dependent viscosity, their analysis suffers from two major drawbacks: They used the uniaxial elongational viscosity, while the one-dimensional film casting process is governed exclusively by planar deformation, and they neglected the time dependence of the viscosity by accounting for the steady values, which are valid only for sufficiently long lasting deformations at constant deformation rate.",
"For these reasons, it is still unclear if strain hardening (strain thinning) always leads to stabilization (destabilization) and why (not).",
"Moreover, the qualitative difference between the stabilizing influence of elasticity as predicted by the UCM model on the one side, and the opposite predictions of other viscoelastic models on the other side has not been explained up to now, which is strongly connected to the minimum requirements for a constitutive equation to yield at least qualitative coincidence with experimental findings.",
"The present study aims to address these questions, as well as to provide a comprehensive and consistent picture of the stability behavior of the commonly most used viscoelastic models.",
"The paper is organized as follows.",
"We start with the derivation of the one-dimensional models following an asymptotic expansion of the flow equations in Sec. .",
"We will use a uniform notation which enables us to use the same equations for both employed constitutive equations by simply tuning the corresponding parameters.",
"Initially, an empirical stress boundary condition will be defined.",
"Then, a steady state analysis is performed in Sec.",
", which primarily serves the purpose of refining the stress boundary condition in a more consistent, physical way.",
"Afterwards, the linear stability equations are derived and the stability results are presented in Sec.",
"for both constitutive equations.",
"We then focus on non-Newtonian, viscous models in Sec.",
"and analyze in particular the effective elongational viscosity.",
"Based on this quantity, we test the possibility to approximate the previously studied viscoelastic models by GNF models, and discuss the limitations of this approach.",
"Finally, the mechanisms underlying the stability behavior of strain hardening and strain thinning materials are revealed in Sec. .",
"The paper closes with brief conclusions and outlook.",
"The infinite width film casting model as presented by [2] is extended to viscoelastic constitutive equations.",
"For this purpose, an asymptotic expansion in a small film parameter is employed to obtain the one-dimensional equations at leading order, following previous studies on Newtonian fluids [6], [7], [22].",
"Figure REF (b) shows a sketch of the film model between inlet and outlet.",
"Assuming an infinite width of the film, it is sufficient to analyze the cross section of the film in the $xy$ -plane, which is described by the film thickness $h(x,t)$ and the casting length $L$ .",
"The velocity field is denoted by $v = (u,v)$ , with axial velocity $u$ and transversal velocity $v$ , and $t$ denotes the time.",
"At the inlet, axial velocity and thickness are prescribed by the extrusion to $u_0$ and $h_0$ , respectively.",
"At the outlet, the axial velocity is fixed by the take-up with the chill roll to $u_L=D\\!r\\,u_0$ , where the draw ratio $D\\!r$ is introduced as primary control parameter.",
"The continuity and momentum conservation equations, given by, respectively, $\\nabla \\cdot v &= 0,\\\\\\nabla \\cdot \\sigma &= 0,$ with $\\nabla = (\\partial _x,\\partial _y)$ and stress tensor $\\sigma $ , are coupled to the general constitutive equation $\\lambda \\,\\overset{\\nabla }{\\tau }+Y(\\tau ) = \\eta _0\\,\\left(\\nabla v + (\\nabla v)^T\\right),$ with upper-convected time derivative $\\overset{\\nabla }{\\tau } = \\partial _t\\tau + v\\cdot \\nabla \\tau - \\tau \\cdot \\nabla v - (\\nabla v)^T\\cdot \\tau $ of the extra stress tensor $\\tau =p1 + \\sigma $ , $p$ denoting the pressure.",
"$\\lambda $ is the relaxation time and $\\eta _0$ the zero-shear viscosity, both assumed to be constant.",
"The auxiliary tensor $Y$ is defined as $Y(\\tau ) &= {\\left\\lbrace \\begin{array}{ll}\\tau + \\alpha \\frac{\\lambda }{\\eta _0}\\tau ^2\\ &\\text{for the Giesekus model},\\\\[10pt]\\exp \\left(\\varepsilon \\frac{\\lambda }{\\eta _0}tr(\\tau )\\right)\\tau &\\text{for the PTT model},\\end{array}\\right.",
"}$ $tr(\\tau )$ denoting the trace of $\\tau $ .",
"The Giesekus and PTT models [23] introduce an additional, nonlinear parameter denoted by $\\alpha $ or, respectively, $\\varepsilon $ .",
"Note that only the so-called simplified PTT model with one nonlinear parameter is treated in this paper.",
"In the limit cases of $\\alpha \\rightarrow 0$ or $\\varepsilon \\rightarrow 0$ , the UCM model is recovered.",
"In the following, we will shorten the notation by combining the two models with the note that one of the nonlinear parameters, $\\alpha $ or $\\varepsilon $ , has to be set to zero depending on the chosen model, i.e., $Y(\\tau ) = \\exp \\left(\\varepsilon \\,\\lambda /\\eta _0\\,tr(\\tau )\\right)\\tau +\\alpha \\,\\lambda /\\eta _0\\,\\tau ^2.$ At the free surfaces $h_{\\pm }$ defined by $y=\\pm h/2$ , the kinematic and stress boundary conditions, $\\partial _t h_\\pm + u|_{h_\\pm }\\partial _xh_\\pm -v|_{h_\\pm } &= 0,\\\\n_\\pm \\cdot \\sigma |_{h_\\pm } &= 0,$ with normal vector $n_\\pm = (-\\partial _xh_\\pm ,1)/\\sqrt{1+(\\partial _xh)^2/4}$ , are imposed."
],
[
"Scaling and asymptotic expansion",
"The system is scaled according to the following transformations, $\\begin{split}x \\rightarrow L\\,x ,\\ \\ y \\rightarrow \\beta \\, L\\, y ,\\ \\ u \\rightarrow u_0\\, u ,\\ \\ v \\rightarrow \\beta \\, u_0\\, v ,\\\\h \\rightarrow \\beta \\, L\\, h ,\\ \\ t \\rightarrow \\frac{L}{u_0}\\, t ,\\ \\ \\tau \\rightarrow \\eta \\,\\frac{u_0}{L}\\, \\tau ,\\ \\ p \\rightarrow \\eta \\,\\frac{u_0}{L}\\, p,\\ \\ \\sigma \\rightarrow \\eta \\,\\frac{u_0}{L}\\, \\sigma ,\\end{split}$ with the film parameter $\\beta =h_0/L$ assumed to be small.",
"From now on, only scaled variables will be used unless explicitly stated differently.",
"The continuity equation (REF ) can then be written, $\\partial _x u + \\partial _y v &= 0,$ and the two components of the momentum conservation () become $\\beta ^2\\,\\partial _x\\sigma _{xx}+\\partial _y\\sigma _{xy}^* &= 0,\\\\\\partial _x\\sigma _{xy}^*+\\partial _y\\sigma _{yy} &= 0,$ with $\\sigma _{xy}^*=\\beta \\,\\sigma _{xy}$ .",
"The components of the stress boundary condition () read $-\\beta ^2\\,\\partial _xh_\\pm \\,\\sigma _{xx}|_{h_\\pm } + \\sigma _{xy}^*|_{h_\\pm } &= 0,\\\\-\\partial _xh_\\pm \\,\\sigma _{xy}^*|_{h\\pm } + \\sigma _{yy}|_{h_\\pm } &= 0,$ and the constitutive equation (REF ) is given by $\\begin{split}D\\!e\\left(\\beta ^2\\left(\\partial _t\\tau _{xx}+u\\,\\partial _x\\tau _{xx}+v\\,\\partial _y\\tau _{xx}\\right)\\right.\\\\\\left.-2\\left(\\beta ^2\\,\\tau _{xx}\\,\\partial _xu+\\tau _{xy}^*\\,\\partial _yu\\right)\\right) + Y_{xx} &= 2\\,\\beta ^2\\,\\partial _xu,\\end{split}\\\\\\begin{split}\\beta ^2\\,D\\!e\\left(\\partial _t\\tau _{yy}+u\\,\\partial _x\\tau _{yy}+v\\,\\partial _y\\tau _{yy}-\\right.\\\\\\left.2\\left(\\tau _{yy}\\,\\partial _yv+\\tau _{xy}^*\\,\\partial _xv\\right)\\right) + Y_{yy} &= 2\\,\\beta ^2\\,\\partial _yv,\\end{split}\\\\\\begin{split}D\\!e\\left(\\partial _t\\tau _{xy}^*+u\\,\\partial _x\\tau _{xy}^*+v\\,\\partial _y\\tau _{xy}^*-\\tau _{xy}^*\\,\\partial _yv\\right.\\\\\\left.-\\beta ^2\\,\\tau _{xx}\\,\\partial _xv-\\tau _{xy}^*\\,\\partial _xu-\\tau _{yy}\\,\\partial _yu\\right) + Y_{xy} &= \\beta ^2\\,\\partial _xv+\\partial _yu,\\end{split}$ where $\\tau _{xy}^* = \\beta \\,\\tau _{xy}$ , and with Deborah number $D\\!e = \\lambda \\,u_0/L$ and $Y_{xx} &= \\beta ^2\\,\\tau _{xx}\\exp \\left[\\varepsilon \\,D\\!e(\\tau _{xx}+\\tau _{yy})\\right] + \\alpha \\,D\\!e\\left(\\beta ^2\\,\\tau _{xx}^2+\\tau _{xy}^{*2}\\right),\\\\Y_{yy} &= \\beta ^2\\,\\tau _{yy}\\exp \\left[\\varepsilon \\, D\\!e\\, (\\tau _{xx}+\\tau _{yy})\\right]+\\alpha \\,D\\!e\\left(\\beta ^2\\,\\tau _{xx}^2+\\tau _{xy}^{*2}\\right),\\\\Y_{xy} &= \\tau _{xy}^*\\exp \\left[\\varepsilon \\, D\\!e\\, (\\tau _{xx}+\\tau _{yy})\\right]+ \\alpha \\,D\\!e\\left(\\beta ^2\\,\\tau _{yy}^2+\\tau _{xy}^{*2}\\right).$ The stresses, pressure, and velocity variables are now expanded in the square film parameter $\\beta ^2$ , i.e., for any variable $\\phi \\in \\lbrace u,v,p,\\tau ,\\sigma \\rbrace $ , $\\phi = \\phi ^{(0)} + \\beta ^2\\,\\phi ^{(1)} + \\mathcal {O}(\\beta ^4).$ Equations (REF ) and (REF ) truncated at $\\mathcal {O}(\\beta ^2)$ then lead to $\\sigma _{xy}^{*(0)}=\\tau _{xy}^{*(0)}=0$ , and therefore Eq.",
"() yields $\\left(D\\!e\\,\\tau _{yy}^{(0)}+1\\right)\\partial _yu^{(0)} = 0$ at leading order.",
"Assuming $\\partial _y u^{(0)}$ being non-zero would directly lead to $\\tau _{yy}^{(0)} = -\\frac{1}{D\\!e},$ which implies that the transversal stress component diverges in the case of the Newtonian limit $D\\!e\\rightarrow 0$ .",
"As this is obviously not the case, the only consistent way to fulfill Eq.",
"(REF ) is $\\partial _yu^{(0)}=0$ , i.e., the axial velocity depends only on $x$ and $t$ at leading order."
],
[
"Averaged equations",
"Averaging the continuity equation (REF ) at leading order over the thickness $h$ , together with employing the kinematic boundary conditionThe kinematic boundary condition remains unchanged by the scaling transformation.",
"(REF ), we obtain the one-dimensional form of the continuity equation, $\\partial _t h + \\partial _x\\left(h\\,u^{(0)}\\right) &= 0.$ Equations () and () yield $\\sigma _{yy}^{(0)}=0$ , which makes $\\sigma _{xx}^{(0)}$ equal to the normal stress difference $\\nu ^{(0)}=\\tau _{xx}^{(0)}-\\tau _{yy}^{(0)}$ at leading order.",
"Averaging the $x$ -component of the momentum equation (REF ) over the film thickness by utilizing the first order solution at the boundaries $\\sigma _{xy}^{*(1)}$ as obtained from Eq.",
"(REF ) finally leads therefore to $\\partial _x\\left(h\\,\\overline{\\nu ^{(0)}}\\right) = 0,$ with the depth-averaged normal stress difference at leading order given by $\\overline{\\nu ^{(0)}} = \\frac{1}{h}\\int _{-h/2}^{h/2}dy\\,\\nu ^{(0)}.$ Finally, the first non-vanishing order of Eqs.",
"(REF ) and () is averaged along the film thickness.",
"For this we have to assume additionally $\\left|\\tau _{ii}^{(0)} - \\overline{\\tau _{ii}^{(0)}}\\right| = \\mathcal {O}(\\beta ),$ as this ensures that we can calculate the average of nonlinear stress terms at leading order by $\\frac{1}{h}\\int _{-h/2}^{h/2}dy\\,\\left(\\tau _{ii}^{(0)}\\right)^2 = \\left(\\overline{\\tau _{ii}^{(0)}}\\right)^2.$ Employing the continuity equation (REF ), one then obtains $\\begin{split}D\\!e\\left[\\partial _t\\overline{\\nu ^{(0)}}+u^{(0)}\\partial _x\\overline{\\nu ^{(0)}}+2\\,\\partial _xu^{(0)}\\left(\\overline{\\nu ^{(0)}}-2\\,\\overline{\\tau _{xx}^{(0)}}\\right)\\right]\\\\ + \\,Z_{\\nu }\\,\\overline{\\nu ^{(0)}} &= 4\\,\\partial _xu^{(0)},\\end{split}\\\\\\begin{split}D\\!e\\left(\\partial _t\\overline{\\tau _{xx}^{(0)}}+u^{(0)}\\partial _x\\overline{\\tau _{xx}^{(0)}}-2\\,\\partial _xu^{(0)}\\overline{\\tau _{xx}^{(0)}}\\right)\\\\ + \\,Z_{xx}\\,\\overline{\\tau _{xx}^{(0)}} &= 2\\,\\partial _xu^{(0)},\\end{split}$ with $Z_{\\nu } &= \\exp \\left[\\varepsilon \\,D\\!e\\,\\left(2\\,\\overline{\\tau _{xx}^{(0)}}-\\overline{\\nu ^{(0)}}\\right)\\right] + \\alpha \\,D\\!e\\,\\left(2\\,\\overline{\\tau _{xx}^{(0)}}-\\overline{\\nu ^{(0)}}\\right),\\\\Z_{xx} &= \\exp \\left[\\varepsilon \\,D\\!e\\,\\left(2\\,\\overline{\\tau _{xx}^{(0)}}-\\overline{\\nu ^{(0)}}\\right)\\right] +\\alpha \\,D\\!e\\,\\overline{\\tau _{xx}^{(0)}},$ where a generalized version of Eq.",
"(REF ) was used to average the exponential term of the PTT model.",
"Equations (REF ), (REF ) and (REF ) determine the evolution of the four state variables $u^{(0)}$ , $h^{(0)}$ , $\\overline{\\nu ^{(0)}}$ and $\\overline{\\tau _{xx}^{(0)}}$ .",
"We will omit the averaging bar and the superscript “$(0)$ ” from now on and simply use $u$ , $h$ , $\\nu $ and $\\tau _{xx}$ for the variables at leading order.",
"Nevertheless, it is important to point out that, in contrast to the Newtonian model, the stress components are not necessarily identical to their averaged quantities.",
"This is usually ignored in the present literature and the equations are mostly derived by assuming a-priori transversal invariance of the variables as well as neglecting off-diagonal stress components."
],
[
"Boundary conditions (empirical approach)",
"Three boundary conditions are given directly by the process setup, $u(x=0,t) = h(x=0,t) &= 1,\\\\u(x=1,t) &= D\\!r.$ With four first-order differential equations (REF ), (REF ) and (REF ), a fourth spatial boundary condition is needed.",
"This one is physically determined by the pre-history of the flow in the die, which defines the stress components at the inlet.",
"A comprehensive analysis including the die flow increases significantly the complexity of the problem, which is why we will follow an alternative approach here.",
"As already pointed out by [24] for fiber spinning, the ratio of transversal stress to the normal stress difference decreases monotonically with increasing elastic effects.",
"At the Newtonian limit, i.e., $D\\!e\\rightarrow 0$ , the value is fixed to $\\tau _{yy}/\\nu = -1/2$ and in the elastic limit, i.e., $D\\!e\\rightarrow \\infty $ , the transversal stress vanishes.",
"[24] proposed a so-called free boundary condition method to define implicitly a stress inlet condition using finite element methods.",
"This boundary condition is equivalent to assuming that the deformation in the die is exactly the same as outside the die, i.e., purely extensional, and leads therefore to smooth stress profiles without boundary layers at the inlet.",
"Here, we will introduce a rather empirical, but more flexible approach, which will be elaborated completely in Sec.",
"REF .",
"For now, a qualitative expression based on the physical considerations above is used to obtain first results, $\\left.\\left(\\frac{\\tau _{xx}-\\nu }{\\nu }\\right)\\right|_{x=0} = -\\frac{1}{2\\,}\\exp \\left({-\\,D\\!e^b}\\right),$ which tunes the transversal stress $\\tau _{yy}=\\tau _{xx}-\\nu $ at the inlet exponentially from the Newtonian limit to zero with increasing Deborah number.",
"The transition can be adjusted by parameter $b$ , as it is depicted in Fig.",
"REF for two values of $b$ .",
"Figure: Empirical stress boundary condition () for b=0.5b=0.5 and 2."
],
[
"Steady state equations",
"The steady state equations, which determine the time-independent solutions, can be obtained by neglecting the time derivatives in Eqs.",
"(REF ), (REF ) and (REF ).",
"This leads to the following system, $u_s^{\\prime } &= \\frac{\\nu _s}{\\tilde{\\mu }_s},\\\\h_s^{\\prime } &= -\\frac{h_s\\nu _s}{u_s\\,\\tilde{\\mu }_s},\\\\\\nu _s^{\\prime } &= \\nu _s\\frac{u_s^{\\prime }}{u_s},\\\\t_s^{\\prime } &= \\frac{1}{D\\!e\\,u_s}\\left[2\\,u_s^{\\prime }(1+D\\!e\\,t_s)-t_s\\,Z_{xx,s}\\right],$ where we introduce the effective elongational viscosity at steady state, $\\tilde{\\mu }_s = \\frac{\\nu _s}{u_s^{\\prime }} = \\frac{4}{Z_{\\nu ,s}}\\,\\left[1 + D\\!e\\left(t_s-\\frac{3}{4}\\,\\nu _s\\right)\\right],$ which will be investigated in more detail in Sec. .",
"The steady state variables are indicated by a subscript `$s$ ' and $t_s=\\tau _{xx,s}$ is used for the sake of convenience.",
"Equations (REF ) are solved by numerical continuation employing the software auto-07p[25], using the Newtonian limit as an initial solution.",
"Figure: Steady state solutions for the Giesekus and PTT models employing the empirical inlet stress boundary condition () for b=0.5b=0.5 and 2 at two Deborah numbers De=0.7D\\!e=0.7 and 3, with Dr=10D\\!r=10.",
"For the Giesekus model α=0.3\\alpha =0.3 and for the PTT model ε=0.3\\varepsilon =0.3."
],
[
"Influence of stress boundary condition",
"Figure REF depicts the steady state solutions for both models for two Deborah numbers and two values of parameter $b$ .",
"The nonlinear parameters are set to $\\alpha =0.3$ and $\\varepsilon =0.3$ , respectively.",
"For both models, the velocity and thickness profiles appear to be rather independent of the inlet stress condition.",
"However, the transversal stress component changes significantly close to the die with varying initial conditions.",
"With increasing $x$ , the solutions approach each other and seem to be independent of the boundary condition.",
"This was already reported before for the case of fiber spinning of an XPP fluid [13] and can be understood as a fading memory of the deformation pre-history in the die, which leads to the particular inlet stress condition.",
"This transition occurs within a characteristic time reasonably linked to the relaxation time of the material, which in turn corresponds to a spatial position, as the time within which a fluid element has already experienced deformation can be linked to its position using the velocity profile, i.e., $dt=dx/v$ .",
"For $x\\gg D\\!e$ , i.e., exceeding the characteristic time, the pre-history of deformation does not influence the transversal stress any more and is exclusively determined by the deformation outside the die.",
"Note that this effect is not visible in the normal stress difference $\\nu _s$ and the axial stress component $t_s$ , which are almost independent of the initial stress condition."
],
[
"Refined stress boundary condition",
"Following the idea that there exists a characteristic length of transition after which the deformation is independent of the initial stress condition, we plot the transversal stress profile as a function of the Weissenberg number $W\\!i = D\\!e\\,\\partial _x u$ , by evaluating both quantities point-wise along $x$ .",
"The result is shown by Fig.",
"REF for the Giesekus model with $\\alpha =0.3$ .",
"Every single curve corresponds to a different draw ratio, with various values of $b$ in the inlet stress condition (REF ) and arrows indicating the direction of increasing $x$ -position.",
"It can be seen that with increasing $x$ , i.e., with the fluid propagating towards the outlet, all curves collapse to a common master curve, which appears to be independent of the draw ratio.",
"Similar behavior is also observed for the PTT model, including various values of the nonlinear parameter.",
"Figure: Ratio of transversal stress to normal stress difference as a function of the Weissenberg number for the Giesekus model with α=0.3\\alpha =0.3, using the empirical boundary condition () with various values of bb.",
"Every data set corresponds to a particular draw ratio and a point-wise evaluation along the xx-axis.",
"The arrows indicate the increasing xx-position, i.e., increasing time.The master curve for a particular model and nonlinear parameter value is obtained by truncating the transition regime, followed by numerical interpolation.",
"Figure REF depicts several of these master curves for the Giesekus and PTT models.",
"Using this master curve, we can now define an initial stress condition based on the physical assumption that the pre-deformation occurring in the die is of the same type as between inlet and outlet, i.e., a planar elongation with varying strain rate.",
"While this assumption may be questionable regarding the different flow properties inside and outside the die, it is still a reasonable one if we want to decouple the effect of die flow as much as possible from the actual casting deformation.",
"In particular, it is identical to the free boundary condition presented by [24].",
"Another possibility would be to follow [26], who used the fully developed flow profile in a rectangular channel to obtain the initial stress condition.",
"This approach, however, increases the computational complexity significantly and introduces additional parameters for the channel.",
"For the rest of this paper, boundary condition (REF ) is replaced by setting the transversal stress to the value obtained from the interpolated master curve.",
"In general, it should be noted that a reasonable derivation of inlet boundary conditions from the physical setup is a highly non-trivial and yet unsolved problem.",
"Moreover, the stability behavior can depend significantly on the choice of boundary conditions, as for instance demonstrated by [27] in the limit of infinite Deborah number of a UCM model.",
"Figure: Master curves of the transversal stress as a function of the Weissenberg number for the Giesekus and PTT model for several nonlinear parameter values.",
"The curves are obtained by numerical interpolation of the data obtained using initial stress condition (), after it is truncated by the transition regime close to the inlet.",
"The limit of the UCM model (α,ε→0\\alpha ,\\varepsilon \\rightarrow 0) is not accessible for large Weissenberg numbers, as no steady state solutions exist."
],
[
"Perturbation equations",
"In order to determine the critical parameters beyond which draw resonance occurs, the following ansatz for the variables is posed, $v(x,t) &= v_s(x)+\\delta \\sum _{n=0}^\\infty \\left(V_{\\!n}(x)\\,e^{\\omega _n t} + V_{\\!n}^*(x)\\,e^{\\omega _n^* t}\\right),$ which splits up the state vector $v = (u,h,\\nu ,t)$ into the steady state and a time dependent perturbation part, which is decomposed into perturbation modes with complex perturbation $V_n = (u_sU_n,h_sH_n,\\nu _sN_n,t_sT_n)$ , and corresponding complex eigenvalue $\\omega _n=\\omega _{n,R} + i\\,\\omega _{n,I}$ .",
"The real part $\\omega _{n,R}$ denotes the growth rate and the imaginary part $\\omega _{n,I}$ the frequency.",
"The asterisk denotes the complex conjugate.",
"The growth rate determines the stabilityAs pointed out by one of the referees, the assumption that the linear stability can be determined from the eigenvalues alone seems to be proven rigorously only in the case of Newtonian fluids [28] and is therefore assumed a-priori in this work.",
"of the mode, as $\\omega _{n,R}>0$ leads to an exponential amplification of an initial perturbation, i.e., unstable behavior, and $\\omega _{n,R}<0$ leads to an exponential damping, i.e., stable behavior.",
"We focus on infinitesimal small perturbations only by setting $\\delta \\ll 1$ .",
"Ansatz (REF ) is plugged in Eqs (REF ), (REF ) and (REF ), keeping only terms linear in $\\delta $ .",
"This linearization enables us to decouple the perturbation modes and each mode is determined by the same system of differential equations, $\\begin{split}U^{\\prime } &= \\frac{u_s^{\\prime }}{u_s}\\left[N-U+\\frac{D\\!e}{Z_{\\nu ,s}}\\left(\\vphantom{\\frac{2}{\\tilde{\\mu }_s}}\\left(\\omega H+\\left(\\omega +3\\,u_s^{\\prime }-\\nu _s\\left(\\alpha + \\varepsilon \\,Z_{\\nu ,s}\\right)\\right) N\\right)\\right.\\right.\\\\&\\,\\,\\,\\, \\left.\\left.",
"-\\,2\\,t_s\\left(\\frac{2}{\\tilde{\\mu }_s}-\\left(\\alpha +\\varepsilon \\,Z_{\\nu ,s}\\right)\\right)T\\right)\\right],\\end{split}\\\\H^{\\prime } &= -\\left(\\frac{\\omega }{u_s}H+U^{\\prime }\\right),\\\\N^{\\prime } &= -H^{\\prime },\\\\\\begin{split}T^{\\prime } &= \\frac{1}{D\\!e\\,u_s\\,t_s}\\left[t_s\\,Z_{xx,s}\\,U+2\\,u_s\\left(1+D\\!e\\,t_s\\right)U^{\\prime }+\\varepsilon \\,D\\!e\\,\\nu _s\\,t_s\\,Z_{xx,s}\\,N\\right.\\\\&\\,\\,\\,\\, \\left.",
"-\\left(2\\,u_s^{\\prime }+D\\!e\\,t_s\\left(\\omega +t_s\\left(\\alpha +2\\,\\varepsilon \\,Z_{xx,s}\\right)\\right)\\right)T\\right],\\end{split}$ which constitutes a homogeneous eigenvalue problem typical for a linear stability analysis.",
"Note that we omit the index `$n$ ' from now on for the sake of convenience.",
"The critical draw ratio $D\\!r_c$ and the frequency at criticality $\\omega _{I,c}$ are determined by the neutral stability condition $\\omega _R=0$ , where we focus on the most unstable mode, i.e., largest $\\omega _R$ .",
"The boundary conditions for system (REF ) are dictated by conditions (REF ) and the prescription of the transversal stress at the inlet.",
"As all these equations are time independent, the corresponding perturbations must vanish at these positions.",
"Additionally, as system (REF ) is homogeneous, we need to specify the amplitude of perturbations.",
"This choice is arbitrary and does not influence the stability predictions.",
"Following previous work [6], [29], we fix the perturbation of the force at the outlet, so that the boundary conditions read $U(0) = H(0) = t_s(0)T(0)-\\nu _s(0)N(0) &= 0,\\\\U(1) &= 0,\\\\H(1) + N(1) &= 0.1.$"
],
[
"Solution methods",
"The perturbation equations are solved by applying two distinct methods.",
"The first consists, analogue to the solution of the steady state equations, in a numerical continuation with auto-07p[25], but this time with three continuation parameters as the real and imaginary parts of the eigenvalue increase the degrees of freedom by two.",
"The procedure is to fix $\\alpha $ and $\\varepsilon $ to the desired values and to start with the solution in the Newtonian limit $D\\!e\\rightarrow 0$ , for which an analytical expression exists [30].",
"The critical conditions are then found by varying $D\\!r$ , $\\omega _R$ , and $\\omega _I$ simultaneously until $\\omega _R=0$ .",
"Afterwards, $\\omega _R=0$ is fixed and the Deborah number is used as primary continuation parameter, together with $D\\!r_c$ and $\\omega _{I,c}$ as secondary parameters, following the neutral stability curve in the parameter space.",
"Despite its simple implementation and high computational efficiency, this method has one major drawback.",
"It is tacitly assumed that the most unstable mode of the Newtonian limit remains the most unstable for all configurations, i.e., the frequency at criticality changes continuously under variation of the control parameters.",
"As shown in the following, however, this is not always the case.",
"We therefore determine additionally the eigenvalue spectrum of Eqs.",
"(REF ) to determine possible changes in the most unstable mode.",
"For this purpose, the system is discretized with a Chebyshev collocation method utilizing the chebfun [31] framework for matlab and the eigenvalues and eigenfunctions are computed with the intrinsic solver.",
"In contrast to the method of numerical continuation, the critical conditions have to be found by manually stepping through the parameter space to find the neutral configuration of zero growth rate.",
"Here we use a minimum incremental step size of $0.1$ for the critical draw ratio, leading to an accuracy of $\\pm 0.05$ .",
"As this procedure increases the computational effort and decreases the accuracy, this method is mostly used to verify, and if necessary correct, the results obtained by numerical continuation."
],
[
"Giesekus model and UCM limit",
"As visualized by Fig.REF , the Giesekus model predicts a mostly destabilizing effect of increasing elasticity, i.e., increasing Deborah number.",
"For small values of $\\alpha \\lesssim 0.1$ , a distinct stability maximum can be observed, while the critical draw ratio decreases monotonically with increasing $D\\!e$ for larger values of $\\alpha $ .",
"The only exception is $\\alpha =0.9$ , where a small stability minimum around $D\\!e\\approx 0.5$ is found.",
"In the UCM limit $\\alpha \\rightarrow 0$ , which is in good agreement with the results presented earlier by [5], a largely stabilizing influence of elasticity is present.",
"The curve exhibits a fold point, leading to unconditional stability for $D\\!e$ values above this point, and to a second critical draw ratio for $D\\!e$ values below this point, above which the system becomes stable again.",
"This second critical draw ratio, however, has never been observed experimentally [1], and it is shown below that the drastic increase of $D\\!r_c$ can be directly correlated to the unphysical divergence of the elongational viscosity of the UCM model.",
"In particular, the spectral analysis for $\\alpha =0.9$ reveals a domain of $D\\!e$ within which the second instability mode with higher frequency becomes unstable first.",
"The two insets in Fig.",
"REF showing the first five complex pairs of eigenvalues at different configurations visualize this effect of leading mode switching, which implies a non-continuous jump in the frequency at criticality $\\omega _{I,c}$ , depicted as well in Fig.",
"REF .",
"Apart from that, the frequency exhibits the same qualitative trend as the critical draw ratio for all values of $\\alpha $ .",
"In the Newtonian limit of small Deborah numbers, all curves converge to $D\\!r_c=20.218$ and $\\omega _{I,c}=14.11$ , as expected.",
"In the elastic limit of large Deborah numbers, the curves approach again a common value of $D\\!r_c=5.6$ and $\\omega _{I,c}=10.5$ .",
"Figure: Critical draw ratio (top) and frequency at criticality (bottom) as a function of the Deborah number for the Giesekus model for α=0,0.1,0.3,0.6,0.9\\alpha =0,\\,0.1,\\,0.3,\\,0.6,\\,0.9, together with comparison to the data of in the UCM limit.",
"The insets show the eigenvalues of the most unstable modes for De=10 -4 D\\!e=10^{-4} (left) and De=0.06D\\!e=0.06 (right) for α=0.9\\alpha =0.9.As can be seen by the right inset in Fig.",
"REF , the real parts of the first two eigenvalues are very close and as a result, the error in $D\\!r_c$ introduced by following exclusively the first mode, as done in numerical continuation, is negligible.",
"The jump in frequencies, however, is significant and can be an indicator of a change in the mechanism underlying the instability.",
"As typical values for the Giesekus model for $\\alpha $ are rather below $0.5$ , we will not further investigate this issue in this work, merely noting that viscoelasticity can lead to switching of the most unstable mode of draw resonance.",
"In general, increasing the nonlinear parameter $\\alpha $ appears to have a destabilizing effect, with a small exception for $0.01<D\\!e<0.1$ , where the neutral stability curves intersect for large values of $\\alpha $ ."
],
[
"PTT model",
"The stability predictions of the PTT model, as visualized by Fig.",
"REF , are in qualitative agreement with those of the Giesekus model.",
"Increasing the Deborah number has a primarily destabilizing effect.",
"For small values of the nonlinear parameter $\\varepsilon \\lesssim 0.1$ , a stability maximum occurs between $D\\!e=0.01$ and $0.1$ .",
"In the elastic limit of high Deborah numbers, all neutral stability curves converge to common values $D\\!r_c = 1.6$ and $\\omega _{I,c}~= 6.7$ , i.e., lower critical draw ratio and frequency as compared to the Giesekus model predictions.",
"Increasing $\\varepsilon $ always leads to destabilization for all Deborah numbers.",
"Similar to the Giesekus model, but more pronounced is the effect of leading mode switching for $\\varepsilon \\gtrsim 0.5$ .",
"For $\\varepsilon =1$ , we find that one of the first six instability modes can become the most unstable one (see bottom insets in Fig.",
"REF (top)), leading to a change in frequency up to a factor of 7, which is also shown by the inset of Fig.",
"REF (bottom).",
"In contrast to the Giesekus model, this mode switching also leads to a significant change in the critical draw ratio, as depicted by the top inset in Fig.",
"REF (top).",
"[19] also observed switching of the most unstable instability mode in their analysis of a multi-relaxation-mode PTT model."
],
[
"Effective elongational viscosity",
"Generalized Newtonain fluid (GNF) models can offer an efficient way to cover the main aspects of viscoelastic fluids, if (non-Newtonian) viscous effects are dominating and purely elastic effects are negligible.",
"In this case, the viscosity is assumed to be a function of the flow properties, mostly the deformation rate.",
"Instead of the shear viscosity $\\eta $ , we use here the steady effective elongational viscosity $\\tilde{\\mu }_s$ as defined by Eq.",
"(REF ), as film casting is dominated by elongational deformation.",
"We evaluate $\\tilde{\\mu }_s$ for the Giesekus and PTT models for various values of $D\\!r$ , $D\\!e$ , and the nonlinear parameters.",
"Analogue to the transversal stress component shown by Fig.",
"REF , the effective elongational viscosity appears to be independent of the draw ratio and different configurations yield parts of a common master curve, which can be interpolated numerically.",
"Using the empirical boundary condition (REF ), the boundary layer effects close to the die, similar to the ones found for the transversal stress (Fig.",
"REF ), can be observed.",
"These effects disappear if the interpolated master curve is used as initial stress boundary condition instead.",
"Figure: Effective viscosities at steady state μ ˜ s \\tilde{\\mu }_s for the (a) Giesekus and (b) PTT models.",
"In both cases, the quantity appears to depend solely on the Weissenberg number WiW\\!i and the shapes of the curves resemble the planar elongational viscosities at constant strain rate, μ ˜ * \\tilde{\\mu }^*, which are depicted as well for comparison.",
"For the PTT model (ε=0.3\\varepsilon =0.3), an empirical fit utilizing Eq.",
"() is indicated.Figure REF depicts the resulting effective elongational viscosities for both models as a function of $W\\!i$ , evaluated using $\\partial _xu_s$ , and compares them to the steady planar elongational viscosity at constant strain rate, $\\tilde{\\mu }^*$ .",
"Note that it may help here to keep in mind that $W\\!i$ can be regarded as a dimensionless strain rate.",
"For all three models, the shape of $\\tilde{\\mu }_s$ is very similar to $\\tilde{\\mu }^*$ .",
"There is, however, a quantitative difference, which is crucial for the draw resonance behavior as shown below.",
"In general it can be stated that the strain hardening effect is diminished as compared to the constant strain rate deformation.",
"For the Giesekus model, all plateau values for large Weissenberg numbers are lowered and a significant strain hardening is present only for small values of $\\alpha $ .",
"Similarly, the strain hardening maximum predicted by the PTT model becomes less pronounced and vanishes for instance already for $\\varepsilon =0.3$ , being still present for $\\tilde{\\mu }^*$ .",
"Comparing now the effective elongational viscosity curves with the neutral stability curves shown in Figs.",
"REF and REF reveals strong similarities.",
"It is striking that a stability maximum is present if and only if the corresponding effective elongational viscosity exhibits strain hardening.",
"Note that this is not true for the planar viscosity at constant strain rate, which for instance shows strain hardening as well for $\\alpha =0.3$ or, respectively, $\\varepsilon =0.3$ , where increasing $D\\!e$ has a purely destabilizing effect.",
"In the UCM limit also shown in Fig.",
"REF (a), the excessive strain hardening clearly resembles the highly stabilizing influence of increasing elasticity.",
"These observations strongly suggest that the effective elongational viscosity plays a key role in the dynamical behavior and in particular in the stability of the process.",
"For this reason, we will analyze in the following the steady state and instability behavior of a GNF model based on the effective elongational viscosity and compare the results to the full viscoelastic model prediction.",
"For the sake of compactness, we focus primarily on the PTT model and refer to the Giesekus model merely in a qualitative way at the end of the section."
],
[
"Reproducibility using a GNF model",
"In absence of strain hardening, it is possible to fit the effective elongational viscosity of the PTT model with the expression of the Carreau-Yasuda (CY) model [32], which is given by $\\tilde{\\mu }_{\\rm cy}(W\\!i) = \\frac{4}{\\left[1+\\left(k\\,W\\!i\\right)^m\\right]^\\frac{1-n}{m}}.$ For $\\varepsilon =0.3$ , the free parameters are determined as $k=0.433$ , $n = 0.162$ , and $m=0.889$ and the corresponding fit is shown in Fig.",
"REF (b).",
"While the effective viscosity is originally obtained exclusively from the steady state solution, the main assumption of the GNF model now is that this viscosity is valid as well for time dependent dynamics.",
"The GNF model is thus given by Eqs.",
"(REF ) and (REF ), supplemented by $\\nu = \\tilde{\\mu }_{\\rm cy}(D\\!e\\,\\partial _xu)\\,\\partial _xu.$ Due to the particular structure of Eq.",
"(REF ), it is impossible to analytically solve the GNF model equation for $\\partial _x u$ , as it occurs with various exponents in the model equations.",
"For this reason, we use $\\partial _x u$ instead of $\\nu $ as a variable, which leads to the following steady state equations, $u_s^{\\prime \\prime } &= \\frac{u_s^{\\prime 2}\\left(1+k\\,D\\!e\\,u_s^{\\prime }\\right)^m}{u_s\\,c},\\\\h_s^{\\prime } &= -\\frac{h_s}{u_s}u_s^{\\prime },$ with the auxiliary variable $c = \\left(1+n\\left(k\\,D\\!e\\,u_s^{\\prime }\\right)\\right)^m.$ The perturbation equations read $\\begin{split}U^{\\prime \\prime } &= \\frac{1}{u_s^2 c^2}\\left[m(1-n)\\left(k\\,D\\!e\\,u_s^{\\prime }\\right)^m u_s^{\\prime 2} U + \\omega \\,c\\left(1+\\left(k\\,D\\!e\\,u_s^{\\prime }\\right)^m\\right)u_s^{\\prime }H\\right.\\\\&\\ \\ \\ \\left.",
"+\\ (1-n)\\left(k\\,D\\!e\\,u_s^{\\prime }\\right)^m\\left(m+2\\,c\\right)u_su_s^{\\prime }U^{\\prime }\\right],\\end{split}\\\\H^{\\prime } &= -\\left(\\omega \\,\\frac{H}{u_s}+U\\right).$ Figure: Comparison between the PTT model, ε=0.3\\varepsilon =0.3, and its GNF approximation based on the effective elongational viscosity, which is modeled by the CY Eq. ().",
"(a,b) Steady state solutions for axial velocity and film thickness for De=10 -2 D\\!e = 10^{-2} and 1 and Dr=10D\\!r=10.",
"(c) Critical draw ratio.",
"(d) Frequency at criticality.Figures REF (a,b) compare the steady states of axial velocity and film thickness as obtained from this GNF approximation to the results of the PTT model for two values of the Deborah number and $D\\!r=10$ , showing a high overall agreement of both models.",
"Small deviations are most likely caused by imperfections in the fitting of the CY model.",
"The neutral stability curves (Fig.",
"REF (c)), however, coincide only for $D\\!e<0.1$ .",
"For larger Deborah numbers, the GNF approximation overpredicts the critical draw ratio and finally diverges at $D\\!e\\approx 0.5$ , while the PTT model predicts a monotonically decreasing stability with increasing $D\\!e$ .",
"Note that the divergent behavior is not caused by a switch of the most unstable mode, as it occurs for the Giesekus and PTT models as discussed above, as higher modes of instability were checked separately.",
"Moreover, the predicted frequency at criticality of the GNF approximation is in good coincidence with the PTT model result until the point of divergence (see Fig.",
"REF (d)).",
"This reveals that the use of GNF models for studying the stability is limited at least to rather low Deborah numbers.",
"On the other hand, in this regime the effective elongational viscosity appears to have a dominant influence on the stabiblity behavior.",
"A likely reason for the limitation to low $D\\!e$ is the missing explicit time dependence of the stress tensor, which appears in the viscoelastic models applied here in the form of the upper-convected time derivative.",
"As a consequence, there exists an additional, yet unexplored mechanism underlying the stability behavior of systems dominated by elastic effects.",
"The presence of such a mechanism is also visible in the stability results of the Giesekus model (Fig.",
"REF ): While the effective elongational viscosity reaches a plateau value in the elastic limit of large $W\\!i$ (Fig.",
"REF (a)), the corresponding critical draw ratio is clearly below the Newtonian limit value of $D\\!r_c = 20.218$ ."
],
[
"Influence of strain hardening and strain thinning",
"Besides this purely elastic effect, the mechanism underlying the (de-)stabilization of non-Newtonian, viscous properties is still unclear.",
"In particular, the question arises why a purely strain thinning material leads to a non-monotonous stability behavior, including a diverging critical draw ratio, as shown in Fig.",
"REF (c)).",
"There exists a supposition for a viscous mechanism underlying draw resonance, which is based on the work of [33], and which was refined later [7], [6], [29].",
"The general idea is to follow the propagation of an initial perturbation of the tension $f = h\\,\\nu $ .",
"According to Eq.",
"(REF ), this tension is constant along $x$ .",
"Evaluation of the continuity equation (REF ) at $x=0$ , where $h$ is time independent and equal to unity, enables us to write $\\partial _xh|_0 = -\\frac{f}{\\tilde{\\mu }|_0},$ where the GNF model expression $\\nu = \\tilde{\\mu }\\,\\partial _x u$ for the normal stress difference was used.",
"This equation reveals that a perturbation of the tension, e.g., at the outlet, instantaneously leads to a perturbation in thickness at the inlet, which then travels downwards, finally causing another perturbation at the outlet and closing the loop of the feedback mechanism.",
"Exploiting the fact that $f$ is constant in space, we can rewrite Eq.",
"(REF ) to better visualize the relation between perturbations at in- and outlet: $\\partial _xh|_0 = -\\frac{\\tilde{\\mu }|_1}{\\tilde{\\mu }|_0}\\,h|_1\\,\\partial _xu|_1.$ Figure: Critical draw ratio of the CY model for n=0.3n=0.3, m=1m=1, and k=1k=1 (blue, solid), together with the ratio of effective elongational viscosities at outlet to inlet (orange, dashed).",
"The dotted lines indicate the extrema of the two curves.Further analysis of the CY model reveals that the neutral stability curve infact does not always diverge, but exhibits for some parameter combinations merely a stability maximum with an overall destabilizing trend for increasing Deborah number.",
"An example is shown in Fig.",
"REF , where $m=k=1$ and $n=0.3$ , and where a stability maximum occurs close to $D\\!e=1$ .",
"Note that $n$ tunes the slope of the power-law regime for $k\\,W\\!i\\gg 1$ , while $m$ sets the sharpness of the transition from Newtonian to power-law behavior.",
"The shown case here basically differs from the one fitted to the PTT viscosity above in the value of $n$ , leading to a less steep power-law slope.",
"Following Eq.",
"(REF ), we now look at the ratio of effective elongational viscosities at outlet to inlet, as this is the quantity which differs from the Newtonian case.",
"It is plotted in Fig.",
"REF as well and exhibits a local minimum close to the stability maximum.",
"With the viscous stability mechanism described above in mind, Eq.",
"(REF ) can be interpreted in the following way.",
"While a perturbation at the outlet leads to a perturbation at the inlet, this effect is scaled by the ratio of effective viscosities.",
"This seems to be reasonable as the mechanism is based on the perturbation of tension, which is proportional to the viscosity.",
"Therefore, a perturbation at the outlet has less effect on the perturbation at the inlet, if the viscosity at the inlet is larger as compared to the outlet.",
"If the ratio of viscosities is below a threshold, the instability is completely suppressed and the critical draw ratio diverges, as observed in Fig.",
"REF (c).",
"Note that a similar argumentation was already successfully used in the past to explain the change in stability behavior caused by the neck-in in film casting [6].",
"As we are employing merely the steady states of the elongational viscosities in this reasoning, we call this mechanism the “static mechanism” underlying non-Newtonian draw resonance.",
"However, according to this explanation one would expect a purely stabilizing effect of a strain thinning fluid like the analyzed one, as the strain rate is expected to increase along $x$ during the casting and the thinning effect is more pronounced for larger $D\\!e$ .",
"Instead, the overall trend appears to be destabilizing with increasing $D\\!e$ and it is known that pure power-law models lead to an increase/decrease of $D\\!r_c$ for strain hardening/thinning materials [9].",
"For this reason, there has to exist a second mechanism, which we will call the “dynamic mechanism”.",
"Figure: Visualization of the dynamic mechanism underlying the destabilizing effect of strain thinning.",
"While ff is constant in space, all other quantities correspond to a position close to the inlet.The dynamic mechanism is visualized in Fig.",
"REF for a strain thinning material and can be understood as follows.",
"According to Eq.",
"(REF ), a perturbation of the tension, e.g., a temporary increase, leads to a stronger (negative) gradient of the film thickness at the inlet and therefore to a smaller film thickness close to the inlet.",
"Due to continuity, this increases the velocity and thus the strain rate close to the inlet as well.",
"For a strain thinning/hardening material, this has the consequence of a decreasing/increasing effective viscosity, which in turn leads to an increase/decrease of the ratio on the right-hand side of Eq.",
"(REF ), which finally amplifies/damps the initial effect.",
"Therefore, the dynamic mechanism leads to a destabilization/stabilization of strain thinning/hardening, while the static mechanism has an opposing effect.",
"Whether the static or the dynamic mechanism is more dominant and governs the stability behavior depends on the particular shape of the effective elongational viscosity.",
"For pure power-law behavior, the dynamic mechanism is clearly stronger, but in the case of the CY model, the situation is more subtle.",
"As long as the ratio of outlet to inlet effective viscosity does not deviate too much from unity, the dynamic mechanism is more pronounced than the static.",
"If, however, the material at the inlet is rather governed by the Newtonian plateau regime, while the material at the outlet enters already the power-law regime, the non-Newtonian effect close to the inlet and thus the dynamic mechanism is weak compared to the static mechanism, and the stability behavior changes qualitatively.",
"It should be noted that it was separately verified that a time-independent GNF model, which depends on the steady state of the strain rate only, yields indeed a purely stabilizing/destabilizing effect for strain thinning/hardening materials, as the dynamic mechanism is absent here.",
"A similar effect was observed by [7], when they found that cooling can have a destabilizing effect if the Stanton number is very large so that the material temperature is actually completely governed by the ambient temperature and therefore time independent."
],
[
"Conclusions and Outlook",
"We studied the influence of viscoelastic, and in particular non-Newtonian, effects on the draw resonance instability in film casting.",
"For this purpose, a comprehensive framework for a linear stability analysis of the Giesekus and PTT models, with the UCM model as limit case, was derived.",
"The boundary condition for the initial stress was chosen in a way that a deformation history caused by a particular flow inside the die is excluded from the analysis, which is consistent with the free boundary condition method of [24].",
"Both the Giesekus and the PTT models show an overall destabilizing trend, except for the UCM limit, where a strong stabilizing effect of elasticity is present.",
"This qualitative difference could be related to the unphysical, diverging elongational viscosity of the UCM model.",
"For this reason, the UCM model is highly not recommended for studies of viscoelastic film casting or related processes like fiber spinning.",
"For large values of the nonlinear parameters of the Giesekus and PTT models, a switch of the most unstable mode, and thus a discrete change of the frequency at criticality can be observed.",
"While it is possible for low values of the Deborah number to reproduce the results of the viscoelastic models using GNF models based on the effective elongational viscosity, which was introduced as a key quantity for the analysis, this approximation fails if elastic effects become too pronounced.",
"Nevertheless, the GNF models, in particular the CY model, enable us to shed light on the two opposing mechanisms underlying the effects of strain hardening and thinning materials on the stability.",
"Apart from new insights in non-Newtonian effects in draw resonance, the present results strongly indicate the existence of an additional, purely elastic stability mechanism.",
"Besides the failure of the GNF approximation for high Deborah numbers, a change in the basic mechanism can also be indicated by the observed switching of the most unstable instability mode.",
"Given that the common approaches to explain draw resonance are based on a viscous material, the question arises whether we can still speak of draw resonance in the high elastic regime, or whether this is another type of instability.",
"This is similar to the transition from draw resonance to the Rayleigh-Plateau instability when surface tension in fiber spinning is increased [29].",
"In a weakly nonlinear stability analysis, [14] found a transition from a supercritical to a subcritical instability if the Deborah number exceeds a threshold value, which is another hint for this hypothesis.",
"However, it has to be noted that this analysis is at least partially wrong, as pointed out by [34].",
"Further work on the nature of the elastic instability is therefore needed, and it is hoped that the present study can serve hereby as a solid basis."
],
[
"Acknowledgments",
"The author thanks Manuel Alves, François Gallaire, Rob Poole, and Benoit Scheid for fruitful discussions and useful tips."
]
] | 2001.03363 |
[
[
"The Faint Host Galaxies of C IV Absorbers at z > 5"
],
[
"Abstract We explore the expected galaxy environments of CIV absorbers at z>5 using the Technicolor Dawn simulations.",
"These simulations reproduce the observed history of reionization, the z~6 galaxy stellar mass function, the Ly$\\alpha$ forest transmission at z>5, and the SiIV column density distribution (CDD) at z~5.5.",
"Nonetheless, the CIV CDD remains underproduced.",
"Comparison with observed CII/SiII equivalent width ratios and the CII line incidence suggests that a low carbon yield accounts for some, but not all, of the CIV discrepancy.",
"Alternatively, a density-bounded escape scenario could harden the metagalactic ionizing background more dramatically even than binary stellar evolution, boosting the CIV CDD into near-agreement with observations.",
"In this case galaxies ionize more efficiently and fewer are required to host a given high-ionization absorber.",
"Absorbers' environments therefore constrain ionizing escape.",
"Regardless of the escape scenario, galaxies correlate with CIV absorbers out to 300 proper kpc (pkpc).",
"The correlation strengthens independently with galaxy luminosity and CIV column density.",
"Around strong systems (log(N$_{\\rm CIV}$/cm$^{-2}$)>14)), the overdensity of galaxies with M$_{\\rm UV}$<-18 or log($L_{{\\rm Ly}\\alpha}$/erg s$^{-1}$) > 41.9 declines from 200-300 within 100 pkpc to 40-60 within 250 pkpc.",
"The previously-suggested association between strong CIV absorbers and Ly$\\alpha$ emitters at z>5 is not expected.",
"It may arise if both populations inhabit large-scale voids, but for different reasons.",
"Although most neighboring galaxies are too faint for HST, JWST will, with a single pointing, identify ~10 neighboring galaxies per strong CIV absorber at z > 5.",
"Ground-based tests of these predictions are possible via deep surveys for Ly$\\alpha$ emission using integral field units."
],
[
"Introduction",
"Over the past twenty years, increasingly sophisticated surveys have uncovered thousands of galaxies that were in place and growing vigorously long before the Epoch of Cosmological Hydrogen Reionization (EOR) ended.",
"The overall abundance and spatial distribution of young galaxies has been measured at luminosities less than 1% of $L_*$ , , , , , , , , , , , yielding critical constraints on star formation and feedback at early times.",
"The emerging consensus that faint galaxies were abundant during the EOR begs the question as to what feedback processes regulated their growth, and what, if anything, they contributed to cosmological hydrogen reionization.",
"Theoretical models indicate that the primary mechanism for regulating the growth of galaxies in dark matter halos with masses $M_h > 10^9M_{\\odot }$ is galactic outflows , .",
"Outflows, in turn, leave signatures in the circumgalactic medium (CGM) that are sensitive probes of kinetic and radiative feedback.",
"For example, hydrodynamic simulations have shown that metal absorbers are severely underproduced if outflows are absent , that the geometric cross section for neutral hydrogen absorption is enhanced by outflows , that the abundance of high-ionization absorbers is sensitive to outflow velocities , and that models in which more stars form tend to produce more metals and therefore stronger metal absorbers .",
"While these ensemble studies leverage well the growing catalog of high-ionization EOR absorbers that have been identified over the last decade, less progress has been made in understanding the relationship between individual EOR galaxies and their respective CGM.",
"What sort of absorbers are found near galaxies of differing luminosities, and what sort of galaxies are expected near absorbers of differing strengths?",
"How do answers to these questions constrain kinetic and radiative feedback?",
"Studies of galaxies' environments at $z=$ 2–3 have shown that bright ($\\sim L_*$ ) galaxies possess enriched CGM whose metal column density falls smoothly with impact parameter .",
"By contrast, environmental studies at $z>5$ suggest that strong $\\mathrm {C\\,IV}$ absorbers are found preferentially around faint galaxies rather than bright ones .",
"Does this apparent conflict reflect the difference between selecting galaxy-absorber pairs based on galaxy luminosity at low redshift versus $\\mathrm {C\\,IV}$ column density at high redshift, or does it indicate that the characteristic host galaxy of strong $\\mathrm {C\\,IV}$ absorbers evolves with time?",
"One suggested explanation is that faint galaxies dominated the metagalactic ionizing ultraviolet background (UVB) during the EOR .",
"As a $\\mathrm {C\\,IV}$ system's column density increases with both metallicity and UVB amplitude, faint galaxies could dominate the environments of strong $\\mathrm {C\\,IV}$ absorbers either by ejecting more metals than bright galaxies do, or by releasing more ionizing light into their environments.",
"If this interpretation is correct, then it supports an outsized role for faint galaxies in driving reionization and UVB evolution.",
"Indeed, encouraging qualitative support for the idea that absorbers trace LyC emission was recently presented by , who found evidence for local-scale Lyman-$\\alpha $ forest opacity fluctuations in the vicinity of strong $\\mathrm {C\\,IV}$ absorbers.",
"The possibility of using the environments of high-ionization metal absorbers to trace ionizing flux from faint galaxies represents an intriguing complement to existing efforts.",
"By far the most popular current approach involves measuring the galaxy luminosity function (LF), estimating the overall Lyman continuum (LyC) emissivity of all galaxies, and comparing it to the predicted recombination rate of the intergalactic medium (IGM).",
"This method has been used to show that, subject to assumptions regarding the extrapolated abundance of faint galaxies, their intrinsic LyC emissivity $\\xi _\\mathrm {ion}$ (that is, the ratio of the ionizing to non-ionizing luminosity), the fraction $f_{\\rm esc}$ of LyC light that escaped into the IGM, and the overall IGM recombination rate , , , star formation in young galaxies had the potential to drive hydrogen reionization to completion , , , , , .",
"Whether the values of $\\xi _\\mathrm {ion}$ and $f_{\\rm esc}$ that result from these analyses are realistic is more difficult to answer observationally .",
"Evidence that young galaxies had the potential to sculpt their environments comes from the strength of their emission lines, which reflect ionizing light from massive young stars that has been re-processed in the interstellar medium (ISM) .",
"The recent detection of strong line emission from galaxies at $z>4$ , , , constrains the parameter combination $\\xi _\\mathrm {ion}(1-f_{\\rm esc})$ to be larger than expected for active, low-metallicity galaxies, but it does not directly trace the amount of ionizing flux escaping into the IGM.",
"Measurement of galaxies' non-ionizing ultraviolet continua, when modeled using stellar population synthesis techniques, can be used to constrain the product $\\xi _\\mathrm {ion}f_{\\rm esc}$ , .",
"Consistent with other studies, these efforts support the possibility that early galaxies packed sufficient firepower to complete reionization.",
"However, results still depend on an extrapolation from the non-ionizing to the ionizing stellar continuum, which is in turn sensitive at the $\\sim $ factor-of-two level to uncertainties in the underlying stellar populations.",
"At $z<4$ , the relatively-transparent IGM allows direct detection of LyC flux , yielding a more direct constraint on the product $\\xi _\\mathrm {ion}f_{\\rm esc}$ .",
"In a comprehensive analysis of ground-based spectroscopic measurements of bright galaxies at $z\\sim 3$ , report a characteristic escape fraction of $f_{\\rm esc}=9\\pm 1\\%$ .",
"A central limitation in these results is that the galaxies at $z\\le 4$ for which leaking ionizing flux is directly detected may not be representative of the faint systems that dominated the UVB at $z\\ge 5$ , particularly if $f_{\\rm esc}<5\\%$ .",
"The idea that bright galaxies, even if somewhat leaky, may not dominate the UVB is further underscored by , who detected a statistical association between transparent regions in the Ly$\\alpha $ forest (LAF) and Lyman Break galaxies (LBGs) at $5.3 \\lesssim z \\lesssim 6.4$ .",
"They find that, while the LBGs themselves cannot provide the necessary flux to ionize the local LAF, faint galaxies that are presumably clustered about them may be able to provided that $f_{\\rm esc}\\ge 8\\%$ .",
"This pathbreaking study provides independent support for the view that, at $z>5$ , the LAF opacity reflects local-scale UVB fluctuations , , .",
"Nonetheless, its result is qualitatively similar to the overall one: an unseen population of galaxies with unknown LyC emissivity must be invoked in order to explain the observed properties of the high-$z$ LAF.",
"The escape fraction $f_{\\rm esc}$ from faint galaxies may be faithfully sampled via followup spectroscopy of long-duration gamma-ray bursts (GRBs), which are associated with core-collapse supernavae .",
", applying a method developed by , have shown that GRB spectra inevitably show evidence for proximate damped Ly$\\alpha $ absorbers (DLAs), which are optically thick to LyC.",
"They estimate a mean $f_{\\rm esc}$ from the regions where GRBs originate of much less than 1%.",
"If this number applies generally to star-forming regions in the EOR, then not only does it conflict with studies that directly detect an association between galaxies, absorbers, and the LAF , , it rules out the galaxy-driven reionization hypothesis.",
"In this work, we explore how deep galaxy surveys near strong $\\mathrm {C\\,IV}$ absorbers trace the release of metals and ionizing flux from faint galaxies.",
"As a by-product, we will show that the next generation of deep followup surveys using Integral Units as well as the James Webb Space Telescope (JWST) will uncover faint galaxies $\\sim 100\\times $ more efficiently than blank-field surveys when they target the environments of strong metal absorbers.",
"While the actual source densities in these areas will be biased, they will nonetheless probe the faint end of the overall LF indirectly through comparison with models that treat the absorber-galaxy relationship realistically.",
"In Section , we review our simulations.",
"In Section , we highlight improvements with respect to our previous work through comparisons between predictions and observations of the galaxy stellar mass function, the history of reionization, the evolution of the intergalactic medium, and the abundance of metal absorbers.",
"We discuss evidence that adjustments either to the assumed ratio of carbon and silicon yields or to the geometry of ionizing escape may be required.",
"We analyze the predicted relationship between galaxies and absorbers in Section .",
"Finally, we summarize in Section ."
],
[
"Simulation",
"Our simulation is an update to the Technicolor Dawn calculations described in .",
"It assumes the same cosmology in which $(\\Omega _M, \\Omega _\\Lambda , \\Omega _b, H_0, X_H) = (0.3089, 0.6911, 0.0486, 67.74, 0.751)$ .",
"However, it incorporates several adjustments to dynamic range and subgrid physics that were motivated by discrepancies with observations as discussed there.",
"Here, we outline those updates and discuss their effects."
],
[
"Adjustments to the Feedback Model",
"Our newest calculation models a $15h^{-1}{\\rm Mpc}$ volume with $2\\times 640^3$ mass resolution elements, and the UVB is modeled using $80^3$ spatial resolution elements (“voxels\").",
"This “pl15n640RT80NF24\" simulation treats roughly twice the cosmological volume as our previous best calculation with the same mass and spatial resolution, enabling us to account more completely for rare, bright galaxies without compromising on our ability to capture faint galaxies and model the post-reionization LAF.",
"We scale down the rate at which star-forming galaxies eject gas and metals by 0.2 dex with respect to .",
"This rate is governed by the mass-loading factor $\\eta $ , which is the ratio of the rate at which galaxies eject gas to their star formation rate.",
"Previously, we adopted this parameter's dependence on stellar mass $\\eta (M_*)$ from the high-resolution simulations of without adjustment.",
"However, note that the normalization of their published calibration carries an uncertainty of 0.2 dex, which roughly matches the discrepancy between our predictions and observations of the galaxy stellar mass and rest-frame UV LFs at $z\\sim 6$ (cf.",
"Figures 3 and 4 of ).",
"For our updated simulations, we therefore adopt $\\eta (M_*) = 0.63 \\times 3.6\\left(\\frac{M_*}{10^{10}M_{\\odot }}\\right)^{-0.35}.$ With this adjustment, the galaxies in our simulation produce slightly more stars and metals.",
"They also produce more ionizing photons because we have not changed the underlying emissivity model, which is based on a modified version of Yggdrasil as described in .",
"In order that it predict roughly the same reionization history while also reproducing recent measurements of the mean transmission in the post-reionization LAF , we adjust our escape fraction model to $f_{\\rm esc}(z) = 0.166 \\left(\\frac{1+z}{6}\\right)^{2.65},$ and we cap $f_{\\rm esc}(z)$ at a maximum value $f_\\mathrm {esc,max}=0.31$ .",
"This $f_{\\rm esc}(z)$ model is slightly lower than our previous high-resolution model at all redshifts, and it is consistent with recent observational inferences.",
"For example, it predicts $f_{\\rm esc}=0.057$ at $z=3$ , which lies below the observationally inferred value for bright galaxies at that redshift ($0.09\\pm 0.01$ ; ).",
"Likewise, it predicts $f_{\\rm esc}=0.13$ at $z=4.5$ , consistent with the upper limit of 0.13 inferred from observations of H$\\alpha $ emission at $z=$ 4–5 .",
"While these agreements support the emerging view that galaxies could readily have driven reionization and dominated the post-reionization UVB , , , , , Equation REF remains an assumption that must be tested through more detailed observations.",
"As a starting point, $f_{\\rm esc}(z)$ is assumed to be energy-independent; we will explore relaxing this assumption below.",
"We now demonstrate that, with the adjustments described in Section REF , our new simulation yields improved agreement with observations of the galaxy stellar mass function, the history of reionization and the post-reionization LAF, and the abundance of metals in the high-redshift CGM.",
"In order to make these comparisons, we apply the methods for identifying simulated galaxies and modeling absorption in the IGM/CGM previously described in , to which the reader is referred for details on post-processing.",
"As in that work, galaxy luminosities $M_{\\rm UV}$ refer to the rest-frame 1500Åluminosity and are computed as AB magnitudes.",
"Figure: Stellar mass functions at z=6z=6 in our previous and updatedsimulations versus observations; the pl15n640RT80 simulation (black)represents our most up-to-date calibration.",
"The vertical tickmarkindicates the 64-star particle mass resolution limit for all simulations.Turquoise pentagons are from and reflect their point-sourceincompleteness corrections; other points represent previous observationsas indicated in the legend , , .We begin with the galaxy stellar mass function (SMF) at $z=6$ .",
"In Figure REF , we compare predictions from two previously-published simulations and our newest one versus observations.",
"Comparing our previous calibration (pl12n512RT64NF24; magenta) versus our most recent one (pl15n640RT80NF24; black) reveals that suppressing outflows by 0.2 dex (Equation REF ) boosts the predicted stellar mass of all galaxies by a similar factor, yielding improved agreement with the deepest available measurements.",
"As our new simulation subtends nearly twice the cosmological volume, it also extends to slighly higher masses, slightly improving overlap with the observed dynamic range.",
"In an earlier work , we presented a smaller, “p7.5n340RT40\" simulation that readily reproduced the observed $\\mathrm {C\\,IV}$ abundance.",
"By comparing with measurements of the $z\\sim 6$ SMF that have been carried out since that work was published, we now see that, while it reproduced the observed $\\mathrm {C\\,IV}$ abundance at $z>5$ , it may have done so in part by overproducing stars and therefore metals .",
"The tendency for high-redshift star-forming galaxies to be strong line emitters opens up the possibility of quantifying the environments of high-redshift metal absorbers using narrow-line selection in addition to broadband selections .",
"As our simulations do not capture the detailed physics associated with emission and diffusion of Ly$\\alpha $ , we model the outcome of these processes in post-processing via an empirical calibration.",
"Figure: The shaded blue region shows the predicted Lyman-α\\alpha luminosityfunction at z=5.75z=5.75 (see text); its width indicates N\\sqrt{N} uncertainty.Orange triangles, turquoise squares, magenta pentagons, and salmon crossesindicate observations by , , ,and , respectively.",
"All observations have been adjusted to ourassumed cosmology.",
"The predicted and observed LFs are in goodagreement where observational incompleteness is not severe.For each simulated galaxy, we obtain the expectation value of its Ly$\\alpha $ equivalent width (EW$_{{\\rm Ly}\\alpha }$ ) from its stellar mass using Equation 22 of .This step is largely an extrapolation because only a few of our simulated galaxies at $z=5.75$ have $M_*>10^8M_{\\odot }$ , the range that dominates the observations.",
"We show the resulting $M_{\\rm UV}$ -EW$_{{\\rm Ly}\\alpha }$ relationship in the inset panel of Fig.",
"REF .",
"The intrinsic scatter in $M_{\\rm UV}(M_*)$ generates scatter in EW$_{{\\rm Ly}\\alpha }(M_{\\rm UV})$ , but broadly the adopted values are in the range 40–80Å.",
"Observations are incomplete in the luminosity range spanned by the model.",
"For example, Fig.",
"6 of indicates that their $z\\approx (5,5.5,6.64)$ Ly$\\alpha $ LFs are 50% complete at $\\log (L_{{\\rm Ly}\\alpha }/{\\rm erg}\\ {\\rm s}^{-1})=(42.0,42.3,42.7)$ .",
"Nonetheless, the satisfactory agreement between the predicted and observed Lyman-$\\alpha $ luminosity functions in Fig.",
"REF is encouraging.",
"As no ad hoc calibration has been applied to this analysis, the tentative agreement can be viewed as a test of the simulated UV LF and $M_{\\rm UV}-M_*$ relations, which, when combined with the observed dependence of the Ly$\\alpha $ equivalent width on stellar mass at $3 < z < 4.6$ , are what yield the prediction in Fig.",
"REF ."
],
[
"Observations of $\\mathrm {H\\,I}$ Reionization",
"The top panel of Figure REF shows that the updated simulation (pl15n640RT80NF24; solid black) predicts roughly the same overall reionization history as the previous one (pl12n512RT64; dashed salmon).",
"In detail, reionization occurs slightly later in the newer run, yielding improved agreement with constraints on the pre-overlap neutral fraction, but the difference is not large compared to uncertainties.",
"By contrast, the bottom panel shows that the predicted mean transmission in the LAF $T_{\\mathrm {Ly}\\alpha }$ during the interval $5 < z < 6$ is in much better agreement with observations , .",
"The disagreement between our previous simulation and observations of $T_{\\mathrm {Ly}\\alpha }$ is unlikely to reflect primarily resolution limitations because detailed convergence studies indicate that, for our choice of mass resolution and simulation volume, $T_{\\mathrm {Ly}\\alpha }$ should be converged to $\\le 10\\%$ , .",
"We conclude that our newest simulation yields a UVB whose pre-overlap growth rate and post-overlap amplitude are suitably realistic for studies of the EOR CGM."
],
[
"Observations of Metal Absorbers",
"The improvements in Figures REF –REF allow us ask to what extent small, observationally-permitted adjustments to the overall star formation efficiency and ionizing escape fraction could bring the $\\mathrm {Si\\,IV}$ and $\\mathrm {C\\,IV}$ CDDs predicted in our previous work into improved agreement with observations.",
"We address these questions directly in Figures REF –REF .",
"Figure REF confirms that, as expected, boosting the star formation efficiency and the UVB amplitude does increase the overall $\\mathrm {C\\,IV}$ production.",
"Note that this comparison benefits from a realistic treatment for observational incompleteness.",
"We assume, following results from simulations by , that the observed $\\mathrm {C\\,IV}$ census is (60, 70, 85, 100)% complete at $\\log (N_\\mathrm {C\\,IV}/{\\rm cm}^{-2}) = 13.3, 13.4, 13.5, 13.6$ and multiply the predicted abundance by this completeness function, interpolating to each column density.",
"Extrapolating this trend so that completeness falls to zero at $\\log (N_\\mathrm {C\\,IV}/{\\rm cm}^2) \\le 12.6$ causes the predicted turnover at low columns.",
"While our new simulation does produce overall more $\\mathrm {C\\,IV}$ than its predecessor, the predicted CDD still falls noticeably short of observations.",
"Solving this problem by further boosting the UVB amplitude would compromise the excellent agreement with $T_{\\mathrm {Ly}\\alpha }$ (Figure REF ).",
"Nor can the overall star formation efficiency be boosted, as this would overproduce the galaxy stellar mass function (Fig.",
"REF ).",
"Could the offset indicate problems with the assumed metal yields?",
"In order to test whether an arbitrary adjustment to the overall metal yieldThe overall metal yield is the ratio between the mass of new metals of all species ejected to the mass of long-lived stars formed .",
"is indicated, we compare in Figure REF the observed and simulated $\\mathrm {Si\\,IV}$ CDDs at a similar redshift.",
"This comparison also accounts for observational incompleteness: we scale the column densities in the $\\mathrm {C\\,IV}$ incompleteness function using the matched optical depth method so that the $\\mathrm {Si\\,IV}$ completeness at any column density is the same as the $\\mathrm {C\\,IV}$ completeness at a column density that is a factor 2.432 higher.",
"Taking this into account, we find that, whereas our previous simulation systematically underproduced $\\mathrm {Si\\,IV}$ , the updated prediction lies within $1\\sigma $ of observations at all columns.",
"Figure: The distribution of C II / Si II \\mathrm {C\\,II}/\\mathrm {Si\\,II} equivalent widths versus C II \\mathrm {C\\,II}(top) and Si II \\mathrm {Si\\,II} (bottom) equivalent width in the simulation at z=6z=6 (dots)versus in observations spanning 5.75<z<6.255.75 < z < 6.25 .",
"Histogramscollapse the predicted distribution along the x-axis after applying aminimum equivalent width of 0.05 Å.",
"The dashed segment indicates theweighted mean observed ratio.",
"Both panels indicate that the simulatedC II / Si II \\mathrm {C\\,II}/\\mathrm {Si\\,II} ratios are biased low.Figures REF –REF suggest that, if the assumed overall metal yield is correct, then our simulations cannot simultaneously match observations of the $\\mathrm {C\\,IV}$ and $\\mathrm {Si\\,IV}$ CDDs: boosting the UVB amplitude or the stellar mass density in order to match the $\\mathrm {C\\,IV}$ CDD would lead us to overproduce $\\mathrm {Si\\,IV}$ .",
"This may, for example, contribute to the result discussed by , whose reference simulation reproduced the observed high-redshift $\\mathrm {C\\,IV}$ CDD but overproduced $\\mathrm {Si\\,IV}$ .",
"We conclude that matching the observed $\\mathrm {Si\\,IV}$ and $\\mathrm {C\\,IV}$ CDDs simultaneously requires adjustments either to the relative carbon and silicon yields, or to the slope of the UVB.",
"In order to explore the former possibility, we select co-patial $\\mathrm {C\\,II}$ and $\\mathrm {Si\\,II}$ absorbers at $z=6$ in simulations and between $5.75 < z < 6.25$ in the observations and compare the distribution of $\\mathrm {C\\,II}/\\mathrm {Si\\,II}$ equivalent width (EW) ratios in Figure REF .",
"As low-ionization absorbers tend to have weak ionization corrections , the EW ratio of co-spatial absorbers traces the underlying metal abundance ratio.",
"The weighted mean observed EW ratio at $z\\sim 6$ is 0.94 (dashed segment in both panels).",
"Selecting simulated systems using a cutoff of 0.05 Å in EW($\\mathrm {C\\,II}$ ,$\\mathrm {Si\\,II}$ ), we find mean predicted ratios of (0.77, 0.54).",
"Given that the measured abundance ratio of C/Si in the solar photosphere is 3.56 [3], the model therefore agrees qualitatively with observations that the high-redshift C/Si abundance ratios are subsolar (see also , Table 6).",
"Quantitatively, however, they are rather too subsolar; the predicted yield ratio of C to Si could be increased by 22-75%.",
"Figure: A comparison with joint constraints on the line incidences ofC II \\mathrm {C\\,II} and C IV \\mathrm {C\\,IV}.",
"Observations of C II \\mathrm {C\\,II} and C IV \\mathrm {C\\,IV} arefrom and , respectively.",
"Without adjustmentin post-processing, the simulation underproduces both ions.",
"Rescalingsimulated C abundances globally boosts C II \\mathrm {C\\,II} into agreement withobservations.",
"A density-bounded escape model improves agreement withC IV \\mathrm {C\\,IV} while exacerbating the C II \\mathrm {C\\,II} underproduction.Boosting the assumed carbon yields would increase the abundances of both $\\mathrm {C\\,II}$ and $\\mathrm {C\\,IV}$ .",
"As the tendency for low- and high-ionization ions to evolve differently at $z>5$ , may encode key insight into early IGM enrichment and reionization, we compare in Fig.",
"REF the predicted and observed line incidences ${\\rm d}N/{\\rm d}X$ of $\\mathrm {C\\,II}$ and $\\mathrm {C\\,IV}$ .",
"For $\\mathrm {C\\,II}$ , we adopt the catalog of low-ionization absorbers spanning $z=5.75$ –6.25 by and select systems with EW$_\\mathrm {C\\,II}> 0.08$ Å.",
"This catalog probes an absorption path length of $\\Delta X=65.34$ in our cosmology.",
"For $\\mathrm {C\\,IV}$ , we obtain the observed line incidence of systems with column densities $13.5 < \\log (N_\\mathrm {C\\,IV}) < 14.5$ from the fit to Fig.",
"19 of .",
"Adjusted to our cosmology, this yields $dN/dX(\\mathrm {C\\,IV}) = 0.464\\pm 0.091$ for $5.3 < z < 6.2$ .",
"The simulated $\\mathrm {C\\,IV}$ and $\\mathrm {C\\,II}$ line incidences are compiled using the same cuts in equivalent width and column density.",
"Incompleteness is modeled in $\\mathrm {C\\,IV}$ as before, while incompleteness in $\\mathrm {C\\,II}$ is modeled using a fit to Fig.",
"2 from .",
"Uncertainties are $\\sqrt{N}$ except in the case of the observed $\\mathrm {C\\,IV}$ line incidence, where we assume it is dominated by the uncertainty in the slope of the observed $\\mathrm {C\\,IV}$ column density distribution.",
"Without adjustments in post-processing, the magenta point labeled “ionization-bounded\" in Fig.REF indicates clearly that the simulation underproduces both $\\mathrm {C\\,II}$ and $\\mathrm {C\\,IV}$ .",
"This supports the view that the assumed metal yield from supernovae and/or evolved stars is too low.",
"Scaling all simulated C mass fractions up by a factor of $1.75\\times $ (green arrow) brings the $\\mathrm {C\\,II}$ line incidence into agreement with observations while alleviating roughly half (in logarithmic units) of the $\\mathrm {C\\,IV}$ discrepancy."
],
[
"A Model for Density-Bounded Escape",
"In this section, we ask whether physically-motivated changes to the way in which ionizing light escapes from galaxies could steepen the predicted UVB enough to yield simultaneous agreement with $\\mathrm {Si\\,IV}$ and $\\mathrm {C\\,IV}$ observations.",
"While it is conventional to model the UVB under the assumption that the escape of ionizing photons from galaxies is an energy-independent scalar $f_{\\rm esc}$ , , , this model is not required a priori.",
"Physically, the assumption of a scalar $f_{\\rm esc}$ corresponds to a scenario in which ionizing flux escapes through transparent “holes\" in the interstellar medium (ISM) which are in turn separated by opaque “walls\" .",
"Equivalently, it may be imagined that a small fraction of massive stars lie “outside\" the ISM , although it has been argued that this particular effect may not dominate ionizing escape .",
"Following , we refer to this scenario as the “ionization-bounded\" escape model (IB).",
"Direct evidence that some LyC flux escapes through optically-thin channels is provided by LyC-leaking galaxies for which Ly$\\alpha $ emission is observed at the systemic velocity , , .",
"This signature is predicted in such scenarios theoretically .",
"In order to explore the possible consequences of this widespread assumption, we consider the opposite extreme.",
"In this “density-bounded\" (DB) scenario, each star is separated from the ISM's boundary by a thin layer of gas whose column density is tuned so that the overall escape of $\\mathrm {H\\,I}$ -ionizing flux matches the IB model, but the optical depth to more energetic photons is lower.Illustrations of these two escape scenarios are found in Figure 1 of and .",
"Evidence that ionizing flux may escape from high-redshift galaxies via density-bounded media comes from observations that high [OIII]/[OII] emission line flux ratios and weak [SII] emission are reliable predictors of LyC emission [1], , , , .",
"Additionally, report that a representative sample of LyC-leaking galaxies at $z\\sim 3$ do not show appreciable Lyman-$\\alpha $ emission at the systemic velocity.",
"We impose the DB scenario in post-processing as an adjustment to the UVB that is predicted on-the-fly.",
"First, we compute the volume-averaged $\\mathrm {H\\,I}$ ionization rate $\\langle \\Gamma _\\mathrm {H\\,I}\\rangle $ : $\\langle \\Gamma _\\mathrm {H\\,I}\\rangle \\equiv \\left\\langle \\int \\frac{4 \\pi J_\\nu }{h \\nu } \\sigma _\\nu \\mathrm {d}\\nu \\right\\rangle $ where $\\langle \\rangle $ indicate a volume-average and the other quantities have their usual meanings.",
"Figure REF insures that this ionization rate, which turns out to be $3.37\\times 10^{-13}{\\rm s}^{-1}$ at $z=5.75$ , is realistic.",
"Next, we rescale the simulated galaxy UVB by the reciprocal of the assumed $f_{\\rm esc}(z=5.75)=0.227$ (ie., $J_\\nu \\rightarrow J_\\nu /f_{\\rm esc}$ ) and compute the $\\mathrm {H\\,I}$ column density that would return the same $\\langle \\Gamma _\\mathrm {H\\,I}\\rangle $ .",
"This is accomplished by solving the equations $\\langle \\Gamma _\\mathrm {H\\,I}\\rangle & \\equiv & \\left\\langle \\int \\frac{4 \\pi J_\\nu }{f_{\\rm esc}(z=5.75) h \\nu } \\exp (-\\tau _\\nu ) \\sigma _\\nu \\mathrm {d}\\nu \\right\\rangle \\\\\\tau _\\nu & = & \\sigma _{\\nu ,\\mathrm {H\\,I}} N_\\mathrm {H\\,I}+ \\sigma _{\\nu ,\\mathrm {He\\,I}} N_\\mathrm {He\\,I}$ for $N_\\mathrm {H\\,I}$ .",
"The obscuring gas column is assumed to be completely neutral with the same helium mass fraction as the simulation.",
"The resulting $\\mathrm {H\\,I}$ column density turns out to be $4.91789\\times 10^{17}{\\rm cm}^{-2}$ .",
"This low column density would manifest observationally in the Lyman-$\\alpha $ emission profile as a small velocity offset and narrow peak separation , , .",
"We then re-scale our simulated UVB by the ratio $J_\\nu \\rightarrow J_\\nu \\exp (-\\tau _\\nu )/f_{\\rm esc}(\\nu )$ at all frequencies and positions.",
"This adjustment is approximate because it assumes that $J_\\nu $ varies linearly with $\\exp (-\\tau _\\nu )$ .",
"In reality, increasing the emissivity decreases the opacity by ionizing more gas, which in turn amplifies the overall boost to $J_\\nu $ .",
"The detailed, superlinear dependence of $J_\\nu $ on the emissivity model can only be established via numerical simulations.",
"Our heuristic model for DB escape is therefore conservative in the sense that small changes to $f_{\\rm esc}(z,\\nu )$ at high energies will have a larger impact than we estimate on $J_\\nu $ .",
"Figure: Top: The volume-averaged galaxy and quasar UVBs.",
"Inan “ionization-bounded\" escape scenario (solid salmon), the galaxy UVB hasan overall redder spectral slope than in a “density-bounded\" scenario(dashed purple) even though the two yield precisely the same overall HI\\mathrm {H\\,I}ionization rate.",
"Bottom: In the ionization-bounded model,f esc =0.22681f_{\\rm esc}=0.22681.",
"By contrast, f esc f_{\\rm esc} increases with energy in thedensity-bounded scenario even though its emissivity-weighted meanis unchanged.We confirm in the top panel of Figure REF that the volume-averaged galaxy UVB in the IB scenario has an overall steeper spectral slope than in the DB scenario.",
"The difference is strongest for energies $<4$ Ryd, where $f_{\\rm esc}$ varies most strongly with energy in the DB scenario.",
"The top panel also indicates the ionization potentials for the ions that we consider.",
"The impact on the LAF of switching between the two scenarios will be small by design, although a harder UVB will inevitably yield a qualitatively hotter IGM, which may eventually be detectable in the LAF .",
"The impact on ions such as $\\mathrm {Si\\,IV}$ and $\\mathrm {C\\,IV}$ , which are more sensitive to high-energy photons, will clearly be much more significant.",
"In the bottom panel of Figure REF , we illustrate how the different spectral slopes in the top curve result from a different dependence of $f_{\\rm esc}$ on energy.",
"In the IB scenario, $f_{\\rm esc}$ is a constant, whereas in the DB scenario it increases with energy, exceeding 50% at 2 Ryd and rising above 90% for $\\mathrm {He\\,II}$ -ionizing energies ($ > 4 $ Ryd).",
"In essence, the DB model swaps low-energy flux for high-energy flux in order to achieve the same $\\langle \\Gamma _\\mathrm {H\\,I}\\rangle $ .",
"We next ask whether the DB scenario can reconcile simulations with observations of the statistics of $\\mathrm {C\\,IV}$ and $\\mathrm {Si\\,IV}$ absorbers.",
"To do this, we re-extract simulated quasar sightlines that are identical to the ones considered previously except that, at all positions, the metal abundance ratios are recomputed in the DB scenario.",
"The quasar contribution to the UVB is retained without modification.",
"We then identify and characterize synthetic metal absorbers in the same way as for the IB scenario.",
"The blue shaded region in Figure REF shows that, by increasing the $\\mathrm {C\\,IV}$ fraction, the DB model roughly doubles each $\\mathrm {C\\,IV}$ absorber's column density, nearly eliminating the $\\mathrm {C\\,IV}$ discrepancy.",
"Similarly, we show in Fig.",
"REF that the DB model boosts the overall $\\mathrm {C\\,IV}$ line incidence into near-agreement with observations.",
"This improvement comes at the expense of the $\\mathrm {C\\,II}$ , whose line incidence is pushed further away from the observed range.As the impact on $\\mathrm {Si\\,II}$ is similar, the predicted distribution of $\\mathrm {C\\,II}/\\mathrm {Si\\,II}$ equivalent with ratios (Figure REF ) is only weakly sensitive to the choice of escape fraction model.",
"The DB model's impact on $\\mathrm {Si\\,IV}$ is weaker (Fig.",
"REF ) because $\\mathrm {Si\\,IV}$ probes softer photons than $\\mathrm {C\\,IV}$ .",
"The comparisons in Figures REF –REF reiterate that cosmological simulations confront grave difficulty in attempting to match simultaneously observations of the UVB, the galaxy stellar mass function, and the $\\mathrm {C\\,IV}$ and $\\mathrm {Si\\,IV}$ CDDs at high redshift.",
"Although the $\\mathrm {C\\,IV}/\\mathrm {Si\\,IV}$ abundance ratio is a tracer of the UVB's spectral hardness that could constrain the relative contributions of galaxies and AGN , , these figures suggest that it could alternatively probe the details of how ionizing light escapes from galaxies.",
"In reality, as pointed out by , the IB and DB scenarios are opposite extremes and young stars will be separated from the CGM by sightlines spanning a distribution of column densities.",
"Hence the question of how much $\\mathrm {C\\,IV}$ is generated by light from galaxies versus quasars depends on the nature of this unknown column density distribution.",
"Figure: Top: The emissivity from a stellar population that has beenforming stars at a constant rate for 100 Myr.",
"The possibility ofdensity-bounded escape is much more important at high energies than binarystellar evolution.",
"Bottom: The escape fraction in the DB model.This uncertainty is much more important for understanding high-ionization CGM ions than the effects of binary stellar evolution, which have been shown to increase the ionizing output of low-metallicity stellar populations and promote galaxy-driven reionization , , .",
"The reason is that spectral filtering associated with DB escape steepens the emerging ionizing continuum more than binary stellar evolution steepens the intrinsic one.",
"To illustrate this point, we compare in Figure REF three hypothetical emerging spectra computed from version 2.2.1 of the Binary Population and Spectral Synthesis libraries .",
"The red solid and blue dotted curves show spectra emerging from model galaxies that have been forming stars with $Z=0.002$ at a constant rate for $100M_{\\odot }\\mbox{ yr}^{-1}$ ; the default initial mass function “imf135_300\" is used in both cases.",
"When normalized to 1 Ryd, the tendency for binary stars to produce more ionizing flux at high energies is reproduced.",
"However, comparison with the dashed magenta curve shows that the effect is dwarfed by that of a purely DB escape scenario: if we position the single-star model behind a neutral screen with a hydrogen column density of $3\\times 10^{17}{\\rm cm}^{-2}$ and account for bound-free absorptions by both $\\mathrm {H\\,I}$ and $\\mathrm {He\\,I}$ , then the escaping continuum steepens much more dramatically.",
"In the bottom panel, we show the corresponding $f_{\\rm esc}(\\nu )$ , which rises from $f_{\\rm esc}=15\\%$ at the Lyman limit to 100% at 10 Ryd.",
"The uncertainty in the UVB joins other well-known uncertainties related to the stellar population and to the possible role of dust.",
"Decreasing the metallicity of stellar populations generically amplifies and hardens their Lyman continua .",
"Meanwhile, the predicted C/Si supernova yield ratio $y_{\\rm C}/y_{\\rm Si}$ in the models varies nonmonotically with metallicity, with an overall minimum of 0.68 and a maximum of 1.59 .",
"The yield ratios are also sensitive at the $<15\\%$ level to our assumption of a 50% hypernova fraction (ibid.).",
"We have not explored the implications of varying the initial mass function (IMF), although it has been shown previously that a more top-heavy IMF can boost both C and Si yields .",
"Finally, dust extinction generically reddens the emerging UV continuum, potentially counteracting the spectral hardening in the DB scenario.",
"These considerations highlight two unexplored avenues for future research.",
"First, as noted by , it is worth revisiting high-resolution simulations of high-redshift galaxies in order to quantify how $f_{\\rm esc}$ varies with energy when both dust and a realistic distribution of ISM column densities are taken into account.",
"Second, it is worth exploring how the predicted $J_\\nu $ changes in synthesis models of the UVB , , if the conventional assumption of an IB escape scenario is relaxed."
],
[
"Detecting Hosts in the Far-UV Continuum",
"Having established that our simulation produces the correct number of galaxies and a UVB whose amplitude at the Lyman limit is realistic, we now turn to our motivating question of how bright the neighboring galaxies of high-redshift $\\mathrm {C\\,IV}$ absorbers are.",
"We will show that the relationship depends on absorber strength, galaxy luminosity, distance, and the dependence of $f_{\\rm esc}$ on energy.",
"To enable this discussion, we identify simulated galaxies and compute their 1500Å continuum luminosities ($M_{\\rm UV}$ ) as described in and then match them in position space with the simulated absorbers.",
"Next, we compute the galaxy-absorber cross-correlation function $\\xi _\\mathrm {g-abs}(r)$ for different combinations of minimum absorber strength and $M_{\\rm UV}$ .",
"The cross-correlation $\\xi _\\mathrm {g-abs}(r)$ is computed as: $\\xi _\\mathrm {g-abs}(r) \\equiv \\frac{1}{n_0}\\frac{\\Delta N(r)}{\\Delta V} -1,$ where $\\Delta N(r)$ is the mean number of galaxies within a finite spherical shell of volume $\\Delta V$ located a distance $r$ from an absorber, and $n_0$ is the mean number density of galaxies averaged over the entire simulation volume.",
"When computing $\\xi _\\mathrm {g-abs}(r)$ , we extract absorbers in the DB scenario because it reproduces the observed $\\mathrm {C\\,IV}$ CDD best; alternative adjustments to the model that boost $\\mathrm {C\\,IV}$ would yield very similar results.",
"We will use “abundance\" and “environment\" interchangeably to refer to the local volume density of galaxies.",
"Figure REF verifies that absorbers are positively correlated with galaxies out to distances of at least 300 proper kpc (pkpc).",
"Moreover, the amplitude of $\\xi _\\mathrm {g-abs}(r)$ increases with both luminosity and absorber strength.",
"In other words, stronger absorbers have more galaxies in their neighborhoods, and bright galaxies cluster more strongly about absorbers than faint ones do.",
"This first hint that, even at early times, $\\mathrm {C\\,IV}$ absorbers form more efficiently around brighter galaxies seems qualitatively inconsistent with observational results that the host galaxies of strong, high-redshift absorbers tend to be faint , , , .",
"However, these results are not necessarily in tension: if the dependence of clustering strength on luminosity is weak whereas the slope of the luminosity function's faint-end is steep, one still expects to find more faint galaxies than bright ones about strong absorbers.",
"Note that, in order to map from the absorber's position in velocity space to configuration space, we neglect its proper motion.",
"For a typical velocity width of $\\sim 50\\;{\\rm km}\\,{\\rm s}^{-1}$ , this simplification could blur the predicted galaxy-absorber relation on scales of up to $\\sim 75$ pkpc.",
"If this effect were severe, it would cause $\\xi _\\mathrm {g-abs}(r)$ to flatten at small $r$ .",
"The absence of such a feature in Fig.",
"REF indicates that this blurring is not a serious issue.",
"Figure: Top: The predicted average number of detectable galaxies within 100 pkpcof a C IV \\mathrm {C\\,IV} absorber at z=5.75z=5.75 as a function of M UV M_{\\rm UV}.",
"Curves correspondingto different combinations of minimum N C IV N_\\mathrm {C\\,IV} and escape fraction scenario areidentified in the legend.",
"Irrespective of the column density or the choice ofescape scenario, theprobability of detecting a single galaxy does not approach unity for surveys thatprobe only to M UV ∼-17M_{\\rm UV}\\sim -17, which is roughly HST's blank-field limit.Vertical lines with references indicate the depths of published surveys.Bottom:The local galaxy overdensity Δ≡N C IV /N field \\Delta \\equiv N_\\mathrm {C\\,IV}/N_{\\rm field}as a function of luminosity.Figure: Same as Fig.",
", but for a maximum radiusof 250 pkpc.",
"More neighboring galaxies are predicted (top panel), but themean galaxy overdensity out to this larger distance is lower (bottom panel).Diamonds indicate the luminosities of candidate host galaxies that have beenidentified within 250 pkpc of a strong C IV \\mathrm {C\\,IV}absorber at z>4.5z>4.5 , , ; their position on thevertical axis is arbitrary.In order to explore how absorber environment may be quantified observationally, we group simulated galaxies that lie near absorbers (in three-dimensional space) into two bins of minimum $\\mathrm {C\\,IV}$ column density.",
"Within each bin, we compute the mean cumulative number of galaxies that are located within 100 pkpc per absorber as a function of their minimum luminosity.",
"For any given minimum luminosity, this number is an integral over the cross-correlation function $\\xi _\\mathrm {g-abs}(r)$ : $N(\\le M_{\\rm UV}, < 100{\\rm kpc}) = \\int _0^{100{\\rm kpc}} 4\\pi r^2 n_0 (1 + \\xi _\\mathrm {g-abs}(r)) \\mathrm {d}r$ The solid blue curve in Fig.",
"REF shows the cumulative luminosity function of neighbors for absorbers with $N_\\mathrm {C\\,IV}> 10^{14}\\ {\\rm cm}^{-2}$ .",
"As in Fig.",
"REF , the local galaxy abundance increases with absorber strength.",
"Importantly, comparing the red short-dashed and green long-dashed curves reveals that local galaxy abundance also depends on the escape fraction model: if the escape fraction in the relevant energy regime is lower (as in the IB model), then more galaxies are expected per $\\mathrm {C\\,IV}$ absorber.",
"This means measurements of absorbers' environments constrain the high-energy emissivity of faint galaxies.",
"In order to quantify how absorbers' environments vary with luminosity, we use the dot-dashed magenta curve in the top panel to show the “field\" LF.",
"This clearly indicates that galaxy abundance is much lower in a randomly-selected region than in the vicinity of strong $\\mathrm {C\\,IV}$ absorbers.",
"By taking the ratio $\\Delta $ of the absorber and field LFs, we recover the result from Fig.",
"REF that galaxy overdensity increases with luminosity.",
"In particular, we show in the bottom panel of Fig.",
"REF that, within 100 pkpc, $\\Delta $ rises from $\\approx 50$ for $M_{\\rm UV}=-15$ to $\\approx 200$ for $M_{\\rm UV}= -18$ .",
"Future simulations incorporating a larger cosmological volume will be necessary to determine whether the predicted local overdensity continues to rise to brighter luminosities.",
"Even though overdensity increases with luminosity, most of the absorbers' neighbors are difficult to detect.",
"The top axis of Fig.",
"REF converts the 1500Å luminosity to the number of HST orbits required to detect a point source at 5$\\sigma $ significance at $z=5.75$ .",
"We computed this conversion using the online Exposure Time Calculatorhttp://etc.stsci.edu/etc/input/acs/imaging/ for the Advanced Camera for Surveys F805LP ($z_{850}$ ) filter assuming that the source radiates as a 25 Myr Simple Stellar Population with 0.4 times solar metallicity.",
"Down to the HST blank-field limit ($M_{\\rm UV}\\sim -17$ ), fewer than one associated galaxy per strong absorber is expected within a 100 pkpc radius despite the strong association between galaxies and absorbers.",
"This radius corresponds to 17\" at $z=5.75$ in our cosmology, hence it is comparable to the field of view of the Keck Cosmic Web Imager (KCWI) and smaller than the or HST/ACS field of view when imaging through its ramp narrowband filters.",
"These instruments will not generally detect neighboring galaxies with a single pointing.",
"By contrast, we show in Figure REF that a wider search radius—in this case, 250 pkpc—readily turns up more neighbors.",
"The expected neighbor count does not vary linearly with the search area because galaxies are less clustered about absorbers at larger distances.",
"This result was previously seen in Figure REF , and it can be recovered by comparing the bottom panels of Figures REF –REF : the mean overdensity for galaxies with $M_{\\rm UV}=-18$ falls from $\\approx 200$ within 100 pkpc to $\\approx 40$ within 250 pkpc (see also Fig.",
"REF ).",
"A single deep HST/ACS pointing (100–150 orbits in $z_{850}$ ) encloses this entire area and is predicted to uncover 2 galaxies.",
"Redshift confirmation would require equally deep imaging in additional filters, so this approach remains impractical.",
"Fortunately, JWST will soon prove extremely efficient at characterizing absorber environments: assuming a blank-field detection limit of $M_{\\rm UV}=-15$ , JWST/NIRSPEC will, with a single deep pointing, identify $\\sim 10$ neighboring galaxies per strong high-redshift $\\mathrm {C\\,IV}$ absorber.",
"recently used an abundance-matching technique to estimate the typical $M_{\\rm UV}$ and $M_h$ of $\\mathrm {C\\,IV}$ hosts at $5.3 < z < 6.2$ .",
"Assuming a one-to-one relation between $\\mathrm {C\\,IV}$ absorbers and dark matter halos and a metal enrichment radius of 100 pkpc, they find that systems with $\\log (N_\\mathrm {C\\,IV}) > 13$ are associated with $>10^{10}M_{\\odot }$ halos and $M_{\\rm UV}< -16$ galaxies.",
"We may compare this estimate directly with our model: In the DB model, the minimum luminosity such that at least one galaxy is contained within 100 pkpc of such an absorber at $z=5.75$ is $M_{\\rm UV}< -14.5$ , 1.5 magnitudes fainter than the estimate.",
"Although the two models differ in many ways, we suspect that the primary difference is the halo-absorber relationship: in our model, $\\mathrm {C\\,IV}$ clouds from individual galaxies overlap in such a way that, although absorbers and galaxies are physically associated out to scales of $\\approx 300$ pkpc (Fig.",
"REF ), the geometric cross-section per $M_h > 10^{10}M_{\\odot }$ halo is smaller.",
"Equivalently, our model predicts an abundant population of even fainter neighboring galaxies."
],
[
"Comparison with Observations",
"To date, the published literature contains only a few robust associated galaxy-$\\mathrm {C\\,IV}$ absorber pairs at $z>5$ .",
"Here we summarize their properties; where necessary we have converted to our simulation's cosmology.",
"Target 1 in confirms a galaxy with $M_{\\rm UV}=-20.66\\pm 0.05$ , $L_{\\mathrm {Ly}\\alpha } = 8.01\\pm 0.55\\times 10^{42}{\\rm erg}\\ {\\rm s}^{-1}$ lying at a transverse separation of 311.4 pkpc from the weak $\\mathrm {C\\,IV}$ absorber identified as System 10 in Table A3 of .",
"search around two strong absorbers and identify a candidate neighboring galaxy as a Ly$\\alpha $ emitter (LAE) 103027+052419, which has a tranverse separation from the $\\mathrm {C\\,IV}$ absorber of 212.3 pkpc .",
"Its UV continuum luminosity is $M_{\\rm UV}=-20.65\\pm 0.52$ .",
"Finally, identified three LAE candidates around four strong $\\mathrm {C\\,IV}$ absorbers.",
"Two of these were not detected in more recent measurements using an integral field unit (IFU) spectrograph (Díaz et al.",
"2020, in press.",
"), but the candidate near the absorber at $z_\\mathrm {C\\,IV}=4.866$ has not yet been followed up using other instrumentation, so for the present we assume it to be genuine.",
"Its transverse separation from the absorber is 210 pkpc, and its continuum and Ly$\\alpha $ line luminosities are $M_{\\rm UV}=-18.72\\pm 0.19$ , and $L_{\\mathrm {Ly}\\alpha } = 2.60\\pm 0.53\\times 10^{42}{\\rm erg}\\ {\\rm s}^{-1}$ .",
"In total, these efforts have surveyed the environments near six strong $\\mathrm {C\\,IV}$ absorbers and identified three confirmed or candidate neighboring galaxies.",
"Two of these lie within a transverse impact parameter of 250 pkpc of a strong absorber; their luminosities are indicated using diamonds in Figures REF and REF .",
"Are these two neighbors expected?",
"We cannot compare these statistics directly to our predictions owing to the fact that our simulation volume is not large enough to form galaxies brighter than $M_{\\rm UV}=-19$ at $z>5$ .",
"However, given that $\\Delta $ varies slowly with luminosity, we may simply assume a fiducial $\\Delta = 40$ (bottom panel of Figure REF ).",
"Integrating the LF of , we find that the mean comoving space density at $z=5.75$ of galaxies brighter than $M_{\\rm UV}=(-19,-20)$ is $(9.25,2.93)\\times 10^{-4}{\\rm Mpc}^{-3}$ .",
"We would therefore expect an average of 0.74 $(\\Delta /40)$ neighboring galaxies brighter than $M_{\\rm UV}= -19$ within 250 pkpc of each strong absorbers, or 4.44 galaxies about the six absorbers for which positive detections in follow-up surveys have been published to date.",
"This order-of-magnitude level of agreement begs for improved constraints, which will be provided in the long term by observations using JWST and in the short term by surveys with IFUs."
],
[
"Detecting Hosts in Lyman-$\\alpha $",
"In Fig.",
"REF , we show that, irrespective of the choice of UVB or minimum $\\mathrm {C\\,IV}$ column density, the neighboring galaxies of strong $\\mathrm {C\\,IV}$ absorbers are expected to be predominantly faint LAEs.",
"The predicted number of neighboring LAEs is sensitive to the absorber column and UVB model in the same ways that emerge from Fig.",
"REF .",
"Comparing the solid blue and short-dashed red curves reveals that the neighboring galaxies of stronger absorbers are brighter.",
"Likewise, for a fixed column density, the density-bounded UVB predicts overall fainter galaxies because those absorbers are in this case associated with less rare systems.",
"As in the case of stellar continuum detections, we find in the bottom panel of Fig.",
"REF that the ratio $\\Delta $ of the predicted number density of LAEs that lie within 100 pkpc of an absorber to the field LF is in the range of 100–500.",
"In the case of the DB model and strong absorbers, $\\Delta $ increases from $\\approx 50$ at the faintest luminosities currently probed ($L_{\\mathrm {Ly}\\alpha }=10^{41} {\\rm erg}\\ {\\rm s}^{-1}$ ) to $\\approx 200$ at the brightest luminosities captured by our simulation.",
"Whether $\\Delta $ continues to grow at higher luminosities can only be established by simulations subtending larger cosmological volumes.",
"It decreases with absorber strength and is overall higher in the IB model.",
"Finally, we confirm in Figure REF that a wider search area readily turns up more neighboring galaxies.",
"In particular, a single deep VLT/MUSE pointing that probes to $10^{41.5}{\\rm erg}\\ {\\rm s}^{-1}$ should uncover roughly three neighboring LAEs per strong $\\mathrm {C\\,IV}$ absorber at $z=5.75$ ."
],
[
"Discussion: The Possible $\\mathrm {C\\,IV}$ -LAE Association",
"Do strong $\\mathrm {C\\,IV}$ absorbers prefer the company of faint galaxies or bright ones?",
"While only a few candidate neighboring galaxies are currently identified in the stellar continuum within 300 pkpc of strong $\\mathrm {C\\,IV}$ absorbers at $z>5$ , an intriguing result from was the suggestion that, on scales of 10–20 comoving $h^{-1}{\\rm Mpc}$ , the surface density of (bright) LBGs is systematically low in the vicinity of strong $\\mathrm {C\\,IV}$ absorbers while the surface density of LAEs is systematically high.",
"Given that LAEs are fainter in the stellar continuum than LBGs, they speculated that $\\mathrm {C\\,IV}$ absorbers may arise preferentially in the vicinity of faint galaxies rather than bright ones.",
"This is not expected.",
"Figures REF –REF all indicate that the environments of strong $\\mathrm {C\\,IV}$ absorbers are expected to be more overdense in brighter galaxies, the opposite to what infer.",
"Subject to the (significant) caveat that our simulation does not sample bright galaxies or large spatial scales adequately to address the observations directly, we may speculate qualitatively as to what the apparent disagreement suggests.",
"One interpretation is that the UVB's spatial fluctuations are in reality much stronger than our model predicts.",
"For example, we assume for simplicity that all galaxies have the same ionizing escape fraction $f_{\\rm esc}(z)$ .",
"If, instead, $f_{\\rm esc}$ decreases with mass [2], , , , then the dependence of overdensity on luminosity could flatten, strengthening the UVB amplitude and $\\mathrm {C\\,IV}$ abundance in voids.",
"However, it is not clear that the dependence could be reversed entirely given the tendency for faint galaxies to cluster about bright ones.",
"An alternative explanation invokes a coincidence whereby $\\mathrm {C\\,IV}$ absorbers and LAEs both arise preferentially in voids, though not for the same reason.",
"Because $\\mathrm {C\\,IV}$ is sensitive to $\\mathrm {He\\,II}$ -ionizing photons, the portion of the UVB that regulates the $\\mathrm {C\\,III}/\\mathrm {C\\,IV}$ ratio remains highly inhomogeneous until $\\mathrm {He\\,II}$ reionization completes ($z<3$ ; , ).",
"At earlier times, it is likely that $\\mathrm {He\\,II}$ reionization is characterized by an outside-in topology such that overdensities are more opaque at high energies than mean-density regions or voids.",
"This topology is often seen in numerical simulations of reionization , , and results from the tendency for ionization fronts to “leak\" out of overdensities into voids.",
"It is in fact predicted by our simulations: at $z=5.75$ , when the volume ionized fraction for $\\mathrm {He\\,III}$ is 50%, the ratio of the volume-averaged to the mass-averaged $\\mathrm {He\\,III}$ fractions is 1.5, clearly indicating that voids are more transparent to $\\mathrm {He\\,II}$ -ionizing photons than overdensities.",
"If this topology is correct, then the $\\mathrm {C\\,III}/\\mathrm {C\\,IV}$ ratio will inevitably be higher in overdense regions where LBGs are found.",
"If, furthermore, $\\mathrm {C\\,III}/\\mathrm {C\\,IV}$ increases more rapidly with overdensity than metallicity, then an anti-correlation between strong $\\mathrm {C\\,IV}$ absorbers and LBGs is expected.",
"In order for this phenomenology to predict an association between $\\mathrm {C\\,IV}$ absorbers and LAEs, it is necessary that the Ly$\\alpha $ EW decreases with increasing overdensity.",
"show that this could follow from quenching processes that occur predominantly in overdensities.",
"Moreover, it is directly observed at $z\\sim 3$ , where clustering measurements indicate that LBGs with Ly$\\alpha $ in emission reside preferentially in the outskirts of overdensities and in the field whereas LBGs that reside in overdensities have Lyman-$\\alpha $ preferentially in absorption .",
"Other studies have found that narrowband-selected LAEs are less clustered than LBGs throughout $3 < z < 7$ , .",
"Likewise, studies of individual protoclusters at $3 < z < 6$ suggest that cluster members have systematically weaker Ly$\\alpha $ emission than field galaxies of comparable luminosity (, ; see, however, , who did not find this in the case of PC 217.96+32.3, which is at $z=3.786$ ).",
"Similarly, note that the large-scale distribution of LAEs in the BOSS1441 field shows a broad central core that is otherwise unexpected for a protocluster.",
"A tendency for LAEs to avoid overdensities may also contribute to the result reported by , who found an underdensity of LAEs centered about a sightline where the LAF is unusually optically thick at $z=5.7$ .",
"They attributed the LAE underdensity to the presence of a large-scale void based on modeling in which the Ly$\\alpha $ equivalent width was assumed to have no explicit environmental dependence .",
"This conclusion received subsequent support from an analogous survey that uncovered a large-scale deficit of LBGs at the same redshift along the sightline .",
"The emerging result that, whereas regions where the LAF is unusually optically thick are associated with large-scale overdensities at $z=$ 2–3 , they instead trace voids at $z>5$ , may encode key insight into how galaxies drive the UVB's evolving spatial inhomogeneity.",
"For the present discussion, however, the sightline may provide only ambiguous support for the hypothesis that LAEs selectively occupy voids.",
"In summary, if we assume that LAEs avoid overdensities at $z\\sim 6$ , then they could be physically associated with strong $\\mathrm {C\\,IV}$ absorbers owing to a coincidence: During the outside-in stage of $\\mathrm {He\\,II}$ reionization, faint galaxies that cluster about LBGs are not visible as LAEs, and their CGM is not visible in $\\mathrm {C\\,IV}$ .",
"By contrast, voids host more of the LAEs and enjoy higher $\\mathrm {He\\,III}$ and lower $\\mathrm {C\\,III}/\\mathrm {C\\,IV}$ ratios owing to their spectrally harder UVB."
],
[
"Summary",
"We use the Technicolor Dawn simulations to study how systematic surveys for the neighboring galaxies of strong metal absorbers can constrain the details of the galaxy-driven reionization hypothesis.",
"We reduce the mass flux in star formation-driven outflows by 37% with respect to in order to improve agreement with observations of the galaxy stellar mass function; this reduction is within the theoretical uncertainty quoted by .",
"By adopting an empirical relationship between stellar mass and Ly$\\alpha $ equivalent width, we also recover the observed Ly$\\alpha $ LF at $z=5.75$ , suggesting that the model predicts a realistic $M_*-M_{\\rm UV}$ relationship.",
"Adjusting $f_{\\rm esc}(z)$ down then enables excellent agreement with observations of the history of reionization and of $T_{\\mathrm {Ly}\\alpha }(z)$ .",
"These calibrations yield satisfactory out-of-the-box agreement with the observed $\\mathrm {Si\\,IV}$ CDD at $z=5.75$ , but the observed $\\mathrm {C\\,IV}$ CDD remains underproduced.",
"Comparisons with joint observations of $\\mathrm {Si\\,II}, \\mathrm {Si\\,IV}, \\mathrm {C\\,II}$ , and $\\mathrm {C\\,IV}$ indicate that the lingering $\\mathrm {C\\,IV}$ discrepancy in Fig.",
"REF cannot be eliminated by re-scaling the overall metal yields or stellar mass density.",
"Nonetheless, scaling up the carbon yield by a factor of $1.75\\times $ simultaneously eliminates disagreements with observations of $\\mathrm {C\\,II}/\\mathrm {Si\\,II}$ equivalent width ratios and of the overall $\\mathrm {C\\,II}$ line incidence.",
"It also alleviates but does not eliminate the $\\mathrm {C\\,IV}$ deficit.",
"As an alternative, we replace in post-processing the traditional ionization-bounded escape scenario in which $f_{\\rm esc}$ is energy-independent with a density-bounded scenario in which $f_{\\rm esc}$ increases with energy.",
"In a somewhat extreme case where $f_{\\rm esc}>80\\%$ for energies above 4 Ryd, we find significantly more $\\mathrm {C\\,IV}$ without overproducing $\\mathrm {Si\\,IV}$ although in this case the $\\mathrm {C\\,II}$ deficit is exacerbated.",
"Having established that our underlying model for galaxy growth and reionization is realistic even if $\\mathrm {C\\,IV}$ remains enigmatic, we turn to the spatial association between galaxies and absorbers and find: Galaxies with $M_{\\rm UV}<-15$ are positively-correlated with $\\mathrm {C\\,IV}$ absorbers out to distances of at least 300 pkpc.",
"The correlation strength increases independently with $\\mathrm {C\\,IV}$ column density and with $M_{\\rm UV}$ and $L_{{\\rm Ly}\\alpha }$ .",
"For both stellar continuum and Ly$\\alpha $ surveys, the mean expected number of neighbors is larger for stronger absorbers, and at fixed absorber strength it is larger under the assumption of the IB escape model because more galaxies are required to create a sufficiently hard local ionizing background.",
"The mean overdensity for galaxies within a few magnitudes of $L_*$ falls from $\\approx $ 200–300 within 100 pkpc of a strong absorber to $\\approx $ 40–60 within 250 pkpc.",
"The most abundant population of neighboring galaxies is too faint to be accessible to HST, but integral field units can already detect more than one galaxy per absorber in Ly$\\alpha $ .",
"Detection will become routine with JWST.",
"We propose that, if the physical association between LAEs and strong $\\mathrm {C\\,IV}$ absorbers at $z>5$ is real, then it could owe to a coincidence rather than to the tendency for LAEs to dominate the UVB.",
"In this scenario, $\\mathrm {He\\,II}$ reionization is more advanced in voids than in overdensities, with the result that the CGM ionization state favors $\\mathrm {C\\,IV}$ in voids and lower ionization states in overdensities.",
"The observed tendency for Ly$\\alpha $ equivalent widths to decrease with overdensity then leads naturally to a strong but coincidental association between LAEs and strong $\\mathrm {C\\,IV}$ absorbers.",
"Our results indicate that it would be interesting to survey the galaxy environments of high-redshift $\\mathrm {Si\\,IV}$ absorbers for two reasons.",
"First, their overall abundance is well-reproduced and relatively insensitive to the choice of escape fraction model (Fig.",
"REF ).",
"Second, $\\mathrm {Si\\,IV}$ has a lower ionization potential than $\\mathrm {C\\,IV}$ .",
"As such, it is a more direct probe than $\\mathrm {C\\,IV}$ of the contribution that galaxies made to the latter stages of $\\mathrm {H\\,I}$ reionization.",
"Our work leaves a number of questions unanswered, among them: How does $f_{\\rm esc}$ vary with frequency?",
"How does this dependence influence the metagalactic UVB?",
"If calibrating our outflow model and $f_{\\rm esc}(z)$ to observations of the galaxy stellar mass function and the mean transmission in the LAF leads naturally to good agreement with the $\\mathrm {Si\\,IV}$ CDD, does the stubborn discrepancy with $\\mathrm {C\\,IV}$ observations imply problems with the metal yield model, binary stellar evolution effects, the relative contribution of AGN, or $f_{\\rm esc}$ ?",
"Does the dependence of galaxy overdensity on luminosity turn over or continue to grow to the higher luminosities that our limited simulation volume does not capture?",
"Do the topology of $\\mathrm {He\\,II}$ reionization at $z>5$ and the dependence of EW$_{{\\rm Ly}\\alpha }$ equivalenth width on overdensity predict the association between LAEs and strong $\\mathrm {C\\,IV}$ absorbers that was suggested by ?"
],
[
"Acknowledgements",
"Some simulations contributing to this work were run on NMSU's discovery cluster; for technical advice and support we thank the NMSU ICT department.",
"Other calculations used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562; we thank M. Tatineni for technical support with those efforts.",
"We also thank V. Eijkhout for his assistance with code optimization, which was made possible through the XSEDE Extended Collaborative Support Service (ECSS) program.",
"KF thanks G. Becker, T. Suarez Noguez, E. Ryan-Weber, C. Steidel, X.-W. Chen, M. Prescott, and C. Doughty for helpful conversations and encouragement.",
"We thank V. D'Odorico for sharing her $\\mathrm {Si\\,IV}$ measurements.",
"KF also thanks the University of Wisconsin Astronomy Department for hosting him during July 2019.",
"We thank the anonymous referee for many helpful suggestions that improved the paper.",
"Our work made use of the WebPlotDigitizer tool (https://automeris.io/WebPlotDigitizer), for which we thank A. Rohatgi, as well as E. L. Wright's online cosmology calculator .",
"This research would have been quite unthinkable without the NASA Astrophysics Data System and the arXiv eprint service.",
"The Cosmic Dawn Center is funded by the Danish National Research Foundation.",
"Kristian Finlator https://orcid.org/0000-0002-0496-1656 C. Gonzalo Díaz https://orcid.org/0000-0001-5146-1358 Zheng Cai https://orcid.org/0000-0001-8467-6478"
]
] | 2001.03498 |
[
[
"Remarks on the notion of homo-derivations"
],
[
"Abstract The purpose of this paper is to study the (different) notions of homo-derivations.",
"These are additive mappings $f$ of a ring $R$ that also fulfill the identity \\[ f(xy)=f(x)y+xf(y)+f(x)f(y) \\qquad \\left(x, y\\in R\\right), \\] or (in case of the other notion) the system of equations \\[ f(xy)=f(x)f(y)\\] \\[f(xy)=f(x)y+xf(y) \\qquad \\left(x, y\\in R\\right).\\] Our primary aim is to investigate the above equations without additivity as well as the following Pexiderized equation \\[ f(xy)=h(x)h(y)+xk(y)+k(x)y.",
"\\] The obtained results show that under rather mild assumptions homo-derivations can be fully characterized, even without the additivity assumption."
],
[
"Introduction and preliminaries",
"The main aim of this paper is to present some characterization theorems concerning homomorphisms, derivations and also homo-derivations.",
"Thus, at first, we list some notions and preliminary results that will be used in the sequel.",
"All of these statements and definitions can be found in Kuczma [7] and in Zariski–Samuel [13] and also in Kharchenko [6]."
],
[
"Homomorphisms and derivations",
"Definition 1 Let $P$ and $Q$ be (not necessarily unital) rings.",
"A function $f\\colon P\\rightarrow Q$ is called a homomorphism (of $P$ into $Q$ ) if it is additive, i.e.",
"$f(x+y)=f(x)+f(y)\\quad \\left(x, y\\in P\\right)$ and also $f(xy)=f(x)f(y)\\qquad \\left(x, y\\in P\\right)$ holds.",
"If moreover, $f$ is one-to-one, then $f$ is called a monomorphism.",
"If $f$ is onto, then $f$ is called an epimorphism.",
"A homomorphism which is a monomorphism and an epimorphism is called an isomorphism.",
"In case $P=Q$ , the function $f$ is termed to be an endomorphism.",
"Definition 2 Let $Q$ be a (not necessarily unital) ring and let $P$ be a subring of $Q$ .",
"A function $f\\colon P\\rightarrow Q$ is called a derivation if it is additive and also satisfies the so-called Leibniz rule, i.e.",
"equation $f(xy)=f(x)y+xf(y)\\quad \\left(x, y\\in P\\right).$ Remark Let $Q$ be a ring and let $P$ be a subring of $Q$ .",
"Functions $f\\colon P\\rightarrow Q$ fulfilling the Leibniz rule only, will be termed Leibniz functions.",
"Among derivations one can single out so-called inner derivations, similarly as in the case of automorphisms.",
"Definition 3 Let $R$ be a ring and $b\\in R$ , then the mapping $\\mathrm {ad}_{b}\\colon R\\rightarrow R$ defined by $\\mathrm {ad}_{b}(x)=\\left[x, b\\right]=xb-bx\\qquad \\left(x\\in R\\right)$ is a derivation.",
"A derivation $f\\colon R\\rightarrow R$ is termed to be an inner derivation if there is a $b\\in R$ so that $f=\\mathrm {ad}_{b}$ .",
"We say that a derivation is an outer derivation if it is not inner.",
"An another fundamental example for derivations is the following.",
"Remark Let $\\mathbb {F}$ be a field, and let in the above definition $P=Q=\\mathbb {F}[x]$ be the ring of polynomials with coefficients from $\\mathbb {F}$ .",
"For a polynomial $p\\in \\mathbb {F}[x]$ , $p(x)=\\sum _{k=0}^{n}a_{k}x^{k}$ , define the function $f\\colon \\mathbb {F}[x]\\rightarrow \\mathbb {F}[x]$ as $f(p)=p^{\\prime },$ where $p^{\\prime }(x)=\\sum _{k=1}^{n}ka_{k}x^{k-1}$ is the derivative of the polynomial $p$ .",
"Then the function $f$ clearly fulfills $\\begin{array}{rcl}f(p+q)&=&f(p)+f(q)\\\\f(pq)&=&pf(q)+qf(p)\\end{array}$ for all $p, q\\in \\mathbb {F}[x]$ .",
"Hence $f$ is a derivation.",
"Clearly, commutative rings admit only trivial inner derivations.",
"At the same time, it not so evident whether commutative rings (or fields) do or do not admit nontrivial outer derivations.",
"To answer this problem partially, here we recall Theorem 14.2.1 from Kuczma [7].",
"Theorem 1 Let $\\mathbb {K}$ be a field of characteristic zero, let $\\mathbb {F}$ be a subfield of $\\mathbb {K}$ , let $S$ be an algebraic base of $\\mathbb {K}$ over $\\mathbb {F}$ , if it exists, and let $S=\\emptyset $ otherwise.",
"Let $f\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ be a derivation.",
"Then, for every function $u\\colon S\\rightarrow \\mathbb {K}$ , there exists a unique derivation $g\\colon \\mathbb {K}\\rightarrow \\mathbb {K}$ such that $g \\vert _{\\mathbb {F}}=f$ and $g \\vert _{S}=u$ .",
"In [4], El Sofy introduced the notion of homo-derivations.",
"After that several results appeared where the authors proved commutativity results for the domain of these mappings, see e. g. [1], [2], [8], [11].",
"It is objectionable however that there have not been made attempt to characterize or to compare these notions.",
"One of the main purpose of this work is to clarify these problems.",
"Definition 4 (El Sofy [4]) Let $Q$ be a ring and let $P$ be a subring of $Q$ .",
"A function $f\\colon P\\rightarrow Q$ is called a homo-derivation if it is additive and also satisfies the equation $f(xy)=f(x)y+xf(y)+f(x)f(y)\\quad \\left(x, y\\in P\\right).$ We remark that there can be found some other ways of introducing homo-derivations.",
"Here we present a further definition as it appears in [10].",
"Definition 5 (Mehdi Ebrahimi–Pajoohesh [10]) Let $Q$ be a ring and let $P$ be a subring of $Q$ .",
"A function $f\\colon P\\rightarrow Q$ is called a homo-derivation if it is a homomorphism and also satisfies the Leibniz rule.",
"As we will see in the second section, the notion of exponential polynomials and the so-called Levi-Cività functional equation will play a distinguished role while proving our results.",
"In view of Theorem 10.1 of [12], if $(G, \\cdot )$ is an Abelian group, then any function $f\\colon G\\rightarrow \\mathbb {C}$ satisfying the so-called Levi-Cività functional equation, that is, $f(x\\cdot y)= \\sum _{i=1}^{n}g_{i}(x)h_{i}(y)\\qquad \\left(x, y\\in G\\right)$ for some positive integer $n$ and functions $g_{i}, h_{i}\\colon G\\rightarrow \\mathbb {C}\\; (i=1, \\ldots , n)$ , is an exponential polynomial of order at most $n$ .",
"At the same time, in equation (REF ) not only the function $f$ , but also the mappings $g_{i}, h_{i}\\colon G\\rightarrow \\mathbb {C}\\; (i=1, \\ldots , n)$ are assumed to be unknown.",
"If either the functions $g_{1}, \\ldots , g_{n}$ or the functions $h_{1}, \\ldots , h_{n}$ are linearly dependent, then the number $n$ appearing in equation (REF ) can be reduced and in this case the general solution of the equation can contain arbitrary functions, we shall call this case degenerate.",
"Alternatively, if the functions $h_{1}, \\ldots , h_{n}$ are linearly independent, then $g_{1}, \\ldots , g_{n}$ are linear combinations of the translates of $f$ , hence they are exponential polynomials of order at most $n$ , too.",
"Moreover, they are built up from the same additive and exponential functions as the function $f$ .",
"Roughly speaking this is Theorem 10.4 of [12] which is the following.",
"Theorem 2 Let $G$ be an Abelian group, $n$ be a positive integer and $f, g_{i}, h_{i}\\colon G\\rightarrow \\mathbb {C}\\; (i=1, \\ldots , n)$ be functions so that both the sets $\\left\\lbrace g_{1}, \\ldots , g_{n}\\right\\rbrace $ and $\\left\\lbrace h_{1}, \\ldots , h_{n}\\right\\rbrace $ are linearly independent.",
"The functions $f, g_{i}, h_{i}\\colon G\\rightarrow \\mathbb {C}\\; (i=1, \\ldots , n)$ form a non-degenerate solution of equation (REF ) if and only if there exist positive integers $k, n_{1}, \\ldots , n_{k}$ with $n_{1}+\\cdots +n_{k}=n$ ; there exist different nonzero complex exponentials $m_{1}, \\ldots , m_{k}$ ; for all $j=1, \\ldots , k$ there exists linearly independent sets of complex additive functions $\\left\\lbrace a_{j, 1}, \\ldots , a_{j, n_{j}-1}\\right\\rbrace ;$ there exist polynomials $P_{j}, Q_{i, j}, R_{i, j}\\colon \\mathbb {C}^{n_{j}-1}\\rightarrow \\mathbb {C}$ for all $i=1, \\ldots , n; j=1, \\ldots , k$ in $n_{j}-1$ complex variables and of degree at most $n_{j}-1$ ; so that we have $f(x)= \\sum _{j=1}^{k}P_{j}\\left(a_{j, 1}(x), \\ldots , a_{j, n_{j}-1}(x)\\right)m_{j}(x)$ $g_{i}(x)= \\sum _{j=1}^{k}Q_{i, j}\\left(a_{j, 1}(x), \\ldots , a_{j, n_{j}-1}(x)\\right)m_{j}(x)$ and $h_{i}(x)= \\sum _{j=1}^{k}R_{i, j}\\left(a_{j, 1}(x), \\ldots , a_{j, n_{j}-1}(x)\\right)m_{j}(x)$ for all $x\\in G$ and $i=1, \\ldots , n$ .",
"Let $G$ be a groupoid and $\\mathbb {F}$ be field.",
"Given $(A_{i, j})_{i, j}\\in {M}_{n\\times n}(\\mathbb {F})$ and $(\\Gamma ^{(k)}_{i, j})_{i, j} \\in {M}_{n\\times n}(\\mathbb {F})$ , in [9] McKiernan studied the following problems.",
"Find all functions $h,f_{i}, g_{i}\\colon G\\rightarrow \\mathbb {F}$ ($i= 1, \\ldots , n$ ) satisfying the equation $h(xy)=\\sum _{i, j=1}^{n} A_{i, j} f_{i}(x)g_{j}(y)\\qquad \\left(x, y\\in G\\right).$ Find all functions $g_{i}\\colon G\\rightarrow \\mathbb {F}$ ($i=1, \\ldots , n$ ) satisfying the system of equations $g_{k}(xy)=\\sum _{i, j=1}^{n}\\Gamma _{i, j}^{(k)}g_{i}(x)g_{j}(y)\\qquad \\left(x, y\\in G, k=1, \\ldots , n\\right)$ The solutions are obtained by showing that the two problems are essentially equivalent, then transforming to a matrix problem and applying one of his earlier results concerning a multiplicative matrix equation.",
"All in all, the main result of [9] is that if $G$ is a (not necessarily commutative) semigroup, $\\mathbb {F}$ is an algebraically closed field with $\\mathrm {char}(F)\\ge (n-1)!$ , $\\det \\left(A_{i, j}\\right)\\ne 0$ , $f_{1}, \\ldots , f_{n}$ are linearly independent, $g_{1}, \\ldots , g_{n}$ are linearly independent, then the solutions of equation (REF ) as well as (REF ) are exponential polynomials.",
"In what follows [12], that is, the statement below will be utilized several times.",
"Theorem 3 Let $G$ be an Abelian group, $\\mathbb {K}$ a field, $X$ a $\\mathbb {K}$ -linear space and ${V}$ a translation invariant linear space of $X$ -valued functions on $G$ .",
"Let $k_{i}$ be nonnegative integers, $n\\ge 1$ , $m_{i}\\colon G\\rightarrow \\mathbb {K}$ different nonzero exponentials, $A_{i}\\colon G^{n_{i}}\\rightarrow X$ symmetric, $k_{i}$ -additive functions and $q_{i}\\colon G\\rightarrow X$ polynomials of degree at most $k_{i}-1$ ($i=1, \\ldots , n$ ).",
"If the function $\\sum _{i=1}^{n}\\left(\\mathrm {diag}(A_{i})+q_{i}\\right)m_{i}$ belongs to ${V}$ , then there exist polynomials $r_{i}\\colon G\\rightarrow X$ of degree at most $k_{i}-1$ such that $\\left(\\mathrm {diag}(A_{i})+r_{i}\\right)m_{i}$ belongs to ${V}$ for $i=1, \\ldots , n$ .",
"From this, with the choice ${V}= \\left\\lbrace 0\\right\\rbrace $ and $X=\\mathbb {K}$ we get the following.",
"Proposition 1 Let $G$ be an Abelian group, $\\mathbb {K}$ be a field and $n$ be a positive integer.",
"Suppose that for each $x\\in G$ $p_{1}(x)m_{1}(x)+\\cdots +p_{n}(x)m_{n}(x)=0$ holds, where $m_{1}, \\ldots , m_{n}\\colon G\\rightarrow \\mathbb {K}$ are different exponentials and $p_{1}, \\ldots , p_{n}\\colon G\\rightarrow \\mathbb {K}$ are (generalized) polynomials.",
"Then for all $i=1, \\ldots , n$ the polynomial $p_{i}$ is identically zero.",
"As the theorem below shows, the notion of homo-derivations (in the sense of El Sofy [4]) can be characterized even without additivity supposition.",
"Theorem 4 Let $P$ and $Q$ be rings such that $P$ is a subring of $Q$ and assume that $\\varepsilon $ is an arbitrary nonzero element of the center of $Q$ .",
"Function $h\\colon P\\rightarrow Q$ fulfills the functional equation $h(xy)= h(x)y+xh(y)+\\varepsilon h(x)h(y)$ for all $x, y\\in P$ if and only if there exists a multiplicative function $m\\colon P\\rightarrow Q$ such that $\\varepsilon h(x)= m(x)-x\\qquad \\left(x\\in P\\right).$ Multiplying equation (REF ) with $\\varepsilon $ leads to $\\varepsilon h(xy)= \\varepsilon h(x)y+x \\varepsilon h(y)+\\varepsilon h(x)\\varepsilon h(y)\\qquad \\left(x, y\\in P\\right).$ If we add $xy$ to both sides of this equation, then $\\varepsilon h(xy)+xy= \\varepsilon h(x)y+x \\varepsilon h(y)+\\varepsilon h(x)\\varepsilon h(y) +xy\\qquad \\left(x, y\\in P\\right),$ follows.",
"Observe however that $\\varepsilon h(x)y+x \\varepsilon h(y)+\\varepsilon h(x)\\varepsilon h(y) +xy=(\\varepsilon h(x)+x)\\cdot \\left(\\varepsilon h(y)+y\\right)\\qquad \\left(x, y\\in P\\right),$ yielding exactly that the mapping $m\\colon P\\rightarrow Q$ defined through $m(x)= \\varepsilon h(x)+x\\qquad \\left(x\\in P\\right)$ is multiplicative.",
"Remark In the case the ring $Q$ is unital, $\\varepsilon =1$ is a nonzero central element of the ring $Q$ .",
"Hence any solution of the above equation can be represented as $h(x)= m(x)-x\\qquad \\left(x\\in P\\right),$ with an appropriate multiplicative function $m$ .",
"As an immediate corollary we get the following.",
"Corollary 1 Let $P$ be a subring of the ring $Q$ and assume that $\\varepsilon $ is an arbitrary nonzero element of the center of $Q$ .",
"The additive mapping $a\\colon P\\rightarrow Q$ fulfills functional equation $a(xy)= a(x)y+xa(y)+\\varepsilon a(x)a(y)$ for all $x, y\\in P$ if and only if there exists a homomorphism $\\varphi \\colon P\\rightarrow Q$ such that $\\varepsilon a(x)=\\varphi (x)-x\\qquad \\left(x\\in P\\right).$ In view of the above result, homo-derivations in the sense of El Sofy [4] on commutative rings can be characterized.",
"Corollary 2 Let $P$ be a subring of the commutative ring $Q$ and $a\\colon P\\rightarrow Q$ be a homo-derivation in the sense of El Sofy [4].",
"Then and only then there exists a homomorphism $\\varphi \\colon P\\rightarrow Q$ such that $a(x)=\\varphi (x)-x\\qquad \\left(x\\in P\\right).$ The proposition below considers the other notion of homo-derivations.",
"Proposition 2 Let $P$ be a subring of the ring $Q$ , the function $f\\colon P\\rightarrow Q$ fulfills the system of equations $\\begin{array}{rcl}f(xy)&=&f(x)f(y)\\\\f(xy)&=&f(x)y+xf(y)\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right)$ if and only if there exists a non-zero constant $\\alpha \\in Q$ such that $\\alpha \\cdot f$ and $f\\cdot \\alpha $ are identically zero.",
"Assume that the function $f\\colon P\\rightarrow Q$ satisfies the above system.",
"Then clearly, functional equation $f(x)f(y)=f(x)y+xf(y)\\qquad \\left(x, y\\in P\\right)$ also holds.",
"After some rearrangement we arrive at $f(x)\\left[f(y)-y\\right]= xf(y)\\qquad \\left(x\\in P\\right).$ Since $f$ must be a Leibniz function, $f(y)-y=0$ cannot hold for all $y\\in P$ .",
"Thus there exists $y^{*}\\in P$ such that $ f(y^{*})-y^{*}\\ne 0$ , from which $f(x)\\left[f(y^{*})-y^{*}\\right]= xf(y^{*})\\qquad \\left(x\\in P\\right).$ In other words, there are constants $\\alpha , \\beta \\in Q$ with $\\alpha \\ne 0$ such that $f(x)\\alpha = x \\beta \\qquad \\left(x\\in P\\right).$ Writing this back to the Leibniz equation $\\beta =0$ follows, yielding that $f\\cdot \\alpha $ is identically zero.",
"Changing the role of $x$ and $y$ in the above argument, $\\alpha \\cdot f \\equiv 0$ also follows.",
"Corollary 3 Let $P$ be a subring of the ring $Q$ , a function $a\\colon P\\rightarrow Q$ is a homo-derivation in the sense of Mehdi Ebrahimi–Pajoohesh [10] if and only if there exists a nonzero constant $\\alpha \\in Q$ such that $\\alpha \\cdot a\\equiv a\\cdot \\alpha \\equiv 0$ .",
"Especially, if $Q$ has no zero-divisors, then $a$ has to be identically zero.",
"In this subsection we determine the solutions $f, h, k\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ of the functional equation $f(xy)=h(x)h(y)+xk(y)+k(x)y\\qquad \\left(x, y\\in \\mathbb {F}\\right).$ From this, the additive solutions of the same equation will follow immediately.",
"Here we suppose that $\\mathbb {K}$ is an algebraically closed field with $\\mathrm {char}(\\mathbb {K})\\ne 2$ and $\\mathbb {F}$ is a subfield of $\\mathbb {K}$.",
"Observe that equation (REF ) is a special Levi-Cività equation.",
"Therefore according to the value $\\dim \\mathrm {lin}\\left(\\mathrm {id}, h, k\\right)$ , we have to distinguish several cases.",
"Clearly, $\\dim \\mathrm {lin}\\left(\\mathrm {id}, h, k\\right)=3$ means that the mappings involved in the right hand side of (REF ) are linearly independent.",
"Thus in the degenerate cases we have $\\dim \\mathrm {lin}\\left(\\mathrm {id}, h, k\\right)<3$ .",
"Firstly, let us assume that $\\dim \\mathrm {lin}\\left(\\mathrm {id}, h, k\\right)=1$ .",
"In this situation there exist constants $\\lambda _{1}, \\lambda _{2}\\in \\mathbb {K}$ such that $h(x)=\\lambda _{1}x\\quad \\text{and}\\quad k(x)=\\lambda _{2}x\\qquad \\left(x\\in \\mathbb {F}\\right).$ Proposition 3 Let $\\lambda _{1}, \\lambda _{2}\\in \\mathbb {K}$ be arbitrarily fixed.",
"Function $f\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ fulfills functional equation (REF ) if and only if $f(x)= \\left(\\lambda _{1}^{2}+2\\lambda _{2}\\right)x\\qquad \\left(x\\in \\mathbb {F}\\right).$ In case $h(x)=\\lambda _{1}x\\quad \\text{and}\\quad k(x)=\\lambda _{2}x\\qquad \\left(x\\in \\mathbb {F}\\right),$ our equation reduces to $f(xy)= \\left(\\lambda _{1}^{2}+2\\lambda _{2}\\right)xy\\qquad \\left(x, y\\in \\mathbb {F}\\right),$ from which the results follows immediately.",
"Secondly, assume that $\\dim \\mathrm {lin}\\left(\\mathrm {id}, h, k\\right)=2$ , which can happen in different ways.",
"If $\\left\\lbrace \\mathrm {id}, h\\right\\rbrace $ are linearly dependent, that is $h(x)= \\lambda x\\qquad \\left(x\\in \\mathbb {F}\\right)$ holds with a certain $\\lambda \\in \\mathbb {K}$ , then we have the following.",
"Proposition 4 Let $\\lambda \\in \\mathbb {K}$ be an arbitrary constant.",
"Functions $f, k\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ fulfill the functional equation $f(xy)=\\lambda ^{2}xy+xk(y)+k(x)y\\qquad \\left(x, y\\in \\mathbb {F}\\right)$ if and only if there exists a Leibniz function $\\delta \\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ such that $\\begin{array}{rcl}k(x)&=&k(1)x+\\delta (x)\\\\f(x)&=&\\left(\\lambda ^{2}+ 2k(1)\\right)x+\\delta (x)\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ Under the above assumptions, equation (REF ) with $y=1$ yields that $f(x)= (\\lambda ^{2}+k(1))x+k(x)\\qquad \\left(x\\in \\mathbb {F}\\right).$ Writing this back into our equation, we get that $k(xy)+(\\lambda ^{2}+k(1))xy= xk(y)+k(x)y +\\lambda ^{2}xy\\qquad \\left(x, y\\in \\mathbb {F}\\right).$ This means that the function $k\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ fulfills $k(xy)+k(1)xy= xk(y)+k(x)y\\qquad \\left(x, y\\in \\mathbb {F}\\right).$ Thus, there exists a Leibniz function $\\delta \\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ such that $\\begin{array}{rcl}k(x)&=&k(1)x+\\delta (x)\\\\f(x)&=&\\left(\\lambda ^{2}+2k(1)\\right)x+\\delta (x)\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ Corollary 4 Let $\\lambda \\in \\mathbb {K}$ be an arbitrary constant.",
"The additive functions $f, k\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ fulfill the functional equation $f(xy)=\\lambda ^{2}xy+xk(y)+k(x)y\\qquad \\left(x, y\\in \\mathbb {F}\\right)$ if and only if there exists a derivation $d \\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ such that $\\begin{array}{rcl}k(x)&=&k(1)x+d(x)\\\\f(x)&=&\\left(\\lambda ^{2}+ 2k(1)\\right)x+d(x)\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ Our second case is when $\\left\\lbrace \\mathrm {id}, k\\right\\rbrace $ are linearly dependent, that is if $k(x)= \\lambda x\\qquad \\left(x\\in \\mathbb {F}\\right)$ with a certain $\\lambda \\in \\mathbb {K}$ .",
"Proposition 5 Let $\\lambda \\in \\mathbb {F}$ be arbitrarily fixed.",
"Functions $f, h\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ fulfill the functional equation $f(xy)=h(x)h(y)+2\\lambda xy\\qquad \\left(x, y\\in \\mathbb {F}\\right)$ if and only if there exists a multiplicative function $m\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ such that $\\begin{array}{rcl}h(x)&=&h(1)m(x)\\\\f(x)&=&h(1)^{2}m(x)+2\\lambda x\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ Define the function $\\widetilde{f}\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ through $\\widetilde{f}(x)=f(x)-2\\lambda x\\qquad \\left(x\\in \\mathbb {F}\\right)$ to deduce that $\\widetilde{f}(xy)= h(x)h(y)\\qquad \\left(x, y\\in \\mathbb {F}\\right).$ This identity with $y=1$ implies that $\\widetilde{f}(x)=h(1)h(x)\\qquad \\left(x\\in \\mathbb {F}\\right).$ Therefore, either $h(1)=0$ from which $f(x)=2\\lambda x\\qquad \\left(x\\in \\mathbb {F}\\right)$ and $h\\equiv 0$ follows; or $h(1)\\ne 0$ from which we get that $\\dfrac{h(xy)}{h(1)}= \\dfrac{h(x)}{h(1)}\\cdot \\dfrac{h(y)}{h(1)}\\qquad \\left(x, y\\in \\mathbb {F}\\right).$ All in all, there exists a multiplicative function $m\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ such that $h(x)=h(1)m(x)\\qquad \\left(x\\in \\mathbb {F}\\right).$ From this we also obtain that $f(x)=h(1)^{2}m(x)+2\\lambda x\\qquad \\left(x\\in \\mathbb {F}\\right).$ Remark Similarly, as previously, the additive solutions of the above equation are of the form $\\begin{array}{rcl}h(x)&=&h(1)\\varphi (x)\\\\f(x)&=&h(1)^{2}\\varphi (x)+2\\lambda x\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right),$ with a certain homomorphism $\\varphi \\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ .",
"Finally, the last possibility is that $\\left\\lbrace h, k\\right\\rbrace $ is linearly dependent.",
"In this case there are constants $\\lambda _{1}, \\lambda _{2}\\in \\mathbb {K}$ not vanishing simultaneously such that $\\lambda _{1}k(x)+\\lambda _{2}h(x)=0$ for all $x\\in \\mathbb {F}$ .",
"Again we have the following alternative.",
"Either $\\lambda _{2}=0$ and then $k\\equiv 0$ .",
"In this case (REF ) is $f(xy)= h(x)h(y)\\qquad \\left(x, y\\in \\mathbb {F}\\right).$ Using Proposition REF we finally get that there exists a multiplicative function $m\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ such that $\\begin{array}{rcl}h(x)&=&h(1)m(x)\\\\f(x)&=&h(1)^{2}m(x)\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ Or $\\lambda _{2}\\ne 0$ and then there exists a constant $\\lambda \\in \\mathbb {K}$ such that $h(x)=\\lambda k(x)\\qquad \\left(x\\in \\mathbb {F}\\right).$ In this case equation (REF ) is of the form $f(xy)=k(x)y+xk(y)+\\lambda ^{2}k(x)k(y)\\qquad \\left(x, y\\in \\mathbb {F}\\right).$ Proposition 6 Let $\\lambda \\in \\mathbb {K}$ .",
"Functions $f, k\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ fulfill the functional equation $f(xy)=xk(y)+k(x)y+\\lambda ^{2}k(x)k(y)\\qquad \\left(x, y\\in \\mathbb {F}\\right)$ if and only if in case $\\lambda =0$ and there exists a Leibniz function $\\delta \\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ such that $\\begin{array}{rcl}k(x)&=&k(1)x+\\delta (x)\\\\f(x)&=&2k(1)x+\\delta (x)\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right),$ in case $\\lambda \\ne 0$ either there exists a constant $\\gamma \\in \\mathbb {K}$ such that $\\begin{array}{rcl}k(x)&=&\\gamma x\\\\f(x)&=&\\left(\\lambda ^{2}\\gamma ^{2}+2\\gamma \\right) x\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ or there exists a multiplicative function $m\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ and a constant $\\gamma \\in \\mathbb {K}$ such that $\\begin{array}{rcl}k(x)&=& -\\dfrac{1}{\\lambda ^{2}}x+\\gamma m(x)\\\\[2mm]f(x)&=&-\\dfrac{1}{\\lambda ^2}x+\\gamma ^{2}\\lambda ^{2}m(x)\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ Observe, that our equation with $y=1$ immediately yields that $f(x)= (1+\\lambda ^{2}k(1))k(x)+k(1)x\\qquad \\left(x\\in \\mathbb {F}\\right).$ If $\\lambda =0$ , then from this we get that the mapping $\\tilde{k}$ defined on $\\mathbb {F}$ by $\\tilde{k}(x)=k(x)-k(1)x\\qquad \\left(x\\in \\mathbb {F}\\right),$ is a Leibniz function.",
"If $\\lambda \\ne 0$ , then define the function $h\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ by $h(x)= x+\\dfrac{\\lambda ^{2}}{2}k(x)\\qquad \\left(x\\in \\mathbb {F}\\right)$ to derive $f(xy)= k(x)h(y)+h(x)k(y)\\qquad \\left(x, y\\in \\mathbb {F}\\right),$ that is the same sine-type equation as in the proof of Proposition REF .",
"Similarly as there, a careful adaptation shows that alternative (a) corresponds to the case when $k$ and $h$ are linearly dependent, while alternative (b) corresponds to the case when $k$ and $h$ are linearly independent.",
"Remark If we would like to describe the additive solutions of the above functional equation, then the mapping $\\delta $ should be replaced by a derivation $d\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ , and the mapping $m$ should be replaced by a homomorphism $\\varphi \\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ .",
"In this subsection we investigate the so-called non-degenerate case.",
"More precisely, the problem to be studied is the following.",
"Let $\\mathbb {K}$ be an algebraically closed field with $\\mathrm {char}(\\mathbb {K})\\ne 2$ zero and $\\mathbb {F}$ be a subfield of $\\mathbb {K}$, and $f, h, k\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ be functions so that the system $\\left\\lbrace \\mathrm {id}, h, k\\right\\rbrace $ is linearly independent.",
"In what follows we determine the solutions of the functional equation $f(xy)=h(x)h(y)+xk(y)+k(x)y\\qquad \\left(x, y\\in \\mathbb {F}\\right).$ Using the results of Székelyhidi [12] and McKiernan [9] delineated in the first section, we derive immediately that the solutions $f, h, k\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ of the above equation are exponential polynomials of degree at most two.",
"Depending on this degree we have three different possibilities, see pages 89-92 of [12] where the description of the functional equation $f(xy)= g_{1}(x)h_{1}(y)+g_{2}(x)h_{2}(y)+g_{3}(x)h_{3}(y)$ can be found.",
"This obviously covers our equation with the choice $\\begin{array}{rcl}g_{1}(x)=h_{1}(x)&=&h(x)\\\\g_{2}(x)&=&x\\\\h_{2}(x)&=&k(x)\\\\g_{3}(x)&=&k(x)\\\\h_{3}(x)&=&x\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ The first possibility is that there exist different nonzero multiplicative functions $m_{1}, m_{2}, m_{3} \\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ and constants $\\alpha _{i}, \\beta _{j}^{(i)}, \\gamma _{i}^{(j)}\\in \\mathbb {K}$ , $i, j=1, 2, 3$ such that $\\begin{array}{rcl}f(x)&=&\\alpha _{1}m_{1}(x)+\\alpha _{2}m_{2}(x)+\\alpha _{3}m_{3}(x)\\\\[2mm]g_{i}(x)&=&\\beta ^{(i)}_{1}m_{1}(x)+\\beta ^{(i)}_{2}m_{2}(x)+\\beta ^{(i)}_{3}m_{3}(x)\\\\[2mm]h_{i}(x)&=&\\gamma ^{(i)}_{1}m_{1}(x)+\\gamma ^{(i)}_{2}m_{2}(x)+\\gamma ^{(i)}_{3}m_{3}(x)\\\\\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ Condition $g_{1}=h_{1}$ implies that $\\beta _{j}^{(1)}= \\gamma _{j}^{(1)}\\qquad \\left(j=1, 2, 3\\right).$ Similarly, from $h_{2}=g_{3}$ we obtain that $\\beta ^{(3)}_{j} =\\gamma _{j}^{(2)}\\qquad \\left(j=1, 2, 3\\right).$ Finally, from $g_{2}(x)=h_{3}(x)=x\\qquad \\left(x\\in \\mathbb {F}\\right)$ we derive that one of the multiplicative functions $m_{1}, m_{2}, m_{3}$ is the identity, say $m_{1}$ and $\\begin{array}{rcl}\\beta _{2}^{(2)}=\\beta _{3}^{(2)}=0& \\; & \\beta _{1}^{(2)}=1\\\\[2mm]\\gamma _{2}^{(3)}=\\gamma _{3}^{(3)}=0& \\; & \\gamma _{1}^{(3)}=1.\\end{array}$ Using this and our functional equation, we get that for the above constants $\\begin{pmatrix}\\beta _{1}^{(1)} & 1 & \\beta _{1}^{(3)}\\\\\\beta _{2}^{(1)} & 0& \\beta _{2}^{(3)}\\\\\\beta _{3}^{(1)} & 0 & \\beta _{3}^{(3)}\\end{pmatrix}\\cdot \\begin{pmatrix}\\beta _{1}^{(1)} & \\beta _{2}^{(1)} & \\beta _{3}^{(1)}\\\\\\beta _{1}^{(3)} & \\beta _{2}^{(3)}& \\beta _{3}^{(3)}\\\\1 & 0 & 0\\end{pmatrix}=\\begin{pmatrix}\\alpha _{1} & 0& 0\\\\0& \\alpha _{2} & 0\\\\0& 0& \\alpha _{3}\\end{pmatrix}$ has to hold.",
"Especially, $\\beta _{2}^{(1)}\\cdot \\beta _{3}^{(1)}=0$ , which yields that the coefficient of $m_{2}$ or that of $m_{3}$ is zero.",
"All in all, there exists a multiplicative function $m\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ and constants such that $\\begin{array}{rcl}f(x)&=&(\\beta _{1}^{2}+2\\gamma _{1})x+\\beta _{2}^{2}m(x)\\\\h(x)&=&\\beta _{1}x+\\beta _{2}m(x)\\\\k(x)&=&\\gamma _{1}x+\\gamma _{2}m(x)\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ In this case however the functions $\\mathrm {id}, h$ and $k$ span a linear space of dimension at most two.",
"Notice that we are interested in the case $\\dim \\mathrm {lin}\\left(\\mathrm {id}, h, k\\right)=3$ .",
"Therefore the above type of solutions does not appear.",
"The second possibility is that there exist different multiplicative functions $m_{1}, m_{2}$ and a logarithmic function $l\\colon \\mathbb {F}^{\\times }\\rightarrow \\mathbb {K}$ and constants such that $\\begin{array}{rcl}f(x)&=&\\left(\\alpha _{1}l(x)+\\alpha _{2}\\right)m_{1}(x)+\\alpha _{3}m_{2}(x)\\\\g_{i}&=&\\left(\\beta _{1}^{(i)}l(x)+\\beta _{2}^{(i)}\\right)m_{1}(x)+\\beta _{3}m_{2}(x)\\\\h_{i}&=&\\left(\\gamma _{1}^{(i)}l(x)+\\gamma _{2}^{(i)}\\right)m_{1}(x)+\\gamma _{3}m_{2}(x)\\\\\\end{array}\\qquad \\left(x\\in \\mathbb {F}, i=1, 2, 3\\right).$ In our case $\\begin{array}{rcl}g_{1}(x)=h_{1}(x)&=&h(x)\\\\g_{2}(x)&=&x\\\\h_{2}(x)&=&k(x)\\\\g_{3}(x)&=&k(x)\\\\h_{3}(x)&=&x\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right),$ thus either $m_{1}$ or $m_{2}$ is the identity.",
"If we would have $m_{2}(x)=x\\qquad \\left(x\\in \\mathbb {F}\\right),$ then after comparing the coefficients $\\begin{array}{rcl}f(x)&=&\\alpha _{1}x+\\alpha _{2}m(x)\\\\h(x)&=&\\beta _{1}x+\\beta _{2}m(x)\\\\k(x)&=&\\gamma _{1}x+\\gamma _{2}m(x)\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ would follow with certain constants.",
"Similarly as above, in this case we would have $\\dim \\mathrm {lin}\\left(\\mathrm {id}, h, k\\right)\\le 2$ , contrary to our assumptions.",
"The fact that $m_{1}(x)=x\\qquad \\left(x\\in \\mathbb {F}\\right),$ means that there exists a multiplicative function $m\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ and a logarithmic function $l\\colon \\mathbb {F}^{\\times }\\rightarrow \\mathbb {K}$ and constants such that $\\begin{array}{rcl}f(x)&=&\\left(\\alpha _{1}l(x)+\\alpha _{2}\\right)x+\\alpha _{3}m(x)\\\\[2mm]h(x)&=&\\left(\\beta _{1}l(x)+\\beta _{2}\\right)x+\\beta _{3}m(x)\\\\[2mm]k(x)&=&\\left(\\gamma _{1}l(x)+\\gamma _{2}\\right)x+\\gamma _{3}m(x)\\\\\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ Inserting this back into our equation the system of equations $\\begin{array}{rcl}\\beta _{1}&=&0 \\\\[2mm]\\alpha _{1}&=&\\gamma _{1}\\\\[2mm]\\alpha _{2}&=& \\beta _{2}^{2}+2\\gamma _{2}\\\\[2mm]\\alpha _{3}&=&\\beta _{3}^{2}\\end{array}$ follow, that is, $\\begin{array}{rcl}f(x)&=&\\left(\\gamma _{1}l(x)+\\beta _{2}^{2}+2\\gamma _{2}\\right)x+\\beta _{3}^{2}m(x)\\\\[2mm]h(x)&=&\\beta _{2}x+\\beta _{3}m(x)\\\\[2mm]k(x)&=&\\left(\\gamma _{1}l(x)+\\gamma _{2}\\right)x+\\gamma _{3}m(x)\\\\\\end{array},$ where additionally $\\gamma _{3}+\\beta _{2}\\beta _{3}=0$ also has to hold.",
"To guarantee the system $\\left\\lbrace \\mathrm {id}, h, k\\right\\rbrace $ to be linearly independent, $\\beta _{3}\\ne 0$ and $\\gamma _{1}\\ne 0$ also has to be supposed.",
"The last possibility is that all the involved functions are exponential polynomials of degree two.",
"Since $g_{2}(x)=h_{3}(x)= x\\qquad \\left(x\\in \\mathbb {F}\\right),$ the corresponding multiplicative function is the identity, that is, we have $\\begin{array}{rcl}f(x)=\\displaystyle \\sum _{p, q=1}^{2}\\alpha _{p, q}l_{p}(x)l_{q}(x)x+\\displaystyle \\sum _{p=1}^{2}\\alpha _{p}l_{p}(x)x+\\alpha x\\\\[2mm]h(x)=\\displaystyle \\sum _{p, q=1}^{2}\\beta _{p, q}l_{p}(x)l_{q}(x)x+\\displaystyle \\sum _{p=1}^{2}\\beta _{p}l_{p}(x)x+\\beta x\\\\[2mm]k(x)=\\displaystyle \\sum _{p, q=1}^{2}\\gamma _{p, q}l_{p}(x)l_{q}(x)x+\\displaystyle \\sum _{p=1}^{2}\\gamma _{p}l_{p}(x)x+\\gamma x\\\\\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right)$ with certain constants and with linearly independent logarithmic functions $l_{1}, l_{2}\\colon \\mathbb {F}^{\\times }\\rightarrow \\mathbb {K}$ .",
"Substituting these representations into our equation, firstly $\\alpha _{p, q}=\\beta _{p, q}=\\gamma _{p, q}=0\\qquad \\left(p, q\\in \\left\\lbrace 1, 2\\right\\rbrace \\right)$ can be concluded, that is, in fact we have that $\\begin{array}{rcl}f(x)=\\displaystyle \\sum _{p=1}^{2}\\alpha _{p}l_{p}(x)x+\\alpha x\\\\[2mm]h(x)=\\displaystyle \\sum _{p=1}^{2}\\beta _{p}l_{p}(x)x+\\beta x\\\\[2mm]k(x)=\\displaystyle \\sum _{p=1}^{2}\\gamma _{p}l_{p}(x)x+\\gamma x\\\\\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ Again, the comparison of the coefficients leads to $\\beta _{1}=0\\qquad \\beta _{2}=0\\qquad \\alpha _{1}=\\gamma _{1}\\qquad \\alpha _{2}=\\gamma _{2}\\qquad \\alpha =\\beta ^{2}+2\\gamma ,$ that is, there exist linearly independent logarithmic functions $l_{1}, l_{2}\\colon \\mathbb {F}^{\\times }\\rightarrow \\mathbb {K}$ and constants $\\gamma _{1}, \\gamma _{2}, \\beta , \\gamma \\in \\mathbb {K}$ such that $\\begin{array}{rcl}f(x)&=&\\gamma _{1}l_{1}(x)x+\\gamma _{2}l_{2}(x)x+ (\\beta ^{2}+2\\gamma ) x\\\\[2mm]h(x)&=&\\beta x\\\\[2mm]k(x)&=&\\gamma _{1}l_{1}(x)x+\\gamma _{2}l_{2}(x)x+\\gamma x\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ Observe that in this case $\\mathrm {id}$ and $h$ are linearly dependent, yielding that this possibility cannot occur in our situation.",
"Summing up, the following statement holds true.",
"Theorem 5 Let $f, h, k\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ be functions such that the system $\\left\\lbrace \\mathrm {id}, h, k\\right\\rbrace $ is linearly independent.",
"The functional equation $f(xy)=h(x)h(y)+xk(y)+k(x)y$ is fulfilled for any $x, y\\in \\mathbb {F}$ if and only if there exists a multiplicative function $m\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ and a logarithmic function $l\\colon \\mathbb {F}^{\\times }\\rightarrow \\mathbb {K}$ and constants $\\beta _{2}, \\beta _{3}, \\gamma _{1}, \\gamma _{2}, \\gamma _{3}\\in \\mathbb {K}$ such that $\\begin{array}{rcl}f(x)&=&\\left(\\gamma _{1}l(x)+\\beta _{2}^{2}+2\\gamma _{2}\\right)x+\\beta _{3}^{2}m(x)\\\\[2mm]h(x)&=&\\beta _{2}x+\\beta _{3}m(x)\\\\[2mm]k(x)&=&\\left(\\gamma _{1}l(x)+\\gamma _{2}\\right)x+\\gamma _{3}m(x)\\\\\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right),$ where additionally $\\gamma _{3}+\\beta _{2}\\beta _{3}=0$ , $\\beta _{3}\\ne 0$ and $\\gamma _{1}\\ne 0$ also have to hold."
],
[
"Further interpretations and open questions",
"The primary aim of this paper was to study (different) notions of homo-derivations on fields (with and without additivity supposition) as well as the Pexiderized version of this definition.",
"At the same time, the results obtained here can be restated with the aid of the notion of alien functional equations.",
"This concept was developed by J. Dhombres in the paper [3].",
"However, the interested reader should also consult the survey paper Ger–Sablik [5].",
"Let $X$ and $Y$ be nonempty sets and $E_{1}(f)=0$ and $E_{2}(f)=0$ be two functional equations for the function $f\\colon X\\rightarrow Y$ .",
"We say that equations $E_{1}$ and $E_{2}$ are alien, if any solution $f\\colon X\\rightarrow Y$ of the functional equation $E_{1}(f)+E_{2}(f)=0$ also solves the system $\\begin{array}{rcl}E_{1}(f)&=&0\\\\E_{2}(f)&=&0.\\end{array}$ Furthermore, equations $E_{1}$ and $E_{2}$ are strongly alien, if any pair $f, g\\colon X\\rightarrow Y$ of functions that solves $E_{1}(f)+E_{1}(g)=0$ also yields a solution for $\\begin{array}{rcl}E_{1}(f)&=&0\\\\E_{2}(g)&=&0.\\end{array}$ The following statement shows that the multiplicative Cauchy equation and the Leibniz rule equation are not strongly alien, this is an easy consequence of Theorem REF .",
"Proposition 7 Let $h, k\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ be functions such that the system $\\left\\lbrace \\mathrm {id}, h, k\\right\\rbrace $ is linearly independent.",
"The functional equation $h(xy)+k(xy)=h(x)h(y)+xk(y)+k(x)y$ is fulfilled for any $x, y\\in \\mathbb {F}$ if and only if there exists a multiplicative function $m\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ and a logarithmic function $l\\colon \\mathbb {F}^{\\times }\\rightarrow \\mathbb {K}$ and constants $\\beta , \\gamma \\in \\mathbb {K}$ , $\\beta \\ne 1$ , $\\gamma \\ne 0$ such that $\\begin{array}{rcl}h(x)&=&\\beta x+(1-\\beta )m(x)\\\\[2mm]k(x)&=&\\left(\\gamma l(x)+\\beta -\\beta ^{2}\\right)x+(\\beta ^{2}-\\beta )m(x)\\\\\\end{array}\\qquad \\left(x\\in \\mathbb {F}\\right).$ As the proposition below shows, the multiplicative Cauchy equation and the Leibniz rule equation are alien, cf.",
"Proposition REF and take $\\lambda =\\mu $ in the corollary below.",
"Corollary 5 Let $\\lambda , \\mu \\in \\mathbb {K}$ be arbitrarily fixed constants not vanishing simultaneously.",
"Function $f\\colon \\mathbb {F}\\rightarrow \\mathbb {K}$ fulfills the functional equation $\\lambda \\left[f(xy)-f(x)y-xf(y)\\right]+\\mu \\left[f(xy)-f(x)f(y)\\right]=0\\qquad \\left(x, y\\in \\mathbb {F}\\right)$ if and only if either $\\lambda =0$ and $f$ is multiplicative; or $\\mu =0$ and $f$ is a Leibniz function; or none of them is zero and $f$ is identically zero or $f(x)= \\dfrac{\\mu -\\lambda }{\\mu }\\cdot x\\qquad \\left(x\\in \\mathbb {F}\\right).$ Remark Under the assumptions of the previous corollary, the additive solutions of equation (REF ) are the following.",
"either $\\lambda =0$ and $f$ is a homomorphism; or $\\mu =0$ and $f$ is a derivation; or none of them is zero and $f$ is identically zero or $f(x)= \\dfrac{\\mu -\\lambda }{\\mu }\\cdot x\\qquad \\left(x\\in \\mathbb {F}\\right).$ Open Problems In this paper equation (REF ) was considered under rather mild assumptions on the domain and also on the range.",
"At the same time, while investigating functional equation (REF ) we always assumed that the range of the involved mappings is a commutative, algebraically closed field $\\mathbb {K}$ with $\\mathrm {char}(\\mathbb {K})\\ne 2$ .",
"The reason for this is that our main tools were Theorem REF and the related results of McKiernan [9].",
"Clearly, the above equations can be studied without these assumptions.",
"We remark that in case of the so-called degenerate cases a careful adaptation of the proofs shows that the same results hold true for commutative rings (at some places we have to assume that the range does not have any zero-divisors).",
"Therefore we can formulate the following open questions.",
"The algebraic closedness of the field $\\mathbb {K}$ essential in our results.",
"Nevertheless, if the field $\\mathbb {K}$ is not algebraically closed, then we may take $\\mathrm {algcl}(\\mathbb {K})$ as the extended range.",
"Using our method, $\\mathrm {algcl}(\\mathbb {K})$ -valued solutions can be described and every $\\mathbb {K}$ -valued solution belongs to the above larger solution space.",
"How can these solutions be recognized in the larger solution space?",
"To apply the results of Székelyhidi [12] and McKiernan [9], the assumption $\\mathbb {F}\\subset \\mathbb {K}$ be a field is sufficient, but may not be necessary.",
"Observe that we only need $\\mathbb {F}$ to be a (commutative) subring of $\\mathbb {K}$ .",
"In case the range $Q$ is only a (commutative) ring, then what else should be supposed about $Q$ , to get the same results?",
"It is worth to note that if $Q$ has no zero divisors, then up to isomorphism there exists a unique field of fractions that we may denote by $\\mathbb {K}$ .",
"In this case $\\mathbb {K}$ -valued functions can be considered and if it is needed we should take the algebraic closure of this field (c.f.",
"part (A)).",
"In case $P\\subset Q$ are (commutative) rings and we consider mappings from $P$ to $Q$ , then are there different solutions of (REF ) then presented here?",
"Acknowledgement.",
"Gergely Kiss was supported by the Hungarian National Foundation for Scientific Research, Grant No.",
"K124749 and the Premium Postdoctoral Fellowship of Hungarian Academy of Science.",
"The research of Eszter Gselmann has been carried out with the help of the project 2019-2.1.11-TÉT-2019-00049, which has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the TÉT funding scheme."
]
] | 2001.03439 |
[
[
"Censored Quantile Regression Forest"
],
[
"Abstract Random forests are powerful non-parametric regression method but are severely limited in their usage in the presence of randomly censored observations, and naively applied can exhibit poor predictive performance due to the incurred biases.",
"Based on a local adaptive representation of random forests, we develop its regression adjustment for randomly censored regression quantile models.",
"Regression adjustment is based on a new estimating equation that adapts to censoring and leads to quantile score whenever the data do not exhibit censoring.",
"The proposed procedure named {\\it censored quantile regression forest}, allows us to estimate quantiles of time-to-event without any parametric modeling assumption.",
"We establish its consistency under mild model specifications.",
"Numerical studies showcase a clear advantage of the proposed procedure."
],
[
"Introduction",
"Censored data exists in many different areas.",
"In economics, policies such as minimum wage and minimum transaction fee result in left-censored data.",
"In biomedical study, researchers cannot always observe the time until the occurrence of an event of interest because the time span of the study is limited or the patient withdraws from the experiment, resulting in right-censored data.",
"Classical statistical approaches always assume an underlying model like accelerated failure time model [27], [37], [36], [45], [46], [24].",
"These methods perform well when the model is correctly specified, but quickly break down when the model assumption is wrong or the error distribution is heteroscedastic.",
"Other non-parametric methods [33], [21] and rank-based methods [26] strive to achieve assumption-lean modeling of the mean.",
"Forest algorithms [9], [17], [34], [4] are non-parametric and allow for flexible modeling of covariate interactions.",
"However, it is non-trivial to adapt forest algorithms to censored data.",
"Random survival forests [25] or bagging survival trees [22], [23] rely on building survival trees using survival function as the splitting criterion, and are only applicable to right-censored data.",
"Moreover, any technique developed for uncensored data cannot be easily extended to censoring scenario.",
"For example, generalized random forest can effectively deal with heteroscedastic data, but applying the same technique to heteroscedastic and censored data is non-trivial.",
"In this paper, we propose a novel method that connects the quantile forest algorithms to the censored data problem.",
"This is done by a carefully designed estimating equation.",
"In this way, any technique developed for quantile forests on uncensored data can be seamlessly applied to censored data.",
"One of the promising applications of the introduced method is in the estimation of heterogeneous treatment effects when the response variable is censored.",
"We will show in the experiments that the introduced methods do achieve the best performance on both simulated and real datasets.",
"Especially on one heteroscedastic data, the proposed method is the only working solution among many other forest algorithms, including random survival forest.",
"The proposed method, censored quantile regression forest, is motivated by the observation that random forests actually define a local similarity metric [32], [30], [4] which is essentially a data-driven kernel.",
"Using this kernel, random forests can be rephrased as locally weighted regressions.",
"We will review the regression adjustments for forests in Section ."
],
[
"Related Work",
"In the case of right censoring, most non-parametric recursive partitioning algorithms rely on survival tree or its ensembles.",
"[25] proposed random survival forest (RSF) algorithm in which each tree is built by maximizing the between-node log-rank statistic.",
"However, it is not directly estimating the conditional quantiles but instead estimating the cumulative hazard.",
"[47] proposed the recursively imputed survival trees (RIST) algorithm with the same splitting criterion for each individual tree but different ensemble scheme.",
"Other similar methods relying on different kinds of survival trees were proposed in [19], [40], [12], [28] and [29].",
"All these methods as mentioned above use splitting rules specifically designed for the right censored data, and they all rely on the proportional hazard assumption and cannot reduce to a loss-based method that might ordinarily be used in the situation with no censoring.",
"[35] proposed a tree method based on the inverse probability censoring [38] weighted (IPCW) loss function which reduces to the full data loss function used by CART in the absence of censoring.",
"[23] then extended the IPCW idea and proposed a forest-type method in which each tree is trained on resampled observations according to inverse probability censoring weights.",
"However, the censored data always get weights zero and hence only uncensored observations will be resampled.",
"As pointed out by [38], the inverse probability weighted estimators are inefficient because of their failure to utilize all the information available on observations with missing or partially missing data."
],
[
"Regression Adjustments for Forests",
"We will briefly review random forest and generalized forest in this section and show that they can be written as weighted regression problems.",
"We also introduce “forest weights\" that is an essential concept in this paper."
],
[
"Random Forest",
"Let $\\theta $ denote the random parameter determining how a tree is grown, and $\\lbrace (X_i, Y_i): i=1,\\ldots ,n\\rbrace \\in \\mathcal {X} \\times \\mathcal {Y} \\subset \\mathbb {R}^p \\times \\mathbb {R}$ denote the training data.",
"For each tree $T(\\theta )$ , let $R_l$ denotes its $l$ -th terminal leaf.",
"We let the index of the leaf that contains $x$ to be $l(x;\\theta )$ .",
"As shown in [34], for any single tree $T(\\theta )$ , the prediction on $x$ can be written as $\\sum _{i=1}^n w(X_i, x; \\theta ) Y_i$ where $w(X_i, x; \\theta ) = \\mathbb {1}_{\\lbrace X_i \\in R_{l(x;\\theta )} \\rbrace } / \\# \\lbrace j: X_j \\in R_{l(x;\\theta )} \\rbrace $ .",
"Then a random forest containing $m$ trees formulates a prediction of $\\mathbb {E}[Y|X=x]$ as $\\sum _{i=1}^n w(X_i, x) Y_i$ where $ w(X_i,x) = \\frac{1}{m} \\sum _{t=1}^m w(X_i, x; \\theta _t).$ From now on, we call the weight $w(X_i,x)$ in equation REF as random forest weight.",
"The above representation of the random forest prediction of the mean can be equivalently obtained as a solution to the least-squares optimization problem $\\min _{\\lambda \\in \\mathbb {R}} \\sum _{i=1}^n w(X_i,x) (Y_i - \\lambda )^2$ .",
"Therefore, a least-squares regression adjustment, as the above, is equivalent to [9] representation of random forests.",
"However, when we move to estimation quantities that are not the mean, the latter representation is very powerful.",
"Namely, a quantile random forest of [34] can be seen as a quantile regression adjustment [30], i.e., as a solution to the following optimization problem $\\min _{\\lambda \\in \\mathbb {R}} \\sum _{i=1}^n w(X_i, x) \\rho _{\\tau }(Y_i - \\lambda ),$ where $\\rho _\\tau $ is the $\\tau $ -th quantile loss function, defined as $\\rho _{\\tau }(u) = u(\\tau - \\mathbb {1}(u < 0))$ .",
"Local linear regression adjustment was also recently utilized in [4] to obtain a smoother and more poweful generalized forest algorithm."
],
[
"Generalized Random Forests",
"[4] proposed to generalize random forest using a more sophisticated splitting criterion which is model-free.",
"The new criterion aims to maximize the in-sample heterogeneity, formally defined as $\\tilde{\\Delta }(C_1, C_2) = \\sum _{j=1}^2 \\frac{1}{|\\lbrace i:X_i \\in C_j\\rbrace |} \\left( \\sum _{\\lbrace i:X_i \\in C_j\\rbrace } \\rho _i \\right)^2$ where $\\rho _i$ is a pseudo-response defined similarly as in Gradient Boosting [16].",
"Note that in the original random forest, we simply have $\\rho _i = Y_i - \\bar{Y}_P$ where $\\bar{Y}_P$ is the mean response in the parent node.",
"The generalized random forest, while applied to quantile regression problem, can deal with heteroscedasticity because the splitting rule directly targets changes in the quantiles of the $Y$ -distribution.",
"Just like the random forest algorithm, the generalized random forest is also an ensemble of trees and hence defines a weight or similarity between two samples using equation REF .",
"The main difference hence lies in how they split the samples into different terminal regions.",
"Therefore, in the following sections, whenever we refer to forest weight, it can be calculated from either random forest or generalized random forest.",
"In the experiment section, we will distinguish them by RF-weights and GRF-weights."
],
[
"Censored Quantile Regression Forest",
"The forest regressions cannot be directly applied to censored data $\\lbrace (X_i, Y_i)\\rbrace $ because the conditional quantile of $Y$ is different from the quantiles of the latent variable $T$ due to the censoring.",
"Moreover, there is no explicitly defined quantile loss function for randomly censored data.",
"In this section, we design a new approach to achieve both tasks.",
"We will motivate and derive our method using sections REF to REF ."
],
[
"Assumptions:",
"We first assume 1. there is no censoring on the data, and 2. the latent variable $T_i$ has the same conditional probability in a neighborhood $R_x$ of $x$ .",
"Following the regression adjustment reasoning, we could estimate the $\\tau $ -th quantile of $T_i$ at $x$ as $q_{\\tau , x} = \\operatornamewithlimits{arg\\,min}_{q \\in \\mathbb {R}} \\sum _{i=1}^n w(X_i, x) \\rho _{\\tau }(T_i - q).$ The above optimization problem has the following estimating equation $ U_n(q; x) &=(1-\\tau ) - \\sum _{i=1}^n w(X_i, x) \\mathbb {1}(T_i > q) \\approx 0.$ Now out of the $n$ data points, assume $\\lbrace X_1, \\cdots , X_k\\rbrace \\subset R_x$ and $w(X_i, x) = k^{-1} \\mathbb {1}\\lbrace X_i \\in R_x\\rbrace $ .",
"Because of the assumption 1, the estimating equation becomes $ U_k(q) &= (1-\\tau ) - \\frac{1}{k} \\sum _{i=1}^k \\mathbb {1}(T_i > q) \\approx 0.$ Now conditional on $\\lbrace x\\rbrace \\cup \\lbrace X_i\\rbrace _{i=1}^k$ , we have the expected estimating equation $\\mathbb {E} \\left[ U_k(q) | x, X_i, i=1,\\ldots ,k \\right] = (1-\\tau ) - \\mathbb {P}(T > q | x)$ which will be zero at $q^*$ where $\\mathbb {P}(T > q^* | x) = 1-\\tau $ , that is, at the true $\\tau $ th quantile at $x$ ."
],
[
"With Censoring and Locally Invariate $T_i$",
"Let's now consider the case that we could only observe $Y_i = \\min \\lbrace T_i, C_i\\rbrace $ and the censoring indicator $\\delta _i = \\mathbb {1}(T_i \\le C_i)$ .",
"Note that the following analysis extends straightforwardly to left censoring.",
"In order to estimate $q_{\\tau ,x}$ , we cannot simply replace $T_i$ with $Y_i$ in equation REF as the $\\tau $ -th quantile of $T_i$ is no longer the $\\tau $ -th quantile of $Y_i$ .",
"However, because $C_i$ and $T_i$ are conditionally independent, we have the following relation: $\\mathbb {P}(Y_i > q_{\\tau ,x}|x) &= \\mathbb {P}(T_i > q_{\\tau ,x}|x) \\mathbb {P}(C_i > q_{\\tau ,x}|x) \\\\& = (1-\\tau ) G(q_{\\tau ,x}|x)$ where $G(u|x)$ is the conditional survival function of $C_i$ at $x$ .",
"That is to say, the $\\tau $ -th quantile of $T_i$ is actually the $1 - (1-\\tau )G(q_{\\tau ,x}|x)$ -th quantile of $Y_i$ at $x$ .",
"Now, if we define a new estimating equation that resembles equation REF as follows $ S^o_k(q; x) = (1-\\tau )G(q|x) - \\frac{1}{k}\\sum _{i=1}^k \\mathbb {1}(Y_i > q) \\approx 0,$ we still have $\\mathbb {E}[S_k^o(q_{\\tau ,x})|x] = 0$ .",
"An intuitive explanation for using equation REF is that because the $\\tau $ -th quantile of $T_i$ is just the $1 - (1-\\tau )G(q_{\\tau ,x}|x)$ -th quantile of $Y_i$ at $x$ , instead of estimating the former which is not available because of the censoring, we could just estimate the later one.",
"The survival function $G(\\cdot |x)$ can be estimated by any consistent estimator, for example, the Kaplan-Meier estimator $\\hat{G}(\\cdot |x)$ using $\\lbrace Y_i\\rbrace _{i=1}^k$ and $\\lbrace \\delta _i\\rbrace _{i=1}^k$ , and we can then solve for $ S_k(q; x) = (1-\\tau )\\hat{G}(q|x) - \\frac{1}{k}\\sum _{i=1}^k \\mathbb {1}(Y_i > q) \\approx 0.$"
],
[
"Full Model",
"In the previous section, we assume that $\\mathbb {P}(T|X) = \\mathbb {P}(T|x)$ for all $X \\in R_x$ .",
"But in reality, this assumption is not always true, and that is why $w(X_i, x)$ plays an important rule in our final estimator, as it “corrects\" the empirical probability of each $T_i$ at $x$ .",
"Intuitively, if $X_i$ is more similar to $x$ than $X_j$ , then $Y_i$ should play a more important role than $Y_j$ on estimating the quantile at $x$ .",
"Now let $w(X_i,x)$ denote a similarity measure between $X_i$ and $x$ .",
"In order for $\\sum _{i=1}^n w(X_i, x) \\mathbb {1}(T_i \\le q)$ to be a proper estimation of $\\mathbb {P}(T\\le q|x)$ , it needs to satisfy two conditions: $ &(1) \\; \\sum _{i=1}^n w(X_i, x) = 1; \\\\&(2) \\; \\sum _{i=1}^n w(X_i, x) \\mathbb {1}(T_i \\le q) \\overset{p}{\\rightarrow } \\mathbb {P}(T \\le q | x) \\; \\forall q.$ One may think that any fixed Kernel weights, $K(X_i, x)$ , could be a suitable choice, but in fact they would not be able to satisfy the second condition for every distribution $\\mathbb {P}(T | x)$ .",
"Fortunately, as shown in [34] and [4], the data-adaptive (generalized) random forest weight $w(X_i, x)$ perfectly satisfies both conditions.",
"Therefore if we define $U_n(q_{\\tau ,x}) = (1-\\tau ) - \\sum _{i=1}^n w(X_i, x) \\mathbb {1}(T_i > q_{\\tau ,x}),$ we have $U_n(q_{\\tau ,x}) \\overset{p}{\\rightarrow } 0$ asymptotically.",
"Then following the same logic of how we get equation REF , a general case estimating equation for censoring data will be $ S_n(q; x) = (1-\\tau )\\hat{G}(q|x) - \\sum _{i=1}^n w(X_i, x)\\mathbb {1}(Y_i > q) \\approx 0.$"
],
[
"Estimators for $G(q|x)$",
"Many consistent estimators for the conditional survival functions exist.",
"For example, the nonparametric estimator [5] $ \\tilde{G}(q|x) = \\prod _{Y_i \\le q} \\left\\lbrace 1 - \\frac{W_i(x, a_n)}{\\sum _{j=1}^n \\mathbb {1}(Y_j \\ge Y_i) W_j(x,a_n)} \\right\\rbrace ^{1-\\delta _i}$ is shown to be consistent [5], [10], [11], [18], [1], [31], [42].",
"Here, $W_i(x,a_n)$ is the Nadaraya-Watson weight.",
"However, since we already have an adaptive version of kernel – the forest weights $w(X_i, x)$ , we propose the following two new estimators for $G(q|x)$ :"
],
[
"Kaplan-Meier using nearest neighbors.",
"We first find the $k$ nearest neighbors of $x$ according to the magnitude of the weights $w(X_i, x)$ , and denote these points as a set $N_x$ .",
"Then we define the Kaplan-Meier estimator on $N_x$ as $ \\prod _{i:X_i \\in N_x, Y_i\\le q} \\left( 1 - \\frac{1}{\\sum _{j=1}^n \\mathbb {1}(Y_j \\ge Y_i) \\mathbb {1}(X_j \\in N_x)} \\right)^{1-\\delta _i}.$ Here, the number of nearest neighbors $k$ will be a tuning hyperparameter."
],
[
"Beran estimator with forest weights.",
"We replace the Nadaraya-Watson weights in equation REF with the forest weights $w(X_i, x)$ : $ \\hat{G}(q|x) = \\prod _{Y_i \\le q} \\left\\lbrace 1 - \\frac{w(X_i, x)}{\\sum _{j=1}^n \\mathbb {1}(Y_j \\ge Y_i) w(X_j, x)} \\right\\rbrace ^{1-\\delta _i}.$ In fact equation REF is a special case of equation REF when the weight $w(X_i, x) = 1/k$ for $X_i \\in R_x$ and 0 otherwise."
],
[
"Algorithm",
"We summarize our algorithm in Algorithm REF .",
"The details for choosing the candidate set $\\mathcal {C}$ is in Section REF .",
"Censored quantile regression forest [2] Input: number of trees: $B$ , minimum node size: $m$ , the number of nearest neighbors: $k$ (if using equation REF ), test set $\\mathcal {A}$ , training set $ \\mathcal {D}=\\lbrace (X_i, Y_i, \\delta _i)\\rbrace _{i=1}^n$ , quantile $\\tau $ Step 0: Train a forest on $(\\mathcal {D})$ with $B$ trees and minimum node size $m$ .",
"$x \\in \\mathcal {A}$ Step 1: Calculate forest weights $w(x, X_i$ ).",
"Step 2: Calculate the survival function estimate $\\hat{G}(q|x)$ .",
"Step 3: Get the quantile estimation: $\\hat{q}(x) \\leftarrow \\operatornamewithlimits{arg\\,min}_{q \\in \\mathcal {C}} |S_n(q; x)|$ $\\mathcal {C}$ is a candidate set as discussed in Section REF $S_n$ is defined in equation REF"
],
[
"Theoretical Develoments",
"In this section, we will show the consistency of the proposed quantile estimator.",
"The time complexity analysis is in the Appendix."
],
[
"Consistency",
"The consistency of random forest has been extensively studied [2], [4], [7], [8], [6], [13], [32], [39], [44], [43].",
"Following the common settings, we also assume the covariate space $\\mathcal {X} = [0,1]^p$ and the parameter $q \\in \\mathcal {B} \\subset \\mathbb {R}$ where $\\mathcal {B}$ is a compact subset of $\\mathbb {R}$ .",
"In our case, since $q$ stands for the quantile, the assumption means that there exists some $r > 0$ such that $q \\in [-r, r] = \\mathcal {B}$ .",
"We also make another standard assumption that the density of $X$ is bounded away from 0 and $\\infty $ .",
"Note that since $\\mathcal {X}$ is a compact support, the density condition holds true for Gaussian distribution and more broadly any symmetric and continuous distribution with unbounded support.",
"Condition 1 (Lipschitz in $x$ ) Denote $F(y|x) = \\mathbb {P}(Y \\le y | x)$ .",
"There exists a constant $L$ such that $F(y|x)$ is Lipschitz continuous with parameter $L$ , that is, for all $x, x^{^{\\prime }} \\in \\mathcal {X}$ , $\\sup _{y} |F(y|x) - F(y|x^{^{\\prime }})| \\le L \\Vert x - x^{^{\\prime }}\\Vert _1.$ This Condition REF appears in all existing work related to quantile regression and inference thereafter.",
"Condition 2 (Identification) For any fixed $x$ , the latent variable $T$ and the censoring variable $C$ are conditionally independent, and the conditional distribution $\\mathbb {P}(T \\le q|x)$ and $\\mathbb {P}(C \\le q|x)$ are both strictly increasing in $q$ .",
"Conditional independence of $T$ and $C$ is a very standard assumption and can be traced back to [37] among other works.",
"Condition 3 (Tree splitting) For each tree splitting, the probability that each variable is chosen for the split point is bounded from below by a positive constant, and every child node contains at least $\\gamma $ proportion of the data in the parent node, for some $\\gamma \\in (0, 0.5]$ .",
"[Quantile forest [34]] The terminal node size $m \\rightarrow \\infty $ and $m/n \\rightarrow 0$ as $n \\rightarrow \\infty $ .",
"[Generalized forest [4]] The forest is honest and built via subsampling with subsample size $s$ satisfying $s/n \\rightarrow 0$ and $s \\rightarrow \\infty $ .",
"The first two requirements of Condition REF are shared in [34] and [4].",
"For quantile random forest [34], they require that the leaf node size of each tree should increase with the sample size $n$ , but at a slower rate.",
"Our experiments also justify that the required leaf node size of Meinshausen's quantile forest is larger than the node size of the generalized forest.",
"In general, using the generalized forest weights give us more stable estimations because the trees are honest and regular [43].",
"Condition 4 (Censoring variable) For any $x \\in \\mathcal {X}$ , $\\hat{G}(q|x)$ is a uniformly consistent estimator of the true conditional survival function $G(q|x)$ for $q \\in \\mathcal {B}$ .",
"Condition REF is satisfied, for example, by the Kaplan-Meier estimator equation REF [11].",
"Please take a look at Figure REF where we compare finite sample properties of the newly introduced estimators equation REF and equation REF .",
"We observe that the new distributional estimators are more adaptive and yet seemingly inherit consistency to that of the traditional KM estimator.",
"We proceed to showcase asymptotic properties of the proposed estimating equations.",
"We begin by illustrating a concentration of measure phenomenon for the introduced score equations.",
"Theorem 1 Define $S(q;\\tau ) = (1-\\tau )G(q|x) - \\mathbb {P}(Y > q).$ Under Conditions REF – REF , for any $x \\in \\mathcal {X}$ , $r > 0$ and $\\tau \\in (0,1)$ , we have $\\sup _{q \\in [-r,r]} | S_n(q; \\tau ) - S(q; \\tau ) | = o_p(1).$ Next, we present our main result that illustrates an asymptotic consistency of the proposed conditional quantile estimator.",
"The proof is given in Appendix.",
"Theorem 2 Under Conditions REF – REF , for fixed $\\tau \\in (0,1)$ and $x \\in \\mathcal {X}$ , define $q^*$ to be the root of $S(q;\\tau ) = 0$ , and $r > 0$ to be some constant so that $q^* \\in [-r, r]$ .",
"Also define $q_n$ to be $\\operatornamewithlimits{arg\\,min}_{q \\in [-r, r]} \\left|S_n(q; \\tau )\\right|$ .",
"Then $\\mathbb {P}(T \\le q^*|x) = \\tau $ , and $q_n \\overset{p}{\\rightarrow } q^*$ as $n \\rightarrow \\infty $ ."
],
[
"Experiments",
"In this section, we will compare the proposed model, censored regression forest (crf), with generalized random forest (grf) [3], quantile random forest (qrf) [34] and random survival forest (rsf) [23] on various simulated and real datasets.",
"On the simulated datasets, we report both censored and oracle results for qrf and grf.",
"To obtain the censored result, we directly apply generalized random forest and quantile random forest to the censored data, and denote the results by grf and qrf respectively.",
"For oracle result, we instead train the models using the oracle responses without censoring (i.e.",
"$T_i$ 's), and call the results grf-oracle and qrf-oracle."
],
[
"Simulation Study",
"In this section, we denote censored regression forest with generalized forest weights as crf-generalized and the one with (quantile) random forest weights as crf-quantile.",
"We first define the evaluation metric used in this section – quantile loss.",
"The $\\tau $ -th quantile loss is defined as follows.",
"Let $\\hat{q}^{\\tau }_i$ be the estimated $\\tau $ -th quantile at $X_i$ , then $L_{quantile}(\\hat{q}^\\tau _1, \\ldots , \\hat{q}^\\tau _n) = \\frac{1}{n} \\sum _{i=1}^n \\rho _{\\tau }(T_i - \\hat{q}^{\\tau }_i).$ We could use this metric because we know the latent responses $T_i$ 's in simulations."
],
[
"Accelerated Failure Time Data",
"In this section, we generate data from a accelerated failure time (AFT) model.",
"We sample $n=1000$ independent and identically distributed examples where $X_i$ is uniformly distributed over $[0, 2]^p$ with $p=20$ , and $T_i$ is conditional on $(X_i)_1$ and $\\log (T_i|X_i) = (X_i)_1 + \\epsilon $ where $\\epsilon \\sim \\mathcal {N}(0, 0.3^2)$ .",
"The censoring variable $C_i \\sim \\text{Exp}(\\lambda =0.08)$ and $Y_i = \\min (T_i, C_i)$ .",
"This results in about 23% censoring level.",
"The other 19 covariates are noise.",
"We estimate the quantiles at $\\tau =0.1$ and $0.9$ , and draw the predicted quantiles in Figure REF for node size 20.",
"Figure: Quantile estimates on AFT model when τ=0.1\\tau = 0.1 and 0.90.9.",
"The minimum node size is 20 and each forest contains 1000 trees.From the results in Figure REF , both grf and qrf are severely biased downwards because of the right censoring.",
"The proposed methods crf-generalized and crf-quantile both provide consistent quantile estimation that is almost identical to the true quantiles.",
"Random survival forest is very unstable on predicting the quantiles and is not able to correct the censoring bias.",
"We increase the node size to from 20 to 80, all the methods except for rsf become more biased and less variant, but rsf is still volatile.",
"We then repeat the above experiment ten times for different node sizes ranging from 10 to 80, and report the average and standard deviation of the quantile losses in Figure REF .",
"We observe that the performances of both crf-generalized and crf-quantile are close to their corresponding oracles, and are much better than grf or qrf on censored data.",
"crf-quantile (qrf) performs slightly better than crf-generalized (grf), implying that original quantile random forest can be more effective when the data is homoscedastic.",
"The random survival forest rsf behaves only slightly better than the biased grf and qrf, but is much worse than the proposed methods."
],
[
"Heteroscedastic Data",
"We test the proposed method on a heteroscedastic dataset.",
"The dataset is taken from [4].",
"We sample $n=2000$ independent and identically distributed examples where $X_i$ is uniformly distributed over $[-1, 1]^p$ with $p=40$ , and $T_i$ is Gaussian conditionally on $(X_i)_1$ and $T_i|X_i \\sim \\mathcal {N}(10, (1 + \\mathbb {1}\\lbrace (X_i)_1 > 0\\rbrace )^2)$ .",
"The censoring variable $C_i \\sim 8 + \\text{Exp}(\\lambda =0.10)$ and $Y_i = \\min (T_i, C_i)$ .",
"The other 39 covariates are noise.",
"The censoring ratio is about 20% in this example.",
"We shift the mean of $T_i|X_i$ to 10 because the random survival forest only allows positive responses.",
"We estimate the quantiles at $\\tau =0.1, \\, 0.5, \\, 0.9$ .",
"The results are in Figure REF .",
"When the data is heteroscedastic, using generalized forest weights (crf-generalized) provides much more accurate quantile estimation than all the other methods.",
"The predicted quantiles by crf-generalized are almost identical to the truths.",
"Our results are inline with [4] that generalized random forest is very effective at dealing with heteroscedasticity.",
"Note that the random survival forest (rsf) also fails to recognize the variance shift.",
"This experiment indicates that our method coupled with generalized forest weights is the most, arguably the only effective method when dealing with heteroscedastic data.",
"We also repeat the above experiment ten times for different node sizes and report the quantile losses in Figure REF .",
"We again observe that crf-generalized achieves almost the same performance of grf-oracle.",
"rsf and crf-quantile have similar performance on this dataset, and are both slightly worse than even grf when $\\tau =0.1$ .",
"This shows that the splitting rule of random survival forest or quantile forest do not work well on heteroscedastic data.",
"Figure: Quantile losses on multi-dimensional AFT data with different node sizes.Figure: Quantile losses on heteroscedastic examples."
],
[
"Conditional Survival Functions",
"In this section, we compare the two proposed conditional survival function estimators equation REF and equation REF .",
"We generate examples from the AFT model, and then choose four test points $\\lbrace x_1=0.4, x_2=0.8, x_3=1.2, x_4=1.6\\rbrace $ to plot the conditional survival function estimations on them.",
"The results are shown in Figure REF .",
"Figure: The results on real datasets.",
"On BostonHousing, we report the quantile losses (the lower the better), and on pbc dataset, we report the C-index (the higher the better).We observe that when $n$ increases, two curves become closer and are both good approximations of the true survival curve.",
"But the first method equation REF has an extra tuning parameter $k$ – the number of nearest neighbors.",
"Therefore, in the experiments, we always choose to use the second estimator equation REF which is parameter free.",
"Note that the estimated survival function will degenerate at the tail of the distribution when the test point $x$ is small.",
"This is a common phenomenon even for the regular KM estimator because there is no censored observations beyond some time point.",
"In the AFT model, when $x$ is small, the conditional mean of $T$ is also small, and hence we could not observe most of the censoring values, leading to degenerated survival curves."
],
[
"Real Data",
"In this section, we compare the proposed method with other forest algorithms on two real datasets, BostonHousing [14] and Primary Biliary Cirrhosis (PBC) Data [15].",
"On the BostonHousing data, we manually generate censoring variables from $\\text{Exp}(\\lambda =1/2 \\bar{y})$ where $\\bar{y}$ is the sample mean of the house prices.",
"The censoring level is about 40%.",
"We evaluate the models using quantile loss because we know the true responses in this case.",
"The PBC data is already right-censored, and the censoring rate is about 60%.",
"On this dataset, we cannot use quantile loss for evaluation because we do not know the true responses of the censored data in the test set.",
"(Note that we can sample uncensored data to form a test set, but these data will be biased.)",
"Instead, we use the Harrell's concordance index (C-index) [20].",
"We repeat the experiment for each node size for 50 times and report the mean and standard deviation of the C-index.",
"For each experiment, we randomly sample 80% of the data for training and the rest for testing.",
"All the forests contain 1000 trees.",
"The results are in Figure REF .",
"Overall, the proposed method with random forest weights crf-quantile has the best performance.",
"It agrees with our observation in the simulations that crf-quantile works better than the other methods if there is no clear heteroscedasticity in the data."
],
[
"Discussion",
"In this article, we introduced censored quantile regression forest, a novel non-parametric method for quantile regression problems that is integrated with the censored nature of the observations.",
"While preserving information carried by the censored observations, the novel estimating equation maintains the flexibility of general forest approaches.",
"One of the promising applications of the introduced method is in the estimation of heterogeneous treatment effects when the response variable is censored.",
"Treatment discovery with right-censored observations is an important and yet poorly understood research area.",
"Equipping this literature with the proposed fully non-parametric approach would lead to a significant broadening of the now more known parametric approaches.",
"We also observe that our estimating equations can be easily replaced with another kind that targets treatment effects directly."
],
[
"Time complexity",
"The step REF in Algorithm REF involves of finding the $q^*$ in a candidate set $\\mathcal {C}$ that sets the estimating equation $S_n(q; \\tau )$ closest to zero.",
"We simply evaluate the function $S_n(q;\\tau )$ for all possible $q$ in $\\mathcal {C}$ and find the minimum point.",
"Note that for any fixed $\\tau $ , $S_n(q; \\tau )$ is a step function in $q$ with jumps at $Y_i$ 's because the discontinuities only happen at $Y_i$ 's for $\\hat{G}(q|x)$ (both equation REF and equation REF ) and $\\sum _{i=1}^n w(X_i, x) \\mathbb {1}(Y_i > q)$ .",
"Therefore, the candidate set $\\mathcal {C} \\subset \\lbrace Y_i\\rbrace _{i=1}^n$ , and $|\\mathcal {C}| = n$ in the worst case.",
"But in fact, for any fixed $x$ , only $Y_i$ 's with the corresponding feature vector $X_i \\in R_x$ equation REF or with $w(X_i, x) > 0$ equation REF will be jump points, and hence, we can refine $\\mathcal {C} = \\lbrace Y_i: X_i \\in R_x\\rbrace $ for equation REF or $\\mathcal {C} = \\lbrace Y_i: w(X_i, x) > 0\\rbrace $ for equation REF .",
"We then have the following theorem.",
"Theorem 3 For a fixed test point $x$ , depending on whether $G(q|X)$ is estimated by equation REF or equation REF , the time complexity for Algorithm REF is $O(n \\max \\lbrace k, \\log (n)\\rbrace )$ or $O(n m \\log (n)^{p-1})$ , respectively.",
"[Proof of Theorem REF ] To get the candidate set $\\mathcal {C}$ , if we use the k-nearest neighbor estimator equation REF , then the first step is to sort $n$ weights and choose the largest $k$ elements.",
"This is in general a $O(n \\log (n))$ procedure.",
"If we use the Beran estimator equation REF , then the time complexity is $O(n)$ because we need to find all the nonzero weights.",
"After we have the candidate set $\\mathcal {C}$ , evaluating $S_n(q;\\tau )$ for all $q \\in \\mathcal {C}$ and finding the minimum is a $O(n |\\mathcal {C}|)$ procedure.",
"For equation REF , $|\\mathcal {C}| = k$ ; and for equation REF , $|\\mathcal {C}|$ is in the order of $m \\log (n)^{p-1}$ by [32]."
],
[
"Proof of Theorem ",
"When the conditions REF to REF are satisfied, by Theorem 3 in [4] or Theorem 1 in [34], we have $\\left| \\sum _{i=1}^n w(X_i, x) \\mathbb {1}\\lbrace Y_i \\le q\\rbrace - \\mathbb {P}(Y \\le q | x) \\right| = o_p(1).$ Note that $\\sum _{i=1}^n w(X_i, x) = 1$ and $0 \\le w(X_i, x) \\le 1/m$ .",
"For convenience, we suppress the dependency on $x$ and denote $F_n(q) = \\sum _{i=1}^n w(X_i, x) \\mathbb {1}\\lbrace Y_i \\le q\\rbrace $ and $F(q) = \\mathbb {P}(Y \\le q | x)$ .",
"Because $F$ is continuous, choose $q_0 < q_1 < \\ldots < q_n$ from $\\mathcal {B}$ such that $F(q_j) - F(q_{j-1}) = 1/n$ .",
"Then for any $q \\in \\mathcal {B}$ , there exists $j \\in \\lbrace 1,\\ldots , n\\rbrace $ such that $q \\in [q_{j-1}, q_j]$ , and hence $F_n(q) - F(q) \\le F_n(q_j) - F(q_{j-1}) = F_n(q_j) - F(q_j) + 1/n$ .",
"Similarly, $F_n(q) - F(q) \\ge F_n(q_{j-1}) - F(q_{j-1}) - 1/n$ .",
"Therefore, we have $\\sup _{q \\in \\mathcal {B}} \\left| F_n(q) - F(q) | x) \\right| \\le \\max _{j=1,\\ldots ,n} \\left| F_n(q_j) - F(q_j) \\right| + 1/n.$ Then by Bonferroni's inequality, we have $\\sup _{q \\in \\mathcal {B}} \\left| F_n(q) - F(q) | x) \\right| = o_p(1).$ Combined with Condition REF , we have the expected result."
],
[
"Proof of Theorem ",
"[Proof of Theorem REF ] By [41], we only need to show for any $\\tau \\in (0,1)$ , $x \\in \\mathcal {X}$ , $\\sup _{q \\in [-r,r]} | S_n(q; \\tau ) - S(q; \\tau ) | = o_p(1)$ .",
"For any $\\epsilon > 0$ , $\\inf \\lbrace |S(q;\\tau )|: |q - q^*| \\ge \\epsilon , q \\in [-r, r] \\rbrace > 0$ .",
"Here, $q^*$ stands for the true $\\tau $ th quantile of $T$ .",
"$S_n(q_n; \\tau ) = o_p(1)$ .",
"Part REF has been proved by Theorem REF .",
"For part REF , note that $S(q;\\tau ) &=& (1-\\tau )G(q|x) - \\mathbb {P}(Y > q|x) \\\\&=& (1-\\tau )G(q|x) - \\mathbb {P}(T > q|x) \\mathbb {P}(C > q|x) \\\\&=& ((1-\\tau ) - \\mathbb {P}(T > q|x)) G(q | x) \\\\&=& (\\mathbb {P}(T \\le q|x) - \\tau ) G(q | x).$ The second equality is because of the conditionally independency between $T$ and $C$ .",
"Fix an $\\epsilon > 0$ , and denote $E = \\lbrace |S(q;\\tau )|: |q - q^*| \\ge \\epsilon , q \\in [-r, r]\\rbrace .$ Since $0 < \\tau < 1$ , by Condition REF , there exists some $l > 0$ such that $G(q|x) \\ge l$ and $|\\mathbb {P}(T \\le q | x) - \\tau | \\ge l$ for $q \\in E$ .",
"Now for part REF , by the definition of $q_n$ , we know $|S_n(q_n; \\tau )| = \\min _{q \\in [-r,r]} |S_n(q;\\tau )|.$ Also by definition of $q^*$ , $0 = |S(q^*;\\tau )| = \\min _{q \\in [-r,r]} |S(q;\\tau )|.$ Then we get $& |S_n(q_n;\\tau )| \\\\& = |S_n(q_n;\\tau )| - |S_n(q^*;\\tau )| + |S_n(q^*;\\tau )| - |S(q^*;\\tau )| \\\\&\\le | S_n(q^*;\\tau ) - S(q^*;\\tau ) | \\\\&\\le \\sup _{q \\in [-r,r]} |S_n(q;\\tau ) - S(q;\\tau )| \\\\&= o_p(1)$ where the first inequality is because of the definition of $q_n$ and the triangular inequality."
],
[
"Prediction Intervals",
"All the forest methods can be used to get $95\\%$ prediction intervals by predicting the $0.025$ and $0.975$ quantiles of the true response variable.",
"Then for any location $x \\in \\mathcal {X}$ , a straightforward confidence interval will be $[Q(x;0.025), Q(x;0.975)]$ .",
"The result is illustrated in Figure REF for the case of univariate censored sine model.",
"For each data set, we bootstrap the data and calculate the $0.025$ and $0.975$ quantile for the out of bag points.",
"Then for each node size, we repeat this process for 20 times and calculate the average coverage rate of the confidence intervals.",
"Figure: Prediction intervals of the univariate censored since model.",
"We observe that in all of the cases, our method crf and qrf-oracle give the coverage closest to 95%95\\%.",
"Both qrf and grf perform much worse on predicting lower quantiles.",
"They tend to under-estimate the lower quantiles and hence make the confidence intervals much wider than the true ones."
],
[
"One-dimensional Sine-curve Model",
"Since the proposed method crf is nonparametric and does not rely on any parametric assumption, it can be used to estimate quantiles for any general model $T = f(X) + \\epsilon $ .",
"Hence we set $f(x) = \\sin (x)$ and $T = 2.5 + \\sin (X) + \\epsilon $ where $X \\sim \\textrm {Unif}(0,2 \\pi )$ and $\\epsilon \\sim \\mathcal {N}(0, 0.3^2)$ .",
"The censoring variable $C \\sim 1 + \\sin (X) + \\textrm {Exp}(\\lambda = 0.2)$ depends on the covariates, and the censoring level is about $25\\%$ .",
"The results are in Figure REF .",
"Again, the proposed model crf produces almost identical quantile predictions compared with grf-oracle.",
"Especially when $\\tau =0.1$ , the $grf$ result (blue dotted curve) severely deviates from the true quantile, while crf still predicts the correct quantile and performs as good as the oracle grf-oracle."
]
] | 2001.03458 |
[
[
"Mobility spaces and their geodesic paths"
],
[
"Abstract We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points.",
"This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers.",
"Mobility algebras consist on a set $A$ together with three constants and a ternary operation.",
"In the case of the closed unit interval $A=[0,1]$, the three constants are 0, 1 and 1/2 while the ternary operation is $p(x,y,z)=x-yx+yz$.",
"A mobility space is a set $X$ together with a map $q\\colon{X\\times A\\times X\\to X}$ with the meaning that $q(x,t,y)$ indicates the position of a particle moving from point $x$ to point $y$ at the instant $t\\in A$, along a geodesic path within the space $X$.",
"A mobility space is thus defined with respect to a mobility algebra, in the same way as a module is defined over a ring.",
"We introduce the axioms for mobility spaces, investigate the main properties and give examples.",
"We also establish the connection between the algebraic context and the one of spaces with geodesic paths.",
"The connection with affine spaces is briefly mentioned."
],
[
"Introduction",
"The purpose of this work is to introduce an algebraic system which can be used to model spaces with geodesics.",
"The main idea stems from the interplay between algebra and geometry.",
"In Euclidean geometry the notion of affine space is well suited for this purpose.",
"Indeed, in an affine space we have scalar multiplication, addition and subtraction and so it is possible to parametrize, for any instant $t\\in [0,1]$ , a straight line between points $x$ and $y$ with the formula $(1-t)x+ty$ .",
"Such a line is clearly a geodesic path from $x$ to $y$ .",
"In general terms, we may use an operation $q=q(x,t,y)$ to indicate the position, at an instant $t$ , of a particle moving in space from a point $x$ to a point $y$ .",
"If the particle is moving along a geodesic path then this operation must certainly verify some conditions.",
"The aim of this project is to present an algebraic structure, $(X,q)$ , with axioms that are verified by any operation $q$ representing a geodesic path in a space between any two of its points.",
"First results concerning this investigation were presented in [5] where a binary operation, obtained by fixing $t$ to a value that positions the particle at half way from $x$ to $y$ , is studied.",
"The whole movement of a particle on a geodesic path is captured when the variable $t$ is allowed to range over a set of values, of which the unit interval is the most natural choice.",
"The investigation of those structures and properties relevant to our study led us to the discovery of a new algebraic structure that was called mobi algebra [7].",
"A mobility algebra (or mobi algebra), besides being a suitable algebraic model to the unit interval, offers an interesting comparison with rings.",
"A slogan may be used to illustrate that comparison: a mobi algebra is to the unit interval in the same way as a ring is to the set of reals.",
"A mobi algebra is an algebraic system $(A,p,0,\\text{{1}{2}},1)$ consisting of a set together with a ternary operation $p$ and three constants.",
"Moreover, every ring in which 2 is invertible has a mobi algebra structure, while a mobi algebra in which the element $\\text{{1}{2}}$ is invertible (as defined in [7]) has a ring structure.",
"Following the analogy with rings, and extending it to modules over a ring, we have considered a new structure, called mobility space (mobi space for short).",
"If $A$ is a mobi (or mobility) algebra then a mobi space, say $(X,q)$ , is defined over a mobi algebra in the sense that $q=q(x,t,y)$ operates on $x,y\\in X$ and $t\\in A$ .",
"The axioms defining a mobi space are similar to the ones defining a mobi algebra and examples show that these axioms are indeed appropriate to capture the features of spaces with geodesic paths between any two of their points.",
"This paper is organized as follows.",
"The notion of mobi algebra [7] is recalled together with its derived operations (Section ).",
"Two important new sorts of examples are added to the list presented in [7].",
"A first sort of examples is constructed (Proposition REF ) from a midpoint algebra by considering a suitable subset of its endomorphisms (Definition REF ).",
"Another sort of examples is generated by considering a unitary ring in which the element 2 is invertible.",
"As a particular case, the set of endomorphisms of an abelian group is canonically equipped with a unitary ring structure and hence, if the map $x\\mapsto 2x$ is invertible, then it gives rise to a mobi algebra structure on its set of endomorphisms (Proposition REF ).",
"These examples will be used to characterize affine mobi spaces which include, among other things, all vector spaces over a field and will be considered in a sequel of this work.",
"Nonetheless, our last section is dedicated to a thoroughly comparison between affine mobi spaces and modules over a ring (Theorem REF and Theorem REF ).",
"In addition, some details are given in the form of examples in subsection REF .",
"Section is totally devoted to the definition of mobi space and a short list of its properties that will be needed in this paper.",
"Further studies are postponed to a future work, namely the transformations between such structures, and the fact that the category of mobi spaces is a weakly Mal'tsev category [5], [12], [13].",
"Or its comparison with the work of A. Kock on affine connections with neighbouring relations [8], [9], [10] and its relation to the work of Buseman [1], [2] on spaces with unique geodesic paths.",
"Section 4 is dedicated to an exhaustive list of worked examples and counter examples that have been used for testing the strength of our axioms.",
"These include simple examples (Example REF ), more complex examples (Example REF ), the particular case of affine mobi spaces (which are comparable to euclidean geometry) and some examples with a physical interpretation (Examples REF and REF ).",
"In particular, we observe that even for other type of physical equations, rather than those for geodesics, there is a procedure which can be followed to include physical phenomenon into a mobility space.",
"The procedure is based on the simple trick of considering time as a geometrical dimension (see Example REF and Corollary REF ).",
"In Section we study the main examples which arise as the solution to the equation for geodesics and extend that result to cover a wide range of cases in which geodesics are not uniquely determined.",
"This is done via the process of identification spaces.",
"In the end we have added a section with a comparison between affine mobi spaces and modules over a unitary ring in which the element 2 is invertible."
],
[
"Mobi algebra",
"In this section we briefly recall the notion of mobi algebra which was introduced in [7] and some of its basic properties.",
"A mobi algebra consists of a set equipped with three constants and one ternary operation.",
"The unit interval $[0,1]$ , with the three constants 0, $\\frac{1}{2}$ , 1 and the formula $p(a,b,c)=a+bc-ba$ has always been the example of a mobi algebra which have driven our intuition.",
"Nevertheless, several other examples of very different nature were presented in [7].",
"At the end of this section we add two new important classes to the list of examples.",
"As we will see in the next section (while introducing the notion of mobi space) a mobility algebra plays the role of scalars (for a mobility space) in the same way as a ring (or a field) models the scalars for a module (or a vector space) over the base ring (or field).",
"In order to have an intuitive interpretation of its axioms we may consider a mobi algebra as a mobi space over itself and use the geometric intuition provided in section .",
"Namely, that the operation $p(x,t,y)$ is the position of a particle moving from a point $x$ to a point $y$ at an instant $t$ while following a geodesic path.",
"Definition 2.1 [7] A mobi algebra is a system $(A,p,0,\\text{{1}{2}},1)$ , in which $A$ is a set, $p$ is a ternary operation and 0, $\\text{{1}{2}}$ and 1 are elements of A, that satisfies the following axioms: $p(1,\\text{{1}{2}},0)=\\text{{1}{2}}$ $p(0,a,1)=a$ $p(a,b,a)=a$ $p(a,0,b)=a$ $p(a,1,b)=b$ $p(a,\\text{{1}{2}},b_1)=p(a,\\text{{1}{2}},b_2)\\Rightarrow b_1=b_2$ $p(a,p(c_1,c_2,c_3),b)=p(p(a,c_1,b),c_2,p(a,c_3,b))$ $p(p(a_1,c,b_1),\\text{{1}{2}},p(a_2,c,b_2))=p(p(a_1,\\text{{1}{2}},a_2),c,p(b_1,\\text{{1}{2}},b_2))$ .",
"Some properties of mobi algebras can be suitably expressed in terms of a unary operation \"$\\overline{()}$ \" and binary operations \"$\\cdot $ \", \"$\\circ $ \" and \"$\\oplus $ \" defined as follows (see [7] for more details).",
"Definition 2.2 Let $(A,p,0,\\text{{1}{2}},1)$ be a mobi algebra.",
"We define: $\\overline{a}&=&p(1,a,0)\\\\a\\cdot b&=&p(0,a,b)\\\\a\\oplus b&=&p(a,\\text{{1}{2}},b)\\\\a\\circ b&=&p(a,b,1).$ We recall a list of some properties of a mobi algebra.",
"If $(A,p,0,\\text{{1}{2}},1)$ is a mobi algebra, then: $ \\overline{\\text{{1}{2}}}&=&\\text{{1}{2}}\\\\ a\\cdot \\text{{1}{2}}=\\text{{1}{2}}\\cdot a&=&0\\oplus a\\\\ \\text{{1}{2}}\\cdot a=\\text{{1}{2}}\\cdot a^{\\prime }&\\Rightarrow & a=a^{\\prime }\\\\ p(\\overline{a},\\text{{1}{2}},a)&=&\\text{{1}{2}}\\\\ \\overline{a}=a &\\Rightarrow & a=\\text{{1}{2}}\\\\ \\overline{p(a,b,c)}&=&p(\\overline{a},b,\\overline{c})\\\\ p(c,b,a)&=&p(a,\\overline{b},c)\\\\ \\overline{a\\circ b}&=&\\overline{b}\\cdot \\overline{a}\\\\\\text{{1}{2}}\\cdot p(a,b,c)&=&(\\overline{b}\\cdot a)\\oplus (b\\cdot c).$ A long list of examples is provided in [7].",
"The following example shows that any non-empty midpoint algebra $(X,\\oplus )$ , with a chosen element $e\\in X$ in it, gives rise to a mobi algebra structure on a suitable subset of endomaps of $X$ .",
"This example will be crucial for the characterization of affine mobi spaces (see Section REF ).",
"Let us recall that a midpoint algebra [3] consists of a set $X$ and a binary operation $\\oplus $ satisfying the following axioms: $\\begin{array}{rl}\\text{(idempotency)} & x\\oplus x= x\\\\\\text{(commutativity)} & x\\oplus y= y \\oplus x\\\\\\text{(cancellation)} & x\\oplus y= x^{\\prime } \\oplus y\\Rightarrow x=x^{\\prime }\\\\\\text{(mediality)}&(x\\oplus y)\\oplus (z\\oplus w)=( x \\oplus z)\\oplus (y\\oplus w) .\\end{array}$ It is remarkable that a certain subset of endomaps of $X$ can always be given the structure of a mobi algebra.",
"Definition 2.3 Let $(X,\\oplus )$ be a non-empty midpoint algebra and $e\\in X$ a fixed chosen element in $X$ .",
"We denote by $End_{e}^{\\oplus }(X)$ the set of endomaps of $X$ consisting of those maps $g\\colon {X\\rightarrow X}$ satisfying the following conditions: $g(x\\oplus y)=g(x)\\oplus g(y)$ , for all $x,y\\in X$ , $g(e)=e$ for all $x\\in X$ , there exists $\\bar{g}(x)\\in X$ such that $\\bar{g}(x)\\oplus g(x)=e\\oplus x$ for all $x,y\\in X$ , there exists $\\tilde{g}(x,y)\\in X$ such that $\\bar{g}(x)\\oplus g(y)=e\\oplus \\tilde{g}(x,y)$ Let us observe that when $g\\in End_e^{\\oplus }(X)$ then $\\bar{g}$ , defined as in $(iii)$ , is also in $End_e^{\\oplus }(X)$ .",
"Moreover, we have $\\bar{\\bar{g}}=g$ and $\\tilde{\\bar{g}}(x,y)=\\tilde{g}(y,x)$ .",
"An example is provided in subsection REF , where affine mobi spaces are considered.",
"Proposition 2.1 Let $(X,\\oplus )$ be a non-empty midpoint algebra and fix an element $e\\in X$ .",
"The system $(End_e^{\\oplus }(X), \\mathbb {p}, \\mathbb {0}, \\frac{\\mathbb {1}}{\\mathbb {2}},\\mathbb {1})$ , with $\\mathbb {p}(x)=\\mathbb {p}_{f,g,h}(x)=\\tilde{g}(f(x),h(x))$ , $\\mathbb {0}(x)=e$ , $\\frac{\\mathbb {1}}{\\mathbb {2}}(x)=e\\oplus x$ and $\\mathbb {1}(x)=x$ is a mobi algebra.",
"First we show that the operation $\\mathbb {p}$ is well defined.",
"To do that we observe that if $g\\in End_{e}^{\\oplus }(X)$ then $\\tilde{g}$ , defined as in $(iv)$ , is such that $\\tilde{g}(x_1\\oplus x_2,y_1\\oplus y_2)&=&\\tilde{g}(x_1,y_1)\\oplus \\tilde{g}(x_2,y_2)\\\\\\tilde{g}(x,x)&=&x$ Now, given $f,g,h\\in End_{e}^{\\oplus }$ we have to show that $\\mathbb {p}_{f,g,h}$ , defined for every $x\\in X$ as $\\tilde{g}(f(x),h(x))$ , belongs to $End_{e}^{\\oplus }$ .",
"The first two conditions $(i)$ and $(ii)$ follow from the properties of $\\tilde{g}$ .",
"Condition $(iii)$ is satisfied with $\\bar{\\mathbb {p}}_{f,g,h}(x)=\\tilde{g}(\\bar{f}(x),\\bar{h}(x))$ , while condition $(iv)$ is satisfied with $\\tilde{\\mathbb {p}}_{f,g,h}(x,y)=\\tilde{g}(\\tilde{f}(x,y),\\tilde{h}(x,y))$ .",
"The fact that the axioms of a mobi algebra (Definition REF ) are verified by $\\mathbb {p}_{f,g,h}(x)$ is then a consequence of Theorem 6.2 from [7].",
"Indeed, the system $(End_e^{\\oplus }(X), \\overline{()},\\oplus ,\\cdot ,\\mathbb {1})$ with $\\overline{(g)}(x)=\\overline{g}(x)$ , $(f\\oplus g)(x)=f(x)\\oplus g(x)$ , $(f\\cdot g)(x)=f(g(x))$ and $\\mathbb {1}(x)=x$ is a IMM* algebra in the sense of Definition 6.1 from [7].",
"Moreover, the equation $\\overline{\\mathbb {1}}\\oplus \\chi =(\\overline{g}\\cdot f)\\oplus (g\\cdot h)$ has a solution in $End_e^\\oplus (X)$ , namely $\\chi (x)=\\tilde{g}(f(x),h(x))$ for every $x\\in X$ .",
"As it was shown in [7], every unitary ring in which the element 2 is invertible gives rise to a mobi algebra structure.",
"A particular case, which will be used in section REF , is the ring of endomorphisms of an abelian group.",
"Let $(X,+,0)$ be an abelian group.",
"It is clear that the map $x\\mapsto x+x$ is invertible if and only if for every $x\\in X$ , there exists $y\\in X$ with $y+y=x$ .",
"For obvious reasons the element $y$ such that $y+y=x$ will be denoted $\\frac{x}{2}$ .",
"As usual, we denote the set of group endomorphisms of $X$ by $End(X)$ .",
"Proposition 2.2 Let $(X,+,0)$ be an abelian group in which the endomorphism sending $x$ to $x+x$ is invertible.",
"The system $(End(X), \\mathbb {p}, \\mathbb {0}, \\frac{\\mathbb {1}}{\\mathbb {2}},\\mathbb {1}),$ with $\\mathbb {p}(x)=\\mathbb {p}_{f,g,h}(x)=f(x)-gf(x)+gh(x)$ , $\\mathbb {0}(x)=0$ , $\\frac{\\mathbb {1}}{\\mathbb {2}}(x)=\\frac{x}{2}$ and $\\mathbb {1}(x)=x$ is a mobi algebra.",
"The result follows directly from Theorem 7.2 in [7] since $End(X)$ is a unitary ring with one half.",
"In spite of these somehow more abstract examples, the reader is kindly reminded that it is useful to keep in mind that the most natural and intuitive mobi algebra is the closed unit interval with $A=[0,1]$ , the three constants 0, $\\frac{1}{2}$ , 1 and the operation $p(a,b,c)=(1-b)a+bc$.",
"The mobi algebra, in this case, is interpreted as the set of scalars over which a mobi space is constructed."
],
[
"Mobi space",
"In this section we give the definition of a mobi space over a mobi algebra.",
"Its main purpose is to serve as a model for spaces with a geodesic path connecting any two points.",
"It is similar to a module over a ring in the sense that it has an associated mobi algebra which behaves as the set of scalars.",
"In Section REF (see also Section at the end of the paper) we show that the particular case of affine mobi space is indeed the same as a module over a ring when the mobi algebra is a ring.",
"Definition 3.1 Let $(A,p,0,\\text{{1}{2}},1)$ be a mobi algebra.",
"An $A$ -mobi space $(X,q)$ , consists of a set $X$ and a map $q\\colon {X\\times A\\times X\\rightarrow X}$ such that: $q(x,0,y)=x$ $q(x,1,y)=y$ $q(x,a,x)=x$ $q(x,\\text{{1}{2}},y_1)=q(x,\\text{{1}{2}},y_2)\\Rightarrow y_1=y_2$ $q(q(x,a,y),b,q(x,c,y))=q(x,p(a,b,c),y)$ The axioms REF to REF are the natural generalizations of axioms REF to REF of a mobi algebra.",
"The natural generalization of REF would be $q(q(x_1,a,y_1),\\text{{1}{2}},q(x_2,a,y_2))=q(q(x_1,\\text{{1}{2}},x_2),a,q(y_1,\\text{{1}{2}},y_2)).$ This condition, however, is too restrictive and is not in general verified by geodesic paths.",
"That is the reason why we do not include it.",
"When (REF ) is satisfied for all $x_1,x_2,y_1,y_2\\in X$ and $a\\in A$ , then we say that the $A$ -mobi space $(X,q)$ is affine and speak of an $A$ -mobi affine space (see Section REF ).",
"Here are some immediate consequences of the axioms for a mobi space.",
"Proposition 3.1 Let $(A,p,0,\\text{{1}{2}},1)$ be a mobi algebra and $(X,q)$ an A-mobi space.",
"It follows that: $q(y,a,x)=q(x,\\overline{a},y)$ $q(y,\\text{{1}{2}},x)=q(x,\\text{{1}{2}},y)$ $q(x,a,q(x,b,y))=q(x,a\\cdot b,y)$ $q(q(x,a,y),b,y)=q(x,a\\circ b,y)$ $q(q(x,a,y),\\text{{1}{2}},q(x,b,y))=q(x,a\\oplus b,y)$ $q(x,\\text{{1}{2}},q(x,a,y))=q(x,a,q(x,\\text{{1}{2}},y))$ $q(q(x,a,y),\\text{{1}{2}},q(y,a,x))=q(x,\\text{{1}{2}},y)$ $q(q(q(x,a,y),b,x),\\text{{1}{2}},q(x,b,q(x,c,y)))\\\\=q(x,\\text{{1}{2}},q(x,p(a,b,c),y))$ $q(x,a,y)=q(y,a,x)\\Rightarrow q(x,a,y)=q(x,\\text{{1}{2}},y)$ $q(x,a,y)=q(x,b,y)\\Rightarrow q(x,p(a,t,b),y)=q(x,a,y)$ , for all $t$.",
"The following proof of REF , bearing in mind (REF ), uses REF , REF and REF : $q(x,\\overline{a},y)&=&q(x,p(1,a,0),y)\\\\&=&q(q(x,1,y),a,q(x,0,y))\\\\&=&q(y,a,x).$ REF follows directly from REF and REF .",
"Begining with (), REF is a consequence of REF and REF : $q(x,a\\cdot b,y)&=&q(x,p(0,a,b),y)\\\\&=&q(q(x,0,y),a,q(x,b,y))\\\\&=&q(x,a,q(x,b,y)).$ Considering (), property REF follows from REF and REF .",
"$q(q(x,a,y),b,y)&=&q(y,\\overline{b},q(y,\\overline{a},x))\\\\&=&q(y,\\overline{b}\\cdot \\overline{a},x)\\\\&=&q(y,\\overline{a\\circ b},x)\\\\&=&q(x,a\\circ b,y).$ Considering $(\\ref {def_oplus})$ , REF is just a particular case of REF .",
"To prove REF , we use REF , () and REF .",
"$q(x,\\text{{1}{2}},q(x,a,y))&=&q(q(x,0,y),\\text{{1}{2}},q(x,a,y))\\\\&=&q(x,p(0,\\text{{1}{2}},a),y)\\\\&=&q(x,p(0,a,\\text{{1}{2}}),y)\\\\&=&q(q(x,0,y),a,q(x,\\text{{1}{2}},y))\\\\&=&q(x,a,q(x,\\text{{1}{2}},y)).$ The following proof of REF is based on REF , REF and (); $q(q(x,a,y),\\text{{1}{2}},q(y,a,x))&=&q(q(x,a,y),\\text{{1}{2}},q(x,\\overline{a},y))\\\\&=&q(x,p(a,\\text{{1}{2}},\\overline{a}),y)\\\\&=&q(x,\\text{{1}{2}},y).$ To prove REF , we start with the important property () of the underlying mobi algebra and then get: $& &q(x,\\text{{1}{2}}\\cdot p(a,b,c),y)=q(x,\\overline{b}\\cdot a\\oplus b\\cdot c,y)\\\\&\\Rightarrow & q(x,p(0,\\text{{1}{2}},p(a,b,c)),y)=q(x,p(\\overline{b}\\cdot a, \\text{{1}{2}},b\\cdot c),y)\\\\&\\Rightarrow & q(x,\\text{{1}{2}},q(x,p(a,b,c),y))=q(q(x,\\overline{b}\\cdot a,y),\\text{{1}{2}},q(x,b\\cdot c,y))\\\\&\\Rightarrow & q(x,\\text{{1}{2}},q(x,p(a,b,c),y))\\\\& &=q(q(x,\\overline{b},q(x,a,y)),\\text{{1}{2}},q(x,b,q(x,c,y))).$ It is easy to see that $\\ref {Y9}$ is a direct consequence of $\\ref {Y5}$ , while $\\ref {Y10}$ is a consequence of REF and REF .",
"The property $\\ref {Y9}$ shows that if someone is travelling from $x$ to $y$ and someone else is travelling from $y$ to $x$ (along a unique geodesic path parametrized by $q$ that connects $x$ and $y$ ) then they meet at $q(x,\\text{{1}{2}},y)$ .",
"The element $\\text{{1}{2}}$ in a mobility algebra is understood as the point at which $p(a,\\text{{1}{2}},b)$ equals $p(b,\\text{{1}{2}},a)$ .",
"When interpreted in terms of a parameter on the parametrized operation $q(x,\\text{{1}{2}},y)$ on a mobility space it has the meaning that, although it may not be one half of the path between $x$ and $y$ in the expected sense (see Examples 2 and 3 in [7]), it is nevertheless the position at half way in the sense of a metric in a metric space.",
"Property $\\ref {Y10}$ shows that if a particle is at the same place at two distinct moments $a$ and $b$ , then it stays there at any instant in-between.",
"A related property, which is perhaps worthwhile studying, is the following one: $\\text{if there exist}\\ a\\ne b \\ \\text{with}\\ q(x,a,y)=q(x,b,y)\\ \\text{then}\\ x=y.$ This behaviour is desirable to model geodesics as shortest paths but it cannot be deduced from the axioms of a mobi space.",
"Indeed, in Example REF , with $h=-1$ , we have $q(x,(t,s),y)=(1-t+s)x+(t-s)y$ and hence, for every appropriate $u$ , $x$ and $y$ , $q(x,(t+u,s+u),y)= q(x,(t,s),y).$ For example, $q(x,(1,0),y)=q(x,(1/2,-1/2),y)=y$ .",
"Nevertheless, this property is satisfied when the mobi algebra is the unit interval and the mobi space is a space with unique geodesic paths.",
"See the remark following Theorem REF ."
],
[
"Examples",
"In this section we give a list of examples and families of examples of mobi spaces over an appropriate mobi algebra."
],
[
"First examples",
"In the list of examples below, the underlying mobi algebra structure $(A, p, 0, \\text{{1}{2}}, 1)$ is the closed unit interval, i.e.",
"$A=[0,1]$ , the three constants are $0, \\frac{1}{2}, 1$ and $p(a,b,c)=(1-b)a+b c,\\ \\textrm {for all}\\ a,b,c\\in A.$ In each case, we present a set $X$ and a ternary operation $q(x,t,y)\\in X$ , for all $x,y\\in X$ , and $t \\in A$ , verifying the axioms of Definition REF .",
"Example 4.1 Mobi spaces $(X,q)$ over the unit interval: Euclidean spaces provide examples of the form $X=\\mathbb {R}^n\\quad (n\\in \\mathbb {N})$ and $q(x,t,y)=(1-t)x+t\\,y.$ The positive real numbers with the metric inspired by the usual euclidean distance after applying a transformation $x\\mapsto \\exp (x)$ provides an example of the form $X=\\mathbb {R}^+$ with $q(x,t,y)=x^{1-t} y^t.$ Another example, still inspired by the euclidean metric but with a different transformation, is $X=\\mathbb {R}^+$ $q(x,t,y)=\\dfrac{x y}{t x+(1-t) y}.$ The previous examples are all particular instances of a family of examples which can be constructed as follows.",
"Start with a function $F$ , invertible on a subset of the real numbers $X\\subseteq \\mathbb {R}$ , and put $q(x,t,y)=F^{-1}((1-t)F(x)+t\\,F(y)).$ For instance, Example REF (REF ) corresponds to $F(x)=\\log x$ while Example REF (REF ) to $F(x)=\\frac{1}{x}$ .",
"Clearly, in Example REF (REF ), $F(x)=x$ ."
],
[
"Other type of examples",
"The canonical formula for euclidean spaces, considered in Example REF (REF ) of the previous subsection, can be adapted to provide examples of mobi spaces in subsets of $\\mathbb {R}^n$ .",
"Example 4.2 Further examples of mobi spaces $(X,q)$ over the unit interval.",
"The two subsets of the plane $X=]-\\pi ,\\pi ]\\times \\mathbb {R}\\ \\textrm {or}\\ X=[0,2\\pi [\\times \\mathbb {R}$ with the formula $q((\\theta _1,z_1),t,(\\theta _2,z_2))=((1-t)\\theta _1+t\\,\\theta _2,(1-t) z_1+t z_2)$ can be viewed as two different choices of geodesic paths on the cylinder $\\lbrace (x,y,z)\\in \\mathbb {R}^3\\mid x=\\cos \\theta , y=\\sin \\theta , (\\theta ,z)\\in X\\rbrace ,$ from the point $(\\cos \\theta _1,\\sin \\theta _1,z_1)$ to the point $(\\cos \\theta _2,\\sin \\theta _2,z_2)$ .",
"For instance, if considering the point in the cylinder $(\\cos (-\\frac{\\pi }{4}),\\sin (-\\frac{\\pi }{4}),0)=(\\cos (\\frac{7\\pi }{4}),\\sin (\\frac{7\\pi }{4}),0)$ and the point $(\\cos (\\frac{\\pi }{4}),\\sin (\\frac{\\pi }{4}),0),$ the two different parametrizations give two different paths between them.",
"Indeed, one path goes through $(1,0,0)$ while the other goes thorough $(-1,0,0)$ .",
"A third choice for a parametrization, which corresponds to the shortest paths for any two points on the cylinder, will be given in subsection REF .",
"Other examples are also possible, such as the set $X=\\mathbb {R}^+\\times \\mathbb {R}$ with the formula $q((x_1,x_2),t,(y_1,y_2))=(x_1+(y_1-x_1)t,x_2+(y_2-x_2)\\frac{y_1\\,t}{(1-t)x_1+t\\,y_1}).$ Or the set $X=\\mathbb {R}^+\\times \\mathbb {R}$ with the formula $q((x_1,x_2),t,(y_1,y_2))=(x_1+(y_1-x_1)t,x_2+\\frac{y_2-x_2}{x_1+y_1}(2\\,x_1 t+(y_1-x_1) t^2)).$ We can also consider the set $X=\\mathbb {R}^2$ and the formula $q((x_1,x_2),t,(y_1,y_2))=$ $(x_1+(y_1-x_1)t,x_2+(y_2-x_2)\\frac{3x_1^2\\,t+3x_1(y_1-x_1)t^2+(y_1-x_1)^2\\,t^3}{x_1^2+x_1y_1+y_1^2}),$ if $(x_1,y_1)\\ne (0,0)$ , and $q((0,x_2),t,(0,y_2))=\\left(0,x_2+t\\,(y_2-x_2)\\right).$ The last three examples are just particular cases of the following family of mobi spaces, where $f$ is an injective real function of one variable and $X\\subseteq \\mathbb {R}^2$ is any set for which the formula $q((x_1,x_2),t,(y_1,y_2))=$ $ \\left(x_1+(y_1-x_1)t,x_2+(y_2-x_2)\\frac{f(x_1+(y_1-x_1) t)-f(x_1)}{f(y_1)-f(x_1)}\\right),$ if $x_1\\ne y_1$ , and $q((x,x_2),t,(x,y_2))=\\left(x,x_2+t\\,(y_2-x_2)\\right),$ defines a map $q\\colon {X\\times [0,1]\\times X\\rightarrow X}$ .",
"Examples REF (REF ), (REF ) and (REF ), are obtained as particular cases of Example REF (REF ), respectively, with $f(x)=\\frac{1}{x}$ , $f(x)=x^2$ and $f(x)=x^3$ .",
"So far we have considered examples of mobi spaces over the unit interval.",
"Here is an example with a different mobi algebra.",
"Example 4.3 For the mobi algebra $(A,p,0,\\text{{1}{2}},1)$ let us use $A=\\left\\lbrace (t_1,t_2)\\in \\mathbb {R}^2\\colon \\vert t_2\\vert \\le t_1 \\le 1-\\vert t_2\\vert \\right\\rbrace $ $\\text{{1}{2}}=\\left(\\frac{1}{2},0\\right)\\,;\\,1=(1,0)\\,;\\,0=(0,0)$ $p(a,b,c)=(a_1-b_1 a_1-b_2 a_2+b_1 c_1+b_2 c_2,\\\\a_2-b_1 a_2-b_2 a_1+b_1 c_2+b_2 c_1).$ And for the mobi space $(X,q)$ , let us use $X=[0,1]$ and $q(x,(t,s),y)= (1-t-h\\,s) x+(t+h\\,s)y,$ with $h=\\pm 1$ .",
"Let us now turn to a special case of mobi spaces, namely affine mobi spaces."
],
[
"Affine mobi spaces",
"Let $(A,p,0,\\text{{1}{2}},1)$ be a mobi algebra.",
"An affine mobi space (over $A$ ) is a mobi space $(X,q)$ for which the condition $q(q(x_1,a,y_1),\\text{{1}{2}},q(x_2,a,y_2))=q(q(x_1,\\text{{1}{2}},x_2),a,q(y_1,\\text{{1}{2}},y_2))$ is satisfied for every $x_1,x_2,y_1,y_2\\in X$ and $a\\in A$ .",
"We observe that the (even) stronger condition $q(q(x_1,a,y_1),b,q(x_2,a,y_2))=q(q(x_1,b,x_2),a,q(y_1,b,y_2))$ for every $x_1,x_2,y_1,y_2\\in X$ and $a,b\\in A$ , is worthwhile studying due to its connection with Proposition 6.4 in [7].",
"If $(X,q)$ is an affine mobi space then we obtain a midpoint algebra $(X,\\oplus )$ by defining $x\\oplus y=q(x,\\text{{1}{2}},y)$ (see [5] for the particulars of the binary operation $\\oplus $ ).",
"In a future work we will investigate the converse.",
"That is, given a midpoint algebra $(X,\\oplus )$ , we will study under which conditions it is obtained from an affine mobi space.",
"Another interesting topic is to study the collection of all mobi space structures which give rise to the same midpoint algebra, developing the concept of homology for mobi affine spaces.",
"For the moment, let us mention that when we choose an origin $e\\in X$ in a given (non-empty) midpoint algebra and assuming that there is an abelian group structure associated to it (in the sense of [5]), then, the set of group endomorphisms on $X$ has a ring structure and, by the use of Proposition REF , we conclude that it gives rise to a mobi algebra.",
"Recall that a midpoint algebra $(X,\\oplus )$ with an associated abelian group is a midpoint algebra with the property that for every $x,y\\in X$ there is an element $(x+y)\\in X$ such that $e\\oplus (x+y)=x\\oplus y$ , then it follows that $(X,+,e)$ is an abelian group with the property that the map $x\\mapsto x+x$ is invertible, with inverse $x\\mapsto e\\oplus x$ .",
"This simple observation gives rise to an important characterization, to be further developed into a future work, which can be stated as follows: The collection of mobi algebra homomorphisms from $A$ to $End(X)$ is in a one-to-one correspondence with the collection of all affine mobi spaces $(X,q)$ such that $\\oplus $ is determined by $q$ .",
"It is remarkable how similar the notion of affine mobi space is from the notion of an affine vector space.",
"In particular, when $A$ has an inverse to $\\text{{1}{2}}$ , say $2\\in A$ , then the operation $+$ always exists and is explicitly given by the formula $x+y=q(e,2,q(x,\\text{{1}{2}},y))$ .",
"In this case, we have precisely the notion of an $A$ -module (see the last Section).",
"Surprisingly, even when $(X,\\oplus )$ does not allow an abelian group structure, a similar characterization is still possible, at the expense of replacing the ring $End(X)$ with a more sophisticated structure, namely $End_e^{\\oplus }(X)$ which is specially designed (Definition REF ) to keep the analogy with the previous result while extending it into a more general setting.",
"The collection of mobi algebra homomorphisms from $A$ to $End_e^{\\oplus }(X)$ is in a one-to-one correspondence with the collection of all affine mobi spaces $(X,q)$ such that $\\oplus $ is determined by $q$ .",
"This and other aspects of affine mobi spaces will be developed thoroughly in the continuation of this study.",
"However, in order to give a glimpse of the kind of results that are expected, we present an example of a mobi space constructed from a midpoint algebra.",
"Example 4.4 Consider the midpoint algebra $(X,\\oplus )$ , with $X=[0,1]$ and $x\\oplus y=\\dfrac{x+y}{2}$ for every $x,y\\in X$ .",
"The fixed element in $X$ is chosen to be $e=0$ .",
"The mobi algebra is $(A,p,(0,0),(\\frac{1}{2},0),(1,0))$ with $A=\\left\\lbrace (x,y)\\in \\mathbb {R}^2\\mid \\vert y\\vert \\le x \\le 1-\\vert y\\vert \\right\\rbrace $ and $p(a,b,c)=(a_1-b_1 a_1-b_2 a_2+b_1 c_1+b_2 c_2,\\\\a_2-b_1 a_2-b_2 a_1+b_1 c_2+b_2 c_1).$ Let us now use Proposition REF to generate an example of a mobi space from a homomorphism of mobi algebras.",
"It is easily checked that, for any $h\\in [-1,1]$ , the map $\\varphi _{h,(a_1,a_2)}\\colon {A\\rightarrow End_e^\\oplus (X)},$ given by $\\varphi _{h,(a_1,a_2)}(x)=(a_1+ h a_2) x$ is well defined in $End_e^\\oplus (X)$ .",
"Indeed, Definition REF is verified with $\\overline{\\varphi }_{h,(a_1,a_2)}(x)=(1-a_1- h a_2) x$ and $\\tilde{\\varphi }_{h,(a_1,a_2)}(x,y)=(1-a_1-h a_2) x+(a_1+h a_2) y.$ Note that $\\overline{\\varphi }_{h,(a_1,a_2)}=\\varphi _{h,\\overline{(a1,a2)}}$ because $\\overline{(a_1,a_2)}=(1-a_1,-a_2)$ follows from (REF ).",
"However, $\\varphi _{h,(a_1,a_2)}$ is an homomorphism of mobi algebras if and only if $h=\\pm 1$ .",
"In these cases, the mobi operations on $X$ , defined by $0\\oplus q(x,(t,s),y)=\\overline{\\varphi }_{\\pm 1,(t,s)}(x)\\oplus \\varphi _{\\pm 1,(t,s)}(y),$ are given by: $q(x,(t,s),y)=(1-t\\mp s) x+(t\\pm s) y.$ The resulting mobi space $(X,q)$ is the same as the one in Example REF .",
"We will now see some examples obtained from physics."
],
[
"Examples with physical interpretation",
"This section ends with Example REF , obtained from the motion of a projectile, some comments on counter-examples, as well as a general example from mechanics.",
"Example 4.5 For any $k\\in \\mathbb {R}$ , we may form a mobi space $(X,q)$ over the unit interval by taking the set $X=\\mathbb {R}^2$ with the formula $q((x_1,x_2),t,(y_1,y_2))=$ $((1-t)\\,x_1+t\\, y_1+k (y_2-x_2)^2 (1-t) t, (1-t)\\, x_2+ t\\, y_2).$ In this example, when $y_2>y_1$ , the operation $q((x_1,y_1),t,(x_2,y_2))$ simply gives the position at instant $t$ of a projectile in one-dimensional classical mechanics with constant acceleration $a_x=-2 k$ , that is moving from position $x_1$ at time $y_1$ to position $x_2$ at time $y_2$ .",
"Example 4.6 If a one-dimensional space $X$ would have been considered to describe the motion of the projectile instead of the Euclidean space-time of Example REF , the ternary operation would have been: $q(x_1,t,x_2)=(1-t)\\,x_1+t\\, x_2+k (1-0)^2 (1-t) t.$ This operation does not verify all the axioms of Definition REF .",
"In particular, if the particle is not at rest, we will never get the idempotency $q(x,t,x)=x$ .",
"Axiom REF is not verified either.",
"Most of the operations $q:X\\times A\\times X \\rightarrow X$ that we can think of will not verify some of the axioms of Definition REF .",
"Let us just point out that a simple example such as $q(x,t,y)=x+t^2(y-x)$ , for $x,y$ in some subset of $\\mathbb {R}^n$ and $t$ in a non trivial subset of $\\mathbb {R}$ , does not in general verify REF .",
"There are also examples where an operation $q$ might verify the axioms of a mobi space but has no mobi space associated (simply because it may not be everywhere defined).",
"That would be the case if Example REF would be generalized to Special Relativity [4].",
"However, in Minkowski space-time, not every two points can be reached from one another if one point is not inside the light cone of the other.",
"Example 4.7 Finally, let us observe that, in general, the solutions of $\\ddot{x}=f(x,\\dot{x},t)$ (where $\\dot{x}$ is the derivative of $x$ with respect to $t$ ), in $\\mathbb {R}^n$ , will not verify the axioms of a mobi space.",
"Nevertheless, in $\\mathbb {R}^{n+1}$ , where the extra dimension is time, and if every two points can be reached from one another, then we can construct a mobi space as explain in Corollary REF .",
"The case $f(x,\\dot{x},t)=-2k$ gives Example REF .",
"To show another example, let us look at a critically damped harmonic oscillator corresponding to $f(x,\\dot{x},t)=-k^2 x-2k \\dot{x}$ .",
"For any $k\\in \\mathbb {R}$ , we obtain the following mobi space $(\\mathbb {R}^2,q)$ over the unit interval with the formula $q((x_1,x_2),t,(y_1,y_2))=$ $((1-t)\\,x_1\\,e^{k t (x_2-y_2)}+t\\, y_1\\,e^{k (1-t) (y_2-x_2)}, (1-t)\\, x_2+ t\\, y_2).$ In the following section we will thoroughly analyse examples occurring from spaces with geodesic paths."
],
[
"Examples from Geodesics",
"In this section we analyse spaces which satisfy the equation for geodesics and observe that they all give rise to a mobi space.",
"We have decided, for simplicity, to express the results within the scope of $\\mathbb {R}^n$ , but it is clear that the same principle will be valid for pseudo-Riemmanian manifolds with appropriate tensor metrics.",
"This would, however, require a more sophisticated level of abstraction which is beyond our current purpose, namely to show the existence of examples of mobi spaces that are obtained from spaces with geodesics.",
"We also show that under suitable choices for appropriate sections, every identification space inherits the mobi space structure from its covering space.",
"We illustrate this concept with the case of the cylinder (see Example REF (REF ) and Example REF )."
],
[
"Spaces with unique geodesics",
"Let $X,V\\subseteq \\mathbb {R}^{n}$ be two open subsets of the $n$ -dimensional space, with $V$ a vector space, and $g\\colon {X\\times V\\rightarrow \\mathbb {R}^{n}}$ a map such that $g(x,\\lambda v)=\\lambda ^2g(x,v),$ for all $x\\in X$ , $v\\in V$ and $\\lambda \\in \\mathbb {R}$ .",
"Moreover, the initial value problem $\\left\\lbrace \\begin{matrix}x^{\\prime }=v &,& x(0)=x_0\\in X\\\\v^{\\prime }=g(x,v) &,& v(0)=v_0\\in V\\end{matrix}\\right.$ is supposed to have a unique solution, denoted by $x(t)=\\pi (x_0,v_0,t),$ for every pair $(x_0,v_0)\\in X\\times V$ of initial conditions.",
"This means that, for fixed $x\\in X$ and $v\\in V$ , given any continuous map $f\\colon {\\mathbb {R}\\rightarrow X}$ , with continuous derivatives $f^{\\prime }\\colon {\\mathbb {R}\\rightarrow V}$ and $f^{\\prime \\prime }\\colon {\\mathbb {R}\\rightarrow \\mathbb {R}^n}$ , if $f(0)=x,\\quad f^{\\prime }(0)=v,\\quad f^{\\prime \\prime }(t)=g(f(t),f^{\\prime }(t))$ for every $t\\in \\mathbb {R}$ , then $f(t)=\\pi (x,v,t).$ See Example REF for an illustration.",
"Of course, the function $\\pi $ has the following properties: $ \\pi (x,v,0)&=&x\\\\ \\pi ^{\\prime }(x,v,0)&=&v\\\\ \\pi ^{\\prime \\prime }(x,v,s)&=&g(\\pi (x,v,s),\\pi ^{\\prime }(x,v,s)),$ for every $s\\in \\mathbb {R}$ , where $\\pi ^{\\prime }(x,v,s)$ stands for the derivative with respect to the variable $s$ .",
"Lemma 5.1 Consider an initial value problem such as $(\\ref {initial value problem})$ .",
"The following conditions hold for every $x\\in X$ , $v\\in V$ , and $s,t, u\\in \\mathbb {R}$ : $\\pi (x,0,t)&=&x\\\\\\pi (x,v,s+u\\,t)&=&\\pi (\\pi (x,v,s),u\\,\\pi ^{\\prime }(x,v,s),t).$ Considering the map $f(t)=x$ for the first case and the map $f(t)=\\pi (x,v,s+ut)$ for the second case, $f$ satisfies the conditions (REF ) for each case.",
"The desired equalities then follow from the uniqueness of solutions with respect to initial conditions, as expressed in condition (REF ).",
"Some useful particular cases of the previous lemma are: $\\pi (x,v,s\\,t)&=&\\pi (x,s\\,v,t)\\\\\\pi (x,v,1-t)&=&\\pi (\\pi (x,v,1),-\\pi ^{\\prime }(x,v,1),t)\\\\\\pi (x,v,t)&=&\\pi (\\pi (x,v,s),(t-s)\\pi ^{\\prime }(x,v,s),1).$ Property () follows from $\\pi (x,v,t)=\\pi (x,v,s+(t-s)\\,1)$ .",
"Lemma 5.2 Consider an initial value problem such as $(\\ref {initial value problem})$ and suppose the existence of a map $\\beta \\colon {X\\times X\\rightarrow V}$ such that $\\pi (x,\\beta (x,y),1)=y,\\\\$ for every $x,y\\in X$ .",
"Then the following two conditions are equivalent: $\\pi (x,v_1,1)=\\pi (x,v_2,1)\\Rightarrow v_1=v_2,\\quad \\forall x\\in X, v_1,v_2\\in V$ the map $\\beta $ with the property $(\\ref {lemmabeta1.1})$ is unique.",
"Moreover, we always have: $\\beta (x,y_1)=\\beta (x,y_2)\\Rightarrow y_1=y_2.$ Assuming condition $(a)$ , if there exists another map with the same property as $\\beta $ , say $\\hat{\\beta }\\colon {X\\times X\\rightarrow V}$ , then, for every $x,y\\in X$ , we have $\\pi (x,\\beta (x,y),1)=y=\\pi (x,\\hat{\\beta }(x,y),1).$ Now, from $(a)$ it immediately follows that $\\beta =\\hat{\\beta }$ .",
"Conversely, let be given any $x\\in X$ and $v_1,v_2\\in V$ such that $\\pi (x,v_1,1)=\\pi (x,v_2,1)$ and let us denote by $y\\in X$ , both $\\pi (x,v_1,1)$ and $\\pi (x,v_2,1)$ .",
"We now consider $\\beta (x,y)\\in V$ and observe that also $\\pi (x,\\beta (x,y),1)=y$ and hence, by the uniqueness of $\\beta $ , we obtain $\\beta (x,y)=v_1=v_2,$ which concludes the first part of the proof.",
"For the remaining part we simply observe that if $\\beta (x,y_1)=\\beta (x,y_2)$ then $y_1=\\pi (x,\\beta (x,y_1),1)=\\pi (x,\\beta (x,y_2),1)=y_2.$ An interesting consequence of Lemma REF , when applied together with property (REF ), is that within an initial value problem such as $(\\ref {initial value problem})$ where there exists a unique $\\beta $ verifying (REF ), we have, for $s\\ne 0$ , $x\\in X$ and $v_1,v_2\\in V$ , $\\pi (x,v_1,s)=\\pi (x,v_2,s)\\Rightarrow \\pi (x,s\\,v_1,1)=\\pi (x,s\\,v_2,1)\\Rightarrow v_1=v_2.$ Lemma 5.3 Consider an initial value problem such as $(\\ref {initial value problem})$ .",
"If there exists a unique $\\beta \\colon {X\\times X\\rightarrow V}$ such that, for all $x, y \\in X$ , $\\pi (x,\\beta (x,y),1)=y$ then we also have: $\\beta (x,x)&=&0\\\\s\\,\\beta (x,y)&=&\\beta (x,\\pi (x,\\beta (x,y),s))\\\\\\beta (\\pi (x,v,s),\\pi (x,v,t))&=&(t-s) \\pi ^{\\prime }(x,v,s).$ for every $x\\in X$ , $v\\in V$ and $t,s\\in \\mathbb {R}$ .",
"From (REF ), we have $\\pi (x,0,1)=x$ and so $\\beta (x,x)=0$ because $\\beta (x,x)$ is the unique element in $V$ with the property $\\pi (x,\\beta (x,x),1)=x.$ Similarly we conclude, from (REF ) with $t=1$ , that () holds, indeed $\\beta (x,\\pi (x,\\beta (x,y),s))$ is the unique element in $V$ such that $\\pi (x,\\beta (x,\\pi (x,\\beta (x,y),s)),1)=\\pi (x,\\beta (x,y),s).$ Finally, we observe that from (REF ) we get: $\\pi (\\pi (x,v,s),\\beta (\\pi (x,v,s),\\pi (x,v,t)),1)=\\pi (x,v,t)$ which, using (), implies $&&\\pi (\\pi (x,v,s),(t-s)\\pi ^{\\prime }(x,v,s),1)\\\\&=&\\pi (\\pi (x,v,s),\\beta (\\pi (x,v,s),\\pi (x,v,t)),1).$ Therefore, the unicity of $\\beta $ proves ().",
"A particular case of the previous lemma is: $\\pi ^{\\prime }(x,\\beta (x,y),1)&=&-\\beta (y,x).$ Theorem 5.4 Consider an initial value problem such as $(\\ref {initial value problem})$ in which there is a unique $\\beta \\colon {X\\times X\\rightarrow V}$ such that $(\\ref {lemmabeta1})$ holds.",
"Then, the structure $(X,q)$ , with $q(x,t,y)=\\pi (x,\\beta (x,y),t)$ is a mobi space over the mobi algebra $([0,1],p,0,\\frac{1}{2},1)$ , where $p(s,u,t)=s+u\\,(t-s).$ REF is simply a consequence of (REF ) and REF of (REF ): $q(x,0,y)=\\pi (x,\\beta (x,y),0)&=&x,\\\\q(x,1,y)=\\pi (x,\\beta (x,y),1)&=&y.$ To prove REF , we use (REF ) and (REF ): $q(x,t,x)=\\pi (x,\\beta (x,x),t)=\\pi (x,0,t)=x.$ Lemma REF and property (REF ) imply REF : $q(x,\\text{{1}{2}},y_1)=q(x,\\text{{1}{2}},y_2)&\\Rightarrow & \\pi (x,\\beta (x,y_1),\\text{{1}{2}})=\\pi (x,\\beta (x,y_2),\\text{{1}{2}})\\\\&\\Rightarrow & \\beta (x,y_1)=\\beta (x,y_2)\\\\&\\Rightarrow & y_1=y_2.$ Using () and (), we obtain REF : $&&q(q(x,s,y),u,q(x,t,y))\\\\&=& \\pi (\\pi (x,\\beta (x,y),s),\\beta (\\pi (x,\\beta (x,y),s),\\pi (x,\\beta (x,y),t)),u)\\\\&=& \\pi (\\pi (x,\\beta (x,y),s),(t-s)\\pi ^{\\prime }(x,\\beta (x,y),s),u)\\\\&=& \\pi (x,\\beta (x,y),s+u\\,(t-s))\\\\&=& q(x,p(s,u,t),y).$ Note that, when $s\\ne 0$ , from $(\\ref {pi-cancel})$ , we also deduce that $q(x,s,y_1)=q(x,s,y_2)$ implies $y_1=y_2$ .",
"Furthermore, in this case, the property (REF ) is verified.",
"Corollary 5.5 If $S\\subseteq \\mathbb {R}^n$ is a Riemann surface with a unique geodesic path between any two points, then $(S,q)$ is a mobi space with $q(x,t,y)$ being the position at an instant $t$ on the geodesic path between $x$ and $y$ .",
"If $\\Gamma _{ij}^{k}\\colon {S\\rightarrow \\mathbb {R}}$ are the Christoffel symbols (see [11], for instance) for the metric, then the function $g$ in the initial value problem (REF ) is of the form $g_k(x,y)=-\\sum _{i,j}y_{i}\\Gamma _{ij}^{k}(x)y_j$ and henceforth satisfies (REF ).",
"The existence of unique geodesic paths between any two points implies the uniqueness of $\\beta $ in Theorem REF .",
"Corollary 5.6 Consider the functions $f:\\mathbb {R}^n\\times \\mathbb {R}^n\\times \\mathbb {R}\\rightarrow \\mathbb {R}^n$ and $x: \\mathbb {R}\\rightarrow \\mathbb {R}^n$ , and let $\\dot{x}$ and $\\ddot{x}$ be the first and second derivatives of $x$ with respect to the variable $t\\in \\mathbb {R}$ .",
"If the following problem $\\left\\lbrace \\begin{matrix}\\ddot{x}=f(x,\\dot{x},t)\\\\x(t_1)=x_1\\\\x(t_2)=x_2\\end{matrix}\\right.$ has a unique solution for any $x_1, x_2\\in \\mathbb {R}^n$ and $t_1, t_2\\in \\mathbb {R}$ , $t_2\\ne t_1$ , expressed as $x(t)=F(x_1,t_1,x_2,t_2,t)$ , then $(\\mathbb {R}^{n+1},q)$ is a mobi space over the mobi algebra $([0,1],p,0,\\frac{1}{2},1)$ , with $p\\left(t_1,s,t_2\\right)=t_1+s(t_2-t_1),$ $q\\left((x_1,t_1),s,(x_2,t_2)\\right)=\\left(F(x_1,t_1,x_2,t_2,p(t_1,s,t_2)),p(t_1,s,t_2)\\right).$ Considering the variable $s$ such that $p(t_1,s,t_2)=t$ , and denoting by $x^{\\prime }$ and $x^{\\prime \\prime }$ the first and second derivatives of $x$ with respect to s, we get $x^{\\prime }=t^{\\prime } \\dot{x}$ and $x^{\\prime \\prime }=t^{\\prime 2} \\ddot{x}$ .",
"With $\\chi =(x,t)\\in \\mathbb {R}^{n+1}$ and $\\omega =(x^{\\prime },t^{\\prime })$ , we then have: $\\left\\lbrace \\begin{matrix}\\chi ^{\\prime }=\\omega \\\\\\omega ^{\\prime }=g(\\chi ,\\omega )\\end{matrix}\\right.\\ , \\textrm {with } g((x,t),(x^{\\prime },t^{\\prime }))=(t^{\\prime 2} f(x,\\frac{x^{\\prime }}{t^{\\prime }},t),0).$ It is obvious that $g(\\chi ,\\lambda \\omega )=\\lambda ^2 g(\\chi ,\\omega )$ .",
"The unicity of a solution of $(\\ref {xdot-problem})$ implies that there exists a initial value problem (at the initial value $s=0$ ), such as $(\\ref {initial value problem})$ , equivalent to $(\\ref {xdot-problem})$ .",
"Therefore, Theorem REF can be applied and proves this Corollary.",
"The mobi spaces of Examples REF and REF are illustrations of the result of Corollary REF .",
"Example 5.1 To illustrate the result of Theorem REF , consider the explicit example where $X=\\mathbb {R}^2$ , $V=\\mathbb {R}^2$ and, with $k\\in R$ , $g:X\\times V &\\rightarrow & \\mathbb {R}^2\\\\((a_1,a_2),(w_1,w_2) )&\\rightarrow & (-2 k w_2^2,0).$ The solution of the following initial value problem, involving the functions $x:\\mathbb {R}\\rightarrow X$ and $v:\\mathbb {R}\\rightarrow V$ , $\\left\\lbrace \\begin{matrix}x^{\\prime }=v &,& x(0)=(x_1,x_2)\\in X\\\\v^{\\prime }=g(x,v) &,& v(0)=(v_1,v_2)\\in V\\end{matrix}\\right.$ is $ \\pi ((x_1,x_2),(v_1,v_2),t)=(x_1+v_1 t-k v_2^2 t^2,x_2+v_2 t)$ It is easy to see that there is a unique $\\beta :X\\times X\\rightarrow V$ such that $\\pi ((x_1,x_2),\\beta ((x_1,x_2),(y_1,y_2)),1)=(y_1,y_2)$ given by: $\\beta ((x_1,x_2),(y_1,y_2))=(y_1-x_1+k (y_2-x_2)^2,y_2-x_2).$ We then obtain the ternary operation $q:X\\times [0,1]\\times X\\rightarrow X$ of the mobi space $(X,q)$ which is given by: $&&q((x_1,x_2),t,(y_1,y_2))=\\pi ((x_1,x_2),\\beta ((x_1,x_2),(y_1,y_2)),t)\\\\&=&(x_1+(y_1-x_1) t+k (y_2-x_2)^2 (t-t^2), x_2+(y_2-x_2)t).$ This is, in fact, Example REF .",
"For an explicit example of geodesics on a two dimensional surface, we refer to [6].",
"Metric spaces in which there is a unique geodesic through any two points have been characterized in [1] (see also [8])."
],
[
"The cylinder as an identification space and its geodesics",
"If the uniqueness of the geodesic paths in a Riemannian surface is not guaranteed, we may still choose appropriate values for the map $\\beta $ as in Proposition REF and obtain a mobi space.",
"For example, although, on a cylinder, there are infinitely many geodesics through two points that do not lie on the same circle, we can construct mobi spaces over the set of points of a cylinder as illustrated in Example REF (REF ) of Section .",
"See, for example [2] for the study of spaces in which a geodesic through two points is not unique.",
"Proposition 5.7 Let $(X,q)$ be a mobi space over the mobi algebra $(A,p,0,\\text{{1}{2}},1)$ and $h\\colon {X\\rightarrow S}$ a map onto a set $S$ .",
"If we can find two maps $s\\colon {S\\rightarrow X}$ and $\\theta \\colon {S\\times S\\rightarrow X}$ , satisfying the following conditions for every $u,v,v^{\\prime }\\in S$ $a_1,a_2,a_3\\in A$ $hs(u) &=& u\\\\h\\theta (u,v) &=& v\\\\\\theta (u,u)&=&s(u)\\\\hq(s(u),\\text{{1}{2}},\\theta (u,v))&=&hq(s(u),\\text{{1}{2}},\\theta (u,v^{\\prime }))\\Rightarrow v=v^{\\prime } \\\\hq(s(u),p(a_1,a_2,a_3),\\theta (u,v))&=& hq(shq(s(u),a_1,\\theta (u,v)),a_2,\\cdots \\\\&& \\theta (hq(s(u),a_1,\\theta (u,v)),hq(s(u),\\cdots \\\\&& a_3,\\theta (u,v))))$ then $(S,q_S)$ is a mobi space over $A$ with $q_S(u,a,v)=hq(s(u),a,\\theta (u,v)),$ for every $u,v\\in S$ and $a\\in A$ .",
"It is clear that the axioms of a mobi space follow from the conditions that are assumed to be satisfied by the three maps.",
"As an example we show how to obtain geodesic paths on a cylinder from geodesic paths on a plane.",
"Example 5.2 let $(X,q)$ be the euclidean plane with the usual geodesic paths over the mobi algebra of the unit interval.",
"Denote by $S$ the set $[0,2\\pi [\\times \\mathbb {R}$ and define three maps: $h$ , $s$ and $\\theta $ .",
"The map $h\\colon {X\\rightarrow S}$ is defined as $h(x,y)=(x\\mod {2}\\pi , y)$ , the map $s\\colon {S\\rightarrow X}$ is the inclusion map and $\\theta \\colon {S\\times S\\rightarrow X}$ is defined as $\\theta ((x_1,y_1),(x_2,y_2))=\\left\\lbrace \\begin{array}{lcl}(x_2+2\\pi ,y_2) & \\text{if} & x_2-x_1< -\\pi \\\\(x_2,y_2) & \\text{if} & -\\pi \\le x_2-x_1\\le \\pi \\\\(x_2-2\\pi ,y_2) & \\text{if} & \\pi < x_2-x_1\\end{array}\\right..$ It can be checked that these maps satisfy the conditions of the previous proposition and hence they define a mobi space on the set $S$ .",
"The resulting mobi space structure in this case is not the same as the ones presented in Example REF (REF ).",
"Indeed, in this case, we obtain a parametrization for the cylinder which induces, via the mobi space, the shortest path between any two points.",
"That was not the case in Example REF (REF )."
],
[
"Comparison with R-modules",
"Consider a unitary ring $(R,+,\\cdot ,0,1)$ .",
"It has been proven [7] that if $R$ contains the inverse of $1+1$ , then it is a mobi algebra and if a mobi algebra $(A,p,0,\\text{{1}{2}},1)$ contains the inverse of $\\text{{1}{2}}$ , in the sense of (), then it is a unitary ring.",
"In this section, we will compare a module over a ring $R$ with a mobi space over a mobi algebra A.",
"First, let us just recall that a module over a ring $R$ is a system $(M,+,e,\\varphi )$ , where $\\varphi :R\\rightarrow End(M)$ is a map from $R$ to the usual ring of endomorphisms, such that $(M,+,e)$ is an abelian group and $\\varphi $ is a ring homomorphism.",
"The following Theorem shows how to construct a mobi space from a module over a ring containing the inverse of 2.",
"Theorem 6.1 Consider a module $(X,+,e,\\varphi )$ over a unitary ring $(A,+,\\cdot ,0,1)$ .",
"If $A$ contains $(1+1)^{-1}=\\text{{1}{2}}$ then $(X,q)$ is an affine mobi space over the mobi algebra $(A,p,0,\\text{{1}{2}},1)$ , with $p(a,b,c)&=&a+b c-b a\\\\q(x,a,y)&=&\\varphi _{1-a}(x)+\\varphi _a(y).$ $(A,p,0,\\text{{1}{2}},1)$ is a mobi algebra by Theorem 7.2 of [7].",
"We show here that the axioms of Definition REF , as well as (REF ), are verified.",
"The first three axioms are easily proved: $q(x,0,y)&=&\\varphi _1(x)+\\varphi _0(y)=x+ e=x\\\\q(x,1,y)&=&\\varphi _0(x)+\\varphi _1(y)=e+ y=y\\\\q(x,a,x)&=&\\varphi _{1-a}(x)+\\varphi _a(x)=\\varphi _{1-a+a}(x)=\\varphi _1(x)=x.$ Axiom REF is due to the fact that $\\text{{1}{2}}+\\text{{1}{2}}=1$ and consequently $\\varphi _\\text{{1}{2}}(y_1)=\\varphi _\\text{{1}{2}}(y_2)&\\Rightarrow & \\varphi _\\text{{1}{2}}(y_1)+\\varphi _\\text{{1}{2}}(y_1)=\\varphi _\\text{{1}{2}}(y_2)+\\varphi _\\text{{1}{2}}(y_2)\\\\ &\\Rightarrow & \\varphi _1(y_1)=\\varphi _1(y_2)\\Rightarrow y_1=y_2.$ Next, we give a proof of Axiom REF .",
"It is relevant to notice that, besides other evident properties of the module $X$ , the associativity of $+$ plays an important part in the proof: $&&q(q(x,a,y),b,q(x,c,y))\\\\&=&\\varphi _{1-b}(\\varphi _{1-a}(x)+\\varphi _a(y))+\\varphi _b(\\varphi _{1-c}(x)+\\varphi _c(y))\\\\&=&\\varphi _{1-b}(\\varphi _{1-a}(x))+\\varphi _{1-b}(\\varphi _a(y))+\\varphi _b(\\varphi _{1-c}(x))+\\varphi _b(\\varphi _c(y))\\\\&=&\\varphi _{(1-b)(1-a)}(x)+\\varphi _{b(1-c)}(x)+\\varphi _{(1-b)a}(y)+\\varphi _{bc}(y)\\\\&=&\\varphi _{1-a+ba-bc}(x)+\\varphi _{a-ba+bc}(y)\\\\&=&\\varphi _{1-p(a,b,c)}(x)+\\varphi _{p(a,b,c)}(y)\\\\&=&q(x,p(a,b,c),y).$ It remains to prove (REF ): $&&q(q(x_1,a,y_1),\\text{{1}{2}},q(x_2,a,y_2))\\\\&=&\\varphi _{\\text{{1}{2}}}(\\varphi _{1-a}(x_1)+\\varphi _a(y_1))+\\varphi _\\text{{1}{2}}(\\varphi _{1-a}(x_2)+\\varphi _a(y_2))\\\\&=&\\varphi _{\\text{{1}{2}}}(\\varphi _{1-a}(x_1)+\\varphi _a(y_1)+\\varphi _{(1-a)}(x_2)+\\varphi _a(y_2))\\\\&=&\\varphi _{\\text{{1}{2}}}(\\varphi _{1-a}(x_1+ x_2)+\\varphi _a(y_1+ y_2))\\\\&=&\\varphi _{(1-a)\\text{{1}{2}}}(x_1+ x_2)+\\varphi _{a\\text{{1}{2}}}(y_1+ y_2)\\\\&=&\\varphi _{(1-a)}(\\varphi _\\text{{1}{2}}(x_1)+\\varphi _\\text{{1}{2}}(x_2))+\\varphi _a(\\varphi _{\\text{{1}{2}}}(y_1)+\\varphi _{\\text{{1}{2}}}(y_2))\\\\&=&\\varphi _{(1-a)}(q(x_1,\\text{{1}{2}},x_2))+\\varphi _{a}(q(y_1,\\text{{1}{2}},y_2))\\\\&=&q(q(x_1,\\text{{1}{2}},x_2)),a,q(y_1,\\text{{1}{2}},y_2)).$ Theorem 6.2 Consider a mobi space $(X,q)$ , with a fixed chosen element $e\\in X$ , over a mobi algebra $(A,p,0,\\text{{1}{2}},1)$ .",
"If $A$ contains 2 such that $p(0,\\text{{1}{2}},2)=1$ then $(X,+,e,\\varphi )$ is a module over the unitary ring $(A,+,\\cdot ,0,1)$ , with $a+b&=&p(0,2,p(a,\\text{{1}{2}},b))\\\\a\\cdot b&=&p(0,a,b)\\\\\\varphi _a(x)&=&q(e,a,x)\\\\x+ y&=&q(e,2,q(x,\\text{{1}{2}},y))=\\varphi _2(q(x,\\text{{1}{2}},y)).$ $(A,+,\\cdot ,0,1)$ is a unitary ring by Theorem 7.1 of [7].",
"We prove here that $(X,+,e,\\varphi )$ is a module over $A$ .",
"First, we observe that, using in particular REF of Proposition REF , we have: $q(e,\\text{{1}{2}},x+y)&=& q(e,\\text{{1}{2}},q(e,2,q(x,\\text{{1}{2}},y)))\\\\&=& q(e,\\text{{1}{2}}\\cdot 2,q(x,\\text{{1}{2}},y))\\\\&=& q(e,1,q(x,\\text{{1}{2}},y))\\\\&=& q(x,\\text{{1}{2}},y).$ Then, the property (REF ) of an affine mobi space is essential to prove the associativity of the operation $+$ of the module: $q(e,\\text{{1}{2}},q(e,\\text{{1}{2}},(x+ y)+ z))&=&q(q(e,\\text{{1}{2}},e),\\text{{1}{2}},q(x+ y,\\text{{1}{2}},z))\\\\&=& q(q(e,\\text{{1}{2}},x+ y),\\text{{1}{2}},q(e,\\text{{1}{2}},z))\\\\&=& q(q(x,\\text{{1}{2}},y),\\text{{1}{2}},q(e,\\text{{1}{2}},z))\\\\&=& q(q(x,\\text{{1}{2}},e),\\text{{1}{2}},q(y,\\text{{1}{2}},z))\\\\&=& q(q(e,\\text{{1}{2}},x),\\text{{1}{2}},q(e,\\text{{1}{2}},y+ z))\\\\&=& q(q(e,\\text{{1}{2}},e),\\text{{1}{2}},q(x,\\text{{1}{2}},y+ z))\\\\&=& q(e,\\text{{1}{2}},q(e,\\text{{1}{2}},x+(y+ z)))\\\\$ Which, by REF , implies that $(x+ y)+ z=x+(y+ z)$ .",
"Commutativity of $+$ and the identity nature of $e$ are easily proved: $q(e,\\text{{1}{2}},e+ x)&=&q(e,\\text{{1}{2}},x)\\Rightarrow e+ x=x\\\\q(e,\\text{{1}{2}},x+ y)&=&q(x,\\text{{1}{2}},y)=q(y,\\text{{1}{2}},x)\\\\&=&q(e,\\text{{1}{2}},y+ x)\\Rightarrow x+ y=y+ x.$ Cancellation is achieved with $-x=q(e,p(1,2,0),x)$ .",
"Indeed: $q(e,\\text{{1}{2}},q(e,p(1,2,0),x)+ x)&=&q(q(e,p(1,2,0),x),\\text{{1}{2}},x)\\\\&=&q(q(e,p(1,2,0),x),\\text{{1}{2}},q(e,1,x))\\\\&=&q(e,p(p(1,2,0),\\text{{1}{2}},1),x)\\\\&=&q(e,p(1,p(2,\\text{{1}{2}},0),0),x)\\\\&=&q(e,p(1,1,0),x)\\\\&=&q(e,0,x)=e=q(e,\\text{{1}{2}},e)\\\\$ To prove that $\\varphi _a(x+y)=\\varphi _a(x)+\\varphi _a(y)$ , we will again need (REF ): $q(e,\\text{{1}{2}},\\varphi _a(x+y))&=& q(e,\\text{{1}{2}},q(e,a,x+ y))\\\\&=& q(q(e,a,e),\\text{{1}{2}},q(e,a,x+ y))\\\\&=& q(q(e,\\text{{1}{2}},e),a,q(e,\\text{{1}{2}},x+ y))\\\\&=& q(e,a,q(x,\\text{{1}{2}},y))\\\\&=& q(q(e,a,x),\\text{{1}{2}},q(e,a,y))\\\\&=& q(\\varphi _a(x),\\text{{1}{2}},\\varphi _a(y))\\\\&=& q(e,\\text{{1}{2}},\\varphi _a(x)+\\varphi _a(y)).$ To prove that $\\varphi _{a+b}(x)=\\varphi _a(x)+\\varphi _b(x)$ , let us first recall that, in a mobi algebra with 2 and $a+b=p(0,2,p(a,\\text{{1}{2}},b))$ , we have the following property: $p(0,\\text{{1}{2}},a+b)=p(a,\\text{{1}{2}},b).$ We then have $q(e,\\text{{1}{2}},\\varphi _{a+b}(x))&=& q(e,\\text{{1}{2}},q(e,a+b,x))\\\\&=& q(q(e,0,x),\\text{{1}{2}},q(e,a+b,x))\\\\&=& q(e,p(0,\\text{{1}{2}},a+b),x)\\\\&=& q(e,p(a,\\text{{1}{2}},b),x)\\\\&=& q(q(e,a,x),\\text{{1}{2}},q(e,b,x))\\\\&=& q(\\varphi _a(x),\\text{{1}{2}},\\varphi _b(x))\\\\&=& q(e,\\text{{1}{2}},\\varphi _a(x)+\\varphi _b(x)).$ Last two properties are easily proved: $\\varphi _{a\\cdot b}(x)=q(e,a\\cdot b,x)=q(e,a,q(e,b,x))=\\varphi _a(\\varphi _b(x))$ $\\varphi _1(x)=q(e,1,x)=x.$ Proposition 6.3 Consider a R-module $(X,+,e,\\varphi )$ within the conditions of Theorem REF and the corresponding mobi space $(X,q)$ .",
"Then the R-module obtained from $(X,q)$ by Theorem REF is the same as $(X,+,e,\\varphi )$ .",
"From $(X,q)$ , we define $x+^{\\prime }y=q(e,2,q(x,\\text{{1}{2}},y))\\ \\textrm {and}\\ \\varphi ^{\\prime }(x)=q(e,a,x)$ and obtain the following equalities: $x+^{\\prime }y&=&e+\\varphi _2(q(x,\\text{{1}{2}},y))\\\\&=&\\varphi _2(\\varphi _\\text{{1}{2}}(x)+\\varphi _\\text{{1}{2}}(y))\\\\&=&\\varphi _{2\\cdot \\text{{1}{2}}}(x)+\\varphi _{2\\cdot \\text{{1}{2}}}(y)\\\\&=&x+y$ $\\varphi ^{\\prime }_a(x)=\\varphi _{1-a}(e)+\\varphi _a(x)=e+\\varphi _a(x)=\\varphi _a(x).$ Proposition 6.4 Consider an affine mobi space $(X,q)$ within the conditions of Theorem REF and the corresponding module $(X,+,e,\\varphi )$ .",
"Then the affine mobi space obtained from $(X,+,e,\\varphi )$ by Theorem REF is the same as $(X,q)$ .",
"From $(X,+,e,\\varphi )$ , we define $ q^{\\prime }(x,a,y)=\\varphi _{1-a}(x)+\\varphi _a(y)$ and obtain the following equalities: $q^{\\prime }(x,a,y)&=& q(e,\\overline{a},x)+q(e,a,y)\\\\&=& q(x,a,e)+q(e,a,y)\\\\&=& q(e,2,q(q(x,a,e),\\text{{1}{2}},q(e,a,y))).$ Now, because we are considering that $(X,q)$ is affine, we get: $q^{\\prime }(x,a,y)&=& q(q(e,a,e),2,q(q(x,\\text{{1}{2}},e),a,q(e,\\text{{1}{2}},y)))\\\\&=& q(q(e,2,q(e,\\text{{1}{2}},x)),a,q(e,2,q(e,\\text{{1}{2}},y)))\\\\&=& q(q(e,2\\cdot \\text{{1}{2}},x),a,q(e,2\\cdot \\text{{1}{2}},y))\\\\&=& q(x,a,y).$ We have completely characterized affine mobi spaces in terms of modules over a unitary ring in which 2 is invertible.",
"In a sequel to this paper we will investigate how to characterize mobi spaces in terms of homomorphisms between mobi algebras.",
"We will also dedicate our attention to the case of non affine mobi spaces.",
"We finish this section by taking a closer look to Example REF and Example REF (REF ).",
"Example REF is an example of an affine mobi space and can be extended to the case where the underlying mobi algebra is $(\\mathbb {R},p,0,\\frac{1}{2},1)$ .",
"Then by Theorem REF we get a module over $(\\mathbb {R},+,\\cdot ,0,1)$ given by $(\\mathbb {R}^2,+,(0,0),\\varphi )$ with $(x_1,x_2)+(y_1,y_2)&=&(x_1+y_1-2 k x_2 y_2,x_2+y_2)\\\\\\varphi _a(x_1,x_2)&=&(a x_1+k (1-a) a x_2^2,a x_2).$ By Theorem REF , we can construct a mobi space from this module and verify that it is the same as Example REF .",
"Of course, in this example, the module is a vector field and there is a homomorphism, namely $f(x_1,x_2)=(x_1+k (x_2^2-x_2),x_2)$ from $(\\mathbb {R}^2,+,(0,0),\\varphi )$ to the usual vector fied in $\\mathbb {R}^2$ .",
"Example REF (REF ) is a mobi space that it is not affine.",
"Indeed (REF ) is not always verified as, for instance, we have: $q(q((0,0),\\frac{1}{3},(1,0)),\\frac{1}{2},q((1,1),\\frac{1}{3},(0,0)))=(\\frac{1}{2},\\frac{19}{189})$ while $q(q((0,0),\\frac{1}{2},(1,1)),\\frac{1}{3},q((1,0),\\frac{1}{2},(0,0)))=(\\frac{1}{2},\\frac{1}{12}).$ We can also extend its underlying mobi algebra to $(\\mathbb {R},p,0,\\frac{1}{2},1)$ .",
"Theorem REF cannot be applied but we can still find the corresponding $+$ operation and $\\varphi _a$ functions.",
"Choosing $e=(0,0)$ , the results are the following: $&&(x_1,x_2)+(y_1,y_2)\\\\&=&\\left(x_1+y_1,\\dfrac{(x_1^2+4x_1 y_1+7 y_1^2) x_2+(7x_1^2+4x_1y_1+y_1^2) y_2}{x_1^2+x_1y_1+y_1^2}\\right)$ for $( x_1,y_1)\\ne (0,0)$ and $(0,x_2)+(0,y_2)=(0,4x_2+4y_2);$ $\\varphi _a(x_1,x_2)=(a x_1,a^3 x_2).$ The operation $+$ is commutative, admits $e$ as a identity and the symmetric element $(-x_1,-x_2)$ for all $(x_1,x_2)\\in \\mathbb {R}^2$ but it is not associative.",
"On the other side, we have: $\\varphi _a(x+y)&=&\\varphi _{a}(x)+\\varphi _{a}(y)\\\\\\varphi _{a+b}(x)&=&\\varphi _{a}(x)+\\varphi _{b}(x)\\\\\\varphi _{a b}(x)&=&\\varphi _a(\\varphi _b(x))\\\\\\varphi _{1}(x)&=&x\\\\\\varphi _{0}(x)&=&e\\\\$ Beginning with the structure $(X,+,e,\\varphi )$ , we can construct a ternary operation $q$ on $X$ given by (REF ) but then $(X,q)$ is not a mobi space because it doesn't verify axiom REF .",
"This example shows that a mobi space is richer than a structure of the type $(X,+,\\varphi )$ ."
]
] | 2001.03441 |
[
[
"Nontrivial effective lower bounds for the least common multiple of some\n quadratic sequences"
],
[
"Abstract This paper is devoted to studying the numbers $L_{c,m,n} := \\mathrm{lcm}\\{m^2+c ,(m+1)^2+c , \\dots , n^2+c\\}$, where $c,m,n$ are positive integers such that $m \\leq n$.",
"Precisely, we prove that $L_{c,m,n}$ is a multiple of the rational number \\[\\frac{\\displaystyle\\prod_{k=m}^{n}\\left(k^2+c\\right)}{c \\cdot (n-m)!\\displaystyle\\prod_{k=1}^{n-m}\\left(k^2+4c\\right)} ,\\] and we derive (as consequences) some nontrivial lower bounds for $L_{c,m,n}$.",
"We prove for example that if $n- \\frac{1}{2} n^{2/3} \\leq m \\leq n$, then we have $L_{c,m,n} \\geq \\lambda(c) \\cdot n e^{3 (n - m)}$, where $\\lambda(c) := \\frac{e^{- \\frac{2 \\pi^2}{3} c - \\frac{5}{12}}}{(2 \\pi)^{3/2} c}$.",
"Further, it must be noted that our approach (focusing on commutative algebra) is new and different from those using previously by Farhi, Oon and Hong."
],
[
"Introduction and Notation",
"Throughout this paper, we let $\\mathbb {N^*}$ denote the set $\\mathbb {N}\\setminus \\lbrace 0\\rbrace $ of positive integers.",
"For $t\\in \\mathbb {R}$ , we let $\\lfloor t\\rfloor $ and $\\lceil t\\rceil $ respectively denote the floor and the ceiling function.",
"We say that an integer $a$ is a multiple of a non-zero rational number $r$ (or equivalently, $r$ is a divisor of $a$ ) if the ratio $a / r$ is an integer.",
"If $m,n,c$ are positive integers such that $m\\le n$ , we set $L_{c,m,n}:=\\mathrm {lcm}\\left\\lbrace m^2+c,(m+1)^2+c,\\dots ,n^2+c \\right\\rbrace $ .",
"For a given polynomial $P\\in \\mathbb {C}[X]$ , we denote by $\\overline{P}$ its polynomial conjugate in $\\mathbb {C}[X]$ , that is the polynomial we get by replacing each coefficient of $P$ with its complex conjugate.",
"It is well-known that the conjugation of polynomials in $\\mathbb {C}[X]$ is compatible with addition and multiplication, in the sens that for every $P,Q\\in \\mathbb {C}[X]$ , we have $\\overline{P+Q}=\\overline{P}+\\overline{Q}$ and $\\overline{P\\cdot Q}=\\overline{P}\\cdot \\overline{Q}$ .",
"Further, we let $I$ , $E_h$ $(h\\in \\mathbb {R})$ and $\\Delta $ denote the linear operators on $\\mathbb {C}[X]$ which respectively represent the identity, the shift operator with step $h$ ($E_hP\\left(X\\right)=P\\left(X+h\\right)$ , $\\forall P\\in \\mathbb {C}[X]$ ) and the forward difference ($\\Delta P\\left(X\\right)=P\\left(X+1\\right)-P\\left(X\\right)$ , $\\forall P\\in \\mathbb {C}[X]$ ).",
"For $n\\in \\mathbb {N}$ , the expression of $\\Delta ^n$ in terms of the $E_h$ 's is easily obtained from the binomial formula, as follows: $\\Delta ^n=\\left(E_1-I\\right)^n=\\sum _{m=0}^{n}(-1)^{n-m}\\binom{n}{m}E_{1}^m=\\sum _{m=0}^{n}(-1)^{n-m}\\binom{n}{m}E_{m}.$ For falling factorial powers, we use Knuth's notation: $X^{\\underline{n}}:=X\\left(X-1\\right)\\left(X-2\\right)\\cdots \\left(X-n+1\\right)~~~~(\\forall n\\in \\mathbb {N}).$ The study of the least common multiple of the first $n$ consecutive positive integers $(n\\in \\mathbb {N^*})$ began with Chebyshev's work [2] in his attempts to prove the prime number theorem.",
"The latter showed that the prime number theorem is equivalent to stating that $\\log \\mathrm {lcm}\\left(1,2,\\dots ,n\\right)\\sim _{+\\infty } n$ .",
"More recently, many authors are interested in the effective estimates of the least common multiple of consecutive terms of some integer sequences.",
"In 1972, Hanson [6] showed (by leaning on the development of the number 1 in Sylvester series) that $\\mathrm {lcm}\\left(1,2,\\dots ,n\\right)\\le 3^n$ $(\\forall n\\in \\mathbb {N^*})$ .",
"In 1982, investigating the integral $\\int _{0}^{1}x^n(1-x)^n\\mathrm {d}x$ , Nair [11] gave a simple proof that $\\mathrm {lcm}\\left(1,2,\\dots ,n\\right)\\ge 2^n$ $(\\forall n\\ge 7)$ .",
"In the continuation, the second author [5] obtained nontrivial lower bounds for the least common multiple of consecutive terms in an arithmetic progression.",
"In particular, he proved that for any $u_0,r,n\\in \\mathbb {N^*}$ such that $\\gcd \\left(u_0,r\\right)=1$ , we have: $\\mathrm {lcm}\\left(u_0,u_0+r,\\dots ,u_0+nr\\right)\\ge u_0\\left(r+1\\right)^{n-1},$ and conjectured that the exponent $(n-1)$ appearing in the right-hand side of (REF ) can be replaced by $n$ , which is the optimal exponent that can be obtained.",
"That conjecture was confirmed by Hong and Feng [7].",
"Furthermore, several authors obtained improvements of (REF ) for $n$ sufficiently large in terms of $u_{0}$ and $r$ (see e.g., [7],[8] and [10]).",
"The second author [5] also obtained nontrivial lower bounds for some quadratic sequences.",
"In particular, he proved that for any positive integer $n$ , we have: $\\mathrm {lcm}\\left(1^2+1,2^2+1,\\dots ,n^2+1\\right)\\ge 0.32(1.442)^n.$ In 2013, Oon [12] managed to improve (REF ) by proving that for any positive integers $c$ and $n$ , we have: $\\mathrm {lcm}\\left(1^2+c,2^2+c,\\dots ,n^2+c\\right)\\ge 2^n.$ Actually, we have the following result a little stronger: Theorem 1.1 (Oon [12]) Let $c,n,m$ be positive integers such that $m\\le \\left\\lceil \\frac{n}{2}\\right\\rceil $ .",
"Then, we have: $L_{c,m,n}\\ge 2^n.$ In the next, Hong et al.",
"[9] successed to generalize Theorem REF for polynomial sequences $\\left(f(n)\\right)_{n\\ge 1}$ , with $f\\in \\mathbb {Z}[X]$ and the coefficients of $f$ are all nonnegative.",
"In another direction, various asymptotic estimates have been obtained by several authors.",
"For example, Bateman [1] proved that for any $h,k\\in \\mathbb {Z}$ with $k>0$ , $h+k>0$ and $\\gcd (h,k)=1$ , we have: $\\log \\mathrm {lcm}\\lbrace h+k,h+2k,\\dots ,h+nk\\rbrace ~\\sim _{+ \\infty }~ \\left(\\frac{k}{\\varphi (k)} \\sum _{{{\\scriptstyle \\begin{matrix} 1\\le m\\le k \\\\ \\gcd (m , k)=1 \\end{matrix}}}}\\frac{1}{m}\\right) n ,$ where $\\varphi $ denotes the Euler totient function.",
"Another asymptotic estimate a little harder to prove is due to Cilleruelo [3] and states that for every irreducible quadratic polynomial $f\\in \\mathbb {Z}[X]$ , we have: $\\log \\mathrm {lcm}\\lbrace f(1),\\dots ,f(n)\\rbrace = n\\log n +Bn+o(n),$ where $B$ is a constant depending on $f$ .",
"In this paper, we use arguments of commutative algebra and complex analysis to find a nontrivial rational divisor of $L_{c,m,n}$ ($c , m , n \\in {\\mathbb {N}}^*$ ).",
"As a consequence, we derive some new nontrivial lower bounds for $L_{c,m,n}$ .",
"The rest of the paper is organized in four parts (subsections).",
"In the first part, we give an algebraic lemma which allows us, on the one hand to re-demonstrate Theorem REF of Oon by an easy and purely algebraic method, and on the other hand to reformulate the problem of bounding from below the number $L_{c,m,n}$ .",
"In that reformulation, we are leaded to introduce a vital arithmetic function, noted $h_c$ , whose multiple provides a divisor for $L_{c,m,n}$ .",
"In the next two parts, we study the arithmetic function $h_c$ and we find for it a simple multiple.",
"In the last part, we use the obtained multiple of $h_c$ to deduce a nontrivial divisor for $L_{c,m,n}$ .",
"Our new nontrivial lower bounds for $L_{c,m,n}$ then follow from that divisor."
],
[
"An algebraic method",
"Although the method used by Oon [12] to obtain his result (i.e., Theorem REF ) is analytic, the ingredients for its success are algebraic in depth, as we will show it below by applying the following fundamental algebraic lemma: Lemma 2.1 Let $\\mathcal {A}$ be an integral domain and $n$ be a positive integer.",
"Let also $u_0,u_1,\\dots ,u_n$ , $a,b$ be elements of $\\mathcal {A}$ .",
"Suppose that $a$ and $b$ satisfy the following conditions: Each of the elements $u_0,u_1,\\dots ,u_n$ of $\\mathcal {A}$ divides $a$ .",
"Each of the elements $\\prod _{{{\\scriptstyle \\begin{matrix}0\\le j\\le n \\\\ j\\ne i\\end{matrix}}}}\\left(u_i-u_j\\right)$ $(i=0,1,\\dots ,n)$ of $\\mathcal {A}$ divides $b$ .",
"Then the product $ab$ is a multiple of the product $u_0u_1\\cdots u_n$ .",
"If the elements $u_0,u_1,\\dots ,u_n$ of $\\mathcal {A}$ are not pairwise distinct, the result of the lemma is trivial, since by its second condition we have $b=0_{\\mathcal {A}}$ .",
"Suppose for the sequel that the $u_i$ 's $(i=0,1,\\dots ,n)$ are pairwise distinct.",
"We use the well-known result that if a polynomial in one indeterminate, with coefficients in an integral domain, has a number of roots (in that domain) greater than its degree then it is zero.",
"Since $a$ is a multiple of each of the elements $u_0,u_1,\\dots ,u_n$ of $\\mathcal {A}$ , then there exist $k_0,k_1,\\dots ,k_n\\in \\mathcal {A}$ such that: $a=k_0u_0=k_1u_1=\\dots =k_nu_n.$ Similarly, since $b$ is a multiple of each of the elements $\\prod _{{{\\scriptstyle \\begin{matrix}0\\le j\\le n \\\\ j\\ne i\\end{matrix}}}}\\left(u_i-u_j\\right) (i=0,1,\\dots ,n)$ , then there exist $\\ell _0,\\ell _1,\\dots ,\\ell _n\\in \\mathcal {A}$ such that: $b=\\ell _i\\prod _{{{\\scriptstyle \\begin{matrix}0\\le j\\le n \\\\ j\\ne i\\end{matrix}}}}\\left(u_i-u_j\\right)~~\\left(\\forall i\\in \\left\\lbrace 0,1,\\dots ,n\\right\\rbrace \\right).$ Now, consider the following polynomial of $\\mathcal {A}[X]$ : $P\\left(X\\right):=\\sum _{i=0}^{n}\\left[\\ell _i\\prod _{{{\\scriptstyle \\begin{matrix}0\\le j\\le n \\\\ j\\ne i\\end{matrix}}}}\\left(X-u_j\\right)\\right]-b.$ Obviously, we have $\\deg P\\le n$ .",
"On the other hand, we have (according to (REF )): $P\\left(u_i\\right)=0~~\\left(\\forall i\\in \\left\\lbrace 0,1,\\dots ,n\\right\\rbrace \\right),$ showing that the number of roots of $P$ in $\\mathcal {A}$ is greater than its degree.",
"So, according to the elementary result of commutative algebra announced above, the polynomial $P$ is zero.",
"In particular, we have $P\\left(0\\right)=0$ ; that is: $b=(-1)^n\\sum _{i=0}^{n}\\ell _i\\left(\\prod _{{{\\scriptstyle \\begin{matrix}0\\le j\\le n \\\\ j\\ne i\\end{matrix}}}}u_j\\right).$ By multiplying the two sides of this last equality by $a$ , we get (according to (REF )): $ab&=(-1)^n\\sum _{i=0}^{n}\\ell _ia\\left(\\prod _{{{\\scriptstyle \\begin{matrix}0\\le j\\le n \\\\ j\\ne i\\end{matrix}}}}u_j\\right)\\\\&=(-1)^n\\sum _{i=0}^{n}\\ell _ik_iu_i\\left(\\prod _{{{\\scriptstyle \\begin{matrix}0\\le j\\le n \\\\ j\\ne i\\end{matrix}}}}u_j\\right)\\\\&=(-1)^n\\left(\\sum _{i=0}^{n}\\ell _ik_i\\right)u_0u_1\\cdots u_n,$ showing that $ab$ is a multiple of $u_0u_1\\cdots u_n$ , as required.",
"This completes the proof.",
"Now, we use Lemma REF to establish a new proof of Theorem REF , which is purely algebraic.",
"[A new proof of Theorem REF ] Since $L_{c,m,n}$ is obviously non-increasing relative to $m$ , then it suffices to prove the result of the theorem for $m=\\left\\lceil \\frac{n}{2}\\right\\rceil $ , that is $L_{c,\\left\\lceil \\frac{n}{2}\\right\\rceil ,n}\\ge 2^n$ .",
"For simplicity, put $m_0=\\left\\lceil \\frac{n}{2}\\right\\rceil $ .",
"So, we have to show that $L_{c,m_0,n}\\ge 2^n$ .",
"For $n\\in \\left\\lbrace 1,2,\\dots ,6\\right\\rbrace $ , this can be easily checked by hand (as is done by Oon).",
"Suppose for the sequel that $n\\ge 7$ .",
"It is well-known and easily proved that for any integer $r\\ge 7$ , we have $\\left\\lceil \\frac{r}{2}\\right\\rceil \\binom{r}{\\left\\lceil \\frac{r}{2}\\right\\rceil }\\ge 2^{r}$ .",
"According to this inequality for $r=n$ , it suffices to show that $L_{c,m_0,n}\\ge m_0\\binom{n}{m_0}$ .",
"More generally, we shall show that: $L_{c,m^{\\prime },n}\\ge m^{\\prime }\\binom{n}{m^{\\prime }}~~~~(\\forall m^{\\prime }\\in \\mathbb {N^*},~m^{\\prime }\\le n).$ Let $m^{\\prime }\\in \\mathbb {N^*}$ such that $m^{\\prime }\\le n$ .",
"To prove (REF ), we apply Lemma REF for $\\mathcal {A}=\\mathbb {Z}[\\sqrt{-c}]$ by taking for the $u_i$ 's the elements $m^{\\prime }+\\sqrt{-c},m^{\\prime }+1+\\sqrt{-c},\\dots ,n+\\sqrt{-c}$ of $\\mathcal {A}$ and for $a$ and $b$ the integers $a=L_{c,m^{\\prime },n}$ and $b=(n-m^{\\prime })!$ .",
"For any $k\\in \\left\\lbrace m^{\\prime },m^{\\prime }+1,\\dots ,n\\right\\rbrace $ , Since $L_{c,m^{\\prime },n}$ is obviously a multiple (in $\\mathbb {Z}$ , so also in $\\mathcal {A}=\\mathbb {Z}[\\sqrt{-c}]$ ) of $k^2+c$ and $k^2+c=\\left(k+\\sqrt{-c}\\right)\\left(k-\\sqrt{-c}\\right)$ is a multiple (in $\\mathbb {Z}[\\sqrt{-c}]$ ) of $k+\\sqrt{-c}$ , then $L_{c,m^{\\prime },n}$ is a multiple (in $\\mathbb {Z}[\\sqrt{-c}]$ ) of $k+\\sqrt{-c}$ .",
"This shows that the first condition of Lemma REF is satisfied.",
"On the other hand, we have for all $k\\in \\left\\lbrace m^{\\prime },m^{\\prime }+1,\\dots ,n\\right\\rbrace $ : $\\prod _{{{\\scriptstyle \\begin{matrix}m^{\\prime }\\le \\ell \\le n \\\\ \\ell \\ne k\\end{matrix}}}}\\left\\lbrace \\left(k+\\sqrt{-c}\\right)-\\left(\\ell +\\sqrt{-c}\\right)\\right\\rbrace =\\prod _{{{\\scriptstyle \\begin{matrix}m^{\\prime }\\le \\ell \\le n\\\\ \\ell \\ne k\\end{matrix}}}}(k-\\ell )=(-1)^{n-k}(k-m^{\\prime })!",
"(n-k)!,$ which divides (in $\\mathbb {Z}$ , so also in $\\mathbb {Z}[\\sqrt{-c}]$ ) the integer $(n-m^{\\prime })!$ (since $\\frac{(n-m^{\\prime })!",
"}{(k-m^{\\prime })!(n-k)!",
"}=\\binom{n-m^{\\prime }}{k-m^{\\prime }}\\in \\mathbb {Z}$ ).",
"This shows that the second condition of Lemma REF is also satisfied.",
"We thus deduce (by applying Lemma REF ) that $L_{c,m^{\\prime },n}(n-m^{\\prime })!$ is a multiple (in $\\mathbb {Z}[\\sqrt{-c}]$ ) of $\\prod _{k=m^{\\prime }}^{n}\\left(k+\\sqrt{-c}\\right)$ .",
"So, there exist $x,y\\in \\mathbb {Z}$ such that: $L_{c,m^{\\prime },n}(n-m^{\\prime })!=\\left(x+y\\sqrt{-c}\\right)\\prod _{k=m^{\\prime }}^{n}\\left(k+\\sqrt{-c}\\right).$ Then, by taking the modulus in $\\mathbb {C}$ on both sides, we get $L_{c,m^{\\prime },n}(n-m^{\\prime })!=\\sqrt{x^2+cy^{2}}\\prod _{k=m^{\\prime }}^{n}\\sqrt{k^2+c}.$ Next, since $x^2+cy^2\\in \\mathbb {N}$ and $x^2+cy^2\\ne 0$ (because $x^2+cy^2=0$ $\\Longrightarrow $ $L_{c,m^{\\prime },n}=0$ , which is false) then $x^2+cy^2\\ge 1$ .",
"Hence $L_{c,m^{\\prime },n}=\\frac{\\sqrt{x^2+cy^2}\\prod _{k=m^{\\prime }}^{n}\\sqrt{k^2+c}}{(n-m^{\\prime })!",
"}\\ge \\frac{\\prod _{k=m^{\\prime }}^{n}\\sqrt{k^2+c}}{(n-m^{\\prime })!",
"}\\ge \\frac{\\prod _{k=m^{\\prime }}^{n}k}{(n-m^{\\prime })!",
"}=m^{\\prime }\\binom{n}{m^{\\prime }},$ as required.",
"This completes the proof of the theorem.",
"Naturally, we have the following question: How could we improve the Oon lower bound $L_{c,m,n}\\ge \\frac{\\prod _{k=m}^{n}\\sqrt{k^2+c}}{(n-m)!",
"}$ ?.",
"To simplify, suppose that $c=1$ and let $m,n\\in \\mathbb {N^*}$ such that $m\\le n$ .",
"According to Formula (REF ), the positive integer $L_{1,m,n}(n-m)!$ is a multiple (in $\\mathbb {Z}[i]$ ) of the Gauss integer $\\prod _{k=m}^{n}(k+i)$ .",
"Next, by taking the conjugates (in $\\mathbb {C}$ ) of both sides of (REF ), we obtain that $L_{1,m,n}(n-m)!$ is also a multiple (in $\\mathbb {Z}[i]$ ) of the Gauss integer $\\prod _{k=m}^{n}(k-i)$ .",
"It follows from those two facts that $L_{1,m,n}(n-m)!$ is a multiple (in $\\mathbb {Z}[i]$ ) of: ${\\mathrm {lcm}}_{\\mathbb {Z}[i]}\\left\\lbrace \\prod _{k=m}^{n}(k+i),\\prod _{k=m}^{n}(k-i)\\right\\rbrace &=\\frac{\\prod _{k=m}^{n}(k+i)\\cdot \\prod _{k=m}^{n}(k-i)}{{\\gcd }_{\\mathbb {Z}[i]}\\left\\lbrace \\prod _{k=m}^{n}(k+i),\\prod _{k=m}^{n}(k-i)\\right\\rbrace }\\\\&=\\frac{\\prod _{k=m}^{n}(k^2+1)}{{\\gcd }_{\\mathbb {Z}[i]}\\left\\lbrace \\prod _{k=m}^{n}(k+i),\\prod _{k=m}^{n}(k-i)\\right\\rbrace }.$ Consequently $L_{1,m,n}\\ge \\frac{\\prod _{k=m}^{n}(k^2+1)}{(n-m)!\\left|{\\gcd }_{\\mathbb {Z}[i]}\\left\\lbrace \\prod _{k=m}^{n}(k+i),\\prod _{k=m}^{n}(k-i)\\right\\rbrace \\right|}.$ Remarkably, the trivial upper bound $\\left|{\\gcd }_{\\mathbb {Z}[i]}\\left\\lbrace \\prod _{k=m}^{n}(k+i),\\prod _{k=m}^{n}(k-i)\\right\\rbrace \\right|\\le \\left|\\prod _{k=m}^{n}(k+i)\\right|\\le \\prod _{k=m}^{n}\\sqrt{k^2+1}$ suffices to establish the Oon lower bound $L_{1,m,n}\\ge \\frac{\\prod _{k=m}^{n}\\sqrt{k^2+1}}{(n-m)!",
"}$ .",
"So, a nontrivial upper bound for the number $\\left|{\\gcd }_{\\mathbb {Z}[i]}\\left\\lbrace \\prod _{k=m}^{n}(k+i),\\prod _{k=m}^{n}(k-i)\\right\\rbrace \\right|$ certainly gives an improvement of the Oon theorem.",
"On the other hand, for $a,b\\in \\mathbb {Z}$ such that $(a,b)\\ne (0,0)$ , we can easily check that ${\\gcd }_{\\mathbb {Z}[i]}\\left(a+bi,a-bi\\right)$ is not far from ${\\gcd }_{\\mathbb {Z}}(a,b)$ .",
"Precisely, we have: ${\\gcd }_{\\mathbb {Z}[i]}\\left(a+bi,a-bi\\right)=\\left(\\alpha +i\\beta \\right){\\gcd }_{\\mathbb {Z}}(a,b),$ where $\\alpha ,\\beta \\in \\left\\lbrace -1,0,1\\right\\rbrace $ and $(\\alpha ,\\beta )\\ne (0,0)$ .",
"So, for the case $c=1$ , we are leaded to study the arithmetic function: $\\begin{array}{rcl}h :~\\mathbb {Z}[i]\\setminus \\left\\lbrace 0\\right\\rbrace &\\longrightarrow &\\mathbb {N^{*}}\\\\a+bi ~~& \\longmapsto &\\gcd (a,b)\\end{array},$ and precisely to find nontrivial upper bounds for the quantities $h\\left(\\prod _{k=m}^{n}(k+i)\\right)$ $(m,n\\in \\mathbb {N^*},~m\\le n)$ .",
"For the general case $(c\\in \\mathbb {N^*})$ , the arithmetic function we need to study is clearly given by: $\\begin{array}{rcl}h_c :~\\mathbb {Z}[\\sqrt{-c}]\\setminus \\left\\lbrace 0\\right\\rbrace &\\longrightarrow &\\mathbb {N^{*}}\\\\a+b\\sqrt{-c}~~& \\longmapsto &\\gcd (a,b)\\end{array}$ and the quantities we need to bound from above are $h_c\\left(\\prod _{k=m}^{n}(k+\\sqrt{-c})\\right)$ $(m,n\\in \\mathbb {N^*},~m\\le n)$ .",
"The following proposition has as objective to replace a specific arithmetic language of the ring $\\mathbb {Z}[\\sqrt{-c}]$ by its analog (more simple) in $\\mathbb {Z}$ .",
"Proposition 2.2 Let $c\\in \\mathbb {N^*}$ and $N,a,b\\in \\mathbb {Z}$ , with $(a,b)\\ne (0,0)$ .",
"Then, $N$ is a multiple $($ in $\\mathbb {Z}[\\sqrt{-c}])$ of $\\left(a+b\\sqrt{-c}\\right)$ if and only if $N$ is a multiple $($ in $\\mathbb {Z})$ of $\\frac{a^2+cb^2}{\\gcd (a,b)}$ .",
"The result of the proposition is trivial for $b=0$ .",
"Suppose for the sequel that $b\\ne 0$ .",
"Suppose that $N$ is a multiple (in $\\mathbb {Z}[\\sqrt{-c}]$ ) of $\\left(a+b\\sqrt{-c}\\right)$ ; that is there exist $x,y\\in \\mathbb {Z}$ such that: $N=\\left(x+y\\sqrt{-c}\\right)\\left(a+b\\sqrt{-c}\\right).$ By identifying the real and imaginary parts of the two hand-sides of this equality, we get $N&=ax-byc, \\\\ 0&=bx+ay.",
"$ Next, putting $d:=\\gcd (a,b)$ , there exist $a^{\\prime },b^{\\prime }\\in \\mathbb {Z}$ , with $b^{\\prime }\\ne 0$ and $\\gcd (a^{\\prime },b^{\\prime })=1$ , such that $a=da^{\\prime }$ and $b=db^{\\prime }$ .",
"By substituting these in (), we obtain (after simplifying): $b^{\\prime }x=-a^{\\prime }y.$ This last equality shows that $b^{\\prime }$ divides $a^{\\prime }y$ .",
"But since $\\gcd (a^{\\prime },b^{\\prime })=1$ , then (according to the Gauss lemma) $b^{\\prime }$ divides $y$ .",
"So there exists $k\\in \\mathbb {Z}$ such that $y=kb^{\\prime }$ .",
"By reporting this in (REF ), we get $x=-ka^{\\prime }$ .",
"Then, by substituting $x=-ka^{\\prime }=-k\\frac{a}{d}$ and $y=kb^{\\prime }=k\\frac{b}{d}$ in (REF ), we finally obtain $N=-k\\frac{a^2+cb^2}{d}=-k\\frac{a^2+cb^2}{\\gcd (a,b)},$ showing that $N$ is a multiple (in $\\mathbb {Z}$ ) of $\\frac{a^2+cb^2}{\\gcd (a,b)}$ , as required.",
"Conversely, suppose that $N$ is a multiple (in $\\mathbb {Z}$ ) of $\\frac{a^2+cb^2}{\\gcd (a,b)}$ .",
"Then, there exists $k\\in \\mathbb {Z}$ such that: $N=k\\frac{a^2+cb^2}{\\gcd (a,b)}=k\\frac{a-b\\sqrt{-c}}{\\gcd (a,b)}\\left(a+b\\sqrt{-c}\\right)=\\left(k\\frac{a}{\\gcd (a,b)}-k\\frac{b}{\\gcd (a,b)}\\sqrt{-c}\\right)\\left(a+b\\sqrt{-c}\\right).$ Since $\\left(k\\frac{a}{\\gcd (a,b)}-k\\frac{b}{\\gcd (a,b)}\\sqrt{-c}\\right)\\in \\mathbb {Z}[\\sqrt{-c}]$ , the last equality shows that $N$ is a multiple (in $\\mathbb {Z}[\\sqrt{-c}]$ ) of $\\left(a+b\\sqrt{-c}\\right)$ , as required.",
"This completes the proof of the proposition.",
"From Proposition REF , we derive the following corollary, which is the first key step to obtaining the results of this paper.",
"Corollary 2.3 Let $c,m,n\\in \\mathbb {N^*}$ such that $m\\le n$ .",
"Then, the positive integer $L_{c,m,n}(n-m)!$ is a multiple $($ in $\\mathbb {Z})$ of the positive integer: $\\frac{\\prod _{k=m}^{n}\\left(k^2+c\\right)}{h_c\\left(\\prod _{k=m}^{n}\\left(k+\\sqrt{-c}\\right)\\right)}.$ Formula (REF ) (obtained during our new proof of Theorem REF ) shows that $L_{c,m,n}(n-m)!$ is a multiple (in $\\mathbb {Z}[\\sqrt{-c}]$ ) of $\\prod _{k=m}^{n}\\left(k+\\sqrt{-c}\\right)$ .",
"But, according to Proposition REF , this last property is equivalent to the statement of the corollary.",
"In view of Corollary REF , to bound from below $L_{c,m,n}$ $(c,m,n\\in \\mathbb {N^*},~m\\le n)$ , it suffices to bound from above $h_c\\left(\\prod _{k=m}^{n}(k+\\sqrt{-c})\\right)$ .",
"Likewise, to find a nontrivial (rational) divisor of $L_{c,m,n}$ , it suffices to find a nontrivial multiple of $h_c\\left(\\prod _{k=m}^{n}(k+\\sqrt{-c})\\right)$ .",
"This is what we will do in what follows."
],
[
"An explicit Bézout identity",
"In the following, let $c\\in \\mathbb {N^*}$ and $k\\in \\mathbb {N}$ be fixed and define $P_{k}\\left(X\\right)&:=\\left(X+\\sqrt{-c}\\right)\\left(X-1+\\sqrt{-c}\\right)\\cdots \\left(X-k+\\sqrt{-c}\\right):=A_{k}\\left(X\\right)+B_{k}\\left(X\\right)\\sqrt{-c}, \\\\ \\overline{P_{k}}\\left(X\\right)&:=\\left(X-\\sqrt{-c}\\right)\\left(X-1-\\sqrt{-c}\\right)\\cdots \\left(X-k-\\sqrt{-c}\\right):=A_{k}\\left(X\\right)-B_{k}\\left(X\\right)\\sqrt{-c},$ where it is understood that $A_{k},B_{k}\\in \\mathbb {Z}[X]$ .",
"In what follows, we find nontrivial multiples for the positive integers $h_{c}\\left(P_{k}(n)\\right)=\\gcd \\left(A_{k}(n),B_{k}(n)\\right)$ $(n\\ge 1)$ .",
"To do so, we look for two polynomial sequences $\\left(a_{k}(n)\\right)_{n}$ and $\\left(b_{k}(n)\\right)_{n}$ so that the polynomial sequence $\\left(a_{k}(n)A_{k}(n)+b_{k}(n)B_{k}(n)\\right)_{n}$ be independent on $n$ .",
"Clearly, this leads to looking for two polynomials $U_{k},V_{k}\\in \\mathbb {Q}[X]$ which satisfy the Bézout identity: $U_{k}\\left(X\\right)A_{k}\\left(X\\right)+V_{k}\\left(X\\right)B_{k}\\left(X\\right)=1.$ Next, since $A_{k}=\\frac{P_{k}+\\overline{P_{k}}}{2}$ and $B_{k}=\\frac{P_{k}-\\overline{P_{k}}}{2\\sqrt{-c}}$ , the problem is equivalent to looking for $\\alpha _{k},\\beta _{k}\\in \\mathbb {Q}\\left(\\sqrt{-c}\\right)[X]$ such that: $\\alpha _{k}\\left(X\\right)P_{k}\\left(X\\right)+\\beta _{k}\\left(X\\right)\\overline{P_{k}}\\left(X\\right)=1.$ Let us first justify the existence of such $\\alpha _k$ and $\\beta _k$ .",
"Denoting by $Z\\left(P\\right)$ the set of all the complex roots of a polynomial $P\\in \\mathbb {C}[X]$ , we have clearly: $Z\\left(P_{k}\\right)=\\left\\lbrace -\\sqrt{-c},1-\\sqrt{-c},\\dots ,k-\\sqrt{-c}\\right\\rbrace ~\\text{and}~Z\\left(\\overline{P_{k}}\\right)=\\left\\lbrace \\sqrt{-c},1+\\sqrt{-c},\\dots ,k+\\sqrt{-c}\\right\\rbrace ,$ showing that $Z\\left(P_{k}\\right)\\cap Z\\left(\\overline{P_{k}}\\right)=\\emptyset $ ; that is $P_{k}$ and $\\overline{P_{k}}$ do not have a common root in $\\mathbb {C}$ .",
"This implies that $P_{k}$ and $\\overline{P_{k}}$ are coprime in $\\mathbb {C}[X]$ ; so coprime also in $\\mathbb {Q}\\left(\\sqrt{-c}\\right)[X]$ (since $P_{k},\\overline{P_{k}}\\in \\mathbb {Q}\\left(\\sqrt{-c}\\right)[X]$ ).",
"It follows (according to Bézout's theorem) that there exist $\\alpha _{k},\\beta _{k}\\in \\mathbb {Q}\\left(\\sqrt{-c}\\right)[X]$ such that: $\\alpha _{k}P_{k}+\\beta _{k}\\overline{P_{k}}=1$ , as required.",
"Now, to find explicitly such $\\alpha _k$ and $\\beta _k$ , we need the following more precise version of Bézout's theorem: Theorem 2.4 Let $\\mathbb {K}$ be a field and $P$ and $Q$ be two non-constant polynomials of $\\mathbb {K}[X]$ such that ${\\gcd }_{\\mathbb {K}[X]}\\left(P,Q\\right)=1$ .",
"Then, there exists a unique couple $\\left(U,V\\right)$ of polynomials of $\\mathbb {K}[X]$ , with $\\deg U<\\deg Q$ and $\\deg V<\\deg P$ , such that: $PU+QV=1.$ Since ${\\gcd }_{\\mathbb {K}[X]}\\left(P,Q\\right)=1$ , then (according to Bézout's theorem) there exist $U_0,V_0\\in \\mathbb {K}[X]$ such that: $PU_0+QV_0=1.$ Next, consider in $\\mathbb {K}[X]$ the euclidean division of $U_0$ by $Q$ and the euclidean division of $V_0$ by $\\left(-P\\right)$ : $U_0&=U_1Q+U\\\\V_0&=V_1\\left(-P\\right)+V,$ where $U_1,V_1,U,V\\in \\mathbb {K}[X]$ , $\\deg U<\\deg Q$ and $\\deg V<\\deg \\left(-P\\right)=\\deg P$ .",
"So, we have $PU+QV = P\\left(U_0-U_1Q\\right)+Q\\left(V_0+V_1P\\right) = PQ\\left(V_1-U_1\\right)+PU_0+QV_0 = PQ\\left(V_1-U_1\\right)+1 .$ If $V_1-U_1\\ne 0$ , then the last equality implies that $\\deg \\left(PU+QV\\right)\\ge \\deg \\left(PQ\\right)$ , which is impossible, since $\\deg U<\\deg Q$ and $\\deg V<\\deg P$ .",
"Thus $V_1-U_1=0$ , which gives $PU+QV=1$ .",
"The existence of the couple $\\left(U,V\\right)$ as required by the theorem is proved.",
"It remains to prove the uniqueness of $\\left(U,V\\right)$ .",
"Let $\\left(U_*,V_*\\right)$ another couple of polynomials of $\\mathbb {K}[X]$ , with $\\deg U_* <\\deg Q$ , $\\deg V_* <\\deg P$ and $PU_*+QV_*=1$ and let us prove that $\\left(U_*,V_*\\right)=\\left(U,V\\right)$ .",
"We have $P\\left(UV_*-U_*V\\right)=\\left(PU\\right)V_*-\\left(PU_*\\right)V=\\left(1-QV\\right)V_*-\\left(1-QV_*\\right)V=V_*-V,$ showing that the polynomial $\\left(V_*-V\\right)$ is a multiple of $P$ in $\\mathbb {K}[X]$ .",
"But since $\\deg \\left(V_*-V\\right)<\\deg P$ (because $\\deg V<\\deg P$ and $\\deg V_*<\\deg P$ ), we have inevitably $V_*-V=0$ ; hence $V_*=V$ .",
"Using this, we get $PU_*=1-QV_*=1-QV=PU$ .",
"Thus $U_*=U$ .",
"Consequently, we have $\\left(U_*,V_*\\right)=\\left(U,V\\right)$ , as required.",
"This completes the proof of the theorem.",
"In our context, the application of Theorem REF gives the following corollary: Corollary 2.5 There exists a unique polynomial $\\alpha _k\\in \\mathbb {C}[X]$ , with degree $\\le k$ , such that: $\\alpha _{k}P_{k}+\\overline{\\alpha _{k}}\\overline{P_{k}}=1.$ According to Theorem REF (applied for $\\mathbb {K}=\\mathbb {C}$ and $\\left(P,Q\\right)=\\left(P_k,\\overline{P_k}\\right)$ ), there exists a unique couple $\\left(\\alpha _k,\\beta _k\\right)$ of polynomials of $\\mathbb {C}[X]$ , with $\\deg \\alpha _k<\\deg \\overline{P_k}=k+1$ and $\\deg \\beta _k<\\deg P_k=k+1$ , such that $\\alpha _{k}P_{k}+\\beta _{k}\\overline{P_{k}}=1$ .",
"By taking the conjugates in $\\mathbb {C}[X]$ of both sides of the last equality, we derive that $\\overline{\\alpha _{k}}\\overline{P_{k}}+\\overline{\\beta _{k}}P_{k}=1$ , that is $\\overline{\\beta _{k}}P_{k}+\\overline{\\alpha _{k}}\\overline{P_{k}}=1$ .",
"Since $\\deg \\overline{\\beta _k}=\\deg \\beta _k<k+1$ and $\\deg \\overline{\\alpha _k}=\\deg \\alpha _k<k+1$ , this shows that the couple $\\left(\\overline{\\beta _k},\\overline{\\alpha _k}\\right)$ satisfies the characteristic property of the couple $\\left(\\alpha _k,\\beta _k\\right)$ .",
"Thus $\\left(\\overline{\\beta _k},\\overline{\\alpha _k}\\right)=\\left(\\alpha _k,\\beta _k\\right)$ , that is $\\beta _k=\\overline{\\alpha _k}$ .",
"Consequently, we have $\\alpha _{k}P_{k}+\\overline{\\alpha _{k}}\\overline{P_{k}}=1$ .",
"This completes the proof of the corollary.",
"Now, we are going to determine the explicit expression of the polynomial $\\alpha _k$ announced by Corollary REF .",
"By replacing, in the identity $\\alpha _{k}\\left(X\\right)P_{k}\\left(X\\right)+\\overline{\\alpha _{k}}\\left(X\\right)\\overline{P_{k}}\\left(X\\right)=1$ , the indeterminate $X$ by the numbers $s+\\sqrt{-c}$ $(s=0,1,\\dots ,k)$ , we get $\\alpha _k\\left(s+\\sqrt{-c}\\right)=\\frac{1}{P_k\\left(s+\\sqrt{-c}\\right)}~~(\\forall s\\in \\left\\lbrace 0,1,\\dots ,k\\right\\rbrace ).$ (since $\\overline{P_{k}}\\left(s+\\sqrt{-c}\\right)=0$ for $s=0,1,\\dots ,k$ ).",
"So the values of $\\alpha _{k}$ are known for $(k+1)$ equidistant points with distance 1.",
"Since $\\deg \\alpha _k\\le k$ , this is sufficient to determine the expression of $\\alpha _{k}\\left(X\\right)$ by using for example the Newton forward interpolation formula.",
"Doing so, we obtain that: $\\alpha _k\\left(X\\right)=\\sum _{\\ell =0}^{k}\\frac{\\left(\\Delta ^{\\ell }\\alpha _k\\right)\\left(\\sqrt{-c}\\right)}{\\ell !",
"}\\left(X-\\sqrt{-c}\\right)^{\\underline{\\ell }}.$ Then, by using (REF ), we derive that: $\\alpha _{k}\\left(X\\right)&=\\sum _{\\ell =0}^{k}\\sum _{j=0}^{\\ell }\\frac{(-1)^{\\ell -j}}{\\ell !",
"}\\binom{\\ell }{j}\\alpha _{k}\\left(j+\\sqrt{-c}\\right)\\left(X-\\sqrt{-c}\\right)^{\\underline{\\ell }}\\\\&=\\sum _{\\ell =0}^{k}\\left\\lbrace \\frac{1}{\\ell !",
"}\\sum _{j=0}^{\\ell }(-1)^{\\ell -j}\\binom{\\ell }{j}\\alpha _{k}\\left(j+\\sqrt{-c}\\right)\\right\\rbrace \\left(X-\\sqrt{-c}\\right)^{\\underline{\\ell }}\\\\&=\\sum _{\\ell =0}^{k}\\left\\lbrace \\frac{1}{\\ell !",
"}\\sum _{j=0}^{\\ell }(-1)^{\\ell -j}\\binom{\\ell }{j}\\frac{1}{P_{k}\\left(j+\\sqrt{-c}\\right)}\\right\\rbrace \\left(X-\\sqrt{-c}\\right)^{\\underline{\\ell }}$ (according to (REF )).",
"So, by setting for all $\\ell \\in \\left\\lbrace 0,1,\\dots ,k\\right\\rbrace $ : $\\Theta _{k,\\ell }:=\\frac{1}{\\ell !",
"}\\sum _{j=0}^{\\ell }(-1)^{\\ell -j}\\binom{\\ell }{j}\\frac{1}{P_{k}(j+\\sqrt{-c})},$ we get $\\alpha _{k}\\left(X\\right)=\\sum _{\\ell =0}^{k}\\Theta _{k,\\ell }\\left(X-\\sqrt{-c}\\right)^{\\underline{\\ell }}.$ It remains to simplify the expressions of the numbers $\\Theta _{k,\\ell }$ $(0\\le \\ell \\le k)$ .",
"To do so, we introduce the rational functions $R_{k,\\ell }$ $(0\\le \\ell \\le k)$ , defined by: $R_{k,\\ell }(z):=\\frac{1}{\\ell !",
"}\\sum _{j=0}^{\\ell }(-1)^{\\ell -j}\\binom{\\ell }{j}\\frac{1}{P_{k}(z+j+\\sqrt{-c})},$ so that we have $\\Theta _{k,\\ell }=R_{k,\\ell }(0)~~~~(\\forall \\ell \\in \\left\\lbrace 0,1,\\dots ,k\\right\\rbrace ).$ The common domain of holomorphy of the functions $R_{k,\\ell }$ $(0\\le \\ell \\le k)$ is clearly the open connected region $D$ of $\\mathbb {C}$ , given by: $D:=\\mathbb {C}\\setminus \\lbrace j - 2 \\sqrt{-c} ~;~ j \\in {\\mathbb {Z}}\\text{ and} -k\\le j\\le k\\rbrace .$ Using the principle of analytical continuation together with the theory of the gamma and beta functions, we can find another expression of $R_{k,\\ell }$ $(0\\le \\ell \\le k)$ , which is simpler than the above.",
"We have the following proposition: Proposition 2.6 For all $\\ell \\in \\mathbb {N}$ , with $\\ell \\le k$ , and all $z\\in D$ , we have: $R_{k,\\ell }(z)=\\frac{(-1)^{k+\\ell }}{z+2\\sqrt{-c}}\\binom{k+\\ell }{\\ell }\\frac{1}{\\left(k-2\\sqrt{-c}-z\\right)^{\\underline{k}}\\left(\\ell +2\\sqrt{-c}+z\\right)^{\\underline{\\ell }}}.$ Let $\\ell \\in \\mathbb {N}$ such that $\\ell \\le k$ .",
"According to the principle of analytical continuation, it suffices to prove Formula (REF ) for $z\\in \\mathbb {C}$ , such that $\\Re (z)>k$ .",
"For a such $z$ , we have: $R_{k,\\ell }(z)&:=\\frac{1}{\\ell !",
"}\\sum _{j=0}^{\\ell }(-1)^{\\ell -j}\\binom{\\ell }{j}\\frac{1}{P_{k}(z+j+\\sqrt{-c})}\\\\&=\\frac{1}{\\ell !",
"}\\sum _{j=0}^{\\ell }(-1)^{\\ell -j}\\binom{\\ell }{j}\\frac{1}{\\left(z+j+2\\sqrt{-c}\\right)\\left(z+j-1+2\\sqrt{-c}\\right)\\cdots \\left(z+j-k+2\\sqrt{-c}\\right)}\\\\&=\\frac{1}{\\ell !",
"}\\sum _{j=0}^{\\ell }(-1)^{\\ell -j}\\binom{\\ell }{j}\\frac{\\Gamma \\left(z+j-k+2\\sqrt{-c}\\right)}{\\Gamma \\left(z+j+1+2\\sqrt{-c}\\right)}\\\\&=\\frac{1}{\\ell !",
"}\\sum _{j=0}^{\\ell }(-1)^{\\ell -j}\\binom{\\ell }{j}\\frac{1}{k!",
"}\\beta \\left(z+j-k+2\\sqrt{-c},k+1\\right)\\\\&=\\frac{1}{k!\\ell !",
"}\\sum _{j=0}^{\\ell }\\left[(-1)^{\\ell -j}\\binom{\\ell }{j}\\int _{0}^{1}t^{z+j-k-1+2\\sqrt{-c}}(1-t)^{k}\\mathrm {d}t\\right] \\\\&=\\frac{1}{k!\\ell !",
"}\\int _{0}^{1} t^{z-k-1+2\\sqrt{-c}}(1-t)^{k}\\left\\lbrace \\sum _{j=0}^{\\ell }(-1)^{\\ell -j}\\binom{\\ell }{j}t^{j}\\right\\rbrace \\mathrm {d}t \\\\&=\\frac{1}{k!\\ell !",
"}\\int _{0}^{1} t^{z-k-1+2\\sqrt{-c}}(1-t)^{k}\\left(t-1\\right)^{\\ell } \\mathrm {d}t \\\\&=\\frac{(-1)^{\\ell }}{k!\\ell !",
"}\\int _{0}^{1} t^{z-k-1+2\\sqrt{-c}}(1-t)^{k+\\ell }\\mathrm {d}t \\\\&=\\frac{(-1)^\\ell }{k!\\ell !",
"}\\beta \\left(z-k+2\\sqrt{-c},k+\\ell +1\\right)\\\\&=\\frac{(-1)^\\ell }{k!\\ell !",
"}\\frac{\\Gamma \\left(z-k+2\\sqrt{-c}\\right)\\Gamma \\left(k+\\ell +1\\right)}{\\Gamma \\left(z+\\ell +1+2\\sqrt{-c}\\right)}\\\\&=(-1)^\\ell \\binom{k+\\ell }{\\ell }\\frac{1}{\\left(z+\\ell +2\\sqrt{-c}\\right)\\left(z+\\ell -1+2\\sqrt{-c}\\right)\\cdots \\left(z-k+2\\sqrt{-c}\\right)}\\\\&=\\frac{(-1)^{k+\\ell }}{z+2\\sqrt{-c}}\\binom{k+\\ell }{\\ell }\\frac{1}{\\left(k-2\\sqrt{-c}-z\\right)^{\\underline{k}}\\left(\\ell +2\\sqrt{-c}+z\\right)^{\\underline{\\ell }}},$ as required.",
"This completes the proof.",
"From Proposition REF , we immediately derive a simpler explicit expression of $\\alpha _k\\left(X\\right)$ .",
"We have the following corollary: Corollary 2.7 We have: $\\alpha _{k}\\left(X\\right)=\\frac{1}{2\\sqrt{-c}\\left(k-2\\sqrt{-c}\\right)^{\\underline{k}}}\\sum _{\\ell =0}^{k}\\frac{(-1)^{k+\\ell }\\binom{k+\\ell }{\\ell }}{\\left(\\ell +2\\sqrt{-c}\\right)^{\\underline{\\ell }}}\\left(X-\\sqrt{-c}\\right)^{\\underline{\\ell }}.$ This immediately follows from Formulas (REF ), (REF ) and (REF )."
],
[
"Nontrivial multiples of some values of $h_c$",
"In this subsection, we preserve the notations of Subsection REF .",
"From Corollary REF , we derive the following theorem: Theorem 2.8 For all $c,n,m\\in \\mathbb {N^*}$ , with $m\\le n$ , we have: $h_c\\left(\\prod _{\\ell =m}^{n}\\left(\\ell +\\sqrt{-c}\\right)\\right)~~\\text{divides}~~c\\prod _{\\ell =1}^{n-m}(\\ell ^2+4c).$ Let $c,n,m\\in \\mathbb {N^*}$ , with $m\\le n$ .",
"Putting $k:=n-m\\in \\mathbb {N}$ and $d:=c\\prod _{\\ell =1}^{n-m}(\\ell ^2+4c)\\in \\mathbb {N^*}$ , we have $\\prod _{\\ell =m}^{n}\\left(\\ell +\\sqrt{-c}\\right)=P_k(n)$ ; so, we have to show that $h_c\\left(P_k(n)\\right)$ divides $d$ .",
"By noting that $2d=\\sqrt{-c}\\cdot 2\\sqrt{-c}\\left(k-2\\sqrt{-c}\\right)^{\\underline{k}}\\left(k+2\\sqrt{-c}\\right)^{\\underline{k}}$ , we derive from Corollary REF that $2d\\alpha _k\\in \\mathbb {Z}[\\sqrt{-c}][X]$ .",
"So, there exist $r_k,s_k\\in \\mathbb {Z}[X]$ such that: $2d\\alpha _k\\left(X\\right)=r_k\\left(X\\right)+s_k\\left(X\\right)\\sqrt{-c}.$ Next, the identity of polynomials $\\alpha _kP_k+\\overline{\\alpha _k}\\overline{P_k}=1$ (given by Corollary REF ) implies that $2d\\alpha _k\\cdot P_k+\\overline{2d\\alpha _k}\\cdot \\overline{P_k}=2d$ .",
"By substituting in this last equality $P_k$ by $\\left(A_k+B_k\\sqrt{-c}\\right)$ and $2d\\alpha _k$ by $\\left(r_k+s_k\\sqrt{-c}\\right)$ , we obtain (in particular) that: $r_kA_k-cs_kB_k=d,$ implying that ${\\gcd }_{\\mathbb {Z}[X]}\\left(A_k,B_k\\right)$ divides $d$ .",
"We then conclude that ${\\gcd }_{\\mathbb {Z}}\\left(A_k(n),B_k(n)\\right)=h_c\\left(P_k(n)\\right)$ divides $d$ , as required."
],
[
"New estimates for the number $L_{c,m,n}$",
"We have the following theorem: Theorem 2.9 Let $c,m,n\\in \\mathbb {N^*}$ such that $m\\le n$ .",
"Then: The positive integer $L_{c,m,n}$ is a multiple of the rational number $\\frac{\\displaystyle \\prod _{k=m}^{n}\\left(k^2+c\\right)}{c\\cdot (n-m)!\\displaystyle \\prod _{k=1}^{n-m}\\left(k^2+4c\\right)}.$ We have $L_{c,m,n}\\ge \\lambda _1(c) \\cdot m^2\\frac{n!^2}{m!^2(n-m)!^3},$ where $\\lambda _1(c):=e^{-\\frac{2\\pi ^{2}}{3} c}/c$ .",
"The first point of the theorem is an immediate consequence of Corollary REF and Theorem REF .",
"The second one follows from the first one and the easy inequalities: $\\frac{\\prod _{k=m}^{n}\\left(k^2+c\\right)}{c\\cdot (n-m)!\\prod _{k=1}^{n-m}\\left(k^2+4c\\right)}&\\ge \\frac{\\prod _{k=m}^{n}k^2}{c\\cdot (n-m)!^3\\prod _{k=1}^{n-m}\\left(1+\\frac{4c}{k^2}\\right)}\\\\&\\ge \\frac{m^2\\left(\\frac{n!}{m!",
"}\\right)^2}{c\\cdot (n-m)!^3e^{\\sum _{k=1}^{+\\infty }\\frac{4c}{k^2}}}\\\\&=\\frac{e^{-\\frac{2\\pi ^{2}}{3} c}}{c}\\cdot m^2\\frac{n!^2}{m!^2(n-m)!^3}.$ This completes the proof.",
"We shall now impose conditions on $m$ (in terms of $n$ ) in order to optimize (resp.",
"simplify) the estimate of the second point of Theorem REF .",
"To do so, we first need to get rid of the factorials in that estimate.",
"We have the following: Corollary 2.10 Let $c,n,m\\in \\mathbb {N^*}$ such that $m<n$ .",
"Then, we have: $L_{c,m,n}\\ge \\lambda _{2}(c)\\cdot \\frac{nm}{\\left(n-m\\right)^{3/2}}\\left( \\frac{m^2}{(n-m)^3}\\right)^{n-m} e^{3(n-m)},$ where $\\lambda _2(c):=\\frac{e^{-\\frac{2\\pi ^{2}}{3} c -\\frac{5}{12}}}{\\left(2\\pi \\right)^{3/2}c}$ .",
"Starting from the lower bound established by Theorem REF for $L_{c , m , n}$ and estimating each of its factorial terms by using the well-known double inequality: $k^{k}e^{-k}\\sqrt{2\\pi k}\\le k!\\le k^{k}e^{-k}\\sqrt{2\\pi k} e^{\\frac{1}{12k}}~~~~(\\forall k\\in \\mathbb {N^*})$ (which can be found in Problem 1.15 of [4]), we get $L_{c , m , n} \\ge \\lambda _1(c) (2 \\pi )^{- 3/2} \\cdot \\frac{n m}{(n - m)^{3 / 2}} \\cdot \\left(\\frac{n}{m}\\right)^{2 n} \\cdot \\left(\\frac{m^2}{(n - m)^3}\\right)^{n - m} e^{n - m} \\cdot e^{- \\frac{1}{6 m} - \\frac{1}{4 (n - m)}} .$ Next, since $e^{- \\frac{1}{6 m} - \\frac{1}{4 (n - m)}} \\ge e^{- \\frac{1}{6} - \\frac{1}{4}} = e^{- \\frac{5}{12}}$ and $\\left(\\frac{n}{m}\\right)^{2 n} = e^{- 2 n \\log (\\frac{m}{n})} \\ge e^{- 2 n (\\frac{m}{n} - 1)} = e^{2 (n - m)}$ , then we deduce that: $L_{c , m , n} \\ge \\lambda _1(c) (2 \\pi )^{- 3/2} e^{- 5 / 12} \\cdot \\frac{n m}{(n - m)^{3 / 2}} \\left(\\frac{m^2}{(n - m)^3}\\right)^{n - m} e^{3 (n - m)} ,$ as required.",
"In the context of Corollary REF , by supposing that $n-m$ is of order of magnitude $n^\\alpha $ for large $n$ (where $0<\\alpha <1$ ), then the dominant part of the lower bound (REF ) for $L_{c , m , n}$ is $\\left(\\frac{m^2}{(n-m)^3}\\right)^{n-m}$ and has order of magnitude $n^{(2-3\\alpha )n^{\\alpha }}$ .",
"So, to have an optimal estimate, we must take $\\alpha $ less than but not too far from $\\frac{2}{3}$ (a study of the function $\\alpha \\longmapsto (2-3\\alpha )n^{\\alpha }$ shows that the best value of $\\alpha $ is $\\alpha =\\frac{2}{3}-\\frac{1}{\\log n}$ ).",
"A concrete result specifying this heuristic reasoning is given by the following theorem: Theorem 2.11 Let $c,m,n\\in \\mathbb {N^*}$ such that $m\\le n - \\frac{1}{2}n^{2/3}$ .",
"Then, we have: $L_{c,m,n}\\ge \\lambda _3(c)\\cdot \\left(n-\\frac{1}{2}n^{2/3}\\right)\\cdot \\left(2e^{3}\\right)^{\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor },$ where $\\lambda _3(c):=\\frac{e^{-\\frac{2\\pi ^2}{3} c -\\frac{5}{12}}}{\\pi ^{3/2} c}$ .",
"A simple calculation shows that the result of the theorem is true for $n < 3$ .",
"Suppose for the sequel that $n \\ge 3$ and let $m_n:=n-\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor < n$ ; so $m \\le m_n$ .",
"From Corollary REF , we have: $L_{c,m_n,n}&\\ge \\lambda _2(c)\\frac{n\\left(n-\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor \\right)}{\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor ^{3/2}}\\left( \\frac{\\left(n-\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor \\right)^2}{\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor ^3}\\right)^{\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor } e^{3\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor }\\\\&\\ge \\lambda _2(c)\\frac{n\\left(n-\\frac{1}{2}n^{2/3}\\right)}{\\left(\\frac{1}{2}n^{2/3}\\right)^{3/2}}\\left(\\frac{\\left(n-\\frac{1}{2}n^{2/3}\\right)^2}{\\left(\\frac{1}{2}n^{2/3}\\right)^3}\\right)^{\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor }e^{3\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor }\\\\&=2^{3/2}\\lambda _2(c)\\left(n-\\frac{1}{2}n^{2/3}\\right)\\left[8\\left(1-\\frac{1}{2n^{1/3}}\\right)^2\\right]^{\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor }e^{3\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor }.$ But since $1-\\frac{1}{2n^{1/3}}\\ge \\frac{1}{2}$ (because $n\\ge 1$ ), we deduce that: $L_{c,m_n,n}\\ge 2^{3/2}\\lambda _2(c)\\left(n-\\frac{1}{2}n^{2/3}\\right)\\left(2e^3\\right)^{\\left\\lfloor \\frac{1}{2}n^{2/3}\\right\\rfloor }.$ The required result follows from the trivial fact that $L_{c,m,n}\\ge L_{c,m_n,n}$ (since $m\\le m_n$ ).",
"In another direction, we derive from Corollary REF the following theorem, which completes (in a way) Theorem REF above.",
"Theorem 2.12 Let $c,m,n\\in \\mathbb {N^*}$ such that $n-\\frac{1}{2}n^{2/3} \\le m \\le n$ .",
"Then, we have: $L_{c,m,n}\\ge \\lambda _2(c)\\cdot ne^{3(n-m)},$ where $\\lambda _2(c)$ is defined in Corollary REF .",
"The result of the theorem is trivial for $m = n$ .",
"Suppose for the sequel that $m < n$ ; so we have $n \\ge 2$ .",
"Now, let $f:\\left[0,n\\right]\\longrightarrow \\mathbb {R}$ be the function defined by $f(x)=x^2-(n-x)^3$ $(\\forall x\\in [0,n])$ .",
"Obviously, $f$ is increasing.",
"Next, we have: $f\\left(n-\\frac{1}{2}n^{2/3}\\right)&=\\left(n-\\frac{1}{2}n^{2/3}\\right)^2-\\left(\\frac{1}{2}n^{2/3}\\right)^3\\\\&=n^2-n^{5/3}+\\frac{1}{4}n^{4/3}-\\frac{1}{8}n^2\\\\&=\\frac{7}{8}n^2-n^{5/3}+\\frac{1}{4}n^{4/3}.$ But since $n^2\\ge \\frac{8}{7}n^{5/3}$ (because $n\\ge 2$ ), it follows that $f\\left(n-\\frac{1}{2}n^{2/3}\\right)\\ge \\frac{1}{4}n^{4/3}>0$ .",
"So, the increase of $f$ insures that $f(m)>0$ (since $m\\ge n-\\frac{1}{2}n^{2/3}$ by hypothesis).",
"Thus $\\frac{m^2}{(n-m)^3}>1$ and $\\frac{m}{(n-m)^{3/2}}>1$ .",
"By reporting these into (REF ), we then conclude that: $L_{c,m,n}\\ge \\lambda _2(c)\\cdot ne^{3(n-m)},$ as required.",
"This completes the proof."
]
] | 2001.03374 |
[
[
"In Defense of Grid Features for Visual Question Answering"
],
[
"Abstract Popularized as 'bottom-up' attention, bounding box (or region) based visual features have recently surpassed vanilla grid-based convolutional features as the de facto standard for vision and language tasks like visual question answering (VQA).",
"However, it is not clear whether the advantages of regions (e.g.",
"better localization) are the key reasons for the success of bottom-up attention.",
"In this paper, we revisit grid features for VQA, and find they can work surprisingly well - running more than an order of magnitude faster with the same accuracy (e.g.",
"if pre-trained in a similar fashion).",
"Through extensive experiments, we verify that this observation holds true across different VQA models (reporting a state-of-the-art accuracy on VQA 2.0 test-std, 72.71), datasets, and generalizes well to other tasks like image captioning.",
"As grid features make the model design and training process much simpler, this enables us to train them end-to-end and also use a more flexible network design.",
"We learn VQA models end-to-end, from pixels directly to answers, and show that strong performance is achievable without using any region annotations in pre-training.",
"We hope our findings help further improve the scientific understanding and the practical application of VQA.",
"Code and features will be made available."
],
[
"Introduction",
"After the introduction of deep learning [9], [44] and attention mechanisms [47], [48] to multi-modal vision and language research, perhaps one of the most significant developments was the discovery of `bottom-up' attention [2].",
"Unlike normal attention that uses `top-down' linguistic inputs to focus on specific parts of the visual input, bottom-up attention uses pre-trained object detectors [33] to identify salient regions based solely on the visual input itself.",
"As a result, images are represented by a collection of bounding box or regionWe use the terms `region' and `bounding box' interchangeably.-based features [2], [39]–in contrast to vanilla grid convolutional feature maps from ConvNets [35], [15]–for follow-up tasks.",
"These region features have since then gained wide popularity and dominated vision and language leader boards [16], [50] for major tasks like visual question answering (VQA).",
"So what makes these region features successful?",
"Naturally, one would assume a major reason is better localization of individual objects, as the regions are direct bounding box outputs from detectors.",
"Another plausible answer is that a number of regions can easily capture both the coarse-level information and fine-grained details in the image – even if they overlap.",
"However, do these potential advantages actually demonstrate that region features are superior to grids?",
"Figure: We revisit grid-based convolutional features for VQA, and find they can match the accuracy of the dominant region-based features from bottom-up attention , provided that one closely follow the pre-training process on Visual Genome .",
"As computing grid features skips the expensive region-related steps (shown in colors), it leads to significant speed-ups (all modules run on GPU; timed in the same environment).Surprisingly, we discovered that grid features extracted from exactly the same layer of the pre-trained detector can perform competitively against their region-based counterparts for VQA.",
"Moreover, with simple modifications during training, the same grid features can be made even more effective and that they consistently achieve comparable and sometimes better VQA accuracy than region features.",
"In fact, our ablative analysis suggests that the key factors which contributed to the high accuracy of existing bottom-up attention features are: 1) the large-scale object and attribute annotations collected in the Visual Genome (VG) [22] dataset used for pre-training; and 2) the high spatial resolution of the input images used for computing features.",
"As for the feature format itself – region or grid – it only affects accuracy minimally.",
"Through a comprehensive set of experiments, we verified that our observations generalize across different network backbones, different VQA models [16], [50], different VQA benchmarks [3], [12], and even to other relevant tasks (image captioning [4]).",
"Our findings have important consequences for the design of future multi-modal vision and language models.",
"The immediate benefit of switching to grids is inference speed, as we can now skip all of the region-related steps in the existing VQA pipeline (Fig.",
"REF ).",
"For example, using a ResNet-50 [15] backbone, we find the overall running time drops from 0.89s to 0.02s per image – 40+ times faster with slightly better accuracy!",
"In fact, extracting region features is so time-consuming that most state-of-the-art models [20], [50] are directly trained and evaluated on cached visual features.",
"This practice not only imposes unnecessary constraints on model designs, but also limits potential applications of existing vision and language systems.",
"Empowered by grid features, we therefore take an initial step to train VQA models end-to-end from pixels directly to answers.",
"Note that end-to-end training with region features is challenging, since fine-tuning region locations likely requires additional grounding annotations [13] that are computationally expensive and difficult to acquire.",
"In contrast, grid features can be readily optimized for the final objective (to answer questions correctly) without extra grounding.",
"The grid-feature pipeline also allows us to explore more effective designs for VQA (pyramid pooling module [54]) and enables networks pre-trained with zero region-level annotations to greatly reduce the gap in accuracy with VG models (trained on bounding boxes) – indicating strong VQA models can be achieved without any explicit notion of regions.",
"These results further strengthen our defense of grid features for VQA.",
"We hope our discovery can open up new opportunities for vision and language research in general."
],
[
"Related Work",
"paragraph4 .5em plus1ex minus.2ex-.5emVisual features for vision and language tasks.",
"Features have played a key role in the advancement of vision and language tasks.",
"For example, deep learning features led to remarkable improvements in image captioning [9], [44], [8].",
"While a complete review of visual features used for vision and language tasks is beyond the scope of this paper, we note that the accuracies of modern VQA models are dependent on the underlying visual features used, including VGG [35] and ResNet [15] grid features, which were later dominated by bottom-up attention region features [2], [39].",
"Today, most state-of-the-art VQA models focus on fusing schemes [51], [20], [50] and are built with region features as-is [49]; whereas our work revisits grid features, and shows that they can be equally effective and lead to remarkable speed-ups – often greater than an order of magnitude!",
"Pre-training for VQA.",
"Most VQA methods use two separately pre-trained models: vision models trained on ImageNet [6] and VG [22]; and word embeddings [31] for linguistic features.",
"As these separately trained features may not be optimal for joint vision and language understanding, a recent hot topic is to develop jointly pre-trained models [24], [29], [38], [37], [55], [5] for vision and language tasks.",
"A common scheme for such methods is to view regions and words as `tokens' for their respective domain, and pre-train a variant of BERT [7], [42] for `masked' token prediction.",
"Complementary to that direction, our work delves specifically into the `format' of visual tokens and can be potentially combined with such methods for mutual benefits (trade-off between speed and accuracy).",
"Regions grids.",
"The debate between region features and grid features carries some inherent connections to object detection: the dominance of the R-CNN based detection models [33], [14] demonstrates that a region (the `R' in R-CNN) based refinement stage is beneficial for object detection.",
"On the other hand, one-stage detectors [26], [28] approach the detection task without the need for explicit region-level computation and show that grid features can be competitive for object detection.",
"In our work, we also use grid features – no regions for the VQA task.",
"To minimize changes from bottom-up attention paper [2], we pre-train the features with Faster R-CNN [33].",
"However, during inference, we discard the region-related steps from the detector and use only the grid convolutional features.",
"This in fact gives us a stronger defense for grids, as we show that VQA can operate on a `single' feature map, instead of feature maps of `multiple' scales that one-stage detectors [26], [28] thrive on.",
"Figure: From regions to grids.",
"Left: We convert the original region feature extractor used by bottom-up attention back to the ResNet grid feature extractor for the same layer (see Sec.",
", weights in blue are transferred), and find it works surprisingly well for VQA .",
"Right: We build a detector based on 1×\\times 1 RoIPool while keeping the output architecture fixed for grid features (see Sec.",
"), and the resulting grid features consistently perform at-par with region features.It is also worth noting that while region features are effective on benchmarks like VQA [3], [11] and COCO captions [4], for benchmarks that diagnose a model's reasoning abilities when answering visual questions (CLEVR [17]), simple methods based on grids [32] have shown strong performance.",
"We hope that our discovery that grid features also work well for the general VQA task can bridge the gap between these two lines of work [34]."
],
[
"From Regions to Grids",
"In this section, we explain our approach to obtaining grid features that are just as effective as region features, with the constraint that they have been pre-trained with the same task.",
"In Sec.",
", we show that the `same pre-training' constraint can be lifted and grid features can still close the gap to regions with end-to-end training on down-stream tasks.",
"We first briefly review the region features from bottom-up attention [2]."
],
[
"Bottom-Up Attention with Regions",
"The bottom-up attention method [2] uses a Faster R-CNN [33] detection model.",
"The detector is trained on a cleaned version of Visual Genome [22], with thousands of object categories and hundreds of attributes with bounding box (region) annotations.",
"In order to obtain bottom-up attention features for tasks like VQA, two region-related steps are needed: Region selection.",
"As Faster R-CNN is a two-stage detector, region selection happens twice in the pipeline.",
"The first is through a region proposal network [33], which deforms and selects prominent candidate `anchors' as Regions of Interest (RoIs).",
"Another selection is done as post-processing to aggregate top $N$ boxes in a per-class manner.",
"In both steps, non-maximal suppression (NMS) is used, which keeps the region with the highest classification score and removes other near-duplicates in a local neighborhood.",
"Region feature computation.",
"Given regions from the first stage (up to thousands), RoIPool operations [33] are used to extract the initial region-level features.",
"Additional network layers then compute the output representation of regions separately.",
"Finally, region features that survive both rounds of selection are stacked together as the bottom-up features to represent an image.",
"It is important to note that due to the complexity of the VG dataset (thousands of classes) and the specific Faster R-CNN detector used [2] (described next), both steps are computationally intensive.",
"In contrast, directly using grid features can skip or accelerate these steps and offer potentially significant speed-ups."
],
[
"Grid Features from the Same Layer",
"The simplest way to convert region features to grids is to see if one can directly compute outputs of the same network layer, but in a shared, fully convolutional manner.",
"To this end, we take a closer look at the specific Faster R-CNN architecture used by the original bottom-up attention [2].",
"The Faster R-CNN is a variant of the c4 model [15] with an extra branch for attribute classification.",
"It divides the weights from a ResNet [15] into two separate sets: given an input image, it first computes feature maps using the lower blocks of ResNet up to $C_4$ .",
"This feature map is shared among all regions.",
"Then, separately, per-region feature computations are performed by applying the $C_5$ block on the 14$\\times $ 14 RoIPool-ed features.",
"The output of $C_5$ is then AvgPool-ed to a final vector for each region as the bottom-up features [2].",
"Since all the final region features are from $C_5$ , it is easy to convert the detector back to the ResNet classifier and take the same $C_5$ layer as our output grid features.",
"Fig.",
"REF (left) illustrates our conversion process.",
"As our experiments will show, directly using the converted $C_5$ output already works surprisingly well.",
"Any performance drop from doing so may be because Faster R-CNN is highly optimized for region-based object detection, and likely not so much for grids.",
"Therefore, we next see if some minimal adjustments to the model can be made to improve grid features."
],
[
"1$\\times $ 1 ",
"Our idea is to simply use 1$\\times $ 1 RoIPool.",
"This means representing each region with a single vector, rather than a three-dimensional tensor in Faster R-CNN.",
"At first glance, it may seem counter-intuitive, as the two additional spatial dimensions (height and width) are useful to characterize different parts of objects in 2D – indeed, we find this modification negatively affects object detection performance on VG.",
"But importantly, using 1$\\times $ 1 RoIPool regions also means each vector on the grid feature map is forced to cover all the information for a spatial region alone, which can potentially result in stronger grid features.",
"However, directly applying 1$\\times $ 1 RoIPool on the original model is problematic, likely because $C_5$ consists of several ImageNet pre-trained convolutional layers that work best with inputs of particular spatial dimensions.",
"To resolve this, we follow recent developments in object detection and use the entire ResNet up to $C_5$ as the backbone for shared feature computation [56]; and for region-level computation place two 1024D fully-connected (FC) layers on the top, which by default accept vectors as inputs.",
"To reduce the effect of low resolutions when training the detector with features pooled from $C_5$ ($C_5$ has stride 32, whereas $C_4$ has 16), the stride-2 layers are replaced with stride-1 layers, and the remaining layers are dilated with a factor of 2 [56].",
"For grid feature extraction, we remove this dilation and convert it back to the normal ResNet.",
"Fig.",
"REF (right) summarizes the changes we made to improved grids.",
"Note that compared to the original model (left), we only made necessary modifications to the region related components during training.",
"Since all such computations are removed during feature extraction, our grid feature extractor is kept untouched during inference.",
"Table: Main comparison.",
"`R' stands for region features as in bottom-up attention .",
"`G' stands for grid features.",
"All results reported on VQA 2.0 vqa-eval.",
"We show that: 1) by simply extracting grid features from the same layer C 5 C_5 of the same model, the VQA accuracy is already much closer to bottom-up attention than ImageNet pre-trained ones (row 1,3 & 5); 2) 1×\\times 1 RoIPool based detector pre-training improves the grid features accuracy while the region features get worse (row 1,2 & 4).",
"Last column is the gap compared to the original bottom-up features (underlined)."
],
[
"Main Comparison: Regions Grids",
"From this section on, we report our experimental results comparing regions with grids.",
"We choose VQA (2.0) [11] as our main task of interest, since it is currently a major benchmark for evaluating joint vision and language understanding and has clear metrics for evaluation.",
"For all our comparisons, we denote methods using region features with the tag `R', and methods using grid features with `G'.",
"In this section, we focus on reporting our main findings from converting regions to grids as described in Sec. .",
"We begin by briefly describing our experimental setups (more details in the supplementary material).",
"Note that our goal here is to make the conclusion meaningful by controlled comparisons, and not necessarily to optimize for absolute performance."
],
[
"Experimental Setup",
"Faster R-CNN.",
"For analysis, we use Faster R-CNN with a ResNet-50 backbone pre-trained on ImageNet by defaulthttps://github.com/facebookresearch/maskrcnn-benchmark.",
"Closely following bottom-up attention [2], the detector is then trained on the VG dataset [22] with region-level annotations for 1600 object categories and 400 attribute classes.",
"For attributes, an additional branch is added with loss weight 0.5.",
"The model is trained with `1x' schedule [14].",
"Notably, input images are resized to have a maximum shorter side of 600 pixels (longest 1000) when keeping aspect ratio fixed.",
"For region features, we set $N$ =100.",
"VQA split.",
"Unless otherwise specified, we use the default train set for training.",
"To assist our analysis, we create a local validation set, vqa-dev, out of the standard val set to select the best model during training for evaluation.",
"It contains randomly sampled 8.4K images and their corresponding questions, with 66K pairs in total.",
"The rest of the original val set (named vqa-eval) is reserved for testing, on which we report results.",
"VQA model.",
"We use the co-attention model [52] implemented in Pythia [16], [36].",
"This model fuses visual features (either region or grid) with textual representations of questions, and outputs the final answer."
],
[
"Main Results",
"Our main results are summarized in Table REF .",
"We make two observations: First, compared with the widely used bottom-up region features (row 1), directly extracting outputs from $C_5$ with the same model (row 3) works surprisingly well (64.29 63.64 accuracy).",
"In contrast, the standard ResNet-50 model pre-trained on ImageNet [6] shows much worse performance – 60.76 accuracy, a gap of more than 3% with the bottom-up features.",
"Figure: VQA accuracy number of features NN as input to the VQA model.",
"We report the average accuracy and standard deviation across 5 independent runs on the VQA 2.0 vqa-eval set.",
"We observe that the VQA accuracy of region features saturates around 200 regions.",
"In contrast, the grid features benefit from a larger NN (translates from a larger input size) and in this case stays better than regions even when NN is the same (608).Second, while our 1$\\times $ 1 RoIPool-based variant hurts the object detection performance (average precision [27] on VG drops from 4.07 to 2.90), it helps VQA – boosting the accuracy by 0.73% (row 3 & 4) and as a result slightly outperforms the original region-based features.",
"On the other hand, our RoI-based variant does not help the region features method and drops the accuracy of region features to 63.94.",
"This indicates the original model used by bottom-up attention favors regions; while our design works better for grids.",
"Thus, we use the setting of the 1st row (best for regions) to represent `R', and the 4th row (best for grids) to represent `G', to perform a more in-depth study and fair comparison between the two through the rest of the paper."
],
[
"Number of Regions",
"Apart from architectural differences in training, another factor that can affect VQA accuracy is the number of feature vectors $N$ used to represent images.",
"Our region model from Pythia [16] has a default setting that uses the top 100 boxes to represent region features, increasing it from the original 36 boxes in [2] to improve the accuracy.",
"On the other hand, since grid features are convolutional feature maps for a pre-set layer, the number of features is determined by the input size to the network.",
"As our largest input size is 600$\\times $ 1000, a 32-stride feature map ($C_{5}$ ) results in 608 grid features – much larger than the number of region features.",
"To understand how these different numbers of region features affect the accuracy, we ran experiments with varying number of features $N$ and show the results in Figure REF .",
"As for the region features, we observe an improvement in accuracy as the number of regions increases from 30 to 200, beyond which the accuracy saturates.",
"Interestingly, our grid features are better even when compared to the highest number of regionsSince NMS is used in selecting regions, the maximum number $N$ varies across images.",
"Therefore we 1) cannot directly set it to the same number as grids and 2) report maximum $N$ instead (zero paddings are used for images with fewer regions)..",
"Thus, the higher number of feature vectors used in our grid method compared to the baseline region method, is not the reason for its improved VQA accuracy.",
"Table: Region grid features on the VQA 2.0 test-dev with accuracy and inference time breakdown measured in milliseconds per image.",
"Our grid features achieve comparable VQA accuracy to region features while being much faster without region feature computation and region selection."
],
[
"Test Accuracy and Inference Time",
"We now report results on the VQA 2.0 test-dev set to quantify the difference in performance between region and grid features.",
"Note that different from previous setups, we use trainval+vqa-eval for training.",
"We report the VQA accuracy and the inference time breakdown in Table REF .",
"Unlike our grid features which directly use convolutional feature maps, region features involve additional operations of region selection and region feature computation.",
"These additional operations take 98.3% of the total inference time for a region-based model.",
"As a result, the VQA model that takes our grid features as input runs 48$\\mathbf {\\times }$ faster than its counterpart using bottom-up region features.",
"Figure: Visualizations of attention maps overlaid on images produced by VQA models .",
"Source images taken from COCO to compare against bottom-up attention on VQA 2.0 .",
"We show questions (Q), ground-truth answers (GT-A), and side-by-side predictions (attention maps, answers) of region (R) and grid (G) features.",
"From left to right: (a) both region and grid features give correct answers, (b) region features give correct answers but grid features fail, (c) region features fail but grid features give correct answers, and (d) both region and grid features fail.",
"Best viewed in color."
],
[
"Qualitative Comparison",
"We visualize attention maps over input images from the top-down attention module [2], together with answers from both regions and grids in Fig.",
"REF .",
"Source images are taken from COCO [27] on which VQA 2.0 [11] benchmark is built.",
"To obtain the attention map, we propagate the attention value of each region or grid to its corresponding pixels, and then average the attention value for each pixel (normalizing them individually to [0, 1]).",
"As can be seen, both types of features are able to capture relevant concepts in input images (, snowfield in the top left).",
"Naturally, attention maps of region features tend to cover object-like regions, while for grid features the attention does not necessarily cover the full area the supporting concept (, the snowfield), which can be used to answer the question.",
"However, both features are able to answer visual questions well, suggesting that localization is important, but accurate object detection of individual objects is not crucial for VQA [11].",
"We show failure cases of region and grid features in Fig.",
"REF (b)(c)(d).",
"In most examples, the models attend to the supporting concepts but still give wrong answers.",
"In the cases where both region and grid features fail, specifically designed modules may be needed (, counting module [53], [41] in the bottom right example) to answer the question correctly.",
"Table: Comparison between the conventional ImageNet pre-trained and our proposed grid features on the VQA 2.0 vqa-eval set.",
"Besides VQA accuracy, we list two major differences between the two: 1) pre-training task and 2) input image size."
],
[
"Why do Our Grid Features Work?",
"As we mentioned in Sec.",
", grid features are not new – in fact, they were widely used in vision and language tasks before the introduction of bottom-up attention features.",
"Compared to the previous attempts at grid features, why do our grid features work well?",
"In Table REF we show the performance of grid-based methods (ResNet-50 $C_{5}$ features) for different settings and find that there are two major factors: 1) input image size; 2) pre-training task.",
"We study both these factors next and report results on the vqa-eval set."
],
[
"Factor 1: Input Image Size",
"The standard image size used during feature extraction for ImageNet pre-trained models is 448$\\times $ 448 [10] discarding the aspect ratio; whereas for VG detection in bottom-up attention [2], the default size is 600$\\times $ 1000 while keeping the aspect ratio intact.",
"Therefore, we experimented with different combinations and reported results for all of them in Table REF .",
"We note that for grid features, a larger input size means more features for the VQA model.",
"From the table, we find that grid features benefit from larger images as input, indicating this factor is indeed important.",
"However, input size has a different effect for models pre-trained on ImageNet VG.",
"For ImageNet models which are pre-trained on smaller images [15], the performance saturates around 600$\\times $ 1000.",
"Interestingly, the performance of VG models improves with the input size and continues to increase even at 800$\\times $ 1333.",
"We still use 600$\\times $ 1000 for the rest of the paper.",
"Table: Impact of input image size on the VQA 2.0 vqa-eval set.",
"Grid features benefit from larger input image sizes.",
"For an ImageNet pre-trained model, the accuracy saturates around 600×\\times 1000 but the VG model makes a better use of larger input image sizes."
],
[
"Factor 2: Pre-Training Task",
"We now study the difference in VQA accuracy due to the pre-training task in the ImageNet (classification) and VG (detection)Strictly speaking, VG also uses ImageNet classification for pre-training, because the detector is fine-tuned from a standard ImageNet pre-trained model.. To understand these differences better, we introduce an additional pre-trained model in each setting.",
"For classification, we include a model trained on YFCC [40], which has 92M images with image tags.",
"For detection, we include a standard model from COCO [27] which only has object annotations (no attributes).",
"All models use a ResNet-50 backbone for fair comparison.",
"The results are shown in Table REF .",
"In the image classification pre-trained setting, the YFCC model (trained on weak image level tags), performs better than the ImageNet model, possibly because it is trained on two orders of magnitude more data.",
"For detection based pre-training, the VG model (trained with objects and attributes) gives better results than the COCO model.",
"The larger number of categories in VG compared to COCO (1600 80) or the additional attribute annotations it has are two possible reasons for the improved performance.",
"We study the impact of attributes next.",
"Table: Choice of pre-training task.",
"We explore the impact of the type of pre-training task on the final performance while keeping the input size fixed at 600×\\times 1000.",
"Results reported on vqa-eval.",
"We broadly characterize the pre-training tasks into two types - object detection (`det') and image classification (`cls').paragraph4 .5em plus1ex minus.2ex-.5emAttributes.",
"Fig.",
"REF shows the impact of the attribute loss weight on VQA accuracy.",
"Setting the attribute loss weight to zero during pre-training on VG, results in a drop in VQA performance.",
"In fact, the VQA accuracy in this case matches the accuracy from a pre-trained COCO model suggesting that attributes in the pre-training task are a major reason for the better performance of VG models.",
"We also note that the grid features consistently outperform the region features for all values of the attribute loss weight.",
"Figure: Analysis on attribute loss weights when pre-training grid features on Visual Genome (VG).",
"All results on VQA 2.0 vqa-eval set."
],
[
"Generalization of Grid Features",
"We now study whether our findings about grid features are more broadly applicable to other tasks and models.",
"In this section, we study generalization across: 1) different backbones; 2) different VQA models; 3) different VQA tasks; 4) other tasks.",
"For all the studies, we set the attribute loss weight to 0.2, and compare both the accuracy and speed.",
"For regions we use top $N$ =100 ones.",
"Detailed hyper-parameters are in the supplementary material.",
"paragraph4 .5em plus1ex minus.2ex-.5emDifferent backbone.",
"We train Faster R-CNN models with ResNeXt-101-32x8d [46] backbone on VG and use the same Pythia setting from Section REF .",
"Results on VQA 2.0 test-dev split are reported in Table REF .",
"We find that our grid features are competitive to the region features even on this more powerful backbone model.",
"Speed-wise, grid features still run substantially faster (23.8$\\times $ ) than region ones.",
"paragraph4 .5em plus1ex minus.2ex-.5emDifferent VQA model.",
"We further test our features obtained from the previous ResNeXt-101 backbone with the state-of-the-art VQA model, MCAN [50] (2019 VQA Challenge winner).",
"We use the open-sourced implementationhttps://github.com/MILVLG/mcan-vqa to train the large version of the model.",
"The results on VQA 2.0 test-dev set are in Table REF , where our own region features perform better than the results reported in [50] due to stronger backbone.",
"On top of that, our grid features work even better than regions, leading to significant improvement over results reported in MCAN [50] (+1.66).",
"This final model reports a state-of-the-art test-std result of 72.71 (single-model performance) for future reference.",
"paragraph4 .5em plus1ex minus.2ex-.5emDifferent VQA task.",
"We use the VizWiz VQA dataset [12], which is a real world dataset of pictures taken with cellphones by visually-impaired users.",
"It is more challenging due to poor image quality, conversation-style questions, and unanswerable questions, .",
"Pythia [16] model is used (2018 challenge winner).",
"Results on the test-dev set of VizWiz are reported in Table REF , where our grid features achieve comparable results to the regions.",
"It is worth pointing out that our grid features run much faster (23$\\times $ ), which provides great potential to be deployed in practice, , on cell phones, to better assist the visually-impaired.",
"Table: Generalizations of grid features.",
"From left to right: (a) Different backbone.",
"We use a ResNeXt-101-32x8d instead of a ResNet-50 as the backbone.",
"(b) Different VQA model.",
"We use MCAN implementation which is the state-of-the-art VQA model.",
"(c) Accuracy on VizWiz using the same VQA models .",
"(d) Image captioning on COCO Karpathy test split.",
"Abbreviations: BLEU4 (B4), METEOR (M), CIDEr (C), and SPICE (S).",
"Our grid features generalize well by achieving results at-par with bottom-up region features while being significantly faster.paragraph4 .5em plus1ex minus.2ex-.5emImage captioning.",
"We train the bottom-up attention model [2] implemented in Pythia [16] taking our features as input for image captioning on COCO [4].",
"No CIDEr [43] optimization [2] is used for fair comparison.",
"Quantitative results on the test set of Karpathy split [18] are reported in Table REF .",
"We use standard evaluation metrics including BLEU4 [30], METEOR [23], CIDEr, and SPICE [1].",
"Similar to the VQA task, our grid features achieve comparable results to bottom-up region ones for image captioning while being significantly faster.",
"Table: Results of end-to-end trained VQA models with grid features on the VQA 2.0 test-dev set.",
"End-to-end learning boosts accuracy for all models and more for ones trained on ImageNet and YFCC.",
"Adding PPM further improves accuracy."
],
[
"Towards End-to-end VQA",
"Although pre-training on VG, ImageNet, or YFCC provides useful feature representations for VQA, there are still potential domain shifts between the pre-training tasks and the downstream tasks.",
"For example, YFCC contains a lot of outdoor images [40], which are not present in the VQA dataset.",
"Instead of using pre-computed fixed feature representations, end-to-end training, where the initial feature representations will be fine-tuned, provides a natural solution to reducing such domain gaps.",
"Empowered by the dramatic simplification of grid features for the VQA pipeline, we take an initial step towards this goal.",
"paragraph4 .5em plus1ex minus.2ex-.5emTraining details.",
"We adopt the 22K learning rate schedule [16] to train both the ResNet-50 model and the Pythia VQA model jointly, with errors from the answering accuracy directly back-propagated to the grid convolutional feature maps.",
"We fix the first two residual blocks and fine-tune the rest of the model.",
"Since the visual representations are computed online (not stored on disk), it allows us to perform data augmentation including color jitter and affine transformation over the input images to reduce chance of over-fitting.",
"For more details see supplementary material.",
"paragraph4 .5em plus1ex minus.2ex-.5emResults.",
"We experiment with three models pre-trained on VG, ImageNet, and YFCC.",
"Note that while VG uses region-level annotations, both ImageNet and YFCC only use image-level ones (human labels or noisy image tags).",
"As can be seen from Table REF , end-to-end training (denoted as `e2e') can boost accuracy for all three pre-trained models, with the biggest improvements for ImageNet models.",
"paragraph4 .5em plus1ex minus.2ex-.5emFlexible network design.",
"As we now have the ability to train our models end-to-end in a simple manner, it allows us to introduce more flexible architectural designs for vision and language tasks [29].",
"Specifically, on top of the grid features from the ResNet-50 model, we add a Pyramid Pooling Module (PPM, a component widely used for semantic segmentation; details in supplementary material) [54], [45] to aggregate visual information from grid features of different spatial resolutions.",
"After adding this module to different pre-trained models (Table REF , `PPM'), the VQA accuracy can be further improved.",
"Remarkably, for ImageNet and YFCC pre-trained models, a combination of end-to-end training and PPM results in close or even better performance than a VG pre-trained model using pre-computed region features.",
"This result is particularly desirable as it indicates good VQA accuracy can be achieved even with zero use of explicit region (bounding box) annotations."
],
[
"Conclusion",
"In this paper, we revisit grid features as an alternative to the widely used bottom-up region features [2] for vision and language tasks.",
"We show they can in fact achieve on-par results in terms of accuracy over different VQA tasks and models and even on captioning.",
"As a result of skipping the computationally expensive region-related bottlenecks in the pipeline, we see remarkable speed-ups – often more than an order of magnitude – to the existing systems that rely on regions.",
"Our experiments show that rather than the `format' of features (region grids), the semantic content that features represent is more critical for their effectiveness.",
"Such effective representation, per our experiment, can be achieved either by pre-training on object and attribute datasets such as VG, or more importantly, by end-to-end training of grid features directly for the end-task.",
"Note that while easy with grid-features, end-to-end training is not trivial with regions.",
"Even with limited exploration in this direction, we already find that given a more flexible design space, grid features pre-trained without any region-level annotations can in fact achieve strong performance on VQA.",
"While we are aware that for tasks like referring expressions [19] where the output itself is a region, modeling region is likely unavoidable, we hope our grid features can potentially offer new perspectives for vision and language research in general.",
"paragraph4 .5em plus1ex minus.2ex-.5emAcknowledgements.",
"We would like to thank Larry Zitnick, Duy-Kien Nguyen, Devi Parikh, and Amanpreet Singh for helpful discussions.",
"ELM acknowledges support from AFRL and DARPA (#FA8750- 18-2-0126).",
"The U.S. government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation thereon.",
"The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the AFRL and DARPA or the U.S. government."
],
[
"Details of Hyperparameters",
"[1]In the MCAN paper, the model is trained for 13 epochs, where each epoch contains 17,967 iterations.",
"[2]We use the implementation provided in [16].",
"Hyper-parameters of different models are summarized in Table REF .",
"For the SGD optimizer, the momentum is 0.9 and weight decay is 0.0001.",
"For the Adamax optimizer, $\\beta _1$ and $\\beta _2$ are 0.9 and 0.999, respectively.",
"No weight decay is used.",
"For the Adam optimizer used in MCAN [50], $\\beta _1$ and $\\beta _2$ are 0.9 and 0.98, respectively.",
"No weight decay is used.",
"We follow the default setting of hyperparameters for most of models.",
"For the image captioning model [2], [16], the default initial learning rate is 0.01.",
"We found 0.002 leads to slightly better results.",
"For the end-to-end trained Pythia (e2e Pythia in the last row), we use an initial learning rate of 0.002 and a larger value of 1 for the gradient clip when fine-tuning the ResNet model for feature extraction."
],
[
"Region Features from FPN",
"In the Pythia implementation [16], of bottom-up attention [2], a Feature Pyramid Network (FPN) model [25] is used to compute region features.",
"This is different from the original Faster R-CNN model [33] used, and it is commonly believed that FPN can offer better object detection quality.",
"Therefore, to reach a more solid conclusion, in this appendix we show extended results from the main paper to compare our grid features with FPN region features.",
"The FPN model uses an entire ResNet model as the backbone, where the multi-scale feature maps of different blocks of the ResNet model are fused in a feature pyramid.",
"Two randomly initialized fully-connect layers (denoted as fc6 and fc7 for simplicity) are added to predict object category, bounding box regression offsets, and attribute labels for each bounding box proposal.",
"We follow the strategy used in [16] to compute region features.",
"Specifically, we use the output of the fc6 layer as input to a VQA or image captioning model, where the fc7 layer is also used and fine-tuned during VQA training.",
"Accuracy on the VQA 2.0 test-dev set and breakdown inference time of the FPN model, using a ResNet50 as the backbone, are summarized in Table REF .",
"Different from the trend observed in object detection [25], we find the FPN model, when used to provide region features for VQA, does not show clear advantage over the original C4 model [2], which in turn gives on-par results to our grid features.",
"Speed-wise, despite the lighter pre-region computation, we find the region-related steps with FPN are still very expensive, and the efficiency advantage of our grid features is even more significant.",
"We also test the top 100 ($N{=}100$ ) regions using different backbones, VQA models, VQA tasks, and image captioning task, as we have done in Section 6 in the paper.",
"Results are reported in Table REF , REF , REF , and REF .",
"For the accuracy on the VQA 2.0 test-dev set and VizWiz, the FPN model's accuracy is lower than the results reported in [16], because grid features (from an ImageNet pre-trained ResNet-152 [15] model) are used in addition to the region features [16].",
"Using the MCAN model [50], the FPN model achieves better results than reported in [50] but still performs worse than C4 and our grid features.",
"Table: This table extends Table 2 in the main paper for speed and accuracy comparisons with added rows for region features with FPN.",
"Results are reported on VQA 2.0 test-dev with accuracy and inference time breakdown measured in milliseconds per image.",
"Despite the advantages which FPN features have that 1) pools features from higher-resolution feature maps; and 2) fine-tunes the fc7 layer when training VQA; our grid features achieve comparable VQA accuracy to all region features and are much faster.Table: This table extends Table 6 in the main paper for generalization experiments.",
"From left to right: (a) Different backbone.",
"We use a ResNeXt-101-32x8d instead of a ResNet-50 as the backbone.",
"(b) Different VQA model.",
"We use MCAN implementation which is the state-of-the-art VQA model.",
"(c) Accuracy on VizWiz using the same VQA models .",
"(d) Image captioning on COCO Karpathy test split.",
"Abbreviations: BLEU4 (B4), BLEU3 (B3), BLEU2 (B2), BLEU1 (B1), ROUGE_L (RL), METEOR (M), CIDEr (C), and SPICE (S).",
"Our grid features generalize well by achieving results at-par with bottom-up region features while being significantly faster."
],
[
"Details of PPM",
"In Section 7 of the main paper, we introduce end-to-end training of the Pythia VQA model [16] with PPM (Pyramid Pooling Module) [54].",
"A detailed illustration of this module is provided in Fig.",
"REF .",
"Given a grid convolution feature map from a ResNet model, adaptive average pooling operations are performed at three different spatial resolutions: 1$\\times $ 1, 4$\\times $ 4, and 8$\\times $ 8.",
"Three separate convolution layers (followed by batch normalization and ReLU) are added, where the kernel sizes are all set to 1 and output dimensions are all 512.",
"Finally, the original grid feature map is concatenated together with the three ones obtained from PPM as the input for VQA.",
"Figure: Illustration of PPM (Pyramid Pooling Module) experimented in the end-to-end model for VQA.",
"See Section for details."
]
] | 2001.03615 |
[
[
"A modular apparatus for use in high-precision measurements of parity\n violation in polarized eV neutron transmission`"
],
[
"Abstract We describe a modular apparatus for use in parity-violation measurements in epithermal neutron-nucleus resonances with high instantaneous neutron fluxes at the Manuel Lujan Jr.\\ Neutron Scattering Center at Los Alamos National Laboratory.",
"This apparatus is designed to conduct high-precision measurements of the parity-odd transmission asymmetry of longitudinally polarized neutrons through targets containing nuclei with p-wave neutron-nucleus resonances in the 0.1-10 eV energy regime and to accommodate a future search for time reversal violation in polarized neutron transmission through polarized nuclear targets.",
"The apparatus consists of an adjustable neutron and gamma collimation system, a \\(^3\\)He-$^{4}$He ion chamber neutron flux monitor, two identical cryostats for target cooling, an adiabatic eV-neutron spin flipper, a near-unit efficiency \\(^6\\)Li-\\(^{7}\\)Li scintillation detector operated in current mode, a flexible CAEN data acquisition system, and a neutron spin filter based on spin-exchange optical pumping of $^{3}$He gas.",
"We describe the features of the apparatus design devoted to the suppression of systematic errors in parity-odd asymmetry measurements.",
"We describe the configuration of the apparatus used to conduct a precision measurement of parity violation at the 0.7 eV p-wave resonance in $^{139}$La which employs two identical $^{139}$La targets, one to polarize the beam on the p-wave resonance using the weak interaction and one to analyze the polarization."
],
[
"Introduction",
"Compound neutron-nucleus resonance reactions have proven to be an excellent laboratory in which to study parity violation [1], [2].",
"The complex, many-body nature of this nuclear system with its very high level density was predicted theoretically [3], [4] and confirmed by experiment [5], [6] to provide a natural amplification mechanism for Parity Violating (PV) neutron-nucleus interactions.",
"Statistical spectroscopy has been used to successfully analyze the widths of the distributions of observed parity-odd asymmetries in several heavy nuclei in the isolated resonance regime corresponding to eV-keV neutron energies [8].",
"The present theory involving the mixing of nearby s-wave and p-wave resonances that accounts for this amplification of parity-odd amplitudes also predicts a similar amplification of Time Reversal Invariance Violating (TRIV) amplitudes [9].",
"Parity- (P) and Time- (T) odd effects in forward transmission provide null tests for time reversal invariance [10].",
"Compound nuclear systems are therefore attractive candidates for T-violation searches.",
"With the advent of high-flux MW-class short-pulsed spallation neutron sources, sensitive tests of TRIV in neutron-nucleon interactions can be realized.",
"Due to the proportionality between potential TRIV amplitudes and PV amplitudes, these future TRIV tests will need more complete and precise information on the properties of the resonances which exhibit large parity violation effects.",
"In addition to new measurements of (n, $\\gamma $ ) angular distributions [11] which can be used to determine the details of the quantum numbers of these $\\ell =1$ (p-wave) resonances, it is also important to repeat previous measurements of P-odd neutron transmission asymmetries [5], [12], [13], [14], [15] with higher precision.",
"By measuring the transmission asymmetry of longitudinally polarized neutrons through an unpolarized target, one can determine the parity-odd helicity dependence of the total cross section.",
"The transmission asymmetry is defined as $A=\\frac{Y^+-Y^-}{Y^++Y^-}=\\tanh (f_nnt\\sigma _pP)where Y^\\pm is the neutron transmission yield for the two neutron helicities, f_n is the polarization of the neutron beam, \\sigma _p is the p-wave cross section for unpolarized neutrons, n is the number density of the target nuclei, t is the target thickness, and P is the parity-odd longitudinal analyzing power, defined by its relationship to the helicity-dependent cross section, \\sigma ^\\pm :{\\equation \\sigma ^\\pm =\\sigma _p(1\\pm f_nP)\\endequation }$ The two most effective and well-known methods to polarize neutron beams [18] in the eV energy range are spin-dependent scattering from a polarized proton target [19] and spin-dependent absorption in a polarized $^{3}$ He target [20].",
"There also exists a third, lesser-known polarization method which is at present practical only for the $^{139}$ La nucleus.",
"The $^{139}$ La nucleus possesses a p-wave resonance near 0.7 eV, whose interference with nearby s-wave resonances results in a 10% parity-odd asymmetry.",
"This is larger than expected on dimensional grounds alone by about a factor of $10^6$ .",
"As demonstrated by C.D.",
"Bowman et al.",
"[21] and later Yuan et al.",
"[22] this parity violating asymmetry is large enough that one can use it to polarize the beam using the (intrinsically parity-odd) weak interaction.",
"By transmitting such a weakly-polarized beam through a second $^{139}$ La target, one can realize a measurement of the P-odd asymmetry in which the $^{139}$ La targets act as both a polarizer and as a polarization analyzer, resulting in the aptly-named `Double Lanthanum' measurement method.",
"Although the beam polarization produced on the parity-odd 0.7 eV resonance in $^{139}$ La is small [21][22], its use in this configuration eliminates certain systematic errors associated with measuring $P$ , especially the absolute knowledge of the neutron beam polarization $f_n$ .",
"In this paper we describe how our apparatus may be modified to accommodate setups for both the `Double Lanthanum' $P$ measurement method as well as a traditional $P$ measurement using a $^{3}$ He neutron spin filter.",
"In addition to its relevance for future time reversal violation experiments, a precise determination of $P$ for $^{139}$ La is of potential use as a standard analyzer in eV neutron scattering and transmission experiments by allowing one to measure the absolute polarization of epithermal neutron beams containing $\\ell =1$ , 0.7 eV neutrons.",
"$^{139}$ La was already used for this purpose in the measurements of P-odd asymmetries in heavy nuclei conducted by the TRIPLE collaboration [2].",
"It was also used in some pioneering measurements of the depolarization of polarized eV neutrons in transmission through magnetic fields in polarized and aligned nuclear targets to extract information on the internal magnetic domain structure [23][24][25].",
"The large P-odd asymmetry on the $0.88$ eV $\\ell =1$ resonance in $^{81}$ Br could also be used for this purpose.",
"The possibility of constructing a compact eV neutron polarization analyzer at these two energies based on neutron absorption in LaBr$_{3}$ scintillating crystals is under analysis.",
"In this paper we describe the apparatus and experimental setup we have realized to perform precision measurements of the parity violation, $P$ , present in the eV neutron energy range as configured for the Double Lanthanum experiment.",
"The Los Alamos Neutron Science Center (LANSCE) beam facility is described in Section .",
"The mechanical apparatus used for the parity violation measurement, including the collimation, cryostats, mechanical rotation stage, and detector shieldhouse, are all described in Section .",
"The custom spin flipper constructed for this experiment and optimized for 0.7 eV neutrons is described in Section .",
"Details of the electronics design for the spin flipper are described in Section .",
"The fast-response current mode $^6$ Li-$^{7}$ Li glass scintillator detector designed for this experiment is discussed in Section .",
"Section describes the data acquisition system, including all measured parameters and their experimental relevance.",
"Section describes the design of the polarized $^{3}$ He neutron spin filter for eV neutrons to be installed for future parity violation measurements.",
"A brief summary and description of possible future measurements which can be conducted with this instrument is given in Section ."
],
[
"The FP12 neutron beam facility at LANSCE",
"The apparatus described in this paper was located in an experimental hutch on Flight Path 12 (FP12) at LANSCE.",
"Neutrons which are delivered to FP12 are produced by the 1L Lujan Target-Moderator-Reflector-Shield (TMRS) Mark-III assembly and are moderated by a partially-coupled cold hydrogen moderator.",
"The flight path is equipped with an $m=3$ , 10 cm$\\times $ 10 cm supermirror neutron guide designed to transport a large fraction of cold neutrons to the experimental hutch [26].",
"The neutron flux and spectrum in the slow neutron regime between 1-80 meV was already measured in the past and described in the literature [28].",
"MCNPX simulations of FP12 were performed to extract the energy spectrum and emission time distributions of 0.7 eV neutrons.",
"The detailed 3D geometry as implemented in the MCNPX model is shown in Figure REF .",
"The model included a detailed implementation of the TMRS Mark-III surrounded by the Bulk Shielding.",
"The neutron guide is surrounded with concrete/steel/polyethylene shielding and the experimental hutch is shielded with borated polyethylene clad in 3/8\" thick steel.",
"The neutron beam terminates in a beam stop made of steel and borated polyethylene.",
"The calculations of the time-emission spectra followed the methodology outlined in [29].",
"The TMRS model used in these simulations was based on the as-built engineering design.",
"The thermal neutron scattering kernels that were used are for hydrogen and deuterium in water, ortho- and para-hydrogen in liquid hydrogen, aluminum in alloys, and iron in steel.",
"All calculations utilized the next-event-estimator (point detector) variance reduction technique [32] to ensure efficient convergence of all extracted observables.",
"The relatively sharp proton pulse (270 ns pulse width) incident on the spallation target at a short-pulsed spallation neutron source like LANSCE is a very valuable feature for precision measurements of parity violation and time reversal violation on neutron-nucleus resonances as it greatly improves the time resolution of the neutron time-of-flight measurements necessary to accurately determine the neutron energy.",
"Figure: Elevation view of the FP12 geometry as implemented in the MCNPX model.Figure: An MCNPX calculation of the neutron energy spectrum for FP12, normalized to the peak flux intensity.",
"This was generated using MCNPX simulations with the most recent FP12 geometry.Figure: A calculation of the time emission spectra for neutrons of energy 0.7 eV, normalized to the peak intensity.The FP12 hutch is a steel enclosure shaped like a trapezoid, as seen in Figure REF .",
"It is approximately 4.73 m long in the beam direction, 2.74 m tall, and 4.61 to 6.22 meters in the transverse direction.",
"The beam pipe is centered 1.37 meters above the floor.",
"The most upstream components are collimators that scrape the neutron beam to a 10 cm diameter and a $^{3}$ He-$^{4}$ He ion chamber to measure the neutron flux.",
"Next downstream is the first cryogenically cooled $^{139}$ La target, an adiabatic spin flipper, and a second $^{139}$ La target.",
"The furthest downstream component is a shielded $^6$ Li-rich scintillator neutron detector buried in shielding designed to attenuate both neutrons and gamma rays to reduce background noise.",
"Figure REF shows a detailed view of these components and their placement.",
"Figure: Top-down view of the FP12 hutch at LANSCE showing the hutch dimensions and general position of the experimental setup.Figure: A cross-section view of the experimental setup in FP12 at LANSCE as viewed from the side.",
"The three components are the collimator stand, the main apparatus containing the cryostats and the spin flipper coil assembly, and the detector/shieldhouse assembly."
],
[
"Mechanical Apparatus Components",
"Previous experiments have measured $P$ on the 0.7 eV resonance in $^{139}$ La to $\\sim 4\\%$ accuracy [22].",
"In order to achieve a measurement with 1% accuracy as needed for the future time reversal and neutron polarimetry physics, we decided to use cryogenic targets to minimize the Doppler broadening of the transmitted resonance profile.",
"To prepare for the future NOPTREX time reversal search experiment described in [10], we mounted these cryostats on a rotating turntable and chose a $^{3}$ He neutron spin filter polarized by spin-exchange optical pumping (SEOP) methods which can later be placed in the position of the upstream cryostat.",
"These decisions strongly influenced the apparatus design, which otherwise possesses several components very similar to an earlier instrument built by the TRIPLE Collaboration [16].",
"The apparatus also possesses some common features with the POLYANA instrument at the IBR-30 pulsed reactor neutron source at Dubna [17].",
"The apparatus can be divided into three major beamline components; the collimator stand; the main apparatus stand (mounted on the rotation stage); and the detector stand/shieldhouse.",
"One key feature of almost all components is the use of non-magnetic materials.",
"Due to the oscillating magnetic fields produced by the neutron spin-flip coils, any magnetic material could be magnetized, leading to stray fields that could affect the neutron spin-flip efficiency as well as cause magnetic-field dependent gain shifts in our detector electronics.",
"Therefore much of the apparatus is constructed from aluminum, brass, and plastic."
],
[
"Collimators and Stand",
"The collimator stand, located furthest upstream at the exit of the beam pipe, supports interchangeable collimators and the $^3$ He-$^4$ He beam monitor.",
"The stand is made from aluminum held together with brass screws and bolts.",
"A 96 cm long rectangular aluminum channel provides a reliable way to interchange up to 16 collimators and ensures that all collimators are coaxial with the beam.",
"Adjustable feet allow for precision alignment of the collimators to the beam axis using a theodolite.",
"The beam monitor rests at the downstream end of the stand and is electrically isolated from the rest of the stand via a polycarbonate plastic base.",
"The collimators define the neutron beam and minimize the contribution of fast neutrons and gamma rays to the radiation backgrounds in the hutch.",
"5.01 cm thick borated polyethylene collimators attenuate fast neutrons outside of the desired beam profile.",
"The borated polyethylene collimator located the furthest upstream has an additional 1 mm layer of $^6$ Li-loaded fluorinated plastic which collimates slow neutrons and produces a very small number of gamma rays per absorbed neutron.",
"After the borated polyethylene collimators, 3.65 cm thick brass collimators attenuate gamma rays produced both by the `gamma flash' characteristic of spallation targets as well as any gamma rays produced by neutrons absorbing in the borated polyethylene collimators.",
"The collimator collection is also draped with $^{6}$ Li fluorinated plastic to absorb high energy neutrons scattered by the hydrogen but not absorbed by the boron in the fast neutron collimators before they can create additional neutron and gamma ray backgrounds signals in the detector.",
"Collimators made of $^{10}$ B-loaded plastic are also placed outside the upstream ends of the cryostat vacuum flanges to sharply define the beam that passes through the targets.",
"The stand is stable enough to also support a calibration resonance target wheel and a local neutron chopper if needed.",
"We measure the neutron time-of-flight spectrum on a pulse-by-pulse basis using a beam monitor, which is a low efficiency ion chamber with $^{3}$ He and $^{4}$ He gas chambers located back-to-back along the neutron beam and operated in current mode.",
"Since the neutron absorption cross section in $^{3}$ He is very large (of order kilobarns) and well known [34], the neutron absorption cross section in $^{4}$ He is extremely small, and the gamma interactions of these two isotopes are essentially identical, the difference signal from these two chambers is directly proportional to the instantaneous neutron flux.",
"We do not elaborate any more on the details of the design for this device as it is identical to the one used by the TRIPLE collaboration as it is already described in great detail in the scientific literature [35]."
],
[
"Main Apparatus and Rotation Stage",
"The main apparatus frame is constructed from extruded aluminum profiles (manufactured by 80/20 Inc, hereafter referred to as “8020\") mounted to a large rotation stage with adjustable feet.",
"The base of the main apparatus is a large aluminum table with large 8020 support legs.",
"On top of this table sits a Franke TSD-830M rotation stage, labeled in Figure REF .",
"The rotation stage has a 57,000 N load rating and allows for 360 degree rotation of the main apparatus.",
"The rotation stage is mounted to adjustable feet which allow for fine-tuned adjustments to center the targets on the beam axis.",
"A large (1.22 m diameter and 2.54 cm thick) circular aluminum plate rests on the rotation stage and supports the main rectangular frame constructed from 8020.",
"The frame supports the cryostat/cryogenic target assemblies, adiabatic spin-flipper coils, and can be modified to include additional equipment such as a $^3$ He polarizer (described in further detail in Section ).",
"The spin-flipper coils are affixed directly to the 8020 aluminum frame using specially-designed plastic clamps.",
"The aluminum housing for the cryostats are supported from the top of the 8020 frame on sliding rails, allowing for easy adjustment of the target location along the neutron beam.",
"The maximum distance between the two targets is 152 cm.",
"The cryogenic housing is mounted to the sliding rails by three V-groove-and-ball kinematic mounts.",
"This allows for reproducible, precision placement of the targets when moving or replacing the cryogenic housing during a target change.",
"In this modular design other components such as a large $^3$ He polarizer can also couple to the sliding rails or the 8020 frame.",
"One nice feature of using 8020 for the frame is the ease with which components can be precisely and repeatably aligned using mechanically-defined reference points.",
"The aluminum table was aligned to the beam using a theodolite.",
"All other components used mechanically-defined reference points to define alignment.",
"Figure: The main apparatus in the Double Lanthanum experimental setup as configured with two cryogenic 139 ^{139}La targets installed.",
"The lefthand diagram shows a downstream view of the apparatus (i.e.",
"looking down the axis of the neutron trajectory) whereas the righthand shows a sidelong, cross-section view."
],
[
"Target Environment",
"Cryo-Torr 8 cryopumps were stripped of their internal cryopumping surfaces and used as mechanical refrigerators to cool the targets to $\\sim $ 15 K. The cryopumps couple to the aluminum housing and then to an aluminum vacuum chamber below.",
"The neutron beam passes through 1.27 mm thick aluminum vacuum windows centered on the beam.",
"Inside the cryostat vacuum chamber, the target cell is thermally coupled to the 15 K stage of the cold head, while a surrounding radiation shield is coupled to the 80 K stage of the cold head.",
"The target cells are contained in 12.03 cm diameter aluminum cans, filled in an inert argon atmosphere in a glovebox and sealed with indium o-rings.",
"This prevents potential oxidation of sensitive or reactive target materials such as lanthanum to preserve the integrity of the target and to promote the safe handling of activated targets.",
"To monitor the temperatures inside of the cryostats, four silicon diode temperature sensors were placed at different stages of the cryostat; one on the radiation shield, one on the coldhead, one at the top of the cell clamp, and one at the bottom of the cell clamp.",
"The temperatures measured by diode thermometers were read using a Lakeshore 218 temperature monitor.",
"The temperature distribution in the cryogenic system under steady state operation was stable at the level of 0.2 K."
],
[
"Detector and Shieldhouse",
"Located furthest downstream from the beam pipe entrance to the hutch is the detector stand and shielding assembly, known as the shieldhouse.",
"The purpose of this shieldhouse is to protect the $^6$ Li neutron detector from any neutrons and gamma rays originating from outside the defined neutron beam (e.g.",
"multiply-scattered neutrons from the hutch or gamma rays produced in neutron capture reactions on materials in the hutch).",
"The support structure frame is made of 8020 and sheets of borated polyethylene.",
"The outermost layer of shielding consists of 15 cm of borated polyethylene, followed by 10 cm of lead bricks and 5 cm of lithium carbonate powder as seen in Figure REF .",
"The detector sits in an aluminum tube and was aligned to the neutron beam axis using a theodolite and crosshairs of fishing line anchored to the aluminum tube using offset set screws."
],
[
"Field Mapper",
"In order to successfully make a precision measurement of the parity violation in the target nuclei, it is critical to understand the dynamics of the neutron spin motion in the fields and determine the spin-flip efficiency.",
"To do so, detailed maps of the magnetic fields produced by the spin flipper array (discussed in more detail in Section ) were needed.",
"One of the challenges presented with the design of the spin flipper coils was a difficulty in mapping the magnetic field.",
"Mounting a magnetic probe to a motorized mapper system was very difficult due to spatial constraints enforced by the coils, their supports, and lead wires.",
"A simple mapping system was constructed by mounting a Lakeshore 460 triple axis probe to a three-dimensional, manually translatable apparatus.",
"Continuous translational range of motion was possible in the $xy$ plane perpendicular to the direction of the neutron beam.",
"In the longitudinal $z$ direction, pairs of threaded holes were machined into a long, metal plank; pairs of screws were then placed in each set of holes and by laying the y-directional translation stage such that an edge was laid flush to both screws, unique points along the $z$ axis were defined.",
"Cross-sectional $x,y$ slices of the magnetic field were then mapped for each of the points defined along the $z$ -axis.",
"12 planar maps were taken for each configuration of the spin flip field.",
"This data, shown in Figure REF , was then reconstructed and interpolated using COMSOL and used in calculating the neutron spin flip efficiency as discussed further in Section .",
"Figure: CAD drawing of the mapper used to map the B-field.",
"Note the pairs of holes machined into the longitudinal plank; these were used to uniquely define points along the zz-axis for maps to be taken."
],
[
"Adiabatic Spin Flipper for eV Neutrons",
"Following the design of Roberson et al.",
"[16], we simulated and built a neutron Adiabatic Spin Flipper (ASF) based on adiabatic spin motion in static magnetic fields.",
"This spin flipper is mounted and aligned to the 8020 frame, ensuring that the ASF longitudinal coils shown in Figure REF are coaxial with both the lanthanum targets and the neutron beam.",
"The coil geometry and currents were optimized for neutron energies near 1 eV as needed for the Double Lanthanum experiment, though the adiabatic condition can be met for a broad range of neutron energies.",
"As in the TRIPLE apparatus, the neutron spin flipper consists of longitudinal and transverse coils that produce static gradient magnetic fields.",
"The combination of these fields produces a total field with approximately constant amplitude $B_0$ that turns the neutron spin over the distance $L$ on the $z$ -axis over which the spin rotation occurs.",
"The spin flipper operates in two field configurations, which we shall call the no-flip and flip configurations.",
"For the no-flip configuration, only the longitudinal $B_z$ coils are turned on.",
"If the direction of the neutron beam is along the $z$ -axis and $x=y=z=0$ is taken to be the center of the spin flipper as shown in Figure REF , then the magnetic field produced by the longitudinal coils in the no-flip state can be approximated as $B_z ={\\left\\lbrace \\begin{array}{ll}B_0 & \\text{if $z<-L/2$} \\\\-B_0\\sin {(\\pi z/L)} & \\text{if $-L/2 \\le z\\le L/2$} \\\\-B_0 & \\text{if $z>L/2$}\\end{array}\\right.",
"}Recall that the magnetic torque \\vec{\\tau } experienced by a particle with a magnetic moment \\vec{\\mu } in an external magnetic field \\vec{B} is{\\equation \\vec{\\tau }=\\vec{\\mu }\\times \\vec{B}\\endequation }$ For the flip configuration, the transverse $B_y$ coils are turned on (in addition to the $B_z$ coils).",
"The component produced by the transverse coils is given by $B_{y}=\\left\\lbrace \\begin{array}{ll}\\pm B_{0}\\cos (\\pi z/L) & \\mbox{if } -L/2\\le z \\le L/2\\\\\\;\\;\\;0 & \\mbox{otherwise}\\end{array}\\right.$ The $\\pm $ sign indicates that the $B_y$ component can be either parallel or anti-parallel to the $y$ -axis, depending on the direction of the current in the transverse coils.",
"The superposition of the $B_z$ and $B_y$ fields produces a field $\\vec{B}_{\\textnormal {total}}$ that is constant in magnitude but rotates by $180^{\\circ }$ in the $zy$ plane over the length $L$ .",
"A positive $B_y$ component will produce a counter-clockwise rotation of the total field in the $zy$ plane, while a negative $B_y$ component produces a clockwise rotation.",
"As the neutron magnetic moment $\\vec{\\mu }_n$ is related to the neutron spin $\\vec{\\sigma }$ by the gyromagnetic ratio $\\gamma _n$ , one can construct the Hamiltonian of the magnetic interaction with an external magnetic field $\\vec{B}$ .",
"Substituting this Hamiltonian into the Schrödinger equation for a time dependent $\\vec{B}$ field, the Larmor equation for the spin can be obtained [7]: $\\frac{d}{dt}\\vec{\\sigma }(t)=\\gamma _n\\vec{\\sigma }(t)\\times {}\\vec{B}(t)$ The change in the neutron spin direction is normal to both $\\vec{B}$ and $\\vec{\\sigma }$ at any given time.",
"For the no-flip state there should not be a change in the neutron spin as the spin and the magnetic field are parallel.",
"If the field as seen in the rest frame of the neutron varies slowly enough (i.e.",
"adiabatically), the neutron spin will precess about the field direction with the Larmor frequency $\\omega _{L}=\\gamma _{n}B$ and will “follow” the direction of the external field from its initial direction $+z$ to the opposite direction $-z$ at the end of the flipper.",
"The rotation frequency $\\omega _B$ of the magnetic field as seen in the rest frame of the neutron is given by $\\omega _B=\\frac{\\pi }{t}=\\frac{\\pi }{L/v}$ where $t$ is the time the neutron spends in the spin flipper and is given by $L/v$ , where $v$ is the neutron speed and $L$ is the length of the spin flip region.",
"The ratio of the field rotation frequency $\\omega _B$ and the Larmor frequency $\\omega _L$ , is defined as the adiabaticity parameter for this system: $\\gamma =\\frac{\\omega _B}{\\omega _L}=\\frac{\\pi v}{\\gamma _n LB}$ If $\\gamma <<1$ the spin direction will undergo several rotations around the $\\vec{B}$ direction for every small variation of the field, keeping its precession axis approximately aligned with $\\vec{B}$ at any given time.",
"In other words, as long as the neutron speed $v$ is below a certain limit, the transport of the spin will be adiabatic.",
"Figure REF shows the magnetic field configuration for the flip and no-flip states of the spin flipper, as well as the total magnetic field and neutron spin projection along the field direction as neutrons propagate across the spin flipper length.",
"Figure: Components of the magnetic field in the a) spin-flip and b) no-spin-flip configurations.",
"The elements of the apparatus producing each component of the magnetic field (longitudinal -red- coils and transverse -blue- coils) are highlighted.",
"Also the total magnetic field B → total \\vec{B}_{\\textnormal {total}} and the projection of the neutron spin 〈σ →〉\\langle \\vec{\\sigma } \\rangle along its axis of precession are shown in the zyzy plane for each case.This simplified description assumes that all of the neutrons travel directly down the longitudinal axis of the coils.",
"Realistic spin flipper efficiency calculations must account for the finite radius of the beam as well as deviations away from the on-axis magnetic field.",
"Bowman, Penttilä, and Tippens [27] investigated these effects for neutrons whose trajectories deviate slightly from the axis of symmetry.",
"One must consider not only the efficiency of the spin flipper in the flip configuration but the efficiency in the no-flip configuration.",
"The total efficiency of the spin flipper is the average of the efficiencies for the flip state and the no-flip state: $\\epsilon _{\\textnormal {flip}}=1-\\frac{1}{\\sqrt{1+\\gamma ^{-2}}}\\left[1-\\cos \\left(\\pi \\sqrt{\\gamma ^{-2}+1}\\right)\\right],$ and the spin-preserving efficiency in the no-flip state is $\\epsilon _{\\textnormal {no-flip}}=1-\\frac{\\pi ^{3}r^{2}}{8\\gamma L^{2}},$ with $r$ the distance of the neutron trajectory from the spin flipper axis.",
"It can be seen that $\\epsilon _{\\textnormal {flip}} \\rightarrow 1$ when $\\gamma \\rightarrow 0$ ; the smaller the adiabaticity parameter, the higher the spin-flip efficiency.",
"Also, $\\epsilon _{\\textnormal {no-flip}}=1$ when $r=0$ ; if all neutrons travel over the beam axis every spin remains unchanged in the no-flip state.",
"It now makes sense to define a `total' efficiency, $\\epsilon _{tot}$ , as a figure of merit for the design which describes both how well the flip configuration flips the spin and how well the no-flip configuration preserves the spin.",
"We therefore define $\\epsilon _{\\textnormal {total}}=\\frac{\\epsilon _{\\textnormal {flip}}+\\epsilon _{\\textnormal {no-flip}}}{2}For an ideal spin flipper that flips all neutron spins in the \\textit {flip} configuration and leaves all neutron spins unchanged in the \\textit {no-flip} configuration, we see \\epsilon _{tot}=1.$ To define the optimum parameters for our spin flipper, we calculated the flip, no-flip and total efficiency for different values of magnetic field amplitude $B_0$ and spin flipper length $L$ using equations REF -REF .",
"We assumed a 10 cm diameter neutron beam and used the square root of the average value of $r^2$ , $\\sqrt{\\langle r^{2} \\rangle }$ =3.54 cm to calculate the spin-preserving efficiency.",
"Figure REF shows these calculations.",
"Considering the constraints in length for the Double Lanthanum experiment at FP12, we chose $L=120$ cm for the flipping length, the maximum allowed in the apparatus.",
"From $z=-60$ cm to $z=60$ cm the field $\\vec{B}$ has the described form for the flip and no-flip states (equations REF and REF ).",
"For 60 cm $<\\vert z\\vert <$ 75 cm we aimed to produce a longitudinal constant field of amplitude $\\pm B_0$ , providing a uniform field in the region of the La targets (see Figure REF ).",
"The choice of $B_0$ values were constrained by practical experimental and safety reasons: it has to be generated by reasonably attainable electric currents and voltages, yet also has to be large enough to assure a good efficiency.",
"We found that for $L=120$ cm, a magnetic field amplitude of $B_0$ =16-17 G can produce a total efficiency between 92%-98%, as seen in Figure REF .",
"Figure: Calculations of the a) spin flipper efficiency and b) spin preserving efficiency as functions of the parameters B 0 B_0 and LL for 1 eV neutrons at r=3.54r=3.54 cm, the value of 〈r 2 〉\\sqrt{\\langle r^2\\rangle } over a 10 cm diameter beam.",
"The total efficiency is shown in c) and d).To establish the position along $z$ and the parameter $NI$ (number of turns of a particular coil multiplied by the current) for each of the coils comprising the longitudinal field array (see Figure REF ), we considered the current $I$ to be fixed, and in an iterative process of varying the position and obtaining the number of turns by the Single Value Decomposition (SVD) method, we obtained the optimum coil configuration.",
"With $I=15$ A and a maximum number of turns that vary from 1-2 turns for the small coils to $\\sim $ 80 for the large coils, the voltage requirement using 10 AWG copper wire is close to 40 V, which is achievable with the system described in Section .",
"Figure: a) Field maps for the longitudinal and transverse components of the spin flipper; 14 xyxy planes were scanned at positions between -50 to 50 cm along the zz-axis.",
"b) Measured magnetic field components along the zz-axis and their comparison with the design model fields.The field maps of the actual spin flipper that was constructed on FP12 were obtained using the mapper described in Section REF .",
"A total of 14 $xy$ planes of 11 cm $\\times $ 11 cm of cross section around the SF axis were scanned in steps of 5 mm.",
"The field maps span the region between -50 to 50 cm in the $z$ -axis, as shown in Figure REF a.",
"A comparison of the measured magnetic fields and the initially calculated magnetic fields along the spin flipper axis is shown in figure REF b; a good agreement is observed in general, however it is important to point out that the longitudinal magnetic field (pictured in red), has a higher amplitude than the transverse magnetic field, producing a total magnetic field that, although performing the desired rotation by 180$^\\circ $ in the $yz$ plane, does not maintain a constant amplitude.",
"The field map was obtained in the middle of the experimental run; therefore the spin flipper was operated with the magnetic fields configuration shown in figure REF .",
"These field maps, in combination with the images of the neutron beam profile and data from a Monte Carlo neutron spin transport simulation, can be used to estimate the actual spin flipper efficiency 0.7 eV neutrons."
],
[
"Neutron Beam Intensity Maps",
"The radial dependence of the neutron beam intensity must be mapped to sufficient accuracy that it can be included with the magnetic field map to calculate the neutron spin flipper efficiency.",
"We determined that an intensity map with a few mm spatial resolution over the 10 cm diameter beam would suffice to determine the spin flip efficiency with an accuracy more than one order of magnitude better than our statistical accuracy goal.",
"This measurement was performed with a commercial imaging plate using a neutron-sensitive Gd-doped film.",
"Its spatial resolution is one order of magnitude better than required for our purposes.",
"Figure REF shows an example of such a map.",
"The neutron imaging plate technology used to produce these intensity maps has been demonstrated to possess a linear response over a dynamic range of about 4 orders of magnitude as determined in the course of careful studies conducted at NIST for a Penning-trap-based neutron lifetime experiment [37]."
],
[
"Mechanical Design of the Adiabatic Spin Flipper and Controls",
"The spin flipper consists of twelve axially concentric longitudinal coils of different radii and winding numbers connected in series to produce the sinusoidally varying longitudinal field along the $z$ -axis (Eq.",
"REF ) and one pair of axially concentric Helmholtz coils that produce the cosinusoidally varying transverse field along the $y$ -axis (Eq.",
"REF ).",
"The coils were designed using COMSOL to determine the coil parameters (coil dimensions, number of turns, positions, and necessary currents) needed to produce the desired fields.",
"In addition, one pair of `shunt' coils was constructed, consisting of two coils identical in construction to the transverse field coils.",
"Because of the relatively large ($\\sim 15$ A) currents needed to produce these fields and the relatively short timescales ($\\sim $ 100 ms) needed for the transverse field on/off transitions, the shunt coils were introduced to the circuit so that instead of turning the current in the transverse coils on and off and potentially producing electronic crosstalk, the current was instead diverted to the shunt coils via a switch box module.",
"The longitudinal and transverse coils were mounted to the frame of the apparatus and the shunt coils were placed in a corner of the experimental hutch as far away as possible from the spin-flip region of interest, approximately 3 meters from the center of the spin flipper.",
"The field produced by the shunt coils at the center of the spin flipper was measured and determined to be negligible."
],
[
"Switch Box Design",
"To control the neutron spin flip coils, we designed a switch box containing an array of FET switches with opto-isolated gate drivers controlled by TTL signals sent from an Arduino Mega 2560 board.",
"The switch box was designed to be able to handle a maximum current of 20 A and to react quickly enough during a change of state that viable neutron pulses were not lost due to a slow slew time for the switch box.",
"The limiting factor for the switching rate was found to be the intrinsic 120 ms settling time that it took for the currents to achieve a steady state in the coils.",
"This characteristic time was due to the large inductance of the coils and the settling time of the eddy currents in the aluminum coil frames.",
"The switch box allowed complete control over the state of the spin flipper.",
"By sending the appropriate logic signal from the Arduino board to the switch box, current to the coils could be turned on/off, reversed in polarity, or diverted to/from the shunt coils.",
"The Arduino board was connected to the external Lujan Center $t_0$ signal as a trigger source, and its output 5 V logic signals were sent through a simple conditioning circuit before entering the switch box.",
"Three LEDs were included in the conditioning circuit to visually confirm that the state of the Arduino and that the states were changing as expected.",
"Figure: A simplified schematic of the spin flipper circuit including the primary transverse coils and the shunt transverse coils.",
"The circuit was controlled by an Arduino Mega 2560 board."
],
[
"Neutron Spin Flip Sequencing",
"The choice of neutron spin flip sequencing is a critical factor in avoiding systematic errors that would create a false asymmetry and compromise the validity of this experiment.",
"For example, the gain of the detector may drift linearly in time due to effects such as temperature-dependence of the detector electronics.",
"Even more dangerous are gain effects that correlate directly to the absolute spin state of the neutron; stray magnetic fields produced by the spin flipper may influence the gain of the PMT dynode chain and cause the detector to have a different gain for each neutron spin state.",
"An important feature of the spin flipper is that it works equally well for both configurations ($\\pm $ ) of the field produced by the transverse coils.",
"Therefore, a careful choice of the pattern with which the neutron spins are flipped can cancel systematic effects.",
"Following the prescription by Roberson et al [16], the base spin sequence pattern F N N F N F F N was chosen, where N and F denote the no-flip and flip states, respectively.",
"A discussion of how the spin state was determined during the data analysis process is discussed in section REF .",
"Because the neutron spin flip behavior is identical for the cases where the transverse field is aligned ($+$ ) or anti-aligned ($-$ ) with the $y$ -axis and recalling that the transverse field off (0) is the no-flip state, we can additionally alternate the polarity of the field used to produce the flip states, making our spin flip sequence + 0 0 - 0 - + 0.",
"One can write the change in the neutron yield as a function of the stray field produced by the spin flipper, $B_s$ , as $Y(B_s)=Y(B_0)+\\frac{\\partial Y}{\\partial B}B_s+\\frac{1}{2}\\frac{\\partial ^2 Y}{\\partial B^2}B^2_s+...$ then the difference in the neutron yields for the flip and no-flip states, $\\Delta Y$ , is found to be [27]: $\\Delta Y = \\overline{Y(F)}-\\overline{Y(N)}=\\frac{1}{2}\\frac{\\partial ^2Y}{\\partial B^2}B^2_s+...We can see here that the dependence on a term linear in \\vec{B_s} vanishes, leaving us to only have to measure the change in the gain of the magnetic field at the detector location to estimate the quadratic term.$"
],
[
"Fast-Response Current Mode $^{6}$ Li Glass Scintillator Detector",
"The current mode detector used in our apparatus was designed to be identical to one used by the TRIPLE Collaboration [36].",
"The reasons for this design choice were the similarities in neutron flux and the sensitivities achieved in the TRIPLE Collaboration's previous experiments [21], [22].",
"The main goal of the design is to convert the neutron flux in a neutron-absorbing scintillator detector into an output current as a function of neutron time-of-flight.",
"When the neutron rate is low, the detector can also resolve individual pulses in pulse counting mode."
],
[
"Scintillator Characteristics",
"The neutron detector shown in Fig.",
"1 contains a 13.3 cm diameter $\\times $ 1 cm thick cylinder of Scintacor GS20 $^{6}$ Li loaded glass optically coupled to a photomultiplier tube (PMT).",
"The GS20 $^{6}$ Li loaded glass has a density of $2.5$ g/cm$^{3}.$ The reaction n + $^{6}$ Li $\\rightarrow $ $^{7}$ Li* $\\rightarrow $ $^{4}$ He + $^{3}$ H + 4.8 MeV is used to detect neutrons.",
"The GS20 glass is 6.6% lithium by weight, enriched to 95% $^{6}$ Li.",
"The 1 cm thickness of lithium glass gives a neutron absorption efficiency of 90% for 1 eV neutrons.",
"Due to the 1/$v_n$ dependence of the n + $^{6}$ Li $\\rightarrow $ $^{4}$ He + $^{3}$ H reaction cross section, the efficiency is lower at higher energies.",
"The cross section of this reaction at 1 eV neutron energy is approximately 147 b [38].",
"The scintillation light from neutron capture in GS20 glass has a fast 18 ns component, a slower 57 ns component, and a 98 ns rise time for the signal to rise from 10% to 90% of its full value.",
"The scintillation light produced by the energy deposited by the $^{4}$ He and $^{3}$ H ions is detected in the PMT.",
"An otherwise identical detector/PMT combination was constructed which uses glass depleted in $^{6}$ Li so that it is very insensitive to neutrons but has an almost identical response to gammas.",
"The attenuation of gammas in the materials used in the scintillator/PMT is low enough that one can place the $^{7}$ Li-rich detector directly behind the $^{6}$ Li-rich detector to measure and subtract out the signal from the gammas in the beam if needed.",
"Based on the high signal/background ratio of our signal at neutron time of flights corresponding to the 1 eV region, we decided not to install the $^{7}$ Li-rich backing detector for the double lanthanum measurements.",
"Figure: Side view of the detector design.",
"From left to right: an aluminum window 18 cm in diameter, the 6 ^{6}Li glass scintillator, the Hamamatsu R1513 PMT with its μ\\mu -metal magnetic shield, the PCB, and the exit window with electrical feedthroughs.",
"Not shown in this cutaway is the LED installed to produce pulsed light used for calibration purposes."
],
[
"PMT and Analog Electronics",
"We chose a Hamamatsu R1513 PMT with a S-20 photocathode for its low photocathode resistivity and corresponding superior performance in current mode [36].",
"The maximum gain for this PMT is around 3.3 $\\times 10^{5}$ .",
"A single photoelectron pulse at the anode of the PMT has a rise-time of around 7 ns.",
"The PMT base circuitry is shown in Figure REF .",
"The photocathode current was specified to output 2 $\\mu $ A with a full scale of 2 mA.",
"The full scale current requires a gain of 1000 within the dynode chain.",
"This gain was achieved with a 1050 V input.",
"The 2 $\\mu $ A current enables the detector to handle an instantaneous rate of 10$^{11}$ neutrons/s striking the detector given that there are typically 100 photoelectrons produced in the PMT when a single neutron is captured.",
"The cathode within the PMT was grounded to ensure stable, low-noise operation of the PMT.",
"The baseline level was designed to be 0.0 V which corresponded to 0 $\\mu $ A.",
"A 2 $\\mu $ A current in the PMT produced an output voltage of -2.0 V. The PMT uses an active voltage divider so that the divider ratio is unaffected by load currents from the PMT.",
"A light emitting diode was also included in the design of the PMT assembly so that testing could be performed with a internal LED.",
"In addition to the high voltage supply needed to run the PMT, a +8V/-8V supply powers a buffering stage operational amplifier.",
"All output signal cables are LEMO connections.",
"Figure: A simplified schematic of the PMT assembly."
],
[
"Data Acquisition",
"This setup was designed to measure the total neutron transmission through matched disks of $^{139}$ La.",
"Because of the high instantaneous flux rates needed for such a transmission measurement, it is not feasible to count individual neutron pulses; instead, the total integrated current output of the detector is measured.",
"Our data acquisition system, like our detector, must be designed to handle the output signal in real time.",
"We used a CAEN V1724 8-channel, 14-bit digitizer with a maximum sampling rate of 100 MS/s to record the signals from the $^3$ He ion chamber, current-mode neutron detector, and 3 Faraday pickup loops (5 signals in total).",
"The facility $t_0$ signal from the 20 Hz Lujan Center accelerator proton current pickup coil was used to simultaneously trigger the spin-flipper controller and the data acquisition.",
"Upon receiving a $t_0$ trigger, 75000 2-byte samples were recorded for each of the 5 input voltage signals and stored in the onboard memory buffers where it was read out via an optical link to the data acquisition computer.",
"In order to reduce the data file size, the respective waveforms were then decimated in software before being written to long-term storage.",
"All further data manipulation and analysis was handled offline."
],
[
"Signal processing",
"Higher-resolution spectroscopy of the 0.7 eV resonance was desired in order to increase the precision with which we could measure the Doppler-narrowed 0.7 eV resonance peak.",
"To do so, each of the 5 signals were sampled every 10 ns by the V1724 digitizer and then decimated on-board by a factor of $2^6=64$ .",
"After being read out by the CAEN board to the data acquisition desktop computer, the data underwent further decimation where it was structured into `windows,' i.e.",
"different regions of the waveform were decimated by different factors in order to keep the data files at a reasonable size.",
"The total data set for the Double Lanthanum experiment was $\\sim 4$ TB.",
"If such a decimation scheme had not been implemented, the data set was expected to be well over 400 TB, which is unwieldy in terms of data storage and manipulation during the analysis process."
],
[
"Determination of Spin State",
"Extreme caution must be exercised when deciding how to record the spin state of the flipper in the datastream.",
"If one chooses to measure a signal that correlates directly with the absolute spin state of the flipper (e.g.",
"using a magnetic field probe to monitor the actual value of the $\\vec{B}$ field in real time), then it is possible that nonzero false asymmetries may creep into the data via insidious means such as electronic cross-talk between channels on the data acquisition modules.",
"Although data acquisition modules are designed to minimize cross-talk, we did not want to risk such an occurrence in a high-statistics, precision experiment.",
"Systematic ucertainties that effectively mimic a physics asymmetry completely compromise the validity of such an experiment.",
"To monitor the spin flipping process, each of the three coil sets (longitudinal, transverse, and shunt) had a respective lead wire threaded through its own pickup coil.",
"Each pickup coil consisted of a toroidal solenoid wrapped around an iron core.",
"The ferromagnetic core ensured that maximal magnetic flux was captured.",
"Because the state of the flipper is determined by the direction of the current through the coils, any change in current will induce a voltage in the pickup coil.",
"We measured these induced voltages to determine spin state transitions, from which the preceding and succeeding spin states could be determined.",
"Because the longitudinal spin transport field is always on, our spin flipper state can be defined by the state of the transverse field coil, of which there are three: off, no transverse field (0); on, positive transverse field (+); on, negative transverse field (-).",
"For some early test runs, a Lakeshore 460 triple axis magnetic field probe was placed at the center of the spin flipper to record the absolute state of the transverse field to be used to test our sorting algorithm (this probe was later removed so that there was no chance of interference with our dataset).",
"Figure REF shows the overlaid field probe readings of a few thousand pulses.",
"We can see that there are three stable magnetic field states, indicated by the three distinct levels in Figure REF , as well as the transitions happening between them.",
"We have developed a method to sort the spins by identifying the pickup coil voltage signature for each transition and used this information to tag each pulse as either flip or no-flip, described below.",
"Given our spin flip pattern of ($+00-0-+0$ ), we can see that we should have six unique transitions between states: $(+\\rightarrow 0), (0\\rightarrow 0), (0\\rightarrow -), (-\\rightarrow 0)$ , $(-\\rightarrow +)$ , and $(0\\rightarrow +)$ .",
"Figure REF shows a few examples of these voltage signatures that were produced when the spin flipper transitioned states, and Figure REF shows these same traces superimposed on one another to show relative lineshapes and amplitudes.",
"However, we have found that all eight transitions (the six unique transitions and the two degenerate $(0\\rightarrow -)$ and $(+\\rightarrow )$ transitions) between states can be uniquely identified by the pickup loop signature–this is due to the small differences in the placement of the pickup coils in relation to the spin flipper, allowing each to capture dissimilar enough magnetic flux changes to produce distinct voltage signatures.",
"Figure: This figure shows the pulses that were identified using the cuts made on the histograms of integrated voltage signals.",
"Note all eight transitions are present, including the `no change in state' transition, shown in yellow.Figure: Voltage signals were produced in the pickup coils by changing the currents flowing through the spin flipper coils.",
"Here we see examples of 6 different spin state transitions.Figure: Induced voltage signals for all 6 spin state transitions, superimposed to allow for comparison.To sort the data pulses by spin state, each voltage signal for a given pickup coil was integrated over the 50 ms pulse to return a single value for each pulse.",
"These values were then histogrammed and clear peaks emerged for each spin state transition.",
"By defining the upper and lower bounds for each peak on the histogram, we were able to determine and sort each spin state by understanding which transition was happening for each pulse and tagging the preceding spin states appropriately.",
"Figure REF shows a sample of some of the peaks, though it is worth noting that there were more than 8 peaks due to the summation of the pulses immediately preceding the initial transitional pulses causing small peaks very near the `no spin state change' peak.",
"Figure: A histogram showing the summation of each voltage signal produced by a spin state transition shows unique values for each of the states.",
"These values were then used to sort pulses into their appropriate spin states."
],
[
"$^3$ He Spin Filter Apparatus and Design",
"A polarized $^3$ He gas neutron spin-filter works by utilizing the spin dependent neutron absorption cross section of polarized $^3$ He.",
"Neutrons with their spins parallel to that of the $^3$ He gas will be transmitted while neutrons with anti-parallel spins will be absorbed.",
"A polarized $^{3}$ He neutron spin filter of sufficient thickness and $^{3}$ He polarization can polarize the neutron energies of interest between 0.1-10 eV with neutron polarization and transmission high enough to conduct several interesting experiments.",
"The efficient removal of the antiparallel spins by absorption rather than scattering as in a neutron spin filter based on polarized proton scattering, combined with the spatial uniformity of the $^{3}$ He gas polarization in the cell, makes the polarized neutron beam phase space highly uniform, This is an advantageous property for an eventual time reversal violation experiment as it suppresses possible sources of systematic error associated with neutron small angle scattering in the polarized target.",
"The energy-dependent neutron transmission measurements possible at a pulsed spallation neutron source like LANSCE allow the neutron beam polarization produced by such a neutron spin filter to be determined to high accuracy by comparing the relative neutron transmission intensity for the $^{3}$ He in the spin filter polarized versus unpolarized [41].",
"Polarized $^3$ He gas is produced using spin-exchange optical pumping (SEOP) [44].",
"SEOP refers to the process of using optical pumping to polarize rubidium vapor and allowing the spin polarization of the Rb electrons to be transferred to $^{3}$ He nuclei via spin exchange during gas-phase collisions.",
"This spin-exchange is mediated by the hyperfine interaction between the Rb valence electron and the $^3$ He nucleus.",
"Below we include a brief description of the physics behind this process and our design of the system.",
"Extensive references to previous work on polarized $^{3}$ He neutron spin filters can be found in a recent review [45]."
],
[
"Rubidium Optical Pumping",
"Optical pumping describes the process by which photons are used to redistribute the occupied states of some collection of atoms.",
"Resonant absorption of light stimulates these states out of thermodynamic equilibrium toward a single desired state.",
"Rubidium, as an alkali metal, is hydrogen-like because its outermost electron is shielded by complete inner shells.",
"The Hamiltonian for the ground state of Rb in a static magnetic field $B_0 \\hat{z}$ is: ${\\cal H} = A_g\\vec{I} \\cdot \\vec{S} + g_s \\mu _B S_z B_0 \\ - \\frac{\\mu _I}{I}I_z B_0$ where $\\vec{I}$ is the Rb nuclear spin, $\\vec{S}$ is the electron spin, $A_g$ is the isotropic magnetic dipole coupling coefficient, $g_s$ is the electron g-factor, $\\mu _B$ is the Bohr magneton, and $\\mu _I$ is the nuclear magnetic moment.",
"The first term in the Hamiltonian is the hyperfine interaction between the nucleus and electron, while the second and third terms are the Zeeman interactions of the electron and nucleus with the external magnetic field.",
"In the weak-field limit, the hyperfine interaction is larger than the Zeeman interaction, producing hyperfine splitting on the order of GHz and Zeeman splitting on the order of MHz.",
"The total angular momentum of the atom is then $\\vec{F} = \\vec{I} + \\vec{J}$ with its projection $m_F$ onto the the $z$ -axis.",
"For rubidium, the ground state $^{2}S_{1/2}$ and excited state $^{2}P_{1/2}$ have an energy separation corresponding to a light wavelength of $\\lambda = 794.8$ nm.",
"These states are separated further into two $F$ states due to hyperfine splitting.",
"Increasing the $B$ -field further separates the $F$ -states into $2F+1$ subdivisions.",
"In the $^3$ He gas cell there are two isotopes of Rubidium: $^{85}$ Rb with nuclear spin $I=5/2$ and hyperfine levels $F=3,2$ and $^{87}$ Rb with nuclear spin $I=3/2$ and hyperfine levels $F=2,1$ .",
"An unpolarized $794.8$ nm photon will cause a $^{2}S_{1/2} \\rightarrow {}^{2}P_{1/2}$ transition according to the selection rules: $\\Delta F = 0,\\pm 1$ and $\\Delta m_F = 0,\\pm 1$ .",
"However, if said photon is circularly polarized with angular momentum pointing in the direction of the static magnetic field, this further restricts the allowed transition.",
"The new selection rules are then: $\\Delta F = 0,\\pm 1$ and $\\Delta m_F =\\pm 1$ .",
"The excited state will then spontaneously decay back into the ground state by emission of a photon with arbitrary polarization and $\\Delta m_F = 0,\\pm 1$ selection rules.",
"Given sufficient time, nearly all the Rb atoms will migrate to the highest $m_F$ value possible because there is no transition out of this state.",
"This $m_F$ value corresponds to a situation where the electron and nuclear magnetic moments point in the same direction as the static magnetic field.",
"The Rb atoms are therefore polarized and in the ground state.",
"To improve the efficiency of this process a small amount of $\\rm {N_2}$ gas is added to the cell.",
"When an excited Rb atom spontaneously decays the emitted photon has a random polarization and can re-excite another Rb atom.",
"Because the photon is not circularly polarized, this allows for $\\Delta m_F = 0,\\pm 1$ transitions.",
"This in turn will slow down the polarization of the Rb atoms.",
"The $N_2$ in the cell circumvents this bottleneck by allowing the Rb atoms to de-excite before they spontaneously decay.",
"This relaxation is achieved by transferring the energy to the rotational and vibrational states of the $N_2$ molecules.",
"This is made possible by the large quenching cross section of $5.8 \\times 10^{-15}$ cm$^{-2}$ for $N_2$ gas [46].",
"The relaxation takes place $\\sim $ 10 times faster than the spontaneous decay."
],
[
"Spin-Exchange",
"The next stage in polarization of $^3$ He is the spin-exchange between the Rb valence electron and the $^3$ He nucleus.",
"From the perspective of the spins, the collision is dominated by two interactions: $ V = \\gamma \\vec{N} \\cdot \\vec{S} + \\alpha \\vec{K} \\cdot \\vec{S}The first interaction is between the Rb valence electron spin \\vec{S} and the rotational angular momentum \\vec{N} of the Rb and ^3He pair.",
"The second interaction is the hyperfine coupling between the Rb valence electron spin \\vec{S} and the nuclear spin \\vec{K} of the ^3He atom.",
"The constants \\gamma and \\alpha are functions of the separation distance R and determine the interaction strength.",
"Spin relaxation of the Rb electron is from the spin-rotation interaction while the spin-exchange is caused by the hyperfine interaction.",
"During a collision, the spin angular momentum of the Rb electron is transferred to the ^3He nucleus.",
"Because this spin exchange is a slow processes, acquiring maximally polarized ^3He gas takes on the order of 10 hours.$"
],
[
"Cell and Oven",
"The $^3$ He gas cell, shown in Figure REF , is 12.1 cm in diameter and 10.8 cm thick in the longitudinal direction.",
"The cell contains $^3$ He gas at 2 bar pressure, 0.09 bar of N$_2$ gas, and solid Rb inside.",
"To turn the solid Rb into a vapor, the cell needs to be heated to an ambient temperature of about 200 $^\\circ $ C. The cell is held in place with a Teflon holder and high temperature polyimide tape.",
"Teflon is used because of its high heat-resistant properties and flexibility.",
"Additionally, the Teflon holder supports two coils used for free induction decay (FID) NMR.",
"The purpose of FID NMR is to measure the $^3$ He polarization in a non-destructive way.",
"Figure: 3 ^3He gas cell in its Teflon holder.The oven used to heat the cell is a $26.7 \\times 26.7 \\times 63.5$ cm$^3$ rectangular box with 3 layers.",
"The first layer is made of 1.27 cm Garolite “G7\" and supports the gas cell and adiabatic fast passage (AFP) NMR coil.",
"The AFP NMR coil is used to flip the $^3$ He polarization direction.",
"The second layer is made of aluminum and holds six 7.6 cm electrical heating cartridges (Omega: CSH 103220).",
"The third layer is also made of 1.9 cm G7, supports the first two layers and provides insulation.",
"The end caps, also made of 1.27 cm G7, hold sapphire glass windows.",
"G7 is a glass-silicon laminate, has a density of 4.57 g/cm$^3$ , tensile strength of 20,000 PSI, and a maximum continuous operating temperature of 221$^\\circ $ C. These properties make it especially appropriate as a support and as an insulator for an oven with a target temperature of 200$^\\circ $ C. The purpose of the aluminum layer is to better distribute the heat created by the cartridge heaters.",
"The outermost layer is supported by an adjustable support to better align the $^3$ He gas cell to the neutron beam.",
"Almost all oven components are made with non-magnetic material such as aluminum and brass.",
"Figure REF shows the $^3$ He cell situated inside of the oven.",
"Figure: Picture of oven with 3 ^3He gas cell and cartridges heaters installed."
],
[
"Laser Optics",
"To excite the Rb atoms, a 50 W fiber-coupled 795 nm laser is used.",
"Circularly polarized light is created by passing linearly polarized light through a quarter wave plate at a 45$^\\circ $ angle.",
"First, the laser fiber is coupled to a beam collimator (ThorLabs: F220SMA-780) that focuses and reduces the spread of the beam.",
"The laser is then incident on a polarizing cube beam-splitter (Edmund Optics: 49-872) that linearly polarizes and splits the beam 90$^\\circ $ into two paths.",
"Both paths are then incident on a 2.54 cm diameter, 795 nm quarter wave plate (QWP).",
"Each QWP is mounted on a precision motorized rotation device (ThorLabs: K10CR1).",
"These devices allow for remote flipping of the direction of circular polarization by rotating the QWPs by 90$^\\circ $ .",
"One path is then reflected by 90$^\\circ $ using a 50 mm diameter dielectric mirror (Edmund Optics: 33-189).",
"Finally, both paths are reflected in the oven with large 8 inch diameter dielectric coated silicon mirrors.",
"The mirrors reflect the laser light, but are nearly transparent to neutrons.",
"The laser light then passes through the sapphire glass windows and into the the $^3$ He gas cell.",
"Figure REF shows a schematic of the optical layout.",
"All optical components and oven are mounted on to a $48.2 \\times 119.3 \\times 1.27$ cm ($19 \\times 47 \\times 0.5$ inch) optical breadboard.",
"Additionally, the breadboard and optics are enclosed in a black-anodized light tight box, shown in Figure REF .",
"All optical components are made as non-magnetic as possible by using brass and aluminum parts.",
"Figure: Diagram showing the optics layout for creating circularly polarized light.Figure: Picture of optics setup in black-anodized light tight box."
],
[
"$\\mu $ -Metal Solenoid and Support",
"A $\\mu $ -metal shielded solenoid provides the uniform magnetic field required for the SEOP process.",
"The $\\mu $ -metal shielded solenoid is approximately 61 cm $\\times $ 61 cm $\\times $ 122 cm in size.",
"The internal structure of of the $\\mu $ -metal shield is made of an 8020 frame of 2.5 cm $\\times $ 2.5 cm profiles.",
"The light tight box (with the optical components and oven mounted to the optical bread board) slides into the $\\mu $ -metal shield on a Teflon sheet.",
"Figure REF shows the shieldhouse that surrounds the solenoid, light tight box, and oven assembly.",
"The $\\mu $ -metal shield is critical due to the $^3$ He spin-filter's close proximity to the adiabatic spin-flipper.",
"Because a very small magnetic field gradient is needed over the length of the $^3$ He gas cell, it is crucial to shield it from external magnetic fields; $\\mu $ -metal works well as a magnetic shield due to its high magnetic permeability.",
"Stray, slowly-varying magnetic fields are distributed around the shielding and away from the center where the $^3$ He cell is located.",
"The solenoid enclosed in the $\\mu $ -metal shield produces a magnetic field along the neutron beam direction that holds the polarization of $^3$ He atoms.",
"In the central 7 cm $\\times $ 16 cm region where the $^3$ He cell is located, the magnetic field has an average gradient $\\langle \\frac{\\nabla B_x}{B_i}\\rangle = 5.55\\times 10^{-4}\\pm 2.66\\times 10^{-4}$ cm$^{-1}$ along the longitudinal polarization direction, and an average gradient $\\langle \\frac{\\nabla B_y}{B_i}\\rangle = 6.03\\times 10^{-4}\\pm 1.68\\times 10^{-4}$ cm$^{-1}$ in the transverse direction.",
"Based on these measurements, the expected relaxation time of the $^3$ He is $~100-200$ hours.",
"The solenoid has a resistance of $R=25.8$ $\\Omega $ , and is driven at 1.00 A with a current stability of 0.1$\\%$ .",
"At this current, the solenoid produces a holding field of 9.26 gauss at its center which, along with an 30 kHz NMR pickup coil, is tuned to the 3.24 kHz/gauss gyromagnetic ratio of $^3$ He[65].",
"The NMR signal is then used to determine the polarization of the $^3$ He gas.",
"Due to the modular nature of the apparatus, this polarized $^{3}$ He neutron spin filter can be installed at the location of the first cryostat (upon removal of the cryostat from the apparatus).",
"It can then be operated continuously on-line during measurements, as was successfully done several years ago using a similar apparatus [39] for the first phase of the NPDGamma neutron-proton weak interaction experiment [40].",
"Figure: Picture of μ\\mu -metal shielded solenoid."
],
[
"Summary and Outlook",
"We have described a flexible, rotatable cryogenic apparatus for the measurement of parity violation in the transmission of longitudinally polarized neutrons in neutron-nucleus resonances in the 0.1-10 eV energy range at a short-pulsed spallation neutron source.",
"The apparatus employs beam monitors and transmission detectors combining both (n, $\\gamma $ ) discrimination and current mode operation, a broadband neutron spin flipper with a flipping mode and sequence designed to suppress systematic errors, and a neutron polarizer based on spin-exchange optical pumping of $^{3}$ He.",
"The apparatus design employs several features used in previous neutron parity violation experiments at LANSCE, especially from work performed by the TRIPLE and NPDGamma collaborations and referred to in the text.",
"The apparatus that we have described also possesses sufficient flexibility to accommodate several modifications that can greatly extend the range of possible experiments beyond the Double Lanthanum experiment that we concentrated on as an example in this paper.",
"The first major addition to the apparatus we envision for the measurement of P-odd transmission asymmetries in other nuclei beyond $^{139}$ La is the addition of the polarized $^{3}$ He neutron spin filter based on spin-exchange optical pumping (described in Section 8) and which was first demonstrated at LANSCE [20].",
"The experimental hutch on the FP12 beamline possesses an access path through the ceiling near the present location of the downstream refrigerator that we plan to use for the operation of cryostats, which require safe venting of cryogenic liquids into a storage volume on top of the hutch, as done in the past for liquid parahydrogen targets operated on FP12 [42], [43].",
"In particular, this implementation would allow us to conduct a new measurement of the parity-odd asymmetry in the 3.2 eV p-wave resonance in $^{131}$ Xe, which is one of the leading candidates for a sensitive time reversal test, using a solid xenon target.",
"Liquid helium supply and venting for the dilution refrigerator required for the polarized nuclear targets to be used in the time reversal test can also be provided through this path.",
"Much care was taken in designing this apparatus to minimize any systematic effects that would affect the uncertainty associated with the P-violation measurements.",
"Because this apparatus was designed to be modified to accommodate several nuclear species, each measurement will have its own associated systematic effects.",
"The most dominant systematic effect associated with the design of the apparatus itself lies in how well one can determine the neutron spin flip efficiency.",
"In this paper, we present a method to calculate this efficiency using flux monitor measurements of the beam intensity convoluted with the image plate beam profiles and measurements of the $B$ -field combined with a Monte Carlo calculation of the neutron intensity-weighted spin flip efficiency integrated over the beam cross sectional area.",
"In this case, the dominant uncertainty will lie in how well one can reconstruct the $B$ -field produced by the spin flipper coils, i.e.",
"the spatial resolution of the $B$ -field maps, which is on the order of a few parts in 10$^{3}$ .",
"If this accuracy does not suffice one can install a $^3$ He polarizer/ analyzer pair and measure the neutron spin flip efficiency directly using the method described by Musgrave et al.",
"in [41].",
"The beamline components can be translated upstream far enough to accept gamma detector arrays downstream of the target to conduct parity violation measurements using (n, $\\gamma $ ) reactions [47] as well as in neutron transmission measurements.",
"The addition of a gamma detector array to this apparatus would enable one to efficiently search for large P-odd asymmetries between 0.1-10 eV in the several isotopes of heavy nuclei which have never been measured.",
"The TRIPLE collaboration concentrated its choice of nuclei for measurement near the peaks of the 3p and 4p p-wave strength functions in an attempt to maximize the number of p-wave resonance per unit energy interval in the isolated resonance eV-keV regime for their statistical analysis of parity violation in heavy nuclei.",
"Since that work was conducted, improved data on the p-wave strength function [48] has resolved an additional small peak near A=160, which is split off from the 4p giant resonance due to nuclear deformation effects as predicted long ago [49], whereas the spherical optical model by contrast predicts a minimum at A=160.",
"It therefore may be worthwhile to search in this A range to see if one can discover more p-wave resonances that amplify parity violation.",
"One could also search for large parity-odd effects in the (n, $\\gamma $ ) channel for nuclei that undergo neutron-induced fission and try to confirm and extend the many large P-odd asymmetries already measured in the fission channel [50], [51], [52], [53], [54], [4], [55], [56], [57], [58], [59], [60].",
"In particular one could measure the P-odd asymmetries in the (n, $\\gamma $ ) channel in very low energy resonances in $^{235}$ U [61], [62], which is a nucleus that can be safely handled and supplied at the LANSCE facility.",
"There is an already-proven method for nuclear polarization of $^{235}$ U using the very large hyperfine fields produced at the nucleus in the low-temperature ferromagnetic phase of uranium sulfide.",
"This phenomenon was exploited in the past to determine the spin assignments in $^{235}$ U using spin-dependent polarized neutron transmission through polarized $^{235}$ U [63].",
"The nonmagnetic composition of the apparatus also lends itself to the use of polarized xenon targets produced by a spin-exchange optical pumping process very similar to that described above for $^{3}$ He.",
"The 10 cm x 10 cm cross section, m=3 supermirror neutron guide presently installed upstream of the LANSCE FP12 hutch [28] could be replaced in the future with other types of neutron guides that can increase the fraction of neutrons transmitted to the apparatus near 1 eV.",
"The dimensions of the FP12 hutch transverse to the neutron beam direction are wide enough to allow for the complete rotation of the relevant components of the apparatus by 180 degrees as envisioned in one of the modes of realization for a precision test of time reversal invariance [10].",
"According to the results of MCNP simulations of the new configuration of the LANSCE target/moderator/reflector system scheduled to be installed in early 2020 [64] we do not expect the projected changes, including the removal of the liquid hydrogen slow neutron moderator, to significantly influence the beam characteristics in our energy range of interest between 1-10 eV and therefore the performance of the components of the apparatus presented above should remain essentially unchanged after this upgrade."
],
[
"Acknowledgements",
"This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Award Number DE-SC-0014622.",
"D. Schaper would also like to acknowledge support by the NSF Graduate Research Fellowship Program under Grant Number 1247392.",
"The work of C. Auton, J. Curole, J. Doskow, W. Fox, H. Lu, W. M. Snow.",
"K. Steffen, J Vanderwerp, B.",
"Short, and G. Visser was supported by NSF PHY-1614545, NSF PHY-1913789, and the Indiana University Center for Spacetime Symmetries.",
"J. Curole also acknowledges support from the GAANN fellowship program of the US Department of Education and the Department of Energy SCGSR program.",
"L. Cole and D. Olivera would like to acknowledge funding from the KY-NSF EPSCoR Research Scholar Program.",
"B.M.",
"Goodson acknowledges funding from NSF (CHE-1905341), DoD (W81XWH-15-1-0272), and a Cottrell SEED Award from the Research Corporation for Science Advancement.",
"We gratefully acknowledge the local support of the LANSCE neutron facility at Los Alamos National Lab where this measurement was performed.",
"Test measurements for the current mode detector electronics used for this work were conducted in part on the NOBORU instrument at the JPARC MLF through proposal 2016B0021, Characterization of High counting rate Epithermal Neutron Detectors"
]
] | 2001.03432 |
[
[
"Associating Ridesourcing with Road Safety Outcomes: Insights from Austin\n Texas"
],
[
"Abstract Improving road safety and setting targets for reducing traffic-related crashes and deaths are highlighted as part of the United Nation's sustainable development goals and vision zero efforts around the globe.",
"The advent of transportation network companies, such as ridesourcing, expands mobility options in cities and may impact road safety outcomes.",
"In this study, we analyze the effects of ridesourcing use on road crashes, injuries, fatalities, and driving while intoxicated (DWI) offenses in Travis County Texas.",
"Our approach leverages real-time ridesourcing volume to explain variation in road safety outcomes.",
"Spatial panel data models with fixed effects are deployed to examine whether the use of ridesourcing is significantly associated with road crashes and other safety metrics.",
"Our results suggest that for a 10% increase in ridesourcing trips, we expect a 0.12% decrease in road crashes (p<0.05), a 0.25% decrease in road injuries (p<0.001), and a 0.36% decrease in DWI offenses (p<0.0001) in Travis County.",
"Ridesourcing use is not associated with road fatalities at a 0.05 significance level.",
"This study augments existing work because it moves beyond binary indicators of ridesourcing presence or absence and analyzes patterns within an urbanized area rather than metropolitan-level variation.",
"Contributions include developing a data-rich approach for assessing the impacts of ridesourcing use on our transportation system's safety, which may serve as a template for future analyses of other US cities.",
"Our findings provide feedback to policymakers by clarifying associations between ridesourcing use and traffic safety, while helping identify sets of actions to achieve safer and more efficient shared mobility systems."
],
[
"Abstract",
"Improving road safety and setting targets for reducing traffic-related crashes and deaths are highlighted as part of the United Nations sustainable development goals and worldwide vision zero efforts.",
"The advent of transportation network companies and ridesourcing expands mobility options in cities and may impact road safety outcomes.",
"We analyze the effects of ridesourcing use on road crashes, injuries, fatalities, and driving while intoxicated (DWI) offenses in Travis County, Texas.",
"Our approach leverages real-time ridesourcing volume to explain variation in road safety outcomes.",
"Spatial panel data models with fixed-effects are deployed to examine whether the use of ridesourcing is significantly associated with road crashes and other safety metrics.",
"Our results suggest that for a 10% increase in ridesourcing trips, we expect a 0.12% decrease in road crashes, a 0.25% decrease in road injuries, and a 0.36% decrease in DWI offenses in Travis County.",
"On the other hand, ridesourcing use is not significantly associated with road fatalities.",
"This study augments existing work because it moves beyond binary indicators of ridesourcing availability and analyzes crash and ridesourcing trips patterns within an urbanized area rather than their metropolitan-level variation.",
"Contributions include developing a data-rich approach for assessing the impacts of ridesourcing use on the transportation system's safety, which may serve as a template for future analyses for other cities.",
"Our findings provide feedback to policymakers by clarifying associations between ridesourcing use and traffic safety and uncover the potential to achieve safer mobility systems with transportation network companies.",
"Ridesourcing is a term used to describe the operation of transportation network companies such as Uber, Lyft, DiDi Chuxing, and RideAustin, where the rider can hail a vehicle from the convenience of their smartphone using a web or smartphone application.",
"Ridesourcing services are offered in more than a hundred cities in the US and hundreds of cities worldwide.",
"As an example, ridesourcing trips have grown exponentially, with Uber providing ten billion trips globally in 2018 [1].",
"Transportation network companies' service can bridge mobility supply gaps, offering a convenient and competitive mode alternative with pooled-service capabilities [2], enhancing urban transportation options [3], potentially reducing private vehicle ownership [4], and even providing emergency services by replacing expensive ambulances [5], [6].",
"However, empirical studies uncover adverse effects of ridesourcing travel, including associations with congestion [7], competition with public transportation modes [3], [8], increased net energy use [9], and environmental externalities [10].",
"Ridesourcing operations could lead to road safety benefits by reducing road crashes, driving while intoxicated (DWI) offenses, and the welfare losses associated with those [11].",
"Road safety improvements are a focus of several initiatives around the world, such as the United Nations' sustainable development goals, and in the United States through vision zero efforts [12].",
"Ridesourcing use in a city could be associated with a reduction of DWI offenses in locations where other transport modes are not provided or are less attractive than driving one's vehicle under the influence of alcohol (e.g., use of ridesourcing due to alcohol consumption was highly rated by a survey’s respondents) [13].",
"On the other hand, ridesourcing use could be positively associated with crashes, based on existing empirical evaluations of the impact of ridesourcing on road safety at the metropolitan level.",
"Existing empirical studies denote that ridesourcing is associated with an overall increase in vehicle miles traveled due to drivers deadheading between offering rides as well as cruising when waiting to be assigned to the next passenger pick-up [9], [14].",
"There is also evidence of induced travel contributing to vehicle miles traveled rise [15] which does not only result in congested city centers but is also related to increased road crash rates [16].",
"In this study, we investigate how the use of ridesourcing influences the rate of vehicular crashes, injuries, fatalities, and DWI offenses, and we capture safety-related externalities of ridesourcing use.",
"To our knowledge, this is the first empirical study that uncovers such associations while accounting for the intensity of ridesourcing use by leveraging actual ridesourcing pick-up and drop-off coordinates.",
"Previous evaluations of ridesourcing impacts on road safety outcomes focused on effects across metropolitan regions and used dichotomous indicators of ridesourcing presence or absence as a predictor variable.",
"Instead, we utilize more granular data on ridesourcing trips per census tract for Travis County, Texas.",
"Spatial panel data models are employed to demonstrate associations between ridesourcing trips and road safety outcomes.",
"Existing studies focus on uncovering associations between ridesourcing entry and road safety outcomes, DWI offenses, and alcohol consumption.",
"As shown in Table REF , almost all empirical studies examine the impact of ridesourcing services entry on the relative change in road crash rates (e.g., [17]).",
"Out of eight economics and epidemiology studies in this field, only one attempts to capture ridesourcing intensity effects.",
"Barrios et al.",
"use Google search count data as a ridesourcing exposure proxy due to the absence of real-world information on this type of trips [16].",
"Even though safety-related literature highlights significant spatio-temporal associations between traffic and road safety outcomes [18], [19], the studies mentioned above do not capture such effects.",
"Our analysis at a local spatial scale is more appropriate for safety management and policy response than an analysis at a metropolitan scale [18].",
"Table: Summary of methods from existing literature.Across studies, safety outcomes examined vary between crash, injury, and fatality rates, as well as DWI offense rates.",
"A subset of studies focuses on modeling alcohol-related fatalities and injuries [11], [20], [21].",
"Difference-in differences models are the primary approach in the existing literature to capture causal relationships between ridesourcing entry and road safety outcomes but a few researchers also conduct time-series analysis to track trends over time [21], [22].",
"Among the studies reviewed, the following variables serve as controls: vehicle miles traveled, annual average daily traffic or population are included as traffic measures, and indicators of socio-demographic and economic characteristics, driving-related laws, vehicle ownership, and public transit use.",
"Time units of such analysis vary between weeks, months, or quarters.",
"The smallest spatial unit of the observations used in such analysis is more commonly a city, a county, or a metropolitan statistical area, primarily due to the lack of more granular data of transportation network companies' travel patterns.",
"Findings are not consistent across studies, even comparing those that follow similar methods and use similar spatio-temporal units of analysis.",
"For example, Brazil and Kirk conclude that transportation network companies entry is not significantly associated with any of the categories of traffic fatalities examined in their work for the US [17]; a similar conclusion is reached by Huang et al.",
"[22] in their South African-focused study.",
"Morrison et al.",
"hypothesize that the resumption of ridesourcing operation in specific US cities is associated with a decrease in alcohol-involved road crashes [21]; their hypothesis is partially supported for the cities of Portland, Oregon and San Antonio, Texas.",
"However, the studies above do not account for actual ridesourcing travel patterns, which are much needed to capture local effects by overlaying ridesourcing pick-up and drop-off locations and safety outcomes.",
"Other studies also report findings that are in general agreement with the aforementioned outcomes.",
"As an example, Dills and Mulholland indicate that Uber's launch is associated with a reduction in fatal traffic crashes and DWI offenses after a certain number of months of operation, uncovering lags [23].",
"On the contrary, Barrios et al.",
"find that road fatalities (including pedestrian and non-vehicle occupants) increase with ridesourcing use and that these trends persist over time [16].",
"They also demonstrate that the road crash rate increase is significant in cities with greater population levels, higher income quartiles, greater vehicle ownership share and public transportation use, as well as higher carpooling use.",
"Barrios et al.",
"is the only work that captures such exposure, using a proxy for adoption intensity, by adopting Google trends search counts [16].",
"We find that the existing literature fails to assess how the intensity of ridesourcing services in smaller geographic areas is related to road safety since it does not leverage actual locations of crashes and other road safety outcomes data.",
"Ridesourcing operation resembles quite a lot that of a taxi service.",
"Evidence from safety research of the operation of taxis examines the impact of fatigue [25], driver behavior, and working conditions [26], as well as taxi driver offenses [27] on taxi crashes.",
"Associations of these factors with taxi crashes and fatalities are examined through Poisson and logistic regression models without taking into account spatial and temporal characteristics.",
"Such models, along with multivariate regressions models (e.g., [28]), seemingly unrelated regressions (e.g., [29]), and geographically weighted regressions (e.g., [30], [31]) are leveraged in a variety of road safety outcomes modeling studies.",
"Note that after systematically searching for evidence, we have not identified studies that examine associations of taxi use and road safety outcomes the way that ridesourcing use relationships have been studied with safety outcomes.",
"Our analysis aims to bridge literature gaps by leveraging granular local-scale data to shed light on the relationship between transportation network companies' operation and road safety outcomes.",
"In our study, these associations are captured by using real-world ridesourcing trip data.",
"It is crucial to uncover these effects since ridesourcing could be promoted or disincentivized by city managers and policymakers to contribute towards traffic crashes and injuries reduction under a vision zero's plan.",
"Prioritizing successful interventions could be crucial to achieving substantial road safety improvements with ridesourcing services integration.",
"To measure the association between ridesourcing travel and road safety, we use spatial panel data models that have been previously applied in transportation safety research [32], [33], [34].",
"The natural logarithm of crashes, injuries, fatalities, and DWI offenses, as well as ridesourcing exposure, addresses the variables' right skewness via normalization [35].",
"We leverage variables on safety outcomes, ridesourcing and traffic volumes, and socio-demographics from January 2012 to the beginning of April 2017 in our study.",
"To conduct the proposed analysis, we aggregate variables by census tract and month-year units.",
"The data used here (described in greater detail in the next section) are longitudinal, containing repeated observations of the same census tract units over time.",
"Cross-sectional data and models suffer from an inability to capture intertemporal dependence of events, which the panel data models used here are expected to capture.",
"Our proposed models with spatial and time fixed-effects enable reducing bias from unobserved factors that are changing over time but are constant over each spatial unit and controls for unobserved factors that change over space but are constant over time [35].",
"Previous studies that attempt to answer similar research questions with empirical data modeling use the difference-in-differences estimation, which compares how the trajectory of road safety outcomes differs before and after the launch of ridesourcing.",
"Due to the absence of a control group in our study, since the RideAustin operation was launched in Travis County in all census tracts simultaneously (June 2016), we cannot deploy a DiD model.",
"Instead, we employ spatial panel fixed-effects lag and error models and Spatial AutoRegressive with additional AutoRegressive error structure (SARAR) models that allow for the disturbances to be generated by a spatial autoregressive process [36].",
"The index of each spatial unit is $i \\in {{1,…,I}}$ , and each time unit is $t \\in {{1,…,T}}$ .",
"A fixed-effects spatial lag specification is presented in Eq (REF ), according to [37]: $y_{it}=\\lambda \\sum _{i\\ne j} w_{ij}y_{jt}+\\beta x_{it}+\\alpha _i+\\gamma _t+u_{it},$ where $y_{it}$ is each road safety outcome (e.g., the number of total crashes that occurred in the spatial unit $i$ during month $t$ , and similar for injuries, fatalities, and DWI offenses), $w_{ij}$ is a spatial weighted matrix constant over time $t$ with diagonal elements equal to zero in which neighborhood relationships are defined between the spatial units of analysis, $\\sum _{i \\ne j} w_{ij}y_{jt}$ is the spatially lagged dependent variable which denotes that the value of $y$ at time $t$ is explained not only by the values of exogenous independent variables but also those $y$ neighboring the spatial unit $i$ , $\\lambda $ is the scalar spatially autoregressive coefficient of the spatially lagged dependent variable, $\\alpha _i$ is the spatial unit fixed-effect, $\\gamma _t$ is the time unit fixed-effect, $\\beta $ the vector of parameters to be estimated, $x_{it}$ a vector of explanatory variables, and the error terms $u_{it} \\sim N(\\theta ,\\sigma ^2)$ .",
"The census tract fixed-effects control for all time-invariant census tract specific factors that are potentially correlated with safety outcomes, such as area.",
"Similar assumptions hold for the time fixed-effects that control for census tract invariant factors that vary by month like travel patterns.",
"For all models, $w_{ij}$ weights are defined based on binary contiguity, where $w_{ij}=1$ when the intersection of the boundaries of $i$ and $j$ spatial units is not empty, otherwise $w_{ij}=0$ .",
"The spatial error with fixed-effects model is presented in Eq (REF ) and Eq (REF ), according to the specification in [38]: $y_{it}=\\beta x_{it}+\\alpha _i+\\gamma _t+u_{it},$ $u_{it}=\\rho \\sum _{i \\ne j} w_{ij} u_{jt}+\\epsilon _{it},$ where the disturbance term $u_{it}$ follows the first order spatial autoregressive process of the equation presented in Eq (REF ), and $\\rho $ is spatial autoregressive coefficient where $|\\rho |<1$ .",
"The rest of the terms have been specified in the previous paragraph.",
"Comparing the spatial lag and spatial error models, the former suggests a diffusion, where road safety crashes in one spatial unit predict an increased likelihood of road crashes in neighboring places; the latter suggests that we have omitted spatially correlated covariates that would affect inference.",
"Last, the SARAR model is defined as presented in Eq (REF ) and Eq (REF ) from [36], [39]: $y_{it}=\\lambda \\sum _{i \\ne j}w_{ij}y_{jt} +\\beta x_{it}+\\alpha _i+\\gamma _t+u_{it},$ $u_{it}=\\rho \\sum _{i\\ne j}w_{ij}u_{jt}+\\epsilon _{it},$ where both $\\lambda $ and $\\rho $ are spatially autoregressive coefficients.",
"SARAR accounts for both neighboring effects and omitted spatially correlated covariates.",
"Maximization of the likelihood function results in the estimation of the unknown coefficients $\\beta $ , per the existing notation for the spatial lag and error models [37] and for the SARAR model [40].",
"We conduct specification tests to confirm that fixed-effects are the most appropriate over random effects.",
"The Hausman test [41] is applied that denotes that a fixed-effects model is at least as consistent as the random-effects specification, and thus, a fixed-effects specification is chosen.",
"Locally robust panel Langrage Multiplier (LM) tests are used to test the absence of each spatial term, for spatial dependence [38].",
"The LM statistics tests are used to test for spatial autocorrelation in the form of an endogenous spatial lag variable and spatially autocorrelated errors [39].",
"Datasets representative of Travis County, Texas are analyzed including detailed historical road safety measures from the Texas Department of Transportation Crash Records Information System [42] and DWI offenses from the Austin Police Department Crime Reports [43].",
"For both road safety and DWI offense records, apart from temporal variability, we have access to their exact locations as longitude and latitude coordinates.",
"Ridesourcing use is captured based on actual, comprehensive trip-level data from RideAustin's open record.",
"RideAustin is a company that operated in Travis County in Texas, without competing with Uber and Lyft between June 2016 and May 2017.",
"RideAustin offered approximately 1.5 million rides [44] during that period.",
"Note that during the period that Uber and Lyft exited the Austin market, RideAustin was launched and other transportation network companies, such as Fare and Fasten [45].",
"We are not aware of the ridership share that the rest of ridesourcing companies attracted, thus we might underestimate the use of ridesourcing services in the region.",
"However, we know that 47% of those that used the new transportation network companies in Austin chose RideAustin from a local survey's findings [46].",
"RideAustin's market share is sufficient to suggest high representativeness of ridesourcing patterns [2].",
"In our analysis, we aim to control for overall traffic in the region after conducting monthly travel demand analyses using the StreetLight Data platform [47].",
"Normalized trip counts for all census tracts within Travis County during the period of interest are used.",
"The normalized vehicular traffic counts data do not include commercial, ridesourcing, and delivery data according to the specifications of the StreetLight Data platform.",
"The normalized trip counts, used here as a proxy for the travel demand in Austin, are calibrated with annual average daily traffic (AADT) data available at the network level for every link in the Austin metropolitan region [49].",
"Unfortunately, the shapefiles of the annually variant daily vehicle miles traveled (DVMT) in the Austin region are only suitable for calibrating the StreetLight Data trips counts and not for running the model with DVMT as a control variable.",
"Using annually variant DVMT would result in losing the monthly granularity that we are interested in maintaining across both the ridesourcing use and the overall transportation network use covariates.",
"Additional controls that capture socio-demographic characteristics and their change over the analysis period are available from processed American Community Survey data.",
"The socio-economic data capture demographic and land use variations across census tracts and years.",
"Variables of interest include median household income, population density, employment density, and percent of zero vehicle ownership [49].",
"The traffic safety analysis period is from January 2012 to April 2017.",
"We showcase in Fig REF the time series of the monthly crash, injury, fatality, and DWI offense rates (per thousand people) averaged by census tract in Travis County.",
"Note that we use the KABCO scale for injury severity [50]; the injuries used to compute the corresponding rate include incapacitating (A), non-incapacitating (B), and possible injuries indicated by behavior but not visible wounds (C) [51].",
"We observe similarities between the crash and injury rates seasonal trends, even though for the former the fitted linear trend over the total period of time is increasing and for the latter decreasing.",
"Figure: Road safety outcome rates in Travis County, Texas.In Fig REF , we overlay the monthly ridesourcing operation timeline in Travis County with traffic safety outcome rates.",
"The period between January 2012 and June 2014 is characterized by no ridesourcing operation.",
"Uber and Lyft launched in the region during June 2014 and operated there until May 9, 2016, when they stopped offering services due to the city of Austin's regulation that required fingerprint background checks on their drivers [52].",
"At the time of Uber/Lyft entry, crashes and injuries rates were declining; while operating and before Uber's and Lyft's exit, crash and injury rates were rising.",
"Other effects might play a role in that, such as the city's population growth and the significant gasoline price drops in 2014, which are positively associated with additional travel demand.",
"RideAustin, a ridesourcing service, was launched in Travis County in June 2016 and operated until May 29, 2017, when Uber and Lyft returned.",
"Due to the unavailability of ridesourcing use data from these major transportation network companies between June 2014 and May 2016, we exclude that period from our analysis.",
"The group of observations before ridesourcing entry in June 2014 until the end of May 2014 are denoted as a \"before ridesourcing\" group and the observations group after RideAustin's launch in June 2016 up to the beginning of April 2017 as the \"after ridesourcing\" group.",
"The monthly timeline presenting the number of RideAustin trips that were conducted in Travis County is presented in Fig REF , using RideAustin data [44].",
"Specifically, the line graph presents the monthly number of rides given (corresponding right y-axis), and the bar graph shows the monthly number of drivers that offered rides (corresponding left y-axis).",
"The number of rides offered between RideAustin entry's and October 2016 is increasing exponentially; then, the number of rides keeps increasing at a lower rate, reaching a peak during the South by Southwest festival and conference season.",
"Partial trip data for a few days in April 2017 are available; we truncate the dataset to include ridesourcing trip origin and destination (OD) counts only up to the beginning of April 2017.",
"Figure: Ridesourcing use in Travis County, Texas.The average monthly crash rate and RideAustin passengers' pick-up and drop-off rate for each census tract in Travis County are portrayed in Fig REF .",
"We are grouping together ridesourcing trips origins and destinations that fall in our spatial units of analysis (i.e., census tracts), without double-counting trips that have both their origin and destination in the same census tract.",
"Due to this grouping we might not be able to capture that loading and dropping off of customers might have different impacts on the congestion across census tracts and can result in traffic stress levels.",
"We present these indicators in Fig REF , denoting their shares before and after the entry of ridesourcing in the region, without accounting for the excluded period shown in Fig REF .",
"In June 2016, RideAustin started offering rides mainly in the downtown Austin region (smaller area census tracts in the center of the map) and then, over the months, expanded their service radius to cover all census tracts within the Travis County.",
"Figure: Road crash rate and ridesourcing rate before and after the introduction of the ridesourcing service in Travis County, Texas census tracts.Left panel: Average crashes per 1000 people.",
"Right panel: Average RideAustin trips per census tract population.With highway networks spanning the suburban Austin region, we observe an increase of average crashes per 1000 people in the aftermath of RideAustin launch there.",
"Fig REF also suggests that after RideAustin's entry, the rate of total crashes increased in the suburbs of Travis County but not in the downtown regions that the ridesourcing operation covered at a higher rate.",
"The crashes increase in the suburbs could be associated with population and firm growth and could reflect changes in travel patterns that resulted from spatial differences due to the region's economic development.",
"Various measures are introduced as control variables in our model.",
"Their selection is based on prior evidence (as shown in Table REF ) and includes median household income [11], [16], percent of zero vehicle ownership [16], population density [11], [16], [23], and percentage of employment [16], [17], [20], [23].",
"All these covariates are available through the American Community Survey for every census tract on an annual basis [49].",
"We make the assumption that such socio-demographic changes would be mainly observed on an annual basis since decisions of household relocation are made on an annual basis (e.g., [53]).",
"Given the land use changes in Travis County the latest years, we control for the monthly spatial variation in trip OD counts per spatial-time unit of analysis using normalized trip count data from the StreetLight Data platform [47].",
"The normalized OD trips, derived from the underlying data sample per census tract per month, is available as an output of the StreetLight Data travel demand modeling.",
"Unfortunately, we only have the ability to estimate traffic exposure from January 2016 to December 2018 within the StreetLight Data platform due to the unavailability of historical datasets.",
"This period slightly overlaps with our analysis period.",
"Time series analysis is performed to estimate traffic exposure historical data (for more information, see S1Appendix).",
"The hypotheses that support the inclusion of the aforementioned control variables are as follows.",
"Population density, employment percentage, and traffic (provided through the StreetLight Data's platform) are expected to capture the market's size.",
"The median household income is controlling the market's wealth [11].",
"Traffic crashes are significantly increased in cities with lower average per capita income [16].",
"The percentage of zero vehicle ownership is expected to capture likely users who do not own a personal automobile and may rely on other modes of transport [3].",
"Note that, commonly, when examining factors contributing to crash risk, exposure measures tend to include roadway length [54].",
"In our case, the number of roadway miles does not vary at the monthly level per each spatio-temporal unit of analysis.",
"Thus, any unobserved effects can be captured with the spatial effects' parameter entered in our proposed models.",
"Table REF presents descriptive statistics for our sample of 8,720 census tract-month-year units, corresponding to 218 different census tracts in Travis County and 40 monthly periods.",
"The mean, standard deviation, and median values of the safety outcomes, as well as the rest of the socio-economic, travel, and transportation specific covariates are presented there.",
"Only road crashes, fatalities, and injuries that involve light-duty vehicles are considered in the analysis (excluding crashes that only involve trucks, buses, and trains).",
"Table: Summary statistics of census tract-month-year units of analysis.Descriptive statistics of our sample of 8,720 census-tract month units, categorized by ridesourcing presence in Travis County, are presented in Table REF .",
"Ridesourcing was operating in 27.50% of the total census tract-month units of our analysis.",
"Crash and injury rates are higher during the months when ridesourcing was offered.",
"On the contrary, DWI offenses rate and fatality rates are lower, even though the averages' difference is much smaller.",
"We observe an increase in the normalized trip counts rate per census tract and month unit, accompanied by an increase of the median household income and a significant decrease in gas prices, as well as an increase of the average population in the region, after the ridesourcing entry.",
"Table: Descriptive Statistics: Safety and other metrics magnitude before and after ridesourcing entry periods.The spatial fixed-effects panel data model estimates, with all controls included, are presented in Table REF , Table REF , and Table REF for the spatial lag, spatial error, and SARAR models, respectively.",
"Lagrange multiplier tests results are provided there.",
"Signs and magnitudes of the $\\beta $ coefficients are consistent across the three models, supporting the robustness of our findings.",
"Table: Spatial fixed-effects panel data modeling results: spatial lag.Table: Spatial fixed-effects panel data modeling results: error lag.Table: Spatial fixed-effects panel data modeling results: SARAR.Ridesourcing use is found significantly associated with three road safety outcomes: crashes, injuries, and DWI offenses.",
"Based on the SARAR model coefficients shown in Table REF , for a 10% increase in ridesourcing use, we expect a 0.12% decrease in road crashes (p$<0.05$ ), a 0.25% decrease in road injuries (p$<0.001$ ), and a 0.36% decrease in DWI offenses (p$<0.0001$ ).",
"Ridesourcing use is not found associated with fatalities at the 0.1 significance level.",
"These results are well aligned with Dills and Mulholland and Morrison et al.",
"findings that associate the entry of ridesourcing with a decrease in DWI offenses and alcohol-involved crashes, respectively [21], [23].",
"However, Morrison et al.",
"find no significant association between ridesourcing entry and road injuries.",
"Our findings are also aligned with studies that found no significant association between ridesourcing entry and fatalities (e.g., [17], [22]), while we also account for the ridesourcing induced demand effects that the aforementioned studies do not capture.",
"The magnitude of the percentage decrease of road safety externalities associated with ridesourcing use can be considered small compared to the effectiveness of other strategic road safety interventions.",
"Seatbelt laws enforcement may result in up to 9% reduction in road injuries [55] while speed limit reduction and traffic calming measures may enable to reach 10-15% decrease in traffic crashes [56].",
"Therefore, promoting the use of ridesourcing as vision zero policy might not be as effective as prioritizing infrastructure improvements, speed limit changes, and other educational or policing vision zero initiatives.",
"However, a ridesourcing initiative to reduce DWI offenses and increase road safety could benefit groups most likely to ride those such as younger drivers [17].",
"Road crashes and fatalities are significantly associated with population density, as expected.",
"Crashes increase with increased population density, but road fatalities decrease.",
"Population density as fatalities' predictor might serve as a proxy for speed's effect, given that roads in lower density environments tend to be characterized by higher speed limits.",
"The percentage of vehicle owners is positively associated with road crashes and injuries in Travis County.",
"This result is aligned with other literature findings [16]; lower levels of vehicle ownership hint at greater public transit and/or active transportation usage.",
"We note, though, that ridesourcing use could endogenously influence vehicle ownership positively by incentivizing more to drive for transportation network companies.",
"In our analysis, ridesourcing use is positively (but weakly) correlated (corr$=0.12$ ) with the percentage of zero vehicle ownership (shown in S1Fig).",
"The number of DWI offenses would decrease when a greater number of alternative transportation options are available, as expected.",
"This finding also suggests that ridesourcing in Travis County may not only serve as a substitute for taxis and other modes of transportation but also for drunk driving.",
"However, this outcome could also be a result of enforcement, educational, and other efforts, which we are unable to capture here.",
"The concurrent launch of RideAustin and the exit of Uber and Lyft from the Austin region [52] during the period of May and June of 2016 might result in skewing our modeling outcomes since the association of ridesourcing use with road crashes, DWI offenses, and other road safety indices might be attributed to the departure of the popular ridesourcing services.",
"We conduct a robustness check to examine whether the magnitude and sign of the ridesourcing use parameter remain the same when using only the last six months of the ride data of our RideAustin treatment period.",
"This enables us to exclude data representative of the initial and transitory RideAustin operation period and focus only on the operation between October 2016 to March 2017, assuming that the impacts of the exit of Uber and Lyft will have subsided.",
"S1Table, S2Table, and S3Table showcase modeling results of the spatial lag, spatial error, and SARAR models in the SI.",
"Even though the significance of the relationship between ridesourcing use and road safety outcomes is weaker in this case, we observe that the sign of the parameter remains negative.",
"In accordance with the base RideAustin operating period definition, we observe that ridesourcing use is associated with a reduction in DWI offenses and traffic injuries but does not seem to be related to road crashes and fatalities.",
"Note that existing literature findings suggest that once Uber and Lyft services depart from Austin, the crashes and DWI offenses decline [16].",
"In such a case, some of the observed declines in road crashes might be attributed to reductions in the number of cars on the road.",
"Therefore, the treatment might be the departure of Uber and Lyft, not the arrival of RideAustin.",
"In order to be able to explore such a notion, we would need information about the trips conducted by ridesourcing services over the excluded period of analysis.",
"We use RideAustin trips to examine the effect of ridesourcing exposure on road safety outcomes, such as road crashes, injuries, fatalities, and DWI offenses.",
"Spatial fixed-effects panel data models are employed to establish that RideAustin use is significantly associated with a decrease in total road crashes, injuries, and DWI offenses in Travis County, Texas.",
"On the contrary, our findings do not demonstrate significant relationships between ridesourcing use and road fatalities.",
"Given the significant costs associated with road safety outcomes and DWI offenses [57], ridesourcing services can be a low-cost option that could assist cities and counties with meeting goals for road injuries and DWI offenses reduction.",
"At the same time, the magnitude of the road safety externalities decrease that is associated with ridesourcing trips is smaller compared to the safety effectiveness that has been documented after the application of other interventions, including seatbelt laws, reduced speeds, and traffic calming design.",
"This outlines the need for determining population segments that ridesourcing-related solutions or policies could be more impactful for improving road safety, like focusing on younger ridesourcing demographics.",
"Our analysis augments existing work in this field by accounting for spatial distributions of ridesourcing use, road safety outcomes, and other socio-economic characteristics in the given region.",
"Instead of testing associations of the launch of ridesourcing with road injuries and the rest of safety outcomes, we account for spatio-temporal characteristics and capture actual ridesourcing use via real-time trip data analytics in Travis County.",
"The spatial panel data modeling outcomes show that spatial dependence is of significance.",
"Thus, granular longitudinal travel, road safety, and socio-demographic panel data can provide transportation and traffic safety agencies that opportunity to uncover associations and plan for appropriate safety interventions.",
"Additional research efforts should be put towards addressing this study's limitations, including 1) testing the effect of alternate travel demand exposure methods, such as vehicle miles traveled instead of OD trips, 2) exploring whether effects might not manifest immediately from ridesourcing use, accounting for lags [11], 3) examining results robustness by performing similar analysis in additional regions in the US and around the world, since the generalizability of the results can be questioned, and 4) fitting conditional autoregressive models and commenting on their performance differences compared to spatial autoregressive ones [58].",
"Future research should also uncover for which populations and subpopulations road safety outcomes can be improved through ridesourcing use by (a) exploring the relationship of ridesourcing and road safety outcomes for different household income and employment percentage panels and (b) identifying critical drivers of where potential public health benefits of ridesourcing utilization can be the greatest.",
"Road crash rate and ridesourcing rates before and after the introduction of the ridesourcing service.",
"The correlation matrix results of the four safety outcome variables, the ridesourcing use, and control variables are presented to showcase associations between those.",
"Covariate raw data.",
"Transportation and socio-demographic data used in the analysis.",
"Traffic time series analysis.",
"Due to unavailability of OD traffic data for each census tract in Travis County for the whole period of the analysis (January 2012-June 2014 and June 2016-April 2017), a time series, autoregressive integrated moving average model was fitted.",
"We have monthly OD trips rate output data from the StreetLight Data platform from January 2016 until December 2018 [47].",
"The ridesourcing trip rates refer to both origins and destinations for trips that start and end at different census tracts.",
"If both trip's origin and destination fall within the same census tract then it is only counted once.",
"We apply the time series model with a transfer function (based on historical gasoline prices) to populate the historical OD trips data for every spatial unit of analysis.",
"The autoregressive order of the model ARIMA$(p,q,d)$ dictates the number of the lagged values $x_{i,t-1}$ that have an impact on $x_{i,t}$ OD trips.",
"The order of the moving average direction $q$ is the number of the lagged error terms used in the model, and $d$ is the fractional integration parameter used to force stationarity.",
"In our analysis $p=1, q=0, and d=1$ .",
"The transfer function is the average monthly $ per gallon gas price.",
"The chosen model was based on comparisons of R squared adjusted and Akaike's Information Criterion.",
"Its results are used to populate the OD Trips data fields of this analysis.",
"Robustness check using shorter RideAustin operational period dataset (October 2016-March 2017): spatial lag fixed-effects model results.",
"Table: NO_CAPTIONRobustness check using shorter RideAustin operational period dataset (October 2016-March 2017): spatial error fixed-effects model results.",
"Table: NO_CAPTIONRobustness check using shorter RideAustin operational period dataset (October 2016-March 2017): SARAR model results.",
"Table: NO_CAPTIONWe thank StreetLight Data for providing Kontou with academic access to their travel demand modeling platform.",
"We acknowledge the feedback that we received for an earlier iteration of this paper from the statistical and data science consulting service coordinated by Prof. James Stephen Marron and students of the Department of Statistics and Operations Research at the University of North Carolina at Chapel Hill.",
"Kontou acknowledges an honorarium she received from Pervasive Technology Institute of Indiana University, which enabled her to use cloud computing resources for this analysis."
]
] | 2001.03461 |
[
[
"Solving inverse-PDE problems with physics-aware neural networks"
],
[
"Abstract We propose a novel composite framework to find unknown fields in the context of inverse problems for partial differential equations (PDEs).",
"We blend the high expressibility of deep neural networks as universal function estimators with the accuracy and reliability of existing numerical algorithms for partial differential equations as custom layers in semantic autoencoders.",
"Our design brings together techniques of computational mathematics, machine learning and pattern recognition under one umbrella to incorporate domain-specific knowledge and physical constraints to discover the underlying hidden fields.",
"The network is explicitly aware of the governing physics through a hard-coded PDE solver layer in contrast to most existing methods that incorporate the governing equations in the loss function or rely on trainable convolutional layers to discover proper discretizations from data.",
"This subsequently focuses the computational load to only the discovery of the hidden fields and therefore is more data efficient.",
"We call this architecture Blended inverse-PDE networks (hereby dubbed BiPDE networks) and demonstrate its applicability for recovering the variable diffusion coefficient in Poisson problems in one and two spatial dimensions, as well as the diffusion coefficient in the time-dependent and nonlinear Burgers' equation in one dimension.",
"We also show that this approach is robust to noise."
],
[
"Introduction",
"Inverse differential problems, where given a set of measurements one seeks a set of optimal parameters in a governing differential equation, arise in numerous scientific and technological domains.",
"Some well-known applications include X-ray tomography [20], [55], ultrasound [83], MRI imaging [36], and transport in porous media [35].",
"Moreover, modeling and control of dynamic complex systems is a common problem in a broad range of scientific and engineering domains, with examples ranging from understanding the motion of bacteria colonies in low Reynolds number flows [70], to the control of spinning rotorcrafts in high speed flights [30], [31].",
"Other applications in medicine, navigation, manufacturing, etc.",
"need estimation of the unknown parameters in real-time, e.g.",
"in electroporation [91], [53] the pulse optimizer has to be informed about tissue parameters in microsecond time.",
"On the other hand, high resolution data-sets describing spatiotemporal evolution of complex systems are becoming increasingly available by advanced multi-scale numerical simulations (see e.g.",
"[53], [52]).",
"These advances have become possible partly due to recent developments in discretization techniques for nonlinear partial differential equations with sharp boundaries (see e.g.",
"the reviews [25], [24]).",
"However, solving these inverse problems poses substantial computational and mathematical challenges that makes it difficult to infer reliable parameters from limited data and in real-time.",
"The problem can be mathematically formulated as follows.",
"Let the values of $u=u(t, x_1, \\ldots , x_n)$ be given by a set of measurements, which may include noise.",
"Knowing that $u$ satisfies the partial differential equation: $\\frac{\\partial u}{\\partial t} = f \\left(t, x_1, \\ldots , x_n; u, \\frac{\\partial u}{\\partial x_1}, \\ldots \\frac{\\partial u}{\\partial x_n}; \\frac{\\partial ^2 u}{\\partial x_1 \\partial x_1}, \\ldots \\frac{\\partial ^2 u}{\\partial x_1 \\partial x_n}; \\ldots ; \\mathbf {c} \\right),$ find the hidden fields stored in $\\mathbf {c}$ , where the hidden fields can be constant or variable coefficients (scalars, vectors or tensors).",
"Deep neural networks have, rather recently, attracted considerable attention for data modeling in a vast range of scientific domains, in part due to freely available modern deep learning libraries (in particular TensorFlow [1]).",
"For example, deep neural networks have shown astonishing success in emulating sophisticated simulations [29], [90], [89], [10], [75], discovering governing differential equations from data [67], [7], [47], [72], as well as potential applications to study and improve simulations of multiphase flows [25].",
"We refer the reader to [60], [61] for a comprehensive survey of interplays between numerical approximation, statistical inference and learning.",
"However, these architectures require massive datasets and extensive computations to train numerous hidden weights and biases.",
"Therefore, reducing complexity of deep neural network architectures for inverse problems poses a significant practical challenge for many applications in physical sciences, especially when the collection of large datasets is a prohibitive task [66].",
"One remedy to reduce the network size is to embed the knowledge from existing mathematical models [79] or known physical laws within a neural network architecture [45], [23].",
"Along these lines, semantic autoencoders were recently proposed by Aragon-Calvo [2], where they replaced the decoder stage of an autoencoder architecture with a given physical law that can reproduce the provided input data given a physically meaningful set of parameters.",
"The encoder is then constrained to discover optimal values for these parameters, which can be extracted from the bottleneck of the network after training.",
"We shall emphasize that this approach reduces the size of the unknown model parameters, and that the encoder can be used independently to infer hidden parameters in real time, while adding interpretability to deep learning frameworks.",
"Inspired by their work, we propose to blend traditional numerical solver algorithms with custom deep neural network architectures to solve inverse PDE problems more efficiently, and with higher accuracy.",
"Recently, the most widely used approach for solving forward and inverse partial differential equations using neural networks has been the constrained optimization technique.",
"These algorithms augment the cost function with terms that describe the PDE, its boundary and its initial conditions, while the neural network acts as a surrogate for the solution field.",
"Depending on how the derivatives in the PDEs are computed, there may be two general classes of methods that we review in the next paragraph.",
"In the first class, spatial differentiations in the PDE are performed exclusively using automatic differentiation, while temporal differentiation may be handled using the traditional Runge-Kutta schemes (called discrete time models) or using automatic differentiations (called continuous time models) [68].",
"In these methods, automatic differentiation computes gradients of the output of a neural network with respect to its input variables.",
"Hence, the input must always be the independent variables, i.e.",
"the input coordinates $\\mathbf {x}$ , time and the free parameters.",
"In this regard, network optimization aims to calibrate the weights and biases such that the neural network outputs the closest approximation of the solution of a PDE; this is enforced through a regularized loss function.",
"An old idea that was first proposed by Lagaris et al.",
"(1998) [43].",
"In 2015, the general framework of solving differential equations as a learning problem was proposed by Owhadi [57], [58], [59] which revived interest in using neural networks for solving differential equations in recent years.",
"Raissi et al.",
"(2017) [68], [69] presented the regularized loss function framework under the name physics informed neural networks or PINNs and applied it to time-dependent PDEs.",
"Ever since, other authors have mostly adopted PINNs, see e.g.",
"[76], [4].",
"The second class of constrained optimization methods was proposed by Xu and Darve [88] who examined the possibility of directly using pre-existing finite discretization schemes within the loss function.",
"An alternative approach for solving PDE systems is through explicit embedding of the governing equations inside the architecture of deep neural networks via convolutional layers, activation functions or augmented neural networks.",
"Below we review some of these methods: A famous approach is PDE-Net [47], [46] which relies on the idea of numerical approximation of differential operators by convolutions.",
"Therefore, PDE-Nets use convolution layers with trainable and constrained kernels that mimic differential operators (such as $\\rm U_x, U_y, U_{xx}, \\cdots $ ) whose outputs are fed to a (symbolic) multilayer neural network that models the nonlinear response function in the PDE system, i.e.",
"the right hand side in $\\rm U_t = F(U, U_x, U_y, U_{xx}, \\cdots )$ .",
"Importantly, PDE-Nets can only support explicit time integration methods, such as the forward Euler method [47].",
"Moreover, because the differential operators are being learned from data samples, these methods have hundreds of thousands of trainable parameters that demand hundreds of data samples; e.g.",
"see section 3.1 in [47] that uses 20 $\\delta t$ -blocks with $17,000$ parameters in each block, and use 560 data samples for training.",
"Berg and Nyström [6] (hereby BN17) proposed an augmented design by using neural networks to estimate PDE parameters whose output is fed into a forward finite element PDE solver, while the adjoint PDE problem is employed to compute gradients of the loss function with respect to weights and biases of the network using automatic differentiation.",
"Even though their loss function is a simple $\\rm L_2$ -norm functional, the physics is not localized in the structure of the neural network as the adjoint PDE problem is also employed for the optimization process.",
"It is important to recognize that in their approach the numerical solver is a separate computational object than the neural network, therefore computing gradients of error functional with respect to the network parameters has to be done explicitly through the adjoint PDE problem.",
"Moreover, their design can not naturally handle trainable parameters in the numerical discretization itself, a feature that is useful for some meshless numerical schemes.",
"In contrast, in BiPDEs the numerical solver is a computational layer added in the neural network architecture and naturally supports trainable parameters in the numerical scheme.",
"For example in the meshless method developed in section we leverage this unique feature of BiPDEs to also train for shape parameters and interpolation seed locations of the numerical scheme besides the unknown diffusion coefficient.",
"Dal Santos et al.",
"[16] proposed an embedding of a reduced basis solver as activation function in the last layer of a neural network.",
"Their architecture resembles an autoencoder in which the decoder is the reduced basis solver and the parameters at the bottleneck “are the values of the physical parameters themselves or the affine decomposition coefficients of the differential operators” [16].",
"Lu et al.",
"[48] proposed an unsupervised learning technique using variational autoencoders to extract physical parameters (not inhomogeneous spatial fields) from noisy spatiotemporal data.",
"Again the encoder extracts physical parameters and the decoder propagates an initial condition forward in time given the extracted parameters.",
"These authors use convolutional layers both in the encoder to extract features as well as in the decoder with recurrent loops to propagate solutions in time; i.e.",
"the decoder leverages the idea of estimating differential operators with convolutions.",
"Similar to PDE-Nets, this architecture is also a “PDE-integrator with explicit time stepping”, and also they need as few as 10 samples in the case of Kuramoto-Sivashinsky problem.",
"In these methods, a recurring idea is treating latent space variables of autoencoders as physical parameters passed to a physical model decoder.",
"This basic idea pre-dates the literature on solving PDE problems and has been used in many different domains.",
"Examples include Aragon-Calvo [2] who developed a galaxy model fitting algorithm using semantic autoencoders, or Google Tensorflow Graphics [82] which is a well-known application of this idea for scene reconstruction."
],
[
"Present work",
"Basic criteria of developing numerical schemes for solving partial differential equations are consistency and convergence of the method, i.e.",
"increasing resolution of data should yield better results.",
"Not only there is no guarantee that approximating differential operators through learning convolution kernels or performing automatic differentiations provide a consistent or even stable numerical method, but also the learning of convolution kernels to approximate differential operators requires more data and therefore yield less data-efficient methods.",
"Therefore it seems reasonable to explore the idea of blending classic numerical discretization methods in neural network architectures, hence informing the neural network about proper discretization methods.",
"This is the focus of the present manuscript.",
"In the present work, we discard the framework of constrained optimization altogether and instead choose to explicitly blend fully traditional finite discretization schemes as the decoder layer in semantic autoencoder architectures.",
"In our approach, the loss function is only composed of the difference between the actual data and the predictions of the solver layer, but contrary to BN17 [6] we do not consider the adjoint PDE problem to compute gradients of the error functional with respect to network parameters.",
"This is due to the fact that in our design the numerical solver is a custom layer inside the neural network through which backpropagation occurs naturally.",
"This is also in contrast to PINNs where the entire PDE, its boundary and its initial conditions are reproduced by the output of a neural network by adding them to the loss function.",
"Importantly, the encoder learns an approximation of the inverse transform in a self-supervised fashion that can be used to evaluate the hidden fields underlying unseen data without any further optimization.",
"Moreover, the proposed framework is versatile as it allows for straightforward consideration of other domain-specific knowledge such as symmetries or constraints on the hidden field.",
"In this work, we develop this idea for stationary and time-dependent PDEs on structured and unstructured grids and on noisy data using mesh-based and mesh-less numerical discretization methods."
],
[
"Novelties and features of BiPDEs",
"A full PDE solver is implemented as a custom layer inside the architecture of semantic autoencoders to solve inverse-PDE problems in a self-supervised fashion.",
"Technically this is different than other works that implement a propagator decoder by manipulating activation functions or kernels/biases of convolutional layers, or those that feed the output of a neural network to a separate numerical solver such as in BN17 which requires the burden of considering the adjoint problem in order to compute partial differentiations.",
"The novelties and features of this framework are summarized below: General discretizations.",
"We do not limit numerical discretization of differential equations to only finite differences that are emulated by convolution operations, our approach is more general and permits employing more sophisticated numerical schemes such as meshless discretizations.",
"It is a more general framework that admits any existing discretization method directly in a decoder stage.",
"Introducing solver layers.",
"All the information about the PDE system is only localized in a solver layer; i.e.",
"we do not inform the optimizer or the loss function with the adjoint PDE problem, or engineer regularizers or impose extra constraints on the kernels of convolutions, or define exotic activation functions as reviewed above.",
"In other words, PDE solvers are treated as custom layers similar to convolution operations that are implemented in convolutional layers.",
"An important aspect is the ability to employ any of the usual loss functions used in deep learning, for example we arbitrarily used mean absolute error or mean squared error in our examples.",
"Blending meshless methods with trainable parameters.",
"Another unique proposal made in this work is the use of Radial Basis Function (RBF) based PDE solver layers as a natural choice to blend with deep neural networks.",
"Contrary to other works, the neural network is not only used as an estimator for the unknown field but also it is tasked to optimize the shape parameters and interpolation points of the RBF scheme.",
"In fact, our meshless decoder is not free of trainable parameters similar to reviewed works, instead shape parameters and seed locations are trainable parameters that define the RBF discretization, this is analogous to convolutional layers with trainable weights/biases that are used in machine learning domain.",
"In fact this presents an example of neural networks complementing numerical discretization schemes.",
"Choosing optimal shape parameters or seed locations is an open question in the field of RBF-based PDE solvers and here we show neural networks can be used to optimally define these discretization parameters.",
"Explicit/implicit schemes.",
"Most of the existing frameworks only accept explicit numerical discretizations in time, however our design naturally admits implicit methods as well.",
"Using implicit methods allows taking bigger timesteps for stiff problems such as the diffusion problem, hence not only providing faster inverse-PDE solvers, but also present more robust/stable inverse PDE solvers.",
"Data efficient.",
"Our design lowers the computational cost as a result of reusing classical numerical algorithms for PDEs during the learning process, which focuses provided data to infer the actual unknowns in the problem, i.e.",
"reduces the load of learning a discretization scheme from scratch.",
"Physics informed.",
"Domain-specific knowledge about the unknown fields, such as symmetries or specialized basis functions, can be directly employed within our design.",
"Inverse transform.",
"After training, the encoder can be used independently as a real-time estimator for unknown fields, i.e.",
"without further optimization.",
"In other words, the network can be pre-trained and then used to infer unknown fields in real-time applications."
],
[
"Blended inverse-PDE network (BiPDE-Net)",
"The basic idea is to embed a numerical solver into a deep learning architecture to recover unknown functions in inverse-PDE problems, and all the information about the governing PDE system is only encoded inside the DNN architecture as a solver layer.",
"In this section we describe our proposed architectures for inverse problems in one and two spatial dimensions."
],
[
"Deep neural networks (DNN)",
"The simplest neural network is a single layer of perceptron that mathematically performs a linear operation followed by a nonlinear composition applied to its input space, $\\mathcal {N} =\\sigma \\big ( \\mathbf {W}\\mathbf {x}+\\mathbf {b}\\big ),$ where $\\sigma $ is called the activation function.",
"Deep neural networks are multiple layers stacked together within some architecture.",
"The simplest example is a set of layers connected in series without any recurrent loops, known as feedforward neural networks (FNN).",
"In a densely connected FNN, the action of the network is simply the successive compositions of previous layer outputs with the next layers, i.e., $\\mathcal {N}_l =\\sigma \\big ( \\mathbf {W}_l\\mathcal {N}_{l-1}(\\mathbf {x})+\\mathbf {b}_l\\big ),$ where $l$ indicates the index of a layer.",
"This compositional nature of NNs is the basis of their vast potential as universal function estimators of any arbitrary function on the input space $\\mathbf {x}$ , see e.g.",
"[80], [15], [13].",
"Another important feature of NNs is that they can effectively express certain high dimensional problems with only a few layers, for example Darbon et al.",
"[17] have used NNs to overcome the curse of dimensionality for some Hamilton-Jacobi PDE problems (also see [27], [76]).",
"Most machine learning models are reducible to composition of simpler layers which allows for more abstract operations at a higher level.",
"Common layers include dense layers as described above, convolutional layers in convolutional neural networks (CNNs) [44], [42], Long-short term memory networks (LSTM) [33], Dropout layers [77] and many more.",
"In the present work, we pay particular attention to CNNs owing to their ability to extract complicated spatial features from high dimensional input datasets.",
"Furthermore, we define custom PDE solver layers as new member of the family of pre-existing layers by directly implementing numerical discretization schemes inside the architecture of deep neural networks."
],
[
"Custom solver layers",
"A layer is a high level abstraction that plays a central role in existing deep learning frameworks such as TensorFlowFor example see TensorFlow manual page at $\\rm https://www.tensorflow.org/guide/keras/custom\\_layers\\_and\\_models$ [1], Keras API [12], PyTorch [62], etc.",
"Each Layer encapsulates a state, i.e.",
"trainable parameters such as weights/biases, and a transformation of inputs to outputs.",
"States in a layer could also be non-trainable parameters in which case they will be excluded from backpropagation during training.",
"We implement different explicit or implicit numerical discretization methods as custom layers that transform an unknown field, initial data and boundary conditions to outputs in the solution space.",
"Solver layers encapsulate numerical discretization schemes with trainable (e.g.",
"shape parameters and seeds in meshless methods) or non-trainable (e.g.",
"the finite difference methods) state parameters.",
"Interestingly, solver layers with trainable parameters are new computational objects analogous to pre-existing convolutional layers with trainable kernel parameters.",
"An important aspect of layer objects is that they can be composed with other layers in any order.",
"Particularly, this offers an interesting approach for solving inverse problems given by systems of partial differential equations with several unknown fields that can be modeled with neural layers.",
"We will explore this avenue in future work.",
"In the remainder of this manuscript we will only focus on different inverse-PDE examples given by a single PDE equation and one unknown field."
],
[
"Blended neural network architectures",
"BiPDE is a two-stage architecture, with the first stage responsible for learning the unknown coefficients and the second stage performing numerical operations as in traditional numerical solvers (see figure REF ).",
"To achieve higher performance, it is essential to use GPU-parallelism.",
"We leverage the capability provided by the publicly available library TensorFlow [1] by implementing our PDE-solver as a custom layer into our network using the Keras API [12].",
"Details of this includes vectorized operations to build the linear system associated by the PDE discretization.",
"We propose a semantic autoencoder architecture as proposed by Aragon-Calvo (2019) [2] with hidden parameters being represented at the bottleneck of the autoencoder.",
"Figure REF illustrates the architecture for the proposed semantic autoencoder.",
"Depending on static or time dependent nature of the governing PDE, one may train this network over pairs of input-output solutions that are shifted $\\rm p$ steps in time, such that for a static PDE we have $\\rm p=0$ while dynamic PDEs correspond to $\\rm p\\ge 1$ .",
"We call this parameter the shift parameter, which will control the accuracy of the method (cf.",
"see section ).",
"Figure: Architecture of the BiPDE to infer unknown parameters of hidden fields.",
"Here the loss function is the mean squared error between data and output of the autoencoder, however other choices for loss function may be used depending on the nature of data.An important aspect is that the input to BiPDE is the solution data itself.",
"In other words the neural network in a BiPDE is learning the inverse transform, $\\rm \\mathcal {NN}:~ \\lbrace u\\rbrace \\rightarrow \\textrm {hidden\\ field},$ where $\\lbrace u\\rbrace $ indicates an ensemble of solutions, e.g.",
"solutions obtained with different boundary conditions or with different hidden fields.",
"Note that in other competing methods such as PINNs the input is sanctioned to be the coordinates in order for automatic differentiation to compute spatial and temporal derivatives; as a consequence PINNs can only be viewed as surrogates for the solution of the differential problem defined on the space of coordinates.",
"However, we emphasize that semantic autoencoders are capable to approximate the inverse transformation from the space of solutions to the space of hidden fields, a feature that we exploit in section REF .",
"Essentially different numerical schemes can be implemented in the decoder stage.",
"We will blend examples of both mesh-based and mesh-less numerical discretizations and present numerical results and comparisons with PINNs.",
"We will show how BiPDEs can handle data on unstructured grids and data with added noise.",
"In section , we demonstrate performance of mesh-based BiPDEs on inverse problems in two spatial dimensions by using a finite difference discretization and Zernike expansion of the non-homogeneous hidden field, we will consider both stationary and dynamic PDE problems in this section.",
"Then in section , we develop a mesh-less BiPDE and consider a dynamic nonlinear inverse partial differential problem.",
"We consider a variable coefficient Poisson problem in one and two spatial dimensions as well as the one dimensional nonlinear Burger's equation as an example of a nonlinear dynamic PDE problem with a scalar unknown parameter."
],
[
"Stationary Poisson problem",
"We consider the governing equation for diffusion dominated processes in heterogeneous media: $&\\nabla \\cdot \\big (D(\\mathbf {x}) \\nabla u\\big )=-f(\\mathbf {x}), &\\mathbf {x}\\in \\Omega \\\\&u(\\mathbf {x})=u_0(\\mathbf {x}), &\\mathbf {x}\\in \\partial \\Omega $ Here we consider a rectangular domain with Dirichlet boundary conditions.",
"Discretization.",
"In our architecture, we use the standard 5-point stencil finite difference discretization of the Poisson equation in the solver layer, i.e.",
"$&\\frac{D_{i-1/2, j} u_{i-1, j} - (D_{i-1/2,j} + D_{i+1/2,j})u_{i,j} + D_{i+1/2,j}u_{i+1,j}}{\\Delta x^2}+\\\\&\\frac{D_{i, j-1/2} u_{i, j-1} - (D_{i,j-1/2} + D_{i,j+1/2})u_{i,j} + D_{i,j+1/2}u_{i,j+1}}{\\Delta y^2} + f_{i,j}=0,$ and we use the linear algebra solver implemented in TensorFlow to solve for the solution field, i.e.",
"we used tf.linalg.solve method that is a dense linear system solver.",
"Of course, this can be improved by implementing a sparse linear solver.",
"Hidden Model.",
"We decompose the hidden field into a finite number of eigenfunctions and search for their optimal coefficients.",
"This is also advantageous from a physics point of view, because domain's knowledge of hidden fields can be naturally formulated in terms of basis functions into this framework.",
"One such family of series expansions are the moment-based methods that have been largely exploited in image reconstruction [40], [5], [64], [3].",
"In particular, Zernike moments [84] provide a linearly independent set of polynomials defined on the unit circle/sphere in two/three spatial dimensions.",
"Zernike moments are well-suited for such a task and are commonly used for representing optical aberration in astronomy and atmospheric sciences [65], for image reconstruction and for enhanced ultrasound focusing in biomedical imaging [19], [50], [39].",
"Zernike moments are advantageous over regular moments in that they intrinsically provide rotational invariance, higher accuracy for irregular patterns, and are orthogonal, which reduces information redundancy in the different coefficients.",
"Zernike polynomials capture deviations from zero mean as a function of radius and azimuthal angle.",
"Furthermore, the complete set of orthogonal bases provided by Zernike moments can be obtained with lower computational precision from input data, which enhances the robustness of the reconstruction procedure.",
"Odd and even Zernike polynomials are given as a function of the azimuthal angle $\\theta $ and the radial distance $\\rho $ between 0 and 1 measured from the center of image, $&\\begin{bmatrix} Z_{nm}^o(\\rho , \\theta ) \\\\ Z_{nm}^e(\\rho , \\theta ) \\end{bmatrix}=R_{nm}(\\rho ) \\begin{bmatrix} \\sin (m\\theta )\\\\ \\cos (m\\theta )\\end{bmatrix},$ with $R_{nm}(\\rho )&={\\left\\lbrace \\begin{array}{ll}\\sum _{l=0}^{(n-\\vert m\\vert )/2}\\frac{(-1)^l (n-l)!}{l!",
"[(n+\\vert m\\vert )/2 -l]!",
"[(n-\\vert m\\vert )/2 -l]!",
"}\\rho ^{n-2l} & \\textrm {for n-m even,} \\\\0 & \\textrm {for n-m odd,}\\end{array}\\right.",
"}$ where $n$ and $m$ are integers with $n\\ge \\vert m\\vert $ .",
"A list of radial components is given in table REF (from [85]).",
"For an extensive list of Zernike polynomials in both two and three spatial dimensions, we refer the interested reader to [51].",
"Table: First 15 odd and even Zernike polynomials according to Noll's nomenclature.",
"Here, the ordering is determined by ordering polynomial with lower radial order first, cf.",
".Furthermore, each Zernike moment is defined by projection of the hidden field $f(x,y)$ on the orthogonal basis, $& \\begin{bmatrix} A_{nm}\\\\ B_{nm}\\end{bmatrix}=\\frac{n+1}{\\epsilon _{mn}^2\\pi }\\int _x \\int _y f(x,y) \\begin{bmatrix} Z_{nm}^o(x,y) \\\\ Z_{nm}^e(x,y) \\end{bmatrix} dx dy, \\quad x^2 + y^2\\le 1,$ where for $m=0,~ n\\ne 0$ we defined $\\epsilon _{0n}=1/\\sqrt{2}$ and $\\epsilon _{mn}=1$ otherwise.",
"Finally, superposition of these moments expands the hidden field in terms of Zernike moments: $\\hat{f}(x,y)=\\sum _{n=0}^{N_{max}}\\sum _{\\vert m\\vert =0}^{n}\\big [ A_{nm}Z_{nm}^o (r,\\theta ) + B_{nm}Z_{nm}^e (r,\\theta ) \\big ].",
"$ In order to identify the coefficients in the Zernike expansion (REF ) for hidden fields, we use a semantic autoencoder architecture with Zernike moments being represented by the code at the bottleneck of the autoencoder.",
"Figure REF illustrates the architecture for the proposed semantic autoencoder.",
"Figure: Architecture of the semantic autoencoder to infer hidden fields.",
"Zernike moments are discovered at the bottleneck of the architecture.Architecture.",
"Even though a shallow neural network with as few neurons as the number of considered Zernike terms suffices to estimate values of the unknown Zernike moments in each of the problems considered in this section, however we will use a deep convolutional neural network (detailed below) in order to achieve our ultimate goal of approximating the inverse transform for the Poisson problem in a broad range of diffusion coefficient fields.",
"Therefore we design one deep neural network and uniformly apply it to several problems in this section.",
"In the training process of a CNN, the kernels are trained at each layer such that several feature maps are extracted at each layer from input data.",
"The CNN is composed of 3 convolutional blocks with $32,~ 64,~ 128$ channels respectively and kernel size $3\\times 3$ .",
"Moreover, we use the MaxPooling filter with kernel size $(2,2)$ after each convolutional block to downsample the feature maps by calculating the maximum values of each patch within these maps.",
"We use the ReLU activation function [26], i.e.",
"a piecewise linear function that only outputs positive values: ${\\rm ReLU}(x)=\\max (0,x)$ , in the convolutional layers followed by a Sigmoid activation in dense layers and a scaled Sigmoid activation at the final layer, $\\tilde{\\sigma }(x)&=D_{\\min } + (D_{\\max } - D_{\\min })\\sigma (x),$ such that the actual values of the diffusion coefficient are within the range $(D_{\\min }, D_{\\max })$ , known from domain specific knowledge.",
"After each dense layer, we apply Dropout layers with a rate of $0.2$ to prevent overfitting [32], [77] (a feature that is most useful in estimating the inverse transform operator) and avoid low quality local minima during training.",
"Case I.",
"A tilted plane.",
"In the first example we consider a linear diffusion model given by $&D(x,y)=\\sqrt{2} + 0.1(y-x)$ where the boundary condition function $u_{BC}$ and the source field $f$ are given by $&u_{BC}(x,y)=0.01\\cos (\\pi x)\\cos (\\pi y) \\qquad \\textrm {and} \\qquad f(x,y)=\\sin (\\pi x)\\cos (\\pi y)$ Figure: Error in learned diffusion coefficient.In this experiment we only use a single solution field for training.",
"Even though in our experiments the method succeeded to approximate the hidden field even with a single grid point to compute the loss function, here we consider all the grid points in the domain to obtain improved accuracy in the results.",
"We trained the network for $\\rm 30$ epochs using an Adam optimizer [41] that takes 170 seconds on a Tesla T4 GPU available on a free Google Colaboratory accounthttps://colab.research.google.com/.",
"Figure REF depicts the results obtained by the proposed scheme.",
"The diffusion map is discovered with a maximum relative error of only $2\\%$ , while the error in the solution field is $1\\%$ .",
"It is noteworthy to mention that the accuracy of the results in this architecture are influenced by the accuracy of the discretizations used in the solver layer.",
"While we used a second-order accurate finite difference discretization, it is possible to improve these results by using higher order discretizations instead.",
"We leave such optimizations to future work.",
"Figure: Mean absolute error in solution vs. epochs for the two dimensional tilted plane (case I).Table: Influence of architecture of the decoder stage on mean absolute error MAE D ≡∑|D(𝐱)-D ^(𝐱)|/N\\rm MAE_D \\equiv \\rm \\sum \\vert D(\\mathbf {x}) - \\hat{D}(\\mathbf {x})\\vert /N and maximum error L D ∞ \\rm L^\\infty _D in the discovered hidden field in case I.",
"Double vertical lines correspond to MaxPooling2D() layers and triple vertical lines correspond to Flatten() layer.",
"C(o) and D(o) stand for conv2D(filters) and Dense(neurons) layers respectively.",
"There are 3 neurons at the bottleneck not shown in the table.Influence of architecture.",
"Table REF tabulates the mean absolute error in the discovered tilted plane diffusion coefficient for different architectures of the encoder stage.",
"No significant improvement is observed for deeper or shallower encoder network for the example considered here.",
"Case II.",
"superimposed Zernike polynomials.",
"We consider a more complicated hidden diffusion field given by $&D(x,y)= 4 + a_0 + 2a_1x + 2 a_2 y + \\sqrt{3}a_3 (2 x^2 + 2y^2 - 1).$ The boundary condition function $u_{BC}$ and the source field $f$ are given by $&u_{BC}(x,y)=\\cos (\\pi x)\\cos (\\pi y)\\qquad \\textrm {and} \\qquad f(x,y)=x+ y.$ Figure REF illustrates the performance of the proposed Zernike-based network using a mean absolute error measure for the loss function.",
"We trained the network for $\\rm 100$ epochs using an Adam optimizer [41].",
"Figure: Error in learned diffusion coefficient.Resilience to noise.",
"We also assess the performance of our scheme on noisy datasets.",
"We consider a zero-mean Gaussian noise with standard deviation $0.025$ superimposed on the solution field.",
"Figure REF depicts the solution learned from a noisy input image.",
"The network succeeds in discovering the diffusion field with comparable accuracy as in the noise-free case.",
"Note that this architecture naturally removes the added noise from the learned solution, a feature that is similar to applying a low-pass filter on noisy images.",
"Figure: Error in learned diffusion coefficient.Figure: L 2 \\rm L_2 loss vs. epoch for 2D inverse transform."
],
[
"Learning the inverse transform",
"In the previous sections, we have applied BiPDE to find the variable diffusion coefficient from a single input image.",
"Another interesting feature of the proposed semantic autoencoder architecture is its ability to train neural networks in order to discover the inverse transform for the underlying hidden fields in a self-supervised fashion.",
"In this scenario, the trained encoder learns the inverse transform function that approximates the hidden parameters given a solution field to its input.",
"Note that even though the same task could be accomplished by supervised learning of the hidden fields, i.e.",
"by explicitly defining loss on the hidden fields without considering the governing equations, BiPDEs substitute the data labels with a governing PDE and offer comparable prediction accuracy.",
"In this section we train BiPDEs over ensembles of solution fields to estimate hidden Zernike moments of diffusion coefficients underlying unseen data.",
"One dimensional inverse transform Figure: Regression quality is R 2 =0.9834912\\rm R^2=0.9834912.We build a one dimensional semantic autoencoder using 3 layers with $100,~ 40$ , and 2 neurons respectively.",
"We used the ReLU activation function for the first two layers and a Sigmoid activation function for the last layer representing the hidden parameters.",
"A linear solver is then stacked with this encoder that uses the second order accurate finite difference discretization, i.e.",
"$&\\frac{D_{i-1/2} u_{i-1} - (D_{i-1/2} + D_{i+1/2})u_{i} + D_{i+1/2}u_{i+1} }{\\Delta x^2} + f_{i}=0, & D_{i+1/2}=\\frac{D_i + D_{i+1}}{2}$ However, the diffusion map is internally reconstructed using the hidden parameters before feeding the output of the encoder to the solver.",
"As a test problem, we consider the one dimensional Poisson problem with a generic linear form for the diffusion coefficient, $D(x)=1 + a_0 + a_1 x.$ We consider identical left and right Dirichlet boundary conditions of 0.2 for all images and let the source term be $f(x)=\\sin (\\pi x)$ .",
"We consider random diffusion coefficients $a_0$ and $a_1$ with a uniform distribution in $[0.25, 0.75]$ and we generate 1000 solutions over the domain $x\\in [-1,1]$ .",
"We train BiPDE over 900 images from this dataset and validate its performance over the remaining 100 images using a mean squared error loss function for 1000 epochs.",
"Each image is generated on a uniform grid with $\\rm N_x=160$ grid points.",
"We used a batch size of 100 in these experiments using the Adam optimizer.",
"Figure REF shows loss versus epochs in this experiment.",
"Figure REF compares learned and true coefficients over two independent test samples containing 1000 solutions, with and without a zero-mean Gaussian noise with standard deviation $0.025$ , i.e.",
"amounting to $\\sim 13\\%$ added noise over the images.",
"In figure REF , we expanded the range of unknown parameters $a_0,a_1\\in [0.15, 0.85]$ in our test sample to assess performance of trained encoder over unseen data that are outside the range of training data (as a measure of generalizability of BiPDEs).",
"In this figure blue points correspond to new images whose true unknown parameters fall inside the training range, and red data points correspond to those outside the training range.",
"We observe that the encoder is able to predict the unknown parameters even outside of its training range, although its accuracy gradually diminishes far away from the training range.",
"Note that the predicted values for $a_0$ and $a_1$ exhibit a systematic error towards the lower and upper bounds of the Sigmoid activation function, indicative of the influence of the Sigmoid activation function used in the last layer.",
"This emphasizes the significance of properly designing activation functions at the bottleneck.",
"Using the $\\rm R^2$ statistical coefficient as a measure of accuracy for the trained encoder, we assess effects of sample size and grid points on the performance of BiPDEs and report the results in table REF .",
"Effect of sample size: First, we fix number of grid points and vary sample size.",
"We find that increasing sample size improves accuracy of the predictions in the case of clean data, however in the case of noisy data the accuracy does not show significant improvement by enlarging sample size.",
"Effect of grid points: Second, we fix the sample size and gradually increase number of grid points.",
"We find that accuracy of predictions on noisy data is strongly correlated with number of grid points, however this dependence is weaker for clean data.",
"Table: R 2 \\rm R^2 coefficient for predicted Zernike coefficients of the one dimensional Poisson problem at different training sample size and number of grid points.Two dimensional inverse transform We consider an example of variable diffusion coefficients parameterized as $D(x,y)=4 + 2a_2 y + \\sqrt{3}a_3(2x^2 + 2y^2 - 1)$ , with unknown coefficients randomly chosen in range $a_2,a_3\\in [0.25,0.75]$ .",
"The equations are solved on a square domain $\\Omega \\in [-\\frac{1}{\\sqrt{2}}, \\frac{1}{\\sqrt{2}}]^2$ governed by the Poisson equation: $\\nabla \\cdot \\big ([4 + 2a_2 y + \\sqrt{3}a_3(2x^2 + 2y^2 - 1)]\\nabla u\\big ) + x + y=0, &\\qquad (x,y)\\in \\Omega , \\\\u_{BC}=\\cos (\\pi x) \\cos (\\pi y), &\\qquad (x,y)\\in \\partial \\Omega .$ The encoder is composed of two convolutional layers with 32 and 64 channels followed by a $2\\times 2$ average pooling layer and a dense layer with 128 neurons, at the bottleneck there are 2 neurons representing the two unknowns.",
"All activation functions are ReLU except at the bottleneck that has Sigmoid functions.",
"An Adam optimizer is used on a mean squared error loss function.",
"We trained BiPDE over 900 generated solutions with randomly chosen parameters $a_2,a_3$ and validated its performance on 100 independent solution fields for 300 epochs, evolution of loss function is shown in figure REF .",
"Then we tested the trained model over another set of 1000 images with randomly generated diffusion maps independent of the training dataset.",
"Furthermore, we repeated this exercise over 1000 images with added zero-mean Gaussian noise with standard deviation $0.025$ .",
"In each case, the learned parameters are in good agreement with the true values, as illustrated in figure REF .",
"Moreover, we performed a sensitivity analysis on the accuracy of the predictions with respect to sample size.",
"We measured quality of fit by the $\\rm R^2$ statistical coefficient.",
"Results are tabulated in table REF and indicate training over more samples leads to more accurate predictions when applied to clean data, while noisy data do not show a strong improvement by increasing sample size.",
"Figure: Regression quality is R 2 =0.9915149\\rm R^2=0.9915149.Table: R 2 \\rm R^2 coefficient for predicted Zernike coefficients of the two dimensional Poisson problem by increasing training sample size.",
"Number of grid points are fixed at 30×30\\rm 30\\times 30.In this section, we demonstrate the applicability of BiPDEs on time-dependent nonlinear partial differential equations, and we use those results to illustrate the consistency and accuracy of the proposed framework.",
"Similar to previous works [69], we consider the nonlinear Burgers' equation in one spatial dimension, $&\\frac{\\partial u}{\\partial t} + u\\frac{\\partial u}{\\partial x} = \\nu \\frac{\\partial ^2 u}{\\partial x^2} &x\\in [-1,1],~~ t\\in [0,1) \\\\&u(-1,t)=u(1,t)=0 &u(x,0)=-\\sin (\\pi x)$ where $\\nu =1/Re$ with $Re$ being the Reynolds number.",
"Notably, Burgers' equation is of great practical significance for understanding evolution equations as it is nonlinear.",
"Burgers' equation has been used as a model equation for the Navier-Stokes equation and by itself can be used to describe shallow water waves [18], turbulence [9], traffic flow [54], and many more.",
"Discretization.",
"In our design we adopted the 6th-order compact finite difference scheme proposed by Sari and Gurarslan (2009) [71] for its simplicity of implementation, its high accuracy and because it leads to a linear system with narrow band and subsequently ensures computational efficiency.",
"This scheme combines a tridiagonalTridiagonal systems of equations can be obtained in $\\mathcal {O}(N)$ operations.",
"sixth-order compact finite difference scheme (CFD6) in space with a low-storage third-order accurate total variation diminishing Runge-Kutta scheme (TVD-RK3) for its time evolution ([74]).",
"In particular, the high-order accuracy associated with this discretization provides highly accurate results on coarse grids.",
"This is an important aspect of BiPDEs as a data-efficient inverse solver, which stems from their capacity to seamlessly blend highly accurate and sophisticated discretization methods with deep neural networks.",
"The first-order spatial derivatives are given at intermediate points by $&\\alpha u^{\\prime }_{i-1} + u^{\\prime }_i + \\alpha u^{\\prime }_{i+1} = b \\frac{u_{i+2} - u_{i-2}}{4h} + a\\frac{u_{i+1} - u_{i-1}}{2h}, &i=3,\\cdots , N-2$ where $&a=\\frac{2}{3}(\\alpha +2), &b=\\frac{1}{3}(4\\alpha -1),$ and $h=x_{i+1} - x_i$ is the mesh size, with grid points identified by the index $i=1,2,\\cdots , N$ .",
"For $\\alpha =1/3$ we obtain a sixth order accurate tridiagonal scheme.",
"The boundary points (for non-periodic boundaries) are treated by using the formulas [22], [71], $u^{\\prime }_1 + 5 u^{\\prime }_{2} &= \\frac{1}{h} \\bigg [ -\\frac{197}{60}u_1 - \\frac{5}{12}u_2 + 5u_3 - \\frac{5}{3} u_4 + \\frac{5}{12} u_5 -\\frac{1}{20}u_6\\bigg ]\\\\\\frac{2}{11}u^{\\prime }_1 + u^{\\prime }_2 + \\frac{2}{11}u^{\\prime }_3 &= \\frac{1}{h}\\bigg [ -\\frac{20}{33} u_1 -\\frac{35}{132}u_2 + \\frac{34}{33}u_3 - \\frac{7}{33}u_4 + \\frac{2}{33}u_5 -\\frac{1}{132}u_6\\bigg ] \\\\\\frac{2}{11}u^{\\prime }_{N-2} + u^{\\prime }_{N-1} + \\frac{2}{11}u^{\\prime }_{N}&=\\frac{1}{h}\\bigg [ \\frac{20}{33}u_N + \\frac{35}{132}u_{N-1} - \\frac{34}{33}u_{N-2} + \\frac{7}{33}u_{N-3} - \\frac{2}{33}u_{N-4} + \\frac{1}{132}u_{N-5} \\bigg ]\\\\5u^{\\prime }_{N-1} + u^{\\prime }_{N}&=\\frac{1}{h}\\bigg [ \\frac{197}{60}u_N + \\frac{5}{12} u_{N-1} - 5 u_{N-2} + \\frac{5}{3}u_{N-3} -\\frac{5}{12}u_{N-4} + \\frac{1}{20} u_{N-5} \\bigg ]$ This can be easily cast in the matrix form $BU^{\\prime }=AU$ where $U=[u_1, u_2, \\cdots , u_N]^T$ is the vector of solution values at grid points.",
"Furthermore, second order derivatives are computed by applying the first-order derivatives twiceFrom implementation point of view this is a very useful feature of this scheme, because $A$ and $B$ are constant matrices that do not change through training it is possible to pre-compute them using numpy's [56] basic data structures, and then simply import the derivative operators into TensorFlow's custom solver layer using tf.convert_to_tensor() command., $BU^{\\prime \\prime } = AU^{\\prime }$ Burgers' equation are thus discretized as: $&\\frac{dU}{dt}=\\mathcal {L}U, &\\mathcal {L}U=\\nu ~ U^{\\prime \\prime } - UU^{\\prime },$ where $$ is the element-wise multiplication operator and $\\mathcal {L}$ is a nonlinear operator.",
"We use a low storage TVD-RK3 method to update the solution field from time-step $k$ to $k+1$ , $U^{(1)}&=U_k + \\Delta t\\mathcal {L}U_k\\\\U^{(2)}&=\\frac{3}{4}U_k + \\frac{1}{4}U^{(1)} + \\frac{1}{4}\\Delta t \\mathcal {L}U^{(1)}\\\\U_{k+1}&=\\frac{1}{3} U_k + \\frac{2}{3}U^{(2)} + \\frac{2}{3}\\Delta t\\mathcal {L}U^{(2)}$ with a CFL coefficient of 1.",
"Note that this method only requires two storage units per grid point, which is useful for large scale scientific simulations in higher dimensions.",
"Training protocol.",
"For training, we first solve Burgers' equation for $M$ time-steps, then we construct two shifted solution matrices that are separated by a single time-step, i.e., $&\\mathcal {U}^{-1}=\\begin{bmatrix}U^1 & U^2 & \\cdots & U^{M-1}\\end{bmatrix} &\\mathcal {U}^{+1}=\\begin{bmatrix}U^{2} & U^{3} & \\cdots & U^M\\end{bmatrix}$ Basically, one step of TVD-RK3 maps a column of $\\mathcal {U}^{-1}$ to its corresponding column in $\\mathcal {U}^{+1}$ given an accurate prediction for the hidden parameter.",
"Hence, a semantic BiPDE is trained with $\\mathcal {U}^{-1}$ and $\\mathcal {U}^{+1}$ as its input and output respectively.",
"The unknown diffusion coefficient is discovered by the code at the bottleneck of the architecture.",
"Numerical setup.",
"To enable direct comparison with PINNs, we also consider a second parameter $\\gamma $ in Burger's equation.",
"In these current experiments we train for 2 unknown parameters $(\\nu , \\gamma )$ in the Burger's equation given by $&u_t + \\gamma u u_x - \\nu u_{xx}=0, & t\\in [0,1], ~ ~~ x\\in [-1,1]$ Similar to Raissi et al.",
"[69] we consider $\\nu =0.01/\\pi $ and $\\gamma =1.0$ .",
"For completeness we also recall the loss function used in PINN that encodes Burger's equation as a regularization, $MSE=\\frac{1}{N}\\sum _{i=1}^N \\bigg \\vert u(t^i_u, x_u^i) - u^i \\bigg \\vert ^2 +\\frac{1}{N}\\sum _{i=1}^N \\bigg \\vert u_t(t_u^i, x_u^i) + \\gamma u(t_u^i, x_u^i) u_x(t_u^i, x_u^i) - \\nu u_{xx}(t_u^i, x_u^i) \\bigg \\vert ^2$ where $(t^i_u, x^i_u, u^i)$ constitute training data with $N=2000$ observation points in the spatio-temporal domain.",
"In this experiment PINN is composed of 9 layers with 20 neurons per hidden layer.",
"It is worth mentioning that we are reporting BiPDE results by considering solutions in a narrower time span $t\\in [0,0.2]$ .",
"Architecture.",
"Obviously, one can choose a single neuron to represent an unknown parameter $\\nu $ or $\\gamma $ and in a few iterations an approximate value can be achieved.",
"However, our goal is to train a general purpose encoder that is capable of predicting the unknown value from an input solution pair with arbitrary values of $\\nu $ and $\\gamma $ and without training on new observations (cf.",
"see part REF ).",
"Therefore, we consider a BiPDE that is composed of a conv1D layer with 128 filters and a kernel size of 10 with tanh activation function, followed by AveragePooling1D with a pool size of 2 that after flattening is fed to two Dense layers with 20 and 2 neurons respectively that apply Sigmoid activation function.",
"We used the Adam optimizer to minimize the mean absolute error measure for the loss function.",
"Accuracy test.",
"First, we train for two unknowns in Burger's equation, namely $\\nu $ and $\\gamma $ .",
"We perform a sensitivity analysis for 200 epochs with different numbers of spatial grid points, as well as different time-steps.",
"In each case, we measure the error between the learned values of $\\nu $ and $\\gamma $ with their true value $\\nu _{\\rm true}=0.01/\\pi $ and $\\gamma _{\\rm true}=1.0$ .",
"Convergence results of this experiment are tabulated in table REF and shown in figure REF .",
"We find that increasing the number of grid points (i.e.",
"the resolution) improves the accuracy up to almost 700 grid points before accuracy in $\\nu $ (but not $\\gamma $ ) starts to deteriorate.",
"Note the decrease in time-step size does not have a significant effect on accuracy unless large number of grid points $N_x>160$ are considered where decreasing time-step clearly improves results.",
"Figure: Error in γ\\gamma - PINN.Table: Discovering two unknown values of ν\\nu and γ\\gamma in Burger's equation.",
"These values are plotted in figure .For comparison purposes we report numerical results from table 1 of Raissi et al.",
"(2017) [69] in our figures REF –REF .",
"Here we only presented noise-less results of BiPDE, therefore only the $0\\%$ added noise case of PINN is comparable, i.e.",
"the solid red line in figures REF –REF .",
"Even though the two test cases have significant differences and much care should be taken to directly compare the two methods, however BiPDEs have a clear advantage by exhibiting convergence in the unknown values, i.e.",
"more data means better results.",
"Table: Discovering one unknown parameter in Burger's equation, values for ν\\nu .In a second experiment, we fix the value of $\\gamma =1.0$ and only train for the unknown diffusion coefficient $\\nu $ .",
"Similar to previous test we trained the network for 200 epochs and figure REF shows the error in the discovered value of $\\nu $ at different levels of resolution.",
"In this case decreasing time-step size does not seem to have a significant effect on accuracy.",
"A curious observation is the absolute value of error for $\\nu $ is two orders of magnitude more precise when the network is trained for both parameters $\\nu $ and $\\gamma $ than when only tuning for $\\nu $ .",
"Again, convergence in unknown parameter is retained in this experiment.",
"Figure: Sensitivity analysis in training only one parameter ν\\nu .",
"True value of ν true =0.01/π\\nu _{\\rm true}=0.01/\\pi is sought in Burgers' equation at different levels of resolution.",
"Rightmost data points correspond to N x =700N_x=700 grid points."
],
[
"Mesh-less BiPDE: Multi-Quadratic Radial Basis Functions",
"Not only are direct computations of partial derivatives from noisy data extremely challenging, in many real world applications, measurements can only be made on scattered point clouds.",
"Tikhonov regularization type approaches have been devised to avoid difficulties arising from high sensitivity of differencing operations on noisy data [14], [11], [78]; for neural network based approaches, see [49], [73].",
"Recently, Trask et al.",
"[81] have proposed an efficient framework for learning from unstructured data that is based on the Generalized Moving Least Squares (GMLS) technique.",
"They show performance of GMLS-Nets to identify differential operators and to regress quantities of interest from unstructured data fields.",
"Another interesting approach had been put forth in the late 80s by [8], [63] that designed neural networks based on Radial Basis Functions (RBF) to perform functional interpolation tasks.",
"In these networks, the activation function is defined as the radial basis function of interest and the training aims to discover the weights of this network, which interestingly coincide with the coefficients in the corresponding radial basis expansion.",
"Since the early 70s, RBFs have been used for highly accurate interpolation from scattered data.",
"In particular, Hardy [28] introduced a special kind of RBF called the multiquadratic series expansion, that provides superior performance in terms of accuracy, stability, efficiency, simplicity and memory usage [21].",
"Kansa (1990) [37], [38] pioneered the use of radial basis functions to solve time dependent partial differential equations through deriving a modified multi-quadratic scheme.",
"In 1998, Hon and Mao [34] applied multiquadratics as a spatial discretization method for the nonlinear Burgers' equation and solved it for a wide range of Reynolds numbers (from 0.1 to 10,000).",
"Their scheme was later enhanced to second-order accuracy in time by Xie and Li (2013) [87] via introducing a compact second-order accurate time discretization.",
"Interestingly, the accuracy of these mesh-free methods can be improved by fine-tuning distributions of collocation points or their shape parameters.",
"For example, Hon and Mao devised an adaptive point to chase the peak of shock waves, which improved their results.",
"Fortunately, such fine-tuning of parameters can be automated using BiPDE networks; we demonstrate this in this section.",
"Discretization.",
"We chose to blend the second-order accurate method of Xie and Li, briefly described next and we leave further details to their original paper [87].",
"Initially, one can represent a distribution $u(\\mathbf {x})$ in terms of a linear combination of radial basis functions, $&u(\\mathbf {x})\\approx \\sum _{j=0}^{N_s} \\lambda _j \\phi _j(\\mathbf {x}) + \\psi (\\mathbf {x}), &\\mathbf {x}\\in \\Omega \\subset \\mathcal {R}^{dim},$ where $\\phi (\\mathbf {x})$ is the radial basis function that we adopt, $&\\phi _j(\\mathbf {x})=\\sqrt{r_j^2 + c_j^2}, &r_j^2 = \\vert \\vert \\mathbf {x} - \\mathbf {x}_j\\vert \\vert _2^2,$ and $c_j$ is the shape parameter that has been experimentally shown to follow $c_j = Mj + b$ with $j=0, 1, \\cdots , N_s$ ($N_s$ is number of seed points).",
"Moreover $M$ and $b$ are tuning parameters.",
"In equation (REF ), $\\psi (\\mathbf {x})$ is a polynomial to ensure solvability of the resulting system when $\\phi _j$ is only conditionally positive definite.",
"To solve PDEs, one only needs to represent the solution field with an appropriate form of equation (REF ).",
"In the case of Burgers' equation the solution at any time-step $n$ can be represented by $&u^n(x)\\approx \\sum _{j=0}^{N_s} \\lambda _j^n \\phi _j(x) + \\lambda _{N_s+1}^n x + \\lambda _{N_s+2}^n $ over a set of reference points for the basis functions that are given by $x_j=j/N_s$ , $j=0,1,\\cdots , N_s$ .",
"Xie and Li derived the following compact second-order accurate system of equations $\\big [1 + \\frac{\\Delta t}{2} u_x^n(\\hat{x}_j) \\big ] u^{n+1}(\\hat{x}_j) + \\frac{\\Delta t}{2} u^n(\\hat{x}_j) u^{n+1}_x(\\hat{x}_j) - \\frac{\\nu \\Delta t}{2}u_{xx}^{n+1}(\\hat{x}_j) = u^n(\\hat{x}_j) + \\frac{\\nu \\Delta t}{2}u_{xx}^n(\\hat{x}_j) $ over a set of $N_d+1$ distinct collocation points $\\hat{x}_j=(1+j)/(N_d+2)$ with $j=0,1,\\cdots , N_d$ .",
"Two more equations are obtained by considering the left and right boundary conditions $u^{n+1}(x_{L}) = u^{n+1}(x_{R})=0$ .",
"Note that spatial derivatives are directly computed by applying derivative operator over equation (REF ).",
"At every time-step, one solves for the $N+3$ coefficients $\\lambda ^n_0,\\cdots , \\lambda ^n_{N+2}$ , while the spatial components of the equations remain intact (as long as points are not moving).",
"The solution is obtained over the initial conditions given by $u^0(\\hat{x}_j)$ .",
"For implementation purposes, we represent the system of equations (REF ) in a matrix notation that is suitable for tensorial operations in TensorFlow.",
"To this end, we first write equation (REF ) as $U^n(\\hat{x})= A(\\hat{x}) \\Lambda ^n, $ where $&U^n_{(N_d+1)\\times 1} = \\begin{bmatrix}u^n(\\hat{x}_0) \\\\ u^n(\\hat{x}_1) \\\\ \\vdots \\\\ u^n(\\hat{x}_{N_d})\\end{bmatrix}, & \\Lambda ^n_{(N_s+1)\\times 1} = \\begin{bmatrix}\\lambda _0^n \\\\ \\lambda _1^n \\\\ \\vdots \\\\ \\lambda _{N_s}^n\\end{bmatrix},$ $&\\bigg [A_{ij}(\\hat{\\mathbf {x}})\\bigg ]_{(N_d+1) \\times (N_s+1)}=\\bigg [\\phi _{j}(\\hat{x}_i) - \\phi _{j}(x_L) - \\frac{ \\phi _{j}(x_R) - \\phi _j(x_L)}{x_R -x_L}(\\hat{x}_i - x_L) \\bigg ],$ with $i=0,1,\\cdots , N_d$ and $j=0,1,\\cdots , N_s$ .",
"Note that we already injected the homogeneous boundary conditions into equation (REF ).",
"Therefore, equation (REF ) can be written as, $\\bigg [ A + (\\mathbf {g}_x~\\mathbf {1}^T)A + (\\mathbf {g}~ \\mathbf {1}^T)A_x - \\frac{\\nu \\Delta t}{2}A_{xx}\\bigg ] \\Lambda ^{n+1} = \\bigg [A + \\frac{\\nu \\Delta t}{2}A_{xx}\\bigg ] \\Lambda ^n, $ where $\\mathbf {1}^T=[1,~ 1, ~\\cdots , ~1]_{1\\times (N_s+1)}$ , $$ is component-wise multiplication, and $&\\mathbf {g}= \\frac{\\Delta t}{2}A \\Lambda ^n, &\\mathbf {g}_x= \\frac{\\Delta t}{2}A_x \\Lambda ^n, \\\\&(A_x)_{ij}=\\phi ^{\\prime }_{j}(\\hat{x}_i) - \\frac{ \\phi _{j}(x_R) - \\phi _j(x_L)}{x_R -x_L}, &(A_{xx})_{ij}=\\phi ^{\\prime \\prime }_{j}(\\hat{x}_i).$ Note that in case of training for two parameters $(\\nu ,\\gamma )$ , expression for $\\mathbf {g}$ in equation REF needs to be modified by letting $\\mathbf {g}=\\frac{\\gamma \\Delta t}{2}A\\Lambda ^n$ .",
"Architecture.",
"Note that both the collocation points and the interpolation seed points can be any random set of points within the domain and not necessarily a uniform set of points as we chose above.",
"In fact, during training we allow BiPDE to find a suitable set of interpolation points as well as the shape parameters on its own.",
"The input data is calculated using aforementioned finite difference method over uniform grids and later interpolated on a random point cloud to produce another sample of solutions on unstructured grids for training.",
"Thus, in our architecture the last layer of the encoder has $2N_s + 1$ neurons with sigmoid activation functions representing the $2N_s$ shape parameters and seed points, as well as another neuron for the unknown diffusion coefficient.",
"Note that for points to the left of origin, in the range $x\\in [-1, 0]$ , we simply multiplied the output of $N_s$ activation functions by $``-1^{\\prime \\prime }$ within the solver layer (because output of Sigmoid function is always positive).",
"We use the mean squared error between data and predicted solution at time-step $n+p$ as the loss function.",
"We used the Adam optimizer to minimize the loss function.",
"Training protocol.",
"As in the previous case, we apply successive steps of MQ-RBF scheme to march the input data forward to a future time-step.",
"Not surprisingly, we observed that taking higher number of steps improves the results because erroneous guess of the diffusion coefficient leads to more pronounced discrepancy after longer periods of time.",
"Hence, we map the data $\\mathcal {U}^{-p}$ to $\\mathcal {U}^{+p}$ , which is $p$ time-steps shifted in time, $&\\mathcal {U}^{-p}=\\begin{bmatrix}U^1, U^2, \\cdots , U^{M-p}\\end{bmatrix} &\\mathcal {U}^{+p}=\\begin{bmatrix}U^{1+p}, U^{2+p}, \\cdots , U^{M}\\end{bmatrix}$ In our experiments a value of $ p=10$ was sufficient to get satisfactory results at the absence of noise.",
"However, at the presence of Gaussian noise and for smaller values of the diffusion coefficient (such as for $\\nu _{\\rm true}=0.01/\\pi $ ) we had to increase the shift parameter to $ p=100$ .",
"Numerical setup.",
"Once again, we let $\\nu _{\\rm true}=0.01/\\pi \\approx 0.00318$ and integrate Burgers' equation up to $t_f=0.2$ with a fixed time-step of $\\Delta t=0.001$ .",
"We use the finite difference method of the previous section to generate the datasets.",
"We then interpolate the solution on 80 data points, uniformly distributed in the range $(-1,1)$ with 20 interpolation seed points.",
"For this experiment, we set the batch size to 1.",
"We trained the network using Adam optimizer.",
"The results after 50 epochs are given in figure REF .",
"Figure: Evolution of mean squared error during training.Interestingly, for every pair of input-output, the network discovers a distinct value for the diffusion coefficient that provides a measure of uncertainty for the unknown value.",
"We report the average value of all diffusion coefficients as well as the probability density of these values.",
"We observe that for all pairs of solutions, the predicted value for the diffusion coefficient is distributed in the range $0.00305\\le \\hat{\\nu } \\le 0.00340$ with an average value of $<\\hat{\\nu }>=0.00320$ , which is in great agreement with the true value, indeed with $0.6\\%$ relative error.",
"Interestingly, we observe that the BiPDE network has learned to concentrate its interpolation seed points around the origin where the solution field varies more rapidly.",
"Furthermore, around $x=\\pm 0.5$ , the interpolation points are more sparse, which is in agreement with the smooth behavior of the solution field at these coordinates.",
"Therefore, this network may be used as an automatic shock tracing method to improve numerical solutions of hyperbolic problems with shocks and discontinuities as was shown by Hon and Mao.",
"Resilience to noise on unstructured grids.",
"We consider several cases to assess robustness to noise.",
"In each case, we pick 80 randomly distributed points along the $x$ -axis and linearly interpolate the solution field on this set of points.",
"Then, we add a Gaussian noise with a given standard deviation.",
"This noisy and unstructured data field is then fed into the MQ-RBF based BiPDE of this section.",
"We use a batch size of 10, with $10\\%$ of each sample for validation during training.",
"A summary of our results follows: Let $\\nu _{\\rm true}=0.1/\\pi $ , $p=10$ , $N_d=80$ , $N_s=20$ , $\\Delta t=0.001$ , and consider a Gaussian noise with a standard deviation of $1\\%$ .",
"After 100 epochs, we obtain the results in figure REF .",
"Figure: Evolution of mean squared error versus number of epochs.",
"Let $\\nu _{\\rm true}=0.1/\\pi $ , $p=100$ , $N_d=200$ , $N_s=20$ , $\\Delta t=0.001$ , and consider a Gaussian noise with a standard deviation of $5\\%$ .",
"After 150 epochs, we obtain the results in figure REF .",
"Figure: Evolution of mean squared error versus number of epochs.",
"Let $\\nu _{\\rm true}=0.01/\\pi $ , $p=100$ , $N_d=80$ , $N_s=20$ , $\\Delta t=0.001$ , and consider a Gaussian noise with a standard deviation of $1\\%$ .",
"After 200 epochs, we obtain the results in figure REF .",
"Figure: Probability density of diffusion coefficients.",
"Let $\\nu _{\\rm true}=0.01/\\pi $ , $p=100$ , $N_d=200$ , $N_s=20$ , $\\Delta t=0.001$ , and consider a Gaussian noise with a standard deviation of $5\\%$ .",
"After 150 epochs, we obtain the results in figure REF .",
"Figure: Probability density of diffusion coefficients.",
"We observe that this architecture is generally robust to noise.",
"However, at higher noise values require more tuning of hyperparameters, as well as longer training.",
"Accuracy tests.",
"We report the values of the discovered diffusion coefficients in the Burgers' equation for different grid sizes and different time-steps.",
"We use the same setting as that detailed in the numerical setup part in this section.",
"Particularly, the interpolation seeds are determined by the network and the training data is on a uniformly distributed set of points computed by the finite difference method of the previous section.",
"We consider three different time-steps, $\\Delta t=0.001, ~0.0005, ~0.00025$ , and two diffusion coefficients of $\\nu _{\\rm true}=0.01/\\pi ,~0.1/\\pi $ over grids of size $N_x=80,~ 160$ .",
"At each time-step, for all experiments with different grid sizes, we stop the training when the mean squared error in the solution field converges to a constant value and does not improve by more epochs; this roughly corresponds to $50, ~ 25,~12$ epochs for each of the training time-steps, respectively.",
"This indicates that by choosing smaller time steps less number of epochs are needed to obtain the same level of accuracy in the unknown parameter.",
"Furthermore, we use an Adam optimizer with a learning rate of 0.001.",
"The results of the accuracy tests are tabulated in tables REF –REF .",
"We observe, for all experiments, that the discovered coefficient is in great agreement with the true values.",
"Due to adaptivity of the interpolation seed points and their shape parameters for different experiments, the observed error values do not seem to follow the trend of traditional finite difference methods, as depicted in previous sections.",
"This could also be due to lower order of accuracy of the MQ-RBF method, i.e.",
"being a second-order accurate method, compared to the higher-order accurate finite difference method used in the previous section.",
"Table: Discovered values of the diffusion coefficient for ν true =0.03183\\nu _{\\rm true}=0.03183 at different time-steps and grid sizes.Table: Discovered values of the diffusion coefficient for ν true =0.003183\\nu _{\\rm true}=0.003183 obtained with different time-steps and grid sizes."
],
[
"Learning the inverse transform",
"As we emphasized before, a feature of BiPDE is to produce self-supervised pre-trained encoder models for inverse differential problems that are applicable in numerous applications where hidden values should be estimated in real-time.",
"We train an encoder over a range of values $\\nu \\in [0.1/\\pi ,~ 1/\\pi ]$ and assess the performance of the trained model on new data with arbitrarily chosen $\\nu $ values.",
"We choose 50 diffusion coefficients that are distributed uniformly in this range, then integrate the corresponding Burgers' equation up to $t_f=0.2$ with a constant time-step of $\\Delta t=0.0005$ on a grid with $N_x=100$ grid points using the aforementioned finite difference method.",
"There are 4000 time-steps in each of the 50 different realizations of Burgers' equation.",
"For a fixed value of $p=20$ , we draw 10 solution pairs for each value of $\\nu $ at uniformly distributed time instances and discard the first two instances to improve convergence of the network.",
"Hence, the training data uniformly samples the space of solutions over a $8\\times 50$ grid of $(t, \\nu )$ , as illustrated in figure REF .",
"We use the resulting 400 pairs in training of a semantic BiPDE, with 320 pairs used for training and 80 pairs for validation.",
"Figure: Topology of data points for training and testing of the semantic BiPDE.",
"Along the ν\\nu dimension, we depict 10 (out of 50) of the selected data points, while along the time dimension we illustrate the actual 8 data points.",
"Training pairs of 𝒰 -p \\mathcal {U}^{-p} and 𝒰 +p \\mathcal {U}^{+p} are color coded by black and orange dots, respectively; testing pairs are depicted by blue and red crosses.",
"On the right panel, we illustrate the training data for three nominal values of the diffusion coefficient, highlighted by green shades.",
"Green arrows indicate the direction of time.Architecture.",
"Given an arbitrary input, the signature of the hidden physical parameters will be imprinted on the data in terms of complex patterns spread in space and time.",
"We use a CNN layer as a front end unit to transform the input pixels to internal image representations.",
"The CNN unit has 32 filters with kernel size of 5.",
"The CNN is followed by max pooling with pool size of 2, which is then stacked with another CNN layer of 16 filters and kernel size of 5 along with another max pooling layer.",
"The CNN block is stacked with two dense layers with 100 and 41 neurons, respectively.",
"CNN and dense layers have ReLU and Sigmoid activation functions, respectively.",
"Overall, there are $42,209$ trainable parameters in the network.",
"Conceptually, the CNN extracts features on every snapshot that characterizes the evolution of the solution field through time-steps with a proper physical parameter.",
"This parameter is enforced to be the diffusion coefficient through the PDE solver decoder stage.",
"We train this network for 500 epochs using an Adam optimizer.",
"Resilience to noise.",
"Even though the encoder is trained on ideal datasets, we demonstrate a semantic BiPDE still provides accurate results on noisy datasets.",
"In contrast to other methods, we pre-train the network in a self-supervised fashion on clean data and later we apply the trained encoder on unseen noisy dataNote that the network could also be trained on noisy data as we showed before; however training would take longer in that case..",
"Figure: Performance of the encoder on unseen data with Gaussian noise with standard deviation 0.010.01.In figure REF , we provide the performance of this network on training as well as on unseen clean/noisy data-sets.",
"Furthermore, the network determines optimal parameters of the MQ-RBF method by evaluating interpolation seed points as well as their corresponding shape parameters to obtain the best approximation over all input data."
],
[
"Conclusion",
"We introduced BiPDE networks, a natural architecture to infer hidden parameters in partial differential equations given a limited number of observations.",
"We showed that this approach is versatile as it can be easily applied to arbitrary static or nonlinear time-dependent inverse-PDE problems.",
"We showed the performance of this design on multiple inverse Poisson problems in one and two spatial dimensions as well as on the non-linear time-dependent Burgers' equation in one spatial dimension.",
"Moreover, our results indicate BiPDEs are robust to noise and can be adapted for data collected on unstructured grids by resorting to traditional mesh-free numerical methods for solving partial differential equations.",
"We also showed the applicability of this framework to the discovery of inverse transforms for different inverse-PDE problems.",
"There are many areas of research that could be further investigated, such as considering diffusion maps with discontinuities across subdomains, using more sophisticated neural network architectures for more complex problems, using higher-order numerical solvers and finally tackle more complicated governing PDE problems with a larger number of unknown fields or in higher dimensions."
],
[
"Acknowledgment",
"This research was supported by ARO W911NF-16-1-0136 and ONR N00014-17-1-2676.",
"tocsectionBibliography"
]
] | 2001.03608 |
[
[
"Deep Interactive Reinforcement Learning for Path Following of Autonomous\n Underwater Vehicle"
],
[
"Abstract Autonomous underwater vehicle (AUV) plays an increasingly important role in ocean exploration.",
"Existing AUVs are usually not fully autonomous and generally limited to pre-planning or pre-programming tasks.",
"Reinforcement learning (RL) and deep reinforcement learning have been introduced into the AUV design and research to improve its autonomy.",
"However, these methods are still difficult to apply directly to the actual AUV system because of the sparse rewards and low learning efficiency.",
"In this paper, we proposed a deep interactive reinforcement learning method for path following of AUV by combining the advantages of deep reinforcement learning and interactive RL.",
"In addition, since the human trainer cannot provide human rewards for AUV when it is running in the ocean and AUV needs to adapt to a changing environment, we further propose a deep reinforcement learning method that learns from both human rewards and environmental rewards at the same time.",
"We test our methods in two path following tasks---straight line and sinusoids curve following of AUV by simulating in the Gazebo platform.",
"Our experimental results show that with our proposed deep interactive RL method, AUV can converge faster than a DQN learner from only environmental reward.",
"Moreover, AUV learning with our deep RL from both human and environmental rewards can also achieve a similar or even better performance than that with the deep interactive RL method and can adapt to the actual environment by further learning from environmental rewards."
],
[
"Introduction",
"In recent years, the role of autonomous underwater vehicle (AUV) in ocean exploration has become more and more important.",
"Equipped with a series of chemical and biological sensors, AUV can conduct continuous operation without human intervention in the ocean environment.",
"In addition, it can work independently adjusting to the changes of marine environment to complete the ocean observation task.",
"Because of the less investment, good maneuverability and flexible control, AUV has been widely applied in many fields, such as scientific observation, resource investigation, oil and gas engineering, military applications etc.",
"However, today's marine applications put forward higher and higher requirements for the autonomy of AUV.",
"Existing AUVs usually do not have good autonomy and are generally limited to pre-planning or pre-programming tasks.",
"They work well in known and structured environments, but not in uncertain and dynamic ones.",
"Therefore, to realize the autonomy of AUV, it is necessary for it to have strong abilities of environmental perception and understanding, online adjustment of control policies, and task planning.",
"The path planning and following of AUV, which determines the application prospect of AUV in the marine field, can only be realized with accurate control technology, in consideration of its energy consumption, motion characteristics, speed constraints, etc.",
"Therefore, autonomous control that can adapt to the changes of marine environment is the core technology to realize the autonomy of AUV.",
"PID control is the most popular traditional control methods and has been successfully applied to AUV [1].",
"However, the traditional control methods cannot respond and adjust to unpredictable environmental changes in real time, and cannot meet the autonomy of AUV.",
"On the other hand, robot learning based on reinforcement learning (RL) [2] has been introduced into the AUV design and research to improve its autonomy [3].",
"Reinforcement learning is a method for a robot controller to learn optimal control policy through interaction with the environment.",
"The policy defines which action the controller should take when the robot is in a certain environmental state.",
"Under the current policy, after the controller tries to select and execute an action in a certain state, it will receive a reward signal provided by the reward function defined in advance by the designer in the environment.",
"This reward signal reflects the quality of the actions performed by the controller and is used to update the control policy.",
"The goal of the controller is to learn a policy that maximizes the total cumulative reward.",
"Recently, Yu et al.",
"[4] applied the latest deep reinforcement learning (DRL) developed by researchers at Google DeepMind [5] to AUV path following task.",
"Deep reinforcement learning combines the advantages of deep learning (DL) [6] and reinforcement learning, and can realize the end-to-end autonomous learning and control with the raw high-dimensional environment perception information input to the behavior action.",
"Yu et al.",
"claimed that AUV with DRL can achieve better control effect than a PID controller in simulation experiments.",
"However, because it is difficult to define an effective reward function and it usually provides very sparse reward signals, the robot needs a lot of time and samples to explore and test before learning an effective control policy.",
"Therefore, the traditional reinforcement learning and deep reinforcement learning methods are still difficult to apply directly to the actual AUV system.",
"In order to speed up the robot learning, researchers propose interactive reinforcement learning (IRL) [7] based on reward shaping [8] in traditional reinforcement learning methods.",
"Interactive reinforcement learning allows designers and even non-technical personnel to train robots by evaluating their behavior.",
"In this way, human experience knowledge can be embedded into autonomous learning of robot to speed up its learning.",
"Therefore, in this paper, we combine the advantages of deep reinforcement learning and interactive RL by proposing deep interactive reinforcement learning for AUV path following tasks.",
"In addition, since the human trainer cannot provide human rewards for AUV and AUV needs to adapt to a changing environment when it is running in the ocean, we propose a deep reinforcement learning method that learns from both human rewards and environmental rewards at the same time.",
"We test our methods in two tasks—straight line and sinusoids curve following of AUV by simulating in Gazebo.",
"The experimental results show that AUV with our deep interactive RL method learns much faster than with a DQN learner from environmental reward.",
"Moreover, our deep RL learning from both human and environmental rewards can also achieve a similar or even better performance than the deep interactive RL method and adapt to the actual environment by learning from environmental rewards."
],
[
"Related Work",
"PID controller is the most popular traditional controller for AUV and has been successfully applied in many control engineering tasks [1].",
"The disadvantage of PID controller is that its effect will be affected by the disturbance in the complex environment.",
"However, due to the simple structure, the control of many underwater robots is still designed with PID controller.",
"Although the traditional control methods such as PID controller can basically meet the control requirements of AUV, they cannot adapt to unpredictable environmental changes in real time and cope with the uncertainty of external environment.",
"Therefore, they cannot realize the autonomy of AUV.",
"Reinforcement learning (RL) [2] has been successfully applied in many robot tasks [3], and has been introduced into the autonomous control of AUV.",
"Compared with the traditional control methods of AUV, a robot with reinforcement learning can achieve online parameter adjustment and can well cope with environmental changes and uncertainties.",
"Even in the absence of accurate system model or high coupling system, root learning with RL can also obtain good control effect.",
"For example, Yuh proposed an adaptive control algorithm based on neural network [9].",
"In the algorithm, a cost function is designed as reward function and the reinforcement learning method is used to realize adaptive control.",
"Different from the traditional adaptive control method, Yuh and other simulation experiments show that AUV with reinforcement learning can not only deal with the environmental changes and uncertainties, but also has good robustness to the unmodeled system dynamics.",
"El Fakdi and Carreras applied the policy gradient based reinforcement learning method to the underwater robot in the submarine cable tracking task, and showed good tracking results [10], [11].",
"In addition, similar to the Actor-Critic algorithm [12] in reinforcement learning, Cui et al.",
"[13] designed two neural network models using reinforcement learning: one is used to evaluate the long-term control performance of the system; the other is used to compensate the unknown dynamic of the system, such as unknown nonlinear characteristics and interference.",
"The simulation results of Cui et al.",
"show that their proposed reinforcement learning algorithm based on the dual neural network converges faster than the general neural network model and PD controller, and has better stability and control effect.",
"In addition, in order to solve the problem of high-dimensional sensory information input, Yu et al.",
"[4] applied the deep reinforcement learning method [5] to AUV path following task, which can achieve better control effect than PID control in their simulation experiments.",
"However, in traditional reinforcement learning and deep reinforcement learning, the reward function is pre-defined by the agent designer before robot learning, which determines the quality of the final learned control policy and the learning speed to a large extent.",
"To define an effective reward function is not easy, which often needs many debugging and is very time-consuming.",
"More importantly, much experience and knowledge are difficult to be embedded in an effective reward function.",
"An inefficient reward function means sparse reward signal and low learning efficiency.",
"The controller needs a lot of learning samples and time to test and explore before learning an optimal policy, which seriously limits the application of the reinforcement learning method to the actual AUV control system.",
"Based on traditional reinforcement learning, researchers proposed interactive reinforcement learning method [7], which allows people to observe and evaluate the robot's behavior, and use the evaluation as a reward signal to teach the robot how to perform tasks.",
"A robot with interactive reinforcement learning does not need to define reward function in advance, but allows a human trainer to provide reward signal by evaluating the robot's behavior according to one's own experience and knowledge.",
"The human reward signal reflects the quality of robot's behavior, and is used by the robot to improve its behavior.",
"Therefore, the robot can easily use people's experience and knowledge to accelerate its learning.",
"Thomaz and Breazeal [14] implemented a tabular Q-learning [15] agent which can learn from both human and environmental rewards by maximizing the accumulated discounted sum of them.",
"They also show that an agent's performance can be further improved if the human teacher was allowed to provide action advice besides human rewards.",
"Knox and Stone [16], [17] proposed the TAMER framework—a typical interactive reinforcement learning method.",
"The TAMER framework trains a regression model to represent a reward function that gives a reward consistent with the human's feedback.",
"TAMER has been tested in many simulation domains [17], such as Cart Pole, Grid World, Tetris etc., and even in a real robot navigation task [18].",
"The results showed that agents with TAMER can learn faster than those with traditional reinforcement learning.",
"In order to make full use of the information in human feedback, Loftin et al.",
"[19] interpreted human reward as a kind of categorical feedback strategy.",
"In addition, unlike TAMER and the work of Loftin et al., which take human reward signal as feedback to the optimal control policy expected by the trainer, Macglashan et al.",
"[20] proposed the COACH algorithm by further interpreting human reward signal as feedback to the control policy that the robot is executing, and test it in the robot navigation task.",
"Their results show that COACH can make the robot more effective in autonomous learning.",
"Carrera and Ahmadzadeh et al.",
"[21], [22], [23] used a imitation learning method [24] to achieve the autonomous valve rotation task for AUV.",
"They first asked the trainer to provide a demonstration on how to operate the valve based on their knowledge, and the AUV recorded the operation and learned how to turn the valve using a dynamic primitive method.",
"However, this only maximizes the ability of AUV to perform the task and cannot be applied to other tasks or adapt to changes in the marine environment."
],
[
"BACKGROUND",
"As a branch of machine learning, reinforcement learning is usually modeled as an Markov decision process (MDP) [2].",
"MDP mainly consists of five elements: agent, environment, state, action and reward.",
"In reinforcement learning, an agent interacts with the environment by acquiring the environment state, performing actions and obtaining rewards.",
"The basic framework of reinforcement learning is shown in Figure REF .",
"Suppose that the environmental state at time $t$ is $s_t$ , and the agent performs an action $a_t$ after obtaining the state $s_t$ , and the environmental state is transformed from $s_t$ to $s_{t+1}$ at time $t+1$ .",
"Then the environment generates feedback reward $r_{t+1}$ to the agent in the new state $s_{t+1}$ .",
"The agent will update the learned policy with the reward signal and perform a new action $a_{t+1}$ in the new state.",
"The agent will optimizes the policy by continually interacting with the environment until an optimal policy is learned.",
"The agent's goal is to maximize the long-term cumulative rewards.",
"Figure: Illustration of learning mechanism in reinforcement learning.In reinforcement learning, a general policy $\\pi $ maps states to actions and has Markov property.",
"The probability of taking action $a$ in the current state is only related to the current state, and has nothing to do with other factors.",
"The policy can be formulated as: $\\pi (a|s)=p(a_t=a|s_t=s).$ Maximizing a long-term cumulative rewards means the agent needs to consider the rewards of the current time step as well as the rewards of the future time.",
"Assume at current time step $t$ and long-term cumulative rewards can be formulated as $R=r_t+r_{t+1}+...+r_n$ .",
"However, due to the uncertainty of cumulative rewards, discounted future cumulative rewards $G_t$ is generally used in actual tasks: $G_t=R_{t+1}+\\gamma R_{t+2}+\\gamma ^2 R_{t+3}+...=R_{t+1}+\\gamma G_{t+1},$ where $\\gamma $ is the discount factor, ranging from 0 to 1.",
"A value function with discounted cumulative rewards is usually used to express the degree of goodness or badness for an agent in a certain state.",
"There are mainly two kinds of value functions: state value function and action value function.",
"A state value function $V(s)$ represents the expectation of discounted cumulative rewards for an agent in state $s$ : $V(s)=E[G_t|s_t=s],$ where $G_t$ represents the discount accumulated reward in the future.",
"The action value function $Q(s,a)$ represents the reward expectation of discounted cumulative rewards for an agent performing an action $a_t$ in a certain state $s_t$ and following the policy $\\pi (s)$ thereafter: $Q(s,a)=E[G_t|s_t=s,a_t=a].$ The relation of action value function $Q(s,a)$ and state value function $V(s)$ can be formulated as: $V(s)=\\sum _{a\\in A}\\pi (a|s)Q_\\pi (s,a).$ A Bellman equation is formulated to express the relationship between the state values of state $s_t$ and state $s_{t+1}$ : $V_\\pi (s)=E[R_{t+1}+\\gamma V_\\pi (s_{t+1})|s_t=s].$ The objective of reinforcement learning is to find an optimal policy expressed as $\\pi ^*$ , which can be obtained by maximizing the value function V(s) or $Q(s,a)$ under all policies: $V^*(s)=max_\\pi V(s),$ $Q^*(s,a)=max_\\pi Q(s,a).$ Therefore, once the optimal value function is obtained, the optimal policy can be found by greedily selecting actions according to it.",
"By combining reinforcement learning with deep learning [6], deep reinforcement learning (DRL) [5] fully embodies the perceived advantages of representation learning in deep learning and decision-making in reinforcement learning.",
"In deep reinforcement learning, the problem is still defined by reinforcement learning, and the policy and value function can be represented with deep neural network and optimized based on an objective function."
],
[
"Deep Q-network",
"Mnih et al.",
"proposed the first deep reinforcement learning algorithm — deep Q-network (DQN) and tested in the Atari games [5].",
"It combines deep learning and reinforcement learning to successfully learn control policies directly from raw high-dimensional inputs.",
"Specifically, DQN combines the neural network and Q-learning algorithm [15] to fit the Q value of each action with the deep neural network.",
"The state value function $V_\\pi (s)$ and action value function $Q_\\pi (s,a)$ are approximated as $\\hat{V}(s,\\theta )$ and $\\hat{Q}(s,a,\\theta )$ , where $\\theta $ represents the weights.",
"For the state value function $\\hat{V}(s,\\theta )$ , the input of the neural network is the eigenvector of state $s$ , and the output is the corresponding value function.",
"For the action value function, the input is the feature vector of the state, and the output is the action value function with each action corresponds to an output.",
"In DQN, a value function represented with deep neural network is learned and optimized with state data.",
"The DQN algorithm uses two networks for learning: a prediction network $Q(s,a,\\theta )$ and target network $Q^{^{\\prime }}(s,a,{\\theta }^{^{\\prime }})$ .",
"The prediction network is used to evaluate the current state action and updated at each time step.",
"The target network $Q^{^{\\prime }}(s,a,{\\theta }^{^{\\prime }})$ is used to generate target value.",
"The target network is directly copied from the prediction network every certain number of time steps and does not update its parameters.",
"The objective of introducing the target network in DQN is to keep the target Q value unchanged for a period of time.",
"In this case, the correlation between the field prediction Q value and the target Q value will be correlated to some extent, and the network instability during training can be reduced.",
"Specifically, a DQN agent will select the action with the largest Q value based on the outputted Q value by the network.",
"The experience replay mechanism [25], [26] is also used in DQN.",
"It stores the experience samples $\\left\\lbrace s, a, r, s^{^{\\prime }},ifend\\right\\rbrace $ (ifend flag indicates whether the task is completed) obtained from the interaction between the agent and environment at each time step into the experience replay pool.",
"When training the network, a small batch of samples is randomly selected from the experience replay pool and the current target Q value y is computed as: $y_i=\\left\\lbrace \\begin{array}{ll}R_i, & ifend=true\\\\R_i+\\gamma max_a\\hat{Q^{^{\\prime }}}(s^{^{\\prime }},a^{^{\\prime }},{\\theta }^{^{\\prime }}), & ifend=false\\end{array}.\\right.$ This helps remove the correlation and dependence between samples and makes the network more convergent.",
"With the mean square error loss function $L=\\dfrac{1}{N}\\sum _{i=1}^N(y_i-\\hat{Q}(s,a,\\theta ))^2,$ all parameters $\\theta $ of Q network will be updated through gradient back propagation[27]."
],
[
"Interactive reinforcement learning",
"In reinforcement learning and deep reinforcement learning, the reward function is pre-defined by the controller designer before agent learning.",
"The reward function determines the quality of the final control policy and the learning speed to a large extent.",
"However, it is not easy to define an effective reward function since it often needs many debugging, and much experience and knowledge are difficult to be embedded in an effective reward function.",
"An inefficient reward function means that the controller needs a lot of learning samples and time to test and explore, which seriously limits the application of the traditional reinforcement learning method to the actual robot system.",
"Therefore, on the basis of traditional reinforcement learning, researchers put forward interactive reinforcement learning [7], [14], [16], [17], [19], [28], [29], [20], [30].",
"In interactive reinforcement learning, an agent learns in an MDP without reward function.",
"It allows people to observe and evaluate the agent's behavior, and give a reward signal based on their judgement of agent's action selection.",
"The stand or fall of reward signal reflects the quality of agent's behavior.",
"The agent uses the reward signal delivered by the human trainer to improve its behavior.",
"The learning mechanism of interactive reinforcement learning is shown in Figure REF ."
],
[
"Approach",
"In this paper, we propose deep interactive reinforcement learning to complete the underwater vehicle path following task.",
"In our proposed method, we use the DQN algorithm.",
"However, the robot learns from human reward instead of pre-defined environment reward as in the original DQN.",
"In our proposed method, the trainer provides rewards $R_h$ observing the running state of AUV, and evaluating the actions selected by AUV in the current state according to her knowledge.",
"The learning mechanism of DQN is shown in Figure REF .",
"Figure: Learning mechanism of deep interactive reinforcement learning algorithm.In addition, since the human trainer cannot provide rewards to train AUV all the time and AUV also needs to adapt to the changing ocean environment in the task, we propose to allow our proposed method to learn from both human reward and environment reward at the same time.",
"We test our methods in two path following tasks: straight line and sinusoids curve following.",
"In both tasks, the input to the behavior value function network of DQN is the state of AUV.",
"In the straight line following task, the input state $s$ is represented by the current course of AUV and the shortest distance from AUV to the target path: $S=\\left\\lbrace d\\quad c\\right\\rbrace $ , where $d$ represents the distance from the position of AUV to P(x,y)—the intersection point of the perpendicular line to the horizontal axis at the position of AUV and the target line, $c$ represents the current course angle of AUV, as shown in Figure REF .",
"In the sinusoids curve following task, the slope of the tangent line at the intersection point P(x,y) and desired course angle are added as additional features to represent the state: $S=\\left\\lbrace d\\quad c\\quad k\\quad c_d\\right\\rbrace $ , where $d$ represents the distance from the position of AUV to the intersection point P(x,y), $c$ represents current course angle, $k$ represents the slope of the tangent line at the intersection point P(x,y), $c_d$ represents desired course angle, as shown in Figure REF .",
"The action space is the rudder value of AUV, and different actions correspond to different rudder angle values.",
"The environment reward function is defined as the difference between the current course angle and the desired course angle of AUV, considering the distance from AUV to the target line/curve.",
"The way we calculate the desired course angle is similar to the line of sight (LOS) algorithm [31].",
"For example, in the straight line following task, we choose the current target path $L$ with a fixed value from the intersection point $P(x,y)$ along the target line.",
"Then the destination point $S(x_d,y_d)$ is decided since the length of the current path $L$ is fixed.",
"The desired course angle is computed as the angle between the desired course from the current position of AUV to the destination point $S(x_d,y_d)$ and the horizontal axis, as shown in Figure REF .",
"Figure: The desired course angle in the straight line following task, where LL represents the current target path, and dd represents the distance from AUV to the target line, SS represents the current target point.In the sinusoids curve following task, the computation of the desired course angle is a bit different from the straight line following task.",
"Specifically, we choose current target path $L$ with a fixed length along the tangent line at the intersection point $P(x,y)$ .",
"Then the desired course angle is computed as the angle between the desired course from the current position of AUV to the current target point $S(x_d,y_d)$ and the horizontal axis, as shown in Figure REF .",
"Figure: The desired course angle in the sinusoids curve following task, where LL represents the current target path, dd represents the distance from AUV to the target curve, SS represents the current target point.If the distance between the AUV and the following line or curve is directly added to the reward function, it may lead to a large fluctuation of the reward value.",
"Therefore, we introduce an exponential transformation to define the reward function as: $R=-0.9\\cdot \\left|c_d-c\\right|+0.1\\cdot 2^{2-\\frac{d}{10}},$ where $c$ represents the actual current course angle of AUV, $d$ represents the actual distance from AUV to the target line/curve, $c_d$ indicates the desired course angle.",
"In this case, when the action selected by AUV reduces the difference between the current course and the desired course or the distance between AUV and the target path is smaller, the received reward R will be larger, otherwise R will be smaller.",
"During the learning process, the experience received feedback is stored as a sample in the experience replay pool.",
"A small batch of samples in the experience replay pool is randomly selected to update the parameters of Q network as the DQN algorithm."
],
[
"Experiments",
"To verify the effectiveness of our proposed method, we conducted experiments with an extension to the open-source robot simulator Gazebo in underwater scenarios.",
"The Autonomous Underwater Vehicle (AUV) Simulator [32] was used in our experiment.",
"However, we modified it to make fit with the actual AUV system in our lab, as shown in Figure REF .",
"The AUV simulator uses the robot operating system (ROS) to communicate with the underwater environment.",
"It can also simulate multiple underwater robots and intervention tasks using robotic manipulators.",
"Figure: The autonomous underwater vehicle simulator in the Gazebo robotic platform used in our experiment.In our proposed deep interactive reinforcement learning method, the human trainer needs to evaluate the state and action of AUV.",
"It is not easy to observe the exact action of AUV in the simulated environment.",
"Therefore, we developed a human-machine interaction interface for the human trainer to observe the attitude of AUV in real time using the Rviz display tool, as shown in Figure REF .",
"Rviz is a built-in graphical tool of ROS, which makes it very convenient for users to develop and debug ROS through the graphical interface.",
"Figure: Human-machine interaction interface for displaying of the real-time attitude of AUV.",
"Note that the solid green line represents target path.In our experiments, the AUV simulator selects and executes an action $a$ based on its current state $s$ and learned policy.",
"The human trainer observes the action selected by the AUV simulator in the current state from the visualized interface and evaluates its quality according to her knowledge and experience.",
"The evaluation will be taken as human reward and delivered to the AUV simulator via the developed human-machine interface.",
"The AUV simulator will use it to update the DQN network parameters to improve its control policy.",
"We trained three agents in our experiments: the DQNH agent, a DQN agent learning from only human provided reward as we proposed; the DQNHE agent, a DQN agent learning from both environment reward and human provided reward; the DQNE agent, a baseline agent where a DQN agent learns from only environment reward.",
"For both DQNHE and DQNE agents, the environment reward is provided via Equation REF .",
"For the DQNH and DQNHE agents to learn from human reward, the human trainer will give a reward value +0.8 or +0.5 when she thinks the AUV's movement is good through the developed interface.",
"When she thinks the AUV's action is bad, a reward value -0.8 or -0.5 will be given.",
"Specific reward value can be selected by the trainer according to her experience, as below: $R_h=\\left\\lbrace \\begin{array}{lr}+0.8\\quad or +0.5, & good\\quad action \\\\-0.8\\quad or -0.5, & bad\\quad action.\\end{array}\\right.$ When the DQNHE agent learning, the human reward $R_h$ is added to the environment reward $R$ as the final reward value."
],
[
"Experimental Results and Discussion",
"In this section, we present our experimental results tested in two path following tasks: straight line following and sinusoids curve following.",
"We compare our proposed DQN agent learning from solely human reward, termed DQNH, DQN agent learning from both human reward and environmental reward, termed DQNHE, to the original DQN agent learning from solely environment reward, termed DQNE."
],
[
"Straight Line Following",
"In this experiment, we trained the DQNH, DQNHE and DQNE agents in the straight line following task.",
"Figure REF shows the trajectories of the DQNE agent learning at different learning episodes in the straight line task.",
"From Figure REF we can see that, at Episode 1, the trajectory of the agent fluctuate dramatically along the target line at both sides.",
"At Episode 10, the fluctuation of the learning trajectory was reduced to some extent and at Episode 15, the agent can already follow a line but still far away from the target one.",
"Until the 50th episode, the trajectory of the DQNE agent is quite close to the target line.",
"Figure: Trajectories of DQNE agent learning in the straight line following task.",
"Note that X is the coordinate along the horizontal axis, Y is the coordinate along the vertical axis.Figure REF shows the trajectories of the DQNH agent learning at different learning episodes in the straight line following task.",
"From Figure REF we can see that, at Episode 1, the trajectory of the DQNH agent is similar to the one of the DQNE agent and fluctuate along the target line at both sides.",
"At Episode 5, the fluctuation of the DQNH agent learning trajectory was reduced to a large extent.",
"At Episode 10, the trajectory of the DQNH agent is already quite close to the target line, achieving a similar performance to the DQNE agent at the 50th episode as shown in Figure REF .",
"By comparing the learning trajectories of DQNH agent in Figure REF to those of DQNE agent in Figure REF , we found that a DQN agent learning from solely human reward converges much faster than learning from solely environment reward.",
"Figure: Trajectories of the DQNH agent learning in the straight line following task.",
"Note that X is the coordinate along the horizontal axis, Y is the coordinate along the vertical axis.Figure REF shows the trajectories of the DQNHE agent learning at different learning episodes in the straight line following task.",
"From Figure REF we can see that, at Episode 1, the trajectory of the DQNHE agent is similar to the ones of the DQNE agent and DQNH agent.",
"However, at Episode 5, the fluctuation of the DQNHE agent learning trajectory was dramatically reduced and even much better than that of the DQNH agent.",
"At Episode 10, the trajectory of the DQNHE agent is also close to the target line, achieving a similar performance to the DQNH agent.",
"By comparing the learning trajectories of DQNHE agent in Figure REF to those of DQNH agent in Figure REF , we found that a DQN agent learning from both human reward and environment reward can further improve its convergence speed, compared to learning from solely human reward.",
"Figure: Trajectories of the DQNHE agent learning in the straight line following task.",
"Note that X is the coordinate along the horizontal axis, Y is the coordinate along the vertical axis.We also analyzed the tracking error of the DQNE, DQNH, DQNHE agents along the learning process in the straight line following task, as shown in Figure REF .",
"From Figure REF we can see that, when the DQN agent learning from solely human reward or both from human reward and environment reward, the tracking error was dramatically reduced to a minimum in about 10 episodes, while the DQNE agent learning from solely environment reward achieves a similar performance until around the 40th episode.",
"Figure: Tracking error of the DQNE, DQNH, DQNHE agents along the learning process in the straight line following task (averaged over data collected in two trials).In addition, the cumulative environment rewards obtained by the DQNE and DQNHE agents were analyzed and illustrated in Figure REF .",
"Figure REF indicates that the cumulative rewards obtained by DQNHE agent quickly reach the peak in around 10 episodes for the first time, while it takes about 40 episodes for the DQNE agent to reach a similar level.",
"After that, both agents converged.",
"These results suggest that the additional human reward helps the DQN agent converge faster.",
"Figure: Cumulative environment rewards obtained by the DQNE and DQNHE agents in the straight line following task (averaged over data collected in two trials)."
],
[
"Sinusoids Curve Following",
"In this experiment, we trained the DQNH, DQNHE and DQNE agents in the sinusoids curve following task, which is a much complex one than the straight line following task.",
"Figure REF shows the trajectories of the DQNE agent learning at different learning episodes in the sinusoids curve following task.",
"From Figure REF we can see that, at Episode 1 and 10, the DQNE agent cannot finish the task at all.",
"Until the 60th episode, the fluctuation was reduced and the agent can already follow the sinusoids curve with the trajectory quite close to the target curve.",
"At Episode 100, the agent can almost exactly follow the sinusoids curve.",
"Figure: Trajectories of DQNE agent learning in the sinusoids curve following task.",
"Note that X is the coordinate along the horizontal axis, Y is the coordinate along the vertical axis.Figure REF shows the trajectories of the DQNH agent learning at different learning episodes in the sinusoids curve following task.",
"From Figure REF we can see that, at Episode 1 and 10, the DQNH agent cannot finish the task either.",
"At 20th episode, the DQNH agent can already follow the sinusoids curve with the trajectory quite close to the target curve, achieving a similar performance to the DQNE agent at the 60th episode.",
"At Episode 25, the AUV with DQNH can almost exactly follow the sinusoids curve, achieving a similar performance to the DQNE agent at the 100th episode.",
"Figure: Trajectories of DQNH agent learning in the sinusoids curve following task.",
"Note that X is the coordinate along the horizontal axis, Y is the coordinate along the vertical axis.Figure REF shows the trajectories of the DQNHE agent learning at different learning episodes in the sinusoids curve following task.",
"From Figure REF we can see that, at Episode 1 and 10, the DQNHE agent cannot finish the task either.",
"At the 20th episode, the DQNHE agent can also follow the sinusoids curve with the trajectory quite close to the target curve.",
"At Episode 25, the DQNHE agent can achieve a similar performance to that of the DQNE agent at Episode 100.",
"Figure: Trajectories of DQNHE agent learning in the sinusoids curve following task.",
"Note that X is the coordinate along the horizontal axis, Y is the coordinate along the vertical axis.The tracking errors of the DQNE, DQNH, DQNHE agents along the learning process in the sinusoids curve following task were shown in Figure REF .",
"From Figure REF we can see that, when the DQN agent learning from solely human reward, the tracking error was dramatically reduced to the lowest level in about 20 episodes, while the DQNE agent learning from solely environment reward achieves a similar performance until around the 70th episode.",
"When learning both from human reward and environment reward, the tracking error is reduced the fastest.",
"Figure: Tracking error of the DQNE, DQNH, DQNHE agents along the learning process in the sinusoids curve tracking task (averaged over data collected in two trials)In addition, the cumulative environment rewards obtained by the DQNE and DQNHE agents were analyzed and illustrated in Figure REF .",
"Figure REF indicates that the cumulative rewards obtained by DQNHE agent also quickly reach the peak in around 20 episodes for the first time, while it takes about 80 episodes for the DQNE agent to reach a similar level.",
"After that, both agents converged.",
"Figure: Cumulative environment rewards obtained of the DQNE and DQNHE agents in the sinusoids curve following task (averaged over data collected in two trials).In summary, our results in both the straight line and sinusoids curve following tasks suggest that our proposed deep interactive reinforcement learning method can facilitate a DQN agent to converge much faster than learning solely from environment reward.",
"Moreover, our proposed DQNHE agent learning from both human reward and environment reward can combine the advantages of our proposed deep interactive reinforcement learning from human reward and learning from environment reward.",
"The DQNHE method facilitate AUV to reach a similar performance to the one learning from solely environment reward with the speed same to or even better than the one learning from solely human reward.",
"This allows AUV to keep at the peak performance and adapt to the environment even when the human reward is not available in the ocean."
],
[
"conclusion",
"In this paper, we proposed a deep interactive reinforcement learning method for path following of AUV by combining the advantages of deep reinforcement learning and interactive RL.",
"In addition, since the human trainer cannot provide human rewards for AUV all the time and AUV needs to adapt to a changing environment when it is running in the ocean, we propose a deep reinforcement learning method that learns from both human rewards and environmental rewards at the same time.",
"We test our methods in two tasks—straight line and sinusoids curve following of AUV by simulating in the Gazebo platform.",
"Our experimental results show that with our propose deep interactive RL method AUV can learn much faster than that a DQN learner from only environmental reward.",
"Moreover, our deep RL learning from both human and environmental rewards can also achieve a similar or even better performance than the deep interactive RL method and adapt to the actual environment by learning from environmental rewards.",
"In the future, we would like to extend and test our methods with actual AUV system in ocean observation task.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTION He was a Researcher with Nanyang Technological University, Singapore, from 2000 to 2002.",
"He is currently a Full Professor with Ocean University of China, Qingdao, China.",
"His research interests include SLAM, machine learning, and robotics.",
"[Figure: NO_CAPTION He was a Visiting Researcher with Delft University of Technology, Delft, The Netherlands, and a Research Intern with the Honda Research Institute Japan, Co., Ltd., Wako, Japan.",
"He is currently a Lecturer with Ocean University of China, Qingdao, China.",
"His research interests include reinforcement learning, human agent/robot interaction, and robotics."
]
] | 2001.03359 |
[
[
"On the value of the Immirzi parameter and the horizon entropy"
],
[
"Abstract In Loop Quantum Gravity (LQG) the quantisation of General Relativity leads to precise predictions for the eigenvalues of geometrical observables like volume and area, up to the value of the only free parameter of the theory, the Barbero-Immirzi (BI) parameter.",
"With the help of the eigenvalues equation for the area operator, LQG successfully derives the Bekenstein-Hawking entropy of large black holes with isolated horizons, fixing at this limit the BI parameter as $\\gamma \\approx 0.274$.",
"In the present paper we show some evidence that a black hole with angular momentum $\\hbar$ and Planck mass is an eigenstate of the area operator provided that $\\gamma = \\sqrt{3}/6 \\approx 1.05 \\times 0.274$.",
"As the black hole is extremal, there is no Hawking radiation and the horizon is isolated.",
"We also suggest that such a black hole can be formed in the head-on scattering of two parallel Standard Model neutrinos in the mass state $m_2$ (assuming $m_1 = 0$).",
"Furthermore, we use the obtained BI parameter to numerically compute the entropy of isolated horizons with areas ranging up to $250\\,l_P^2$, by counting the number of micro-states associated to a given area.",
"The resulting entropy has a leading term ${\\cal S} \\approx 0.25\\, {\\cal A}$, in agreement to the Bekenstein-Hawking entropy.",
"As the identification of the above eigenstate rests on the matching between classical areas and quantum area eigenvalues, we also present, on the basis of an effective quantum model for the Schwarzschild black hole recently proposed by Ashtekar, Olmedo and Singh, an expression for the quantum corrected area of isolated horizons, valid for any black hole mass.",
"Quantum corrections are shown to be negligible for a Planck mass black hole, of order $10^{-3}$ relative to the classical area."
],
[
"Introduction",
"As well known, the quantisation of gravity suffers, among others difficulties, from the non-convergence of its perturbative expansions, related to the absence of an adimensional coupling constant, contrary to what happens in other gauge theories [1].",
"This has led to the development of non-perturbative approaches, among which Loop Quantum Gravity (LQG) is probably the most complete from a theoretical viewpoint [2], [3].",
"Another difficulty is related to the absence of empirical facts that could drive the postulation of quantisation rules.",
"In spite of that, LQG has successfully built consistent conjugate operators and their commutation relations, up to a free parameter that fixes a particular quantum representation, the Barbero-Immirzi (BI) parameter.",
"Once this parameter is determined by any experiment, additional tests can rule out the theory, which in this way is falsifiable.",
"The task is to find at least two independent tests in the realm of a so weak interaction and so small scales.",
"Surprisingly enough, the theory has been confronted to the derivation of the horizon entropy of large black holesFor a discussion on the area quantisation of cosmological horizons, see e.g.",
"[4], [5] and references therein., explaining the linear relation between entropy and horizon area and fixing the BI parameter as $\\gamma \\approx 0.274$ in order to have the expected slope of $1/4$ .",
"This is done by counting the number of spin network configurations that generate a given horizon area.",
"In this count the horizon is assumed isolated, as usually done for thermodynamic systems.",
"The main goal of this paper is to explore the possibility of an independent determination of the BI parameter.",
"For that, we will initially identify a candidate for a physical eigenstate of the LQG area operator, constituted by a black hole of angular momentum $\\hbar $ , formed for instance by two interacting particles of spin $1/2$ .",
"The black hole must be extremal in order to avoid Hawking radiation, i.e.",
"it has Planck mass $m_P$ and a Planck horizon radius $l_P$ .",
"Although it looks like a gedanken black hole, we will argue that it has an actual realisation if formed by two parallel neutrinos in the mass state $m_2$ (assuming the mass state $m_1 = 0$ ).",
"Indeed, Dirac neutrinos carry magnetic moments due to vacuum fluctuations [6], and the dipoles repulsion energy at a distance $2l_P$ forms a horizon of radius $l_P$ and mass $m_P$ with $99.9\\%$ precision [7].",
"Both the found eigenstate and its physical realisation suggest that the classical expression for the horizon area is valid at the Planck scale, something that will be verified with the help of an effective model for spherically symmetric black holes.",
"The BI parameter determined in this way differs $5\\%$ from the approximate value derived from the entropy of large horizons.",
"As a consistency test, we perform an exact counting of micro-states associated to small horizon areas running up to $250\\, l_P^2$ .",
"The entropy ${\\cal S}({\\cal A})$ shows a linear leading term whose slope differs by less than $1\\%$ from the Bekenstein-Hawking recipe."
],
[
"A Planck scale black hole",
"The simplest quantum black hole may be that formed in the scattering of two identical, repulsively interacting particles, at a centre-of-mass energy of the order of the Planck scale [8].",
"If the particles, for instance, have spin $1/2$ and carry parallel magnetic moments, the repulsion between the dipoles can lead to the formation of a Kerr black hole with angular momentum $\\hbar $ .For a recent study of the classical Kerr solution in real Ashtekar variables, see [9].",
"Quantum black holes are discussed e.g.",
"in [10], [11], [12], [13].",
"The formed horizon is isolated in the extremal case, when the surface gravity is zero and there is no Hawking radiation.",
"In this case mass $M$ , angular momentum $J$ and horizon radius $r_H$ are related by $a^2 = r_H^2 = J$ , where $a = J/M$ [14].",
"This leads to $M = m_P$ and $r_H = l_P$ .",
"On the other hand, in the extremal limit the horizon area is reduced to ${\\cal A} = 4\\pi (r_H^2 + a^2) = 8\\pi J.$ For $J = 1$ , it can be written as $ {\\cal A} = 8\\pi \\gamma l_P^2 \\sum _{i=1}^4 \\sqrt{j_i(j_i+1)},$ with $j_i = 1/2$ and $\\gamma = \\sqrt{3}/6$ .",
"This is the eigenvalues equation for the area operator of LQG [15], that fixes in this way the BI parameter.",
"It can be interpreted as a horizon pierced by four spin network lines of colour $1/2$ or, equivalently, crossed by two lines, with two punctures per line [7].",
"Nevertheless, we still should find in Nature an actual physical system with the above features.",
"It is noteworthy that it may indeed be formed in the head-on scattering of parallel neutrinos in a suitable mass state.",
"Dirac neutrinos carry the smallest magnetic moment among the known particles of Standard Model, provided they have mass.",
"Their magnetic dipole originates from vacuum fluctuations and its value involves the weak coupling constant, the fine structure constant and the masses of leptons and gauge bosons.",
"At 1-loop approximation and neglecting lepton masses as compared to gauge boson masses, the neutrino magnetic moment (in natural units) is given by [6] $\\mu _{\\nu } &\\approx & \\frac{3eG_Fm_{\\nu }}{8\\sqrt{2}\\pi ^2},$ where $G_F$ is the Fermi constant, $e$ is the elementary charge, and $m_{\\nu }$ is the neutrino mass.",
"Note that 2-loops diagrams lead to relative corrections of order $\\alpha /\\pi \\sim 10^{-3}$ , where $\\alpha $ is the low-energy fine structure constant.",
"In fact, high order corrections to magnetic moments do not depend on the energy scale, involving only powers of $\\alpha /\\pi $ [16].",
"Any scale dependence is absorbed, by renormalisation, in the term $e G_F m_{\\nu }$ of Eq.",
"(REF ), that has dimension of charge/mass.",
"This means that further vacuum polarisation effects are negligible compared to the classical interaction energy $U = \\mu _{\\nu }^2/r^3$ , where $r$ is the neutrinos relative distance.",
"In what follows, we shall assume that quantum gravity corrections are also negligible above the Planck length, an assumption that will be verified in Section IV.",
"The next step is to take $r = 2 l_P$ and $U = M$ , which leads to$2l_P$ is the gravitational radius of an extremal Kerr black hole of event horizon radius $r_H = l_P$ .",
"$M \\approx \\frac{9e^2G_F^2m_{\\nu }^2}{1024\\pi ^4}.$ If $M = m_P$ , we have the Kerr solution described above, i.e.",
"a physical eigenstate of the area operator.",
"The no-hair conjecture, if valid at the Planck scale, assures that the magnetic dipoles are not observable from outside once the horizon is formed.",
"Although we do not know the absolute values of the neutrinos masses, flavor oscillation measurements give with precision the gaps between the squared masses.",
"If we assume normal ordering and set the smallest mass $m_1 = 0$ , the lightest massive state is $m_2 = (8.66 \\pm 0.10) \\times 10^{-3}$ eV ($1\\sigma $ ) [17], [18].",
"From (REF ) we then have $M \\approx 1.001\\, m_P$ .",
"Reversing the argument, we would have an exact eigenstate if $m_2 \\approx 8.654 \\times 10^{-3}$ eV [7].",
"The difference to the measured value has the same order of 2-loops corrections to Eq.",
"(REF ).",
"In spite of such a precision in the determination of both the BI parameter and neutrinos mass, matching classical horizon areas to the eigenvalues of the area operator is an ad hoc procedure that should be properly justified, as will be done in Section IV.",
"On the other hand, our result for $\\gamma $ must be independently confirmed through the evaluation of the horizon entropy, what we do in the next section."
],
[
"The horizon entropy",
"The entropy of a black hole of horizon area ${\\cal A}$ can be found by counting the number $N$ of spin network configurations that satisfy the eigenvalues equation of the area operator [19], $ {\\cal A} = 8\\pi \\gamma l_P^2 \\sum _{i=1}^n \\sqrt{j_i(j_i+1)},$ where $n$ is the number of points on the horizon pierced by spin network lines of colours $j_i \\in {Z}_+/2$ .",
"Furthermore, the condition of horizon isolation imposes to the punctures a set of second labels $m_i$ that must satisfy the “projection constraint” [20], [21], [22] $\\sum _{i=1}^{n} m_i = 0,$ with $m_i \\in \\lbrace -j_i,-j_i+1,...,j_i-1,j_i\\rbrace .$ In the limit of large horizon areas, condition (REF ) can be analytically solved up to terms that vanish in the limit ${\\cal A} \\rightarrow \\infty $ , leading to [23] ${\\cal S} = \\ln N = \\frac{\\tilde{\\gamma }}{4\\gamma } {\\cal A},$ where $\\tilde{\\gamma } \\approx 0.274$ is the root of $1 = \\sum _{k=1}^{\\infty } (k+1) \\exp \\left( -\\pi \\tilde{\\gamma } \\sqrt{k(k+2)} \\right).$ Eq.",
"(REF ) fits the Bekenstein-Hawking entropy for $\\gamma = \\tilde{\\gamma }$ .",
"It is also possible to show that (in the same limit of large areas) the projection constraint (REF ) does not affect the leading term (REF ).",
"For small black holes the entropy can be exactly evaluated with the help of appropriate generating functions [21] or by a direct computation of all permitted micro-states for a given area [24].",
"Although the former allows the generation of larger areas in shorter times, we have initially followed the latter procedure for simplicity, which has allowed us to obtain the number of states for areas running up to $160\\, l_P^2$ .",
"The entropy was then found as ${\\cal S} = \\ln N$ .",
"The computation consisted of the following algorithm: fix a value for ${\\cal A}_0$ (in units of $l_P^2$ ); calculate the maximum number $n_{\\text{max}}$ of punctures for which condition (REF ) is satisfied, given by the integer part of ${\\cal A}_0/(4\\pi \\gamma \\sqrt{3})$ ; generate all vectors [$j_i$ ] of length $n \\le n_{\\text{max}}$ for which ${\\cal A} \\in ({\\cal A}_0-\\delta {\\cal A}, {\\cal A}_0+\\delta {\\cal A})$ for a chosen semi-interval $\\delta {\\cal A}$ , excluding permutations of equals $j_i$ ; for each allowed vector, find all combinations [($j_i$ ,$m_i$ )] satisfying the projection constraint (REF ); vary ${\\cal A}_0$ from ${\\cal A}_{\\text{min}}$ to ${\\cal A}_{\\text{max}}$ with a chosen step.",
"The prohibition of permutations with equals $j_i$ comes from the indistinguishability of punctures with equal labels ($j_i,m_i$ ).",
"Therefore, a given vector [$j_i$ ] of length $n$ with $n_s$ elements $j_s$ will have multiplicity $\\frac{n!",
"}{\\prod _s (n_s!",
")}.$ If we do not impose the projection constraint, each vector will also have an additional multiplicity $\\prod _{s} (2j_s+1)^{n_s}.$ Figure: 𝒮{\\cal S} ×\\times 𝒜{\\cal A} for δ𝒜=0.5l P 2 \\delta {\\cal A} = 0.5\\, l_P^2, without the projection constraint for γ=0.274\\gamma = 0.274 (orange) and with the projection constraint for γ=3/6\\gamma = \\sqrt{3}/6 (purple).We fixed $\\delta {\\cal A} = 0.5\\, l_P^2$ as in [24] and, to avoid superposition of intervals, varied ${\\cal A}_0$ in steps of $2 \\delta {\\cal A}$ .",
"Without imposing the projection constraint, and taking $\\gamma = 0.274$ , we reproduced the findings of [24], i.e.",
"a linear relation between ${\\cal S}$ and ${\\cal A}$ with slope $0.2504$ , in excellent agreement with the large area approximation.",
"The result is the orange curve shown in Fig.",
"REF .",
"When we impose the projection constraint and still take $\\gamma = 0.274$ , the oscillatory behaviour found in [21], [24] is evidenced.",
"The fitting of a straight line gives a slope $0.254$ , in contrast to the result $0.237$ reported in [24].",
"Actually, the computation with the projection constraint is sensitive to the adopted semi-interval $\\delta {\\cal A}$ and to the step of variation of ${\\cal A}_0$ .",
"With $\\gamma = \\sqrt{3}/6$ and without including the projection constraint, a linear relation is recovered with slope $0.238$ , in agreement to the analytic approximation (REF ).",
"When we include the projection constraint the oscillations reappear and the fitting of a straight line gives a slope $0.243$ .",
"The corresponding curve (in purple) is also shown in Fig.",
"REF .",
"One can see that the oscillations are attenuated for larger areas, as expected in the thermodynamic limit.",
"Note as well that, with or without the projection constraint, the slopes for $\\gamma = 0.274$ are $1.05$ times higher than for $\\gamma = \\sqrt{3}/6$ , in accordance to the large areas expression (REF ) and to the ratio between these values of the BI parameter.",
"Figure: Upper panel: 𝒮{\\cal S} ×\\times 𝒜{\\cal A} slope as a function of δ𝒜\\delta {\\cal A} for γ=3/6\\gamma = \\sqrt{3}/6 and γ=0.274\\gamma = 0.274, with and without the projection constraint, for 𝒜 max =150l P 2 {\\cal A}_{\\text{max}} = 150\\, l_P^2.",
"Lower panel: The same for 𝒜 max =250l P 2 {\\cal A}_{\\text{max}} = 250\\, l_P^2.We refined the analysis for $\\gamma = \\sqrt{3}/6$ varying the semi-interval $\\delta {\\cal A}$ in steps of $10^{-4}\\, l_P^2$ , for ${\\cal A}_0$ running from $50\\, l_P^2$ to $150\\, l_P^2$ , and we have found the correlations shown in the upper panel of Fig.",
"REF between the ${\\cal S} \\times {\\cal A}$ slope and the adopted area bin.",
"Without the projection constraint (red line), the slope grows for smaller bins, approaching a maximum around $\\delta {\\cal A} = 0.02\\, l_P^2$ , for which the angular coefficient is $0.246$ .",
"When the projection constraint is imposed (blue line), the slope approaches a maximum around $\\delta {\\cal A} = 0.25\\, l_P^2$ .",
"At this point the angular coefficient is $0.249$ , less than $1\\%$ below the Bekenstein-Hawking value.",
"The correspondent results for $\\gamma = 0.274$ are also shown in Fig.",
"REF .",
"As expected, the curves are shifted by a factor of $\\approx 1.05$ , with a maximum slope of $\\approx 0.263$ .",
"The correlations found may possibly be better understood with the help of the counting presented in [21], based on number theory and combinatory methods.",
"Nevertheless, let us comment that for large area bins the fine grain structure of micro-states distribution is lost even for the smallest areas, which could explain the low values obtained for the slope with $\\delta {\\cal A} \\gtrsim 0.5\\, l_P^2$ , that coincide with the large areas approximation (REF ) when the projection constraint is not considered.",
"On the other hand, for small area bins the entropy oscillations become pronounced even for the largest areas, allowing in this way the fitting of straight lines with lower inclinations.",
"The competition between these two effects, that are stronger when the projection constraint is imposed, may explain the maximum around $\\delta {\\cal A} \\approx 0.25\\, l_P^2$ and the oscillatory pattern observed.",
"If this interpretation is correct, we would expect a drift of the maximum slope to smaller area bins when larger maximum areas are used.",
"In order to verify it, we have extended our computations up to ${\\cal A}_{\\text{max}} = 250\\, l_P^2$ .",
"In this step the computation time was significantly shortened by using a formula for the $m$ -degeneracy given by [21] $\\frac{1}{L} \\left[ \\prod _{i=1}^{n} (2j_i + 1) + \\sum _{l=1}^{L-1} \\prod _{i=1}^{n} \\frac{\\sin [2\\pi (2j_i+1)l/L]}{\\sin (2\\pi l/L)} \\right],$ where $L = 1 + 2 \\sum j_i$ if $\\sum j_i$ is integer, and $L = 2 + 2\\sum j_i$ if $\\sum j_i$ is half-integer.",
"Results are shown in the lower panel of Fig.",
"REF .",
"Now the maximum slope is approached at $\\delta {\\cal A} \\approx 0.002\\, l_P^2$ , given again by $\\approx 0.25$ for $\\gamma = \\sqrt{3}/6$ when the projection constraint is imposed.",
"For $\\gamma = 0.274$ the maximum slope is again $\\approx 0.263$ , which suggests that the large area estimation of $\\gamma $ presents a $5\\%$ imprecision.",
"For completeness, we have also computed the entropy as defined in [21], namely ${\\cal S}_{\\le } ({\\cal A}) = \\ln \\, [1 + N_{\\le } ({\\cal A})],$ where $N_{\\le } ({\\cal A})$ is the number of vectors $[j_i]$ that generate areas in the interval $(0, {\\cal A}]$ .",
"This definition is appropriate in the thermodynamic limit of large areas, and we have reproduced the slopes shown in Fig.",
"REF for the largest values of $\\delta {\\cal A}$ .",
"The resulting ${\\cal S}_{\\le } \\times {\\cal A}$ plots present the characteristic stair profile found in [21], as shown in Fig.",
"REF .",
"Figure: 𝒮 ≤ {\\cal S}_{\\le } ×\\times 𝒜{\\cal A}, with and without the projection constraint, for γ=0.274\\gamma = 0.274 and γ=3/6\\gamma = \\sqrt{3}/6.",
"The slopes found reproduce those shown in Fig.",
"for the largest values of δ𝒜\\delta {\\cal A}."
],
[
"The horizon area",
"Our goal in this section is to understand how can the classical horizon (REF ) correspond to the area operator eigenvalue (REF ), contrary to our intuitive belief that quantum corrections to horizon areas should not be negligible at Planck scales.",
"Using an effective, LQG inspired solution for a Schwarzschild quantum black hole [12], [25], we shall obtain an analytic expression for the quantum-corrected horizon area, and we will show that, for a black hole with Planck mass, the relative correction to the classical area is $\\sim 10^{-3}$ .",
"Of course, in this case the Hawking radiation backreaction cannot be neglected, as the Hawking temperature is $\\approx 4\\%$ of the black hole mass.",
"Nevertheless, it has no role at all if the horizon is assumed isolated, and we would not be extrapolating too much in assuming that quantum corrections are also negligible for a Planck mass Kerr black hole."
],
[
"Non-isolated horizons",
"The interior of the spherically symmetric black hole of Ashtekar, Olmedo and Singh (AOS) is isometric to a vacuum Kantowski-Sachs spacetime with metric given, in Kruskal coordinates, bySee [26] for a critical discussion on the asymptotic structure of the AOS solution.",
"$ ds^2 = - N_{\\tau }^2 d\\tau ^2 + \\frac{p_b^2}{p_c L_o^2} dx^2 + p_c (d\\theta ^2 + \\sin ^2\\theta d\\phi ^2).$ Here, $p_b$ and $p_c$ are canonical momenta conjugate to the respective configuration variables $b$ and $c$ of the dynamical space, and $L_o$ is a (non-observable) infrared cut-off introduced to avoid divergences in the Hamiltonian constraint.",
"The lapse $N_{\\tau }$ and the black hole mass are $ N_{\\tau } = \\frac{\\gamma \\sqrt{p_c} \\delta _b}{\\sin (\\delta _b b)},$ $ m = \\frac{\\sin (\\delta _c c) p_c}{\\gamma L_o \\delta _c},$ where the constants of motion $\\delta _b$ and $\\delta _c$ encode quantum geometry corrections to the classical metric.",
"The solution of the effective dynamical equations is given by $ p_c = 4m^2 \\left( e^{2T} + \\frac{\\gamma ^2 L_0^2 \\delta _c^2}{64m^2} e^{-2T} \\right),$ $ p_b = - \\frac{2\\sin (\\delta _c c)}{\\delta _c} \\frac{\\sin (\\delta _b b)}{\\delta _b} \\frac{p_c}{\\gamma ^2 + \\frac{\\sin ^2(\\delta _b b)}{\\delta _b^2}},$ $ \\cos (\\delta _b b) = b_o \\tanh \\left[ \\frac{b_o T}{2} + \\tanh ^{-1} \\left( \\frac{1}{b_o} \\right) \\right],$ with $b_o \\equiv \\sqrt{1 + \\gamma ^2 \\delta _b^2},$ plus an additional equation for $c$ consistent with (REF ).",
"The time variable $T$ is defined by $\\tau = 2m e^{T}$ , with $T = 0$ corresponding to the event horizon.",
"The conjugate momentum $p_c$ presents a minimum at a transition surface ${\\cal T}$ from which the trapped black hole interior tunnels to an anti-trapped white hole solution, resolving in this way the classical singularity.",
"This minimum is given by $ p_c|_{\\cal {T}} = m \\gamma L_0 \\delta _c,$ and it occurs at $ T_{\\cal T} = \\frac{1}{2} \\ln \\left( \\frac{\\gamma L_o \\delta _c}{8m} \\right).$ The strategy of the AOS effective model consists of matching the minimal areas enclosed by holonomies on the transition surface ${\\cal T}$ to the LQG area gap $4\\pi \\sqrt{3} \\gamma l_P^2$ .",
"Assuming that the links of these minimal plaquettes are proportional to $\\delta _b$ on the $\\theta $ -$\\phi $ 2-spheres and to $\\delta _c$ along the $x$ direction, we obtain [12] $ \\alpha ^2 \\delta _b^2 p_c|_{\\cal {T}} = \\sqrt{3} \\gamma ,$ $ \\alpha ^2 \\delta _c \\delta _b |p_b||_{\\cal {T}} = 2\\sqrt{3} \\gamma ,$ where $\\alpha > 0$ is a factor of proportionality with the order of unityIn Ref.",
"[12] $\\alpha $ is assumed 1.",
"As we will see, this assumption is too restrictive for Planck size horizons.. From (REF ) and (REF ) we have $ L_0 \\delta _c = \\frac{\\sqrt{3}}{m\\alpha ^2 \\delta _b^2}.$ Substituting this result into (REF ), it follows thatWe choose $p_b < 0$ in accordance to the classical solution of the dynamical equations (see footnote 6 of [12]).",
"$ p_b|_{\\cal {T}} = -2 m \\gamma L_0 \\delta _b.$ Using (REF ) and (REF ) in (REF ) we obtain $ \\delta _b^2 = \\frac{\\sin (\\delta _c c) \\sin (\\delta _b b)}{\\gamma ^2 + \\frac{\\sin ^2(\\delta _b b)}{\\delta _b^2}} \\quad \\quad (\\text{on}\\; \\cal {T}).$ On the other hand, from (REF ) and (REF ) we see that $ \\sin (\\delta _c c)|_{\\cal T} = 1,$ and, therefore, $ \\delta _b = \\frac{\\frac{\\sin (\\delta _b b)}{\\delta _b}}{\\gamma ^2 + \\frac{\\sin ^2(\\delta _b b)}{\\delta _b^2}} \\quad \\quad (\\text{on}\\; \\cal {T}).$ Let us now define $ x \\equiv \\frac{\\sin (\\delta _b b)}{\\delta _b}.$ Eq.",
"(REF ) is rewritten as $ \\delta _b x^2 - x + \\delta _b \\gamma ^2 = 0,$ with roots $ x = \\frac{1 \\pm \\sqrt{1 - 4 \\delta _b^2 \\gamma ^2}}{2\\delta _b},$ which leads to $ \\sin (\\delta _b b) = \\frac{1 \\pm \\sqrt{1 - 4 \\delta _b^2 \\gamma ^2}}{2} \\quad \\quad (\\text{on}\\; \\cal {T}).$ Now, if we take Eq.",
"(REF ) at $T = 0$ and use (REF ), we find the horizon area $ {\\cal A} = 4\\pi p_c = 16 \\pi m^2 \\left( 1 + \\frac{3 \\gamma ^2}{64 m^4 \\alpha ^4 \\delta _b^4} \\right).$ In the large mass limit we recover the classical Schwarzschild result.",
"For small $m$ , the correction to the horizon area depends on the value of $\\delta _b$ along the correspondent dynamical trajectory, which is determined by simultaneously solving Eqs.",
"(REF ) and (REF ), using, in the former, Eqs.",
"(REF ) for $T$ and (REF ) for $L_o \\delta _c$ .",
"If we take $m\\alpha = 1$ and $\\gamma = \\sqrt{3}/6$ , the largest root found in a numerical integration is $\\delta _b \\approx 1.44$ , leading to a relative correction to the horizon area of $\\approx 9 \\times 10^{-4}$ .",
"The result of the numerical integration is shown in the upper panel of Fig.",
"REF .",
"The four branches refer to the possible combinations of the two branches of $\\tanh ^{-1}$ in (REF ) and the two signs in (REF ).",
"The solutions for $\\delta _b$ (horizontal axis) correspond to the intersections of the line $y = 0$ .",
"Figure: Numerical solutions for δ b \\delta _b in the non-isolated horizon (upper panel) and isolated horizon (lower panel) cases, for mα=1m\\alpha = 1.",
"The solutions (in the horizontal axes) correspond to the intersections of the lines y=0y = 0.Figure: Numerical solutions for δ b \\delta _b in the isolated horizon case for black hole masses in the interval mα∈[0.3,1]m \\alpha \\in [0.3,1].",
"The solutions (in the horizontal axes) correspond to the intersections of the lines y=0y = 0.",
"The panels correspond to the two branches with real solutions.Figure: Time of transition T 𝒯 T_{\\cal T} and the relative correction to the horizon area as functions of mm for the intermediate roots.",
"The panels correspond to the two branches with real solutions.Figure: Time of transition T 𝒯 T_{\\cal T} and the relative correction to the horizon area as functions of mm for the physical root.Figure: Quantum parameters as functions of the black hole mass for the physical root."
],
[
"Isolated horizons",
"For Planck scale masses, evaporation effects could not, in principle, be neglected in the above computation.",
"If we do not take into account for a moment the radiation backreaction on the metric, the quantum corrected Hawking temperature in the AOS model is given by [25] $T_H = \\frac{1}{8\\pi m} \\frac{1}{(1+\\epsilon _m)},$ where $\\epsilon _m = \\frac{\\gamma ^2 L_o^2 \\delta _c^2}{64m^2}.$ With $\\gamma = \\sqrt{3}/6$ , $m\\alpha = 1$ , $L_o \\delta _c$ given by (REF ) and the root for $\\delta _b$ found above, we have $\\epsilon _m \\sim 10^{-3}$ , and the Hawking temperature is still given by its large mass expression $T_H \\approx 1/(8\\pi m)$ .",
"Note that, due to (REF ), the temperature correction $\\epsilon _m$ is equal to the relative correction to the classical area in (REF ).",
"The smaller the latter, the smaller the former.",
"Anyway, for a Planck mass black hole the temperature is about $4\\%$ of the black hole mass, and the backreaction on the metric should also be estimated.",
"This difficulty can be circumvented if we consider an isolated horizon, as usually done in LQG.",
"In this case, homogeneous slices of the Schwarzschild interior are not space-like as in Kantowski-Sachs spacetime, but time-like.",
"This can be achieved through the replacements $b \\rightarrow i \\tilde{b}$ and $p_b \\rightarrow i \\tilde{p}_b$ [12], by which the lapse becomes $ \\tilde{N}_{\\tau } = \\frac{\\gamma \\sqrt{p_c} \\delta _b}{\\sinh (\\delta _b \\tilde{b})},$ and the interior metric assumes the form $ ds^2 = \\tilde{N}_{\\tau }^2 d\\tau ^2 - \\frac{\\tilde{p}_b^2}{p_c L_o^2} dx^2 + p_c (d\\theta ^2 + \\sin ^2\\theta d\\phi ^2).$ Equations (REF ), (REF ) and (REF ) (and hence (REF )) remain unaltered, while (REF ) is rewritten as $ \\tilde{p}_b = - \\frac{2\\sin (\\delta _c c)}{\\delta _c} \\frac{\\sinh (\\delta _b \\tilde{b})}{\\delta _b} \\frac{p_c}{\\gamma ^2 - \\frac{\\sinh ^2(\\delta _b \\tilde{b})}{\\delta _b^2}}.$ Assuming that the constraints (REF )-(REF ) are still valid, we re-obtain (REF ), Eq.",
"(REF ) changes signNow $\\tilde{p}_b >0$ as in the corresponding classical solution [12]., and (REF ) is replaced by $ \\delta _b x^2 - x - \\delta _b \\gamma ^2 = 0,$ where now $ x \\equiv \\frac{\\sinh (\\delta _b \\tilde{b})}{\\delta _b}.$ Its roots lead to $ \\sinh (\\delta _b \\tilde{b}) = \\frac{1 \\pm \\sqrt{1 + 4 \\delta _b^2 \\gamma ^2}}{2} \\quad \\quad (\\text{on}\\; \\cal {T}).$ Eq.",
"(REF ), by its turn, is replaced by $ \\cosh (\\delta _b \\tilde{b}) = b_o \\tanh \\left[ \\frac{b_o T}{2} + \\tanh ^{-1} \\left( \\frac{1}{b_o} \\right) \\right],$ with $T$ still given by (REF ).",
"For $m\\alpha = 1$ and $\\gamma = \\sqrt{3}/6$ , the largest root of the numerical solution of (REF )-(REF ) is $\\delta _b \\approx 3.75$ .",
"From (REF ) we now have a relative correction to the horizon area of $\\approx 2 \\times 10^{-5}$ .",
"The numerical solution is presented in the lower panel of Fig.",
"REF .",
"There are two further non zero roots (shown in the zoom), very close to $\\delta _b = 0.25$ ($0.249$ and $0.251$ ).",
"For this value of $\\delta _b$ , the relative correction in (REF ) is equal to 1, and from (REF ) and (REF ) we have $T_{\\cal T} = 0$ , that is, the transition surface coincides with the horizon.",
"For these roots (also present in the non-isolated case), quantum fluctuations are so large that the horizon is not formed.",
"We performed the same analysis of isolated horizons varying the black hole mass in the interval $m\\alpha \\in [0.3,1]$ , finding the roots shown in Fig.",
"REF .",
"The panels correspond to the two branches with real solutions.",
"Apart $\\delta _b = 0$ , there are again two intermediate roots (upper and lower panels) with $m\\alpha \\delta _b \\approx 0.25$ , for which $T|_{\\cal T} \\approx 0$ , and a largest root (upper panel).",
"For the intermediate roots, we show in Fig.",
"REF the dependence of the relative correction to the horizon area and of the time of transition $T_{\\cal T}$ on the black hole mass.",
"For one of the branches, the intermediate roots lead to $T_{\\cal T} > 0$ and the horizon is not formed.",
"For the other branch, the relative correction to the area increases with the mass, which is not an expected physical behaviour.",
"Therefore, in this range of masses the only physical root is the largest one.",
"In Fig.",
"REF the relative correction to the horizon area for the physical roots is displayed as a function of mass, showing a decrease with $m$ , while $T_{\\cal T}$ remains negative, as expected.",
"In Fig.",
"REF we show that in this case both $\\delta _b$ and $L_o \\delta _c$ also decrease with the black hole mass.",
"The qualitative behaviours shown in figures REF -REF are the same in the non-isolated horizon case (with different roots).",
"Note that, in our range of masses, $\\delta _b > 1$ for the physical roots, which means that its interpretation as a physical fractional length on $\\theta $ -$\\phi $ 2-spheres (i.e.",
"$\\alpha = 1$ ) [12] is not appropriate in this case.",
"Actually, as $\\delta _b$ decreases with the black hole mass, the value of $\\alpha $ determines a minimal admissible mass, corresponding to $\\alpha \\delta _b = 1$ (when $4\\pi p_c|_{\\cal T}$ equals the area gap, see (REF )).",
"The relative correction to the horizon area for this minimal mass is given, from (REF ), by $\\frac{\\Delta {\\cal A}}{{\\cal A}} = \\frac{1}{256\\, m^4}.$ If we assume, for example, that the minimal mass is the Planck mass (the mass of an extremal Kerr black hole with angular momentum $\\hbar $ ), the relative correction to the horizon area is $\\approx 4 \\times 10^{-3}$ .",
"In this case the value of $\\alpha $ corresponds to the intersection between the hyperbole $\\delta _b = 1/(m\\alpha )$ and the curve $\\delta _b \\times (m \\alpha )$ shown in Fig.",
"REF , and it is $\\alpha \\approx 0.13$ , with $\\delta _b \\approx 7.5$ .",
"On the other hand, for the smallest Schwarzschild isolated horizon of LQG we have $m \\approx 0.5,$For two punctures with $j = 1/2$ we have $m \\approx 0.5$ , whereas a single puncture with $j = 1$ gives $m \\approx 0.45$ .",
"and the relative correction to the area is $\\approx 0.06$ .",
"As we can see, the Planck mass is a frontier between classical horizons, for which quantum corrections to the area are negligible, and quantum horizons for that they grow up to $6\\%$ .",
"For extremal Kerr black holes the Hawking radiation is totally null as the surface gravity is zero.",
"The results obtained here for a spherically symmetric horizon do not apply to that case, of course.",
"Nevertheless, the general conclusion is that quantum corrections to the area of Planck size horizons are not necessarily large, and may also be negligible for rotating black holes, as suggested by the precise correspondence shown above between Eqs.",
"(REF ) and (REF ).",
"Although the AOS model is hardly generalisable to rotating black holes, its extension to extremal charged horizons is straightforward, as we show below.",
"We will see that relative corrections to the horizon area are small (though not negligible) also in this case."
],
[
"Extremal charged horizons",
"The extremal Reisnner-Nordström metric (for which $Q = m$ ) can be written as $ds^2 = -\\left( 1 - \\frac{m}{r} \\right)^2 dt^2 + \\left( 1 - \\frac{m}{r} \\right)^{-2} dr^2 + r^2 d\\Omega ,$ where $d\\Omega = d\\theta ^2 + \\sin ^2\\theta \\, d\\phi ^2$ .",
"The black hole interior can be described through the variable changes $r \\rightarrow \\tau $ , $t \\rightarrow x$ [12], under which the metric is rewritten as $ds^2 = \\left( \\frac{m}{\\tau } - 1 \\right)^{-2} d\\tau ^2 - \\left( \\frac{m}{\\tau } - 1 \\right)^2 dx^2 + \\tau ^2 d\\Omega .$ If we now use the dictionary $\\tilde{N}_{\\tau }^2 &=& \\left( \\frac{m}{\\tau } - 1 \\right)^{-2},\\\\\\tilde{p}_b^2 &=& L_o^2 \\left( \\frac{m}{\\tau } - 1 \\right)^2 \\tau ^2,\\\\p_c &=& \\tau ^2,$ the metric assumes the form (REF ) of an isolated horizon, with the difference that, now, the horizon corresponds to $\\tau = m$ .",
"This result assures that the quantum corrected horizon does not radiate either, which is essential for the identification of an eigenstate of the area operator.",
"Whether this is also valid for an extremal rotating black hole is something deserving further investigation.",
"For the extremal charged horizon, all the results of last section apply, with the change $m \\rightarrow m/2$ .",
"For the horizon area we have $ {\\cal A} = 4 \\pi m^2 \\left( 1 + \\frac{3 \\gamma ^2}{4 m^4 \\alpha ^4 \\delta _b^4} \\right).$ The minimal admissible mass corresponds again to $\\delta _b \\alpha = 1$ .",
"For $\\gamma = \\sqrt{3}/6$ , an isolated horizon pierced twice by a line with $j = 1/2$ has $m \\approx 1$ in this caseFor a single puncture with $j = 1$ we have $m\\approx 0.9$ ..",
"Therefore, from (REF ) we have $\\delta {\\cal A}/{\\cal A} \\approx 0.06$ .",
"We note that for both the Schwarzschild isolated horizon and the charged extremal horizon, for the minimal masses the relative correction to the area is $\\approx 6\\%$ , whereas for the extremal rotating black hole it is an order of magnitude smaller.",
"This is in fact expected, as in the former cases the minimal area corresponds to two area gaps, while in the later it corresponds to four punctures (see (REF )).",
"For a given irreducible representation, the larger the number of lines piercing the horizon, the smaller the relative correction to the classical area.",
"The reader may ask whether a model with spherical symmetry can be used at so small scales.",
"Actually, effective models with homogeneous metrics have also been used at the Planck scale in Loop Quantum Cosmology for resolving the big-bang singularity [27], [28].",
"Furthermore, even when dealing with large black holes, the AOS model makes use of the Kantowski-Sachs metric at the transition surface where quantum effects are not negligible.",
"The use of these approximate models at such scales should be rather verified by evaluating the corrections they predict for classical quantities like the horizon area.",
"Deep inside the horizon quantum effects are surely large, so large that the central singularity disappears.",
"Nevertheless, the horizon remains almost unaffected, what permits the use of the classical theory in the evaluation of its mass and spin, as above."
],
[
"Macroscopic horizons",
"An interesting difference of the above analysis with respect to the original AOS model is that the latter treats with macroscopic black holes, although making use of large quantum corrections at the transition surface to avoid the singularity.",
"In Ref.",
"[12], the authors find four roots for $\\delta _b$ as above, namely zero, two intermediate (and approximately equal) roots and a larger one, with slightly different numerical values due to the value adopted for $\\gamma $ and the non-isolated character of the horizons.",
"However, they dismiss the larger root since it leads to unphysical results in this range of masses, e.g.",
"large quantum corrections in low curvature regions.",
"On the other hand, the intermediate roots present a physical behaviour, contrary to what we have found here.",
"In order to understand this apparent contradiction, we plot in Fig.",
"REF the evolution of $\\Delta {\\cal A}/{\\cal A}$ and $T_{\\cal T}$ with the black hole mass for one of the intermediate roots of the non-isolated horizon case, when the mass ranges up to macroscopic values (for the other roots the scalings are the same as shown in Fig.",
"REF and in the upper panel of Fig.",
"REF ).",
"For masses below the Planck scale, we reproduced the unphysical scalings shown in the lower panel of Fig.",
"REF , with quantum corrections increasing with the mass.",
"For large masses we have, however, a physical behaviour, with the corrections decreasing fast with $m$ .",
"The figure also evidences a discontinuity at the Planck scale.",
"We see, therefore, that macroscopic and microscopic black holes seem to belong to disconnected domains, with the former being described by one of the intermediate roots while the latter are described by the larger one.",
"That discontinuity suggests that, when the mass of a macroscopic black hole decreases (owing, for example, to an adiabatic evaporation process), the horizon is disrupted by large quantum fluctuations when its mass approaches the Planck scale.",
"If this is true, microscopic horizons can only be formed in collision processes like that hypothetically considered in Section .",
"The real conditions for such collisions to form actual horizons are not well understood (in particular for the case of our two neutrinos black hole).",
"The reader can find a detailed discussion on this subject in Ref.",
"[8].See also [29].",
"A study on collisions of relativistic black holes can be found in [30]."
],
[
"Concluding remarks",
"We have obtained three curious numerical coincidences in this paper.",
"Using classical GR, we have shown that a black hole formed by parallel neutrinos in the lowest-mass massive state $m_2$ (assuming $m_1 = 0$ ) is an extremal Kerr black hole with Planck mass, within $99.9\\%$ precision.",
"Its classical horizon is eigenstate of the LQG area operator, provided that the Barbero-Immirzi parameter is $5\\%$ above the large area approximate value.",
"And, with the new BI parameter, we have precisely obtained (less than $1\\%$ error) the Bekenstein-Hawking slope for the leading term of the entropy vs area relation for Planck size horizons.",
"In the absence of a full LQG solution for the Kerr black hole or a compelling effective quantum model that includes matter (but see e.g.",
"[31], [32]), in the derivations above we assumed that Planck size black holes have classical horizons, that is, quantum corrections are negligible at the Planck scale.",
"This assumption seems corroborated by our entropy analysis and was verified within an effective model for spherically symmetric quantum black holes.",
"The eigenstate suggested here has implications that transcend a possible signature of quantum gravity and the determination of the BI parameter.",
"Some assumptions were made and finding $m_2$ with that precision may be a consistency test for those assumptions.",
"In the gravity sector we have assumed, as already pointed out, that quantum gravity corrections to the black holes horizon area and to neutrinos magnetic moments are negligible above the Planck length.",
"We have also taken for granted that the scattered magnetic dipoles are no longer observed from outside after the horizon formation, leading to a Kerr solution characterised only by $J$ and $M$ , as postulated by the classical no-hair conjecture.",
"Furthermore, the value found for the BI parameter agrees with the Ghosh-Mitra count of states [20], in opposition to the Dogamala-Lewandowski original count [33].",
"These two counts lead to the same ${\\cal S} \\times {\\cal A}$ relation in the large area limit, but with different values for $\\tilde{\\gamma }$ in Eq.",
"(REF ) [23].",
"Nevertheless, the most intriguing implication is perhaps the corroboration of some assumptions made in the neutrinos sector.",
"Expression (REF ) for the neutrinos magnetic moment is only valid if they are Dirac neutrinos, because Majorana neutrinos do not carry magnetic moments.",
"In its derivation it is also assumed the minimal extension of the Standard Model needed to accommodate massive neutrinos, with the addition of right-handed singlets [6].",
"Finally, the assumption of normal ordering of the mass states, with $m_1 = 0$ , was necessary.",
"We see that looking for empirical validation of quantum gravity may shed light on other, apparently uncorrelated open questions.",
"Nevertheless, the results presented here may be considered provisory in view of some difficult issues still open.",
"First of all, we must verify that the extremal rotating black hole is solution of the Hamiltonian constraint of LQG.",
"If it does not radiate it represents a stationary state, but this is hard to verify even in the realm of effective models, designed to describe spherically symmetric holes.",
"Its identification as an area operator eigenstate rests exclusively on the correspondence between the classical horizon area and the area operator eigenvalues, correspondence explored with the help of the AOS formalism.",
"Although inspired in LQG, our results may rather be considered effective approximations, still requiring verification on a more fundamental ground.",
"On the other hand, the small discrepancy between our value of the BI parameter and the Gosh-Mitra value evidences the approximate derivation of the latter, that uses the entropy of large horizons.",
"The numerical results of Section III, that represents the kern of this paper, are robust and do not depend on assumptions made in other sections.",
"We have followed the recipe given in Refs.",
"[21], [24], among others, to obtain the entropy $\\times $ area relation from the LQG area operator eigenvalues.",
"The results of Figure 2 suggest that the approximate value of the BI parameter found by Meissner [23] on the basis of the Gosh-Mitra count [20] using large horizons is indeed $5\\%$ below the actual value."
],
[
"Acknowledgements",
"We are thankful to G. A. Mena Marugán and J. Olmedo for valuable comments and suggestions, and to J. C. Fabris, P. C. de Holanda, O. L. G. Peres, A. Saa and J. Zanelli for useful discussions.",
"S.C. is partially supported by CNPq (Brazil) with grant No.",
"307467/2017-1."
]
] | 2001.03440 |
[
[
"Axion detection through resonant photon-photon collisions"
],
[
"Abstract We investigate the prospect of an alternative laboratory-based search for the coupling of axions and axion-like particles to photons.",
"Here, the collision of two laser beams resonantly produces axions, and a signal photon is detected after magnetic reconversion, as in light-shining-through-walls (LSW) experiments.",
"Conventional searches, such as LSW or anomalous birefrigence measurements, are most sensitive to axion masses for which substantial coherence can be achieved; this is usually well below optical energies.",
"We find that using currently available high-power laser facilities, the bounds that can be achieved by our approach outperform traditional LSW at axion masses between $0.5-6$ eV, set by the optical laser frequencies and collision angle.",
"These bounds can be further improved through coherent scattering off laser substructures, probing axion-photon couplings down to $g_{a\\gamma\\gamma}\\sim 10^{-8} {\\text{GeV}^{-1}}$, comparable with existing CAST bounds.",
"Assuming a day long measurement per angular step, the QCD axion band can be reached."
],
[
"Introduction",
"The Standard Model of particle physics allows for charge-parity (CP) violation in the electroweak and strong sectors.",
"While the former is experimentally established, the latter is constrained by neutron electromagnetic dipole measurements to be negligibly small, with no natural explanation.",
"This is the strong CP problem.",
"One of the most well-motivated solutions to this problem is the axion proposal by Peccei and Quinn [1], [2].",
"It has been subsequently noted that such models have a pseudo-Goldstone boson [3], [4], the QCD axion, which can make up a substantial fraction or the entirety of the dark matter abundance [5], [6], [7], and perhaps also be responsible for the initial magnetisation of the Universe [8].",
"Similar pseudoscalars, known as axion-like particles (ALPs), readily arise in the low-energy spectrum of string theory [9], [10].",
"A number of experiments have already placed bounds [11] on the available axion parameter space, with varying degrees of model-dependence; we use the term axion to refer to both the QCD axion and CP-conserving ALPs.",
"An axion couples to Standard Model photons via $\\mathcal {L}_p=-\\frac{1}{4}F^{\\mu \\nu }F_{\\mu \\nu }+\\frac{1}{2}\\left(\\partial _\\mu a\\right)\\left(\\partial ^\\mu a\\right)-\\frac{1}{2}m_a^2 a^2 +\\frac{1}{4}g_{a\\gamma \\gamma }a \\tilde{F}^{\\mu \\nu }F_{\\mu \\nu }$ with $a$ the axion field, $m_a$ the axion mass, $F$ the electromagnetic field-strength tensor and $\\tilde{F}^{\\mu \\nu }=\\frac{1}{2} \\varepsilon ^{\\mu \\nu \\sigma \\rho }F_{\\sigma \\rho }$ its dual.",
"We will be interested in constraining the axion-photon coupling $g_{a\\gamma \\gamma }$ .",
"When interested in a CP-conserving scalar field $s$ , the interaction term is instead $\\mathcal {L}_s\\supset \\frac{1}{4}g_{s\\gamma \\gamma }s F^{\\mu \\nu }F_{\\mu \\nu }.$ The axion-photon coupling term gives rise to a cubic interaction of an axion with two photons, and can be expressed as $g_{a\\gamma \\gamma }a\\mathbf {E}\\cdot \\mathbf {B}$ for pseudoscalars or $g_{a\\gamma \\gamma }a(E^2-B^2)$ for scalars.",
"In the presence of an external magnetic field this leads to possibly observable axion-photon mass mixing [12], [13].",
"We consider the mass range around ${1}{eV}$ , where the strongest bounds, excluding purely astrophysical arguments derived from stellar cooling times, are placed by the CAST experiment [14], a helioscope sensitive to the axion flux produced by the Primakoff effect [15] in the Sun.",
"Above 1eV, Primakoff coherent solar axion-to-photon conversion in crystals place stronger constraints [16].",
"Due to the astrophysical origin of the axions and the consequential lack of control over the production, the possibility of model dependence must be seriously taken into account.",
"The non-zero plasma frequency and high temperature conditions in the Sun affect the axion-photon effective coupling [17].",
"Similar arguments apply to stellar cooling bounds.",
"As such, we focus on complementary, model-independent bounds given by purely terrestrial experiments.",
"Current searches like the OSQAR [18] experiment implement a Sikivie-style LSW scheme [12] to detect the mass mixing.",
"A laser beam propagating in a magnetic field can spontaneously create axions, which, being weakly coupled, pass through an intervening wall.",
"Behind the wall, any axions entering the reconversion magnetic field may mix back into photons and be detected.",
"This mass mixing is largest if the magnetic field modes match the momentum exchanged between the axion and the photon.",
"The large momentum exchange needed for conversion of non-relativistic axions suppresses the rate as the modes are set by the inverse length of the external magnetic field region.",
"Other laboratory axion searches include the PVLAS collaboration [19], looking at induced birefringence as a polarised laser beam traverses a magnetic field.",
"This effect receives contributions from both QED [20] and axion [13] interactions.",
"The leading QED contribution is a one-loop process, which competes against the tree-level axion contribution.",
"The exclusion bounds of PVLAS are thus limited by the requirement that the latter process is the dominant one.",
"Once axion interactions are smaller than the leading order Standard Model prediction, one must be careful of, for instance, hadronic contributions.",
"Given the difficulty this presents in other precision measurements [21], disentangling each independent contribution becomes challenging.",
"PVLAS experiments are close to this limit, suggesting new techniques are required to probe axion couplings below those bounds.",
"Our proposal aims to improve current laboratory searches by adopting an alternative axion-production mechanism.",
"Here, we propose to generate axions via the collision of two laser beams.",
"A similar idea involving photon-electron scattering has been explored in [22], and light-by-light scattering at the LHC has placed bounds on heavy ALPs [23].",
"Lighter axions are better suited to experiments involving a large number of photons, rather than collider searches with large background signals of soft particles.",
"Lasers are ideal for this pursuit, and we expect that the non-linear dependence on the number of photons will allow a better scaling compared to the linear dependence in traditional LSW experiments, as we will discuss further below.",
"Further, resonant production at optical masses allows smaller couplings to be probed than in traditional LSW experiments.",
"Using pulsed lasers allows for the rejection of most background by exploiting coincident timing.",
"This minimal approach can achieve sensitivities comparable to PVLAS.",
"With the implementation of spatial modes on the laser beam, coherent generation of axions can significantly improve our bounds on $g_{a\\gamma \\gamma }$ down to ${e-8}{GeV^{-1}}$ .",
"These are comparable to bounds placed by CAST in the relevant mass region, but without any additional assumptions on the solar plasma.",
"Figure: A diagram of the experimental set-up.",
"Photons from two lasers collide at in an interaction region, producing any hypothetical axions (either scalar or pseudoscalar).",
"These pass through an intervening wall, and then reconvert into photons in the presence of a magnetic field.",
"These photons are the signal detected."
],
[
"Axion production and reconversion",
"We consider two photons, each with energy $\\omega _j$ ($j=1$ , 2), colliding at an angle $\\theta _{12}$ , as depicted in Fig.",
"REF .",
"The two incoming momenta define the scattering plane.",
"The polarisation vectors can be expanded in terms of two basis vectors: one lying in the scattering plane and one perpendicular to it; both basis vectors are perpendicular to the momentum.",
"The non-zero polarised matrix element for axion production is $\\begin{split}\\mathcal {M}^p_{||,\\perp }&=-2g_{a\\gamma \\gamma }\\omega _1\\omega _2\\sin {\\left(\\frac{\\theta _{12}}{2}\\right)^2},\\end{split}$ where a subscript $||$ refers to in-plane and $\\perp $ to out-of-plane polarisation.",
"The corresponding cross-section is $\\sigma =\\frac{\\pi }{4 \\omega _1\\omega _2(\\omega _1+\\omega _2)}\\frac{\\left|\\mathcal {M}\\right|^2}{\\sqrt{2-2\\cos {\\theta _{12}}}}\\delta \\left( \\omega _1+\\omega _2-\\omega _a \\right)$ where $\\omega _a=\\sqrt{m_a^2+\\omega _1^2+\\omega _2^2+2\\omega _1 \\omega _2 \\cos \\theta _{12}}$ is the axion energy and only $\\mathcal {M}=\\mathcal {M}^p_{||,\\perp }$ contributes.",
"The appearance of a delta distribution is characteristic of energy-momentum conservation at a three point vertex.",
"In a real experiment however the cross section is non-singular as the photons are provided by a laser beam of a finite pulse length $\\tau $ , and additionally are drawn from a Gaussian distribution $f_j\\left(\\omega \\right)=\\frac{1}{\\sqrt{2\\pi \\Delta _j^2}}\\exp {\\left[-\\frac{\\left(\\omega -\\omega ^0_j\\right)^2}{2\\Delta _j^2}\\right]}$ with central frequency $\\omega ^0_j$ and spectral width $\\Delta _j \\gtrsim \\tau ^{-1}$ dictated by the laser properties.",
"Note also that in equation (REF ) we have implicitly taken the axion to be in an asymptotic state.",
"This is a reasonable approximation given that its lifetime against stimulated decay to a photon is $\\tau _{a\\gamma \\gamma }=64\\pi /N_j g_{a\\gamma \\gamma }^2 m_a^3$ [24] with $N_j$ photons in the interaction region, which is always much longer than the laser pulse-length, $\\tau \\simeq {230}{eV^{-1}}$ .",
"We thus treat the axion resonance as a delta distribution to be integrated against the Gaussian (REF ).",
"We now define the scattering probability as $P_{\\gamma \\gamma \\rightarrow p}=\\frac{1}{V}{\\sqrt{2-2\\cos {\\theta _{12}}}}\\int _0^{\\tau }\\sigma dt,$ where $V$ is the interaction volume of the two lasers.",
"The angular dependence describes the flux of one beam through the other.",
"Given the probability for pseudoscalar production (REF ), the total number of incoherently produced axions is $N_p=N_1 N_2 P_{\\gamma \\gamma \\rightarrow p},$ where $N_j$ is the total number of photons of beam $j$ in the interaction region.",
"Our proposed scheme for axion production can also be applied to scalars with a coupling given by equation (REF ).",
"The associated non-zero matrix elements are $\\mathcal {M}^s_{\\perp ,\\perp } &=\\mathcal {M}^s_{||,||}/ \\cos {\\theta _{12}}=-2g_{s\\gamma \\gamma }\\omega _1\\omega _2\\sin {\\left(\\frac{\\theta _{12}}{2}\\right)}^2.$ Note that scalars and pseudoscalar couple with opposite photon polarizations, making it possible to separate the two signatures independently.",
"This is challenging to realise with other laser-based searches.",
"The photon to scalar conversion probability is completely analogous to equation (REF ), with $p\\rightarrow s$ .",
"Figure: A standing wave is formed in the circled region by passing a single laser pulse through a 50-50 beam splitter, each part of which then reflects of an off-axis parabolic mirror, as outlined in .The probability of generating axions can be further enhanced by implementing spatial structure in one of the beams.",
"In our proposed scheme, coherence must be achieved for large momentum transfer $q$ , since axions become more non-relativistic as their mass approaches the photon frequency.",
"In this case, a traditional LSW experiment suffers from interference effects which destroy coherence along the length $\\ell $ of the interaction region when $q \\ell \\gtrsim 1$ .",
"This is avoided for the interaction of the probe beam with a standing wave, which can lead to a scattering amplitude somewhat similar to Bragg diffraction off of a light grating [26].",
"In this case, a probe laser interacts with a standing wave produced by the other beam as shown in Fig.",
"REF (see also Ref.",
"[25]).",
"The standing wave is realised by reflecting a split laser beam off two off-axis mirrors such that they are counter-propagating at a common focus.",
"Scattering is then coherent within any single half-period of the standing wave, leading to an overall cubic scaling with photon number (see below).",
"Coherence effects have been used in the conversion of solar axions through Bragg diffraction with the electric field of a crystal [27], [16], but, as the enhancement relies on the solar axions being converted into X-rays, it is not compatible with axions produced by optical lasers considered here.",
"Additionally, in our case the momentum transfer and the spatial structure along the direction of beam propagation are intrinsically linked since they are both given by the frequency of the light in the standing wave, precluding coherence along the whole laser beam without some kind of external temporal modulation.",
"Let us consider a probe photon beam interacting with a standing wave (Fig.",
"REF ).",
"We approximate the probe beam photons as point-like, since the probe wavelength is much smaller than the standing wave wavelength.",
"The structure factor associated with the momentum-transfer ($\\mathbf {q}$ ) is then given by $I(\\mathbf {q}) = \\sum _{i,j=1}^{N_1} e^{i\\mathbf {q}\\cdot (\\mathbf {x}_i-\\mathbf {x}_j)},$ where $\\mathbf {x}_i$ are the positions of the standing wave photons in the interaction region.",
"The momentum transfer $q$ is simply the momentum $\\pm k$ of a photon in beam 1 for our elastic scattering.",
"Generically, the sum over these phases will be incoherent, and so only the diagonal piece of the matrix will contribute, leading to a scaling proportional to $N_1$ .",
"To achieve some form of coherence, the off-diagonal pieces ought to constructively interfere, providing a quadratic scaling.",
"We have $I(\\mathbf {q})=\\left(\\frac{2N_1}{L}\\right)^2\\left|\\int _{-\\frac{L}{2}}^{\\frac{L}{2}}dx \\cos ^2(kx+\\phi )e^{-ikx}e^{-\\frac{x^2}{2w^2}}\\right|^2,$ where $L$ is the width of the probe beam and the Gaussian models the intensity fall-off of the standing wave, governed by its Rayleigh length $w$ .",
"The phase $\\phi $ that the probe photon sees is dependent on where the probe beam falls on the standing wave.",
"To achieve the largest coherence, we must achieve a focus such that $L \\simeq \\lambda _1/2$ .",
"For such a scenario, we have $w \\gg L$ , and we find that the structure factor is rather insensitive to $\\phi $ , varying by a factor of 4 over its range.",
"Once created the axions must be reconverted into photons to be detected.",
"Since the axions are on-shell, light, and weakly coupled, they propagate over a macroscopic distance.",
"Such a decay into a visible photon can in principle be enhanced by stimulated processes, similar to what has been proposed for QED precision tests with three colliding lasers [28].",
"An additional (third) beam is introduced into the interaction region and can induce a stimulated decay into a photon.",
"However, given available photon numbers, stimulated decay is negligible for any interesting coupling constant [24].",
"Hence, we will adopt the usual approach of reconverting the axions in a magnetic field placed after an intervening wall.",
"As for any LSW experiment, the reconversion region is filled by a background magnetic field $B$ over a length $L$ .",
"The mixing probability is given by [12], [29] $P_{a\\rightarrow \\gamma }=\\frac{g_{a\\gamma \\gamma }^2 B^2 L^2}{\\beta _a}\\left(\\frac{\\sin {qL/2}}{qL}\\right)^2,$ with the momentum transfer $q = \\sqrt{\\omega _a^2-m_a^2}-\\omega _a$ , the axion energy $\\omega _a = \\omega _1+\\omega _2$ , and $\\beta _a$ the axion velocity.",
"The sinc function describes the momentum modes present in the magnetic field, arising from taking the spatial Fourier transform of the magnetic field, assumed constant over the length $L$ , and zero otherwise.",
"Adler's photon splitting theorem, which would cause the amplitude to vanish for $L \\rightarrow \\infty $ [30], is thus avoided as the magnetic field can transfer momentum.",
"If the total momentum of the incoming lasers points in the same direction, then the reconversion region can be left unchanged from shot to shot.",
"Due to the spectral width of the incoming beams the cross section of the magnetic region has to be sufficiently large to allow detection of any axion in the emission cone.",
"Finally, to avoid edge effects, the length of the magnetic region should be at least as long as its width.",
"The number of detected photons is finally given by $N_\\gamma = \\int I(q)N_2 P_{\\gamma \\gamma \\rightarrow a} P_{a\\rightarrow \\gamma } f_1\\left(\\omega _1\\right)f_2\\left(\\omega _2\\right)d\\omega _1 d\\omega _2,$ where we have assumed the laser beam frequency distribution is given by equation (REF ), and $\\omega _1$ and $\\omega _2$ are the standing wave and probe frequencies, respectively."
],
[
"Projected bounds",
"The proposed experiment is most sensitive to a hypothetical axion whose mass matches the central frequencies of the two beams, that is, $m_a=\\sqrt{2\\omega _1\\omega _2(1-\\cos {\\theta _{12}})}$ , by 4-momentum conservation.",
"Due to the non-zero bandwidth of the incoming laser beams, axions of different masses can be produced with decreasing probability moving away from the central frequencies of the beams.",
"The width of the sensitivity region, which we define as the mass region within which the coupling bound varies by less than $\\sqrt{2}$ , for the case of two beams of similar spectral distribution, is $\\delta m_a/m_a \\simeq \\Delta _1/\\omega _1+\\Delta _2/\\omega _2$ .",
"Hence in order to scan across a range of possible axion masses, we must take shots in increments of angle $\\delta _{12}$ given by $\\delta _{12} \\simeq 2\\frac{\\delta m_a}{m_a}\\left(\\csc {\\theta _{12}}-\\cot {\\theta _{12}}\\right).$ This angular step size and corresponding mass region is shown in Figure REF .",
"We require that the spread in momentum is small enough so that an axion will enter the finite aperture of the reconversion region.",
"For the values assumed in the paper, the aperture must be $\\phi \\ge {0.03}{rad}$ which would require a magnetic field width $\\le {30}{cm}$ at a realistic detector distance $\\le {10}{m}$ .",
"We now estimate the experimental feasibility of the proposed setup by calculating the expected number of photons produced.",
"We consider a transition-edge sensor (TES) similar to the ALPS II experiment [31], [32] capable of single photon detection.",
"The effective exposure time of our experiment is set by the readout time of the detector, which is around $\\sim {1}{\\mu s}$ .",
"The dark count rate of such detectors over the effective exposure time is $10^{-10}$ and efficiencies are high enough to detect single photons.",
"The dominant background will come from blackbody radiation which can easily be rejected via their energy range, which lies outside the axion energy range, or coincident timing with the laser beam.",
"We see from equation REF that to increase the signal we must maximise the laser intensity.",
"Some of the current or near-commissioning high-power laser facilities can achieve numbers of up to $\\sim 10^{21}$ photons per beam.",
"The Aton 4 (${10}{PW}$ ) laser at ELI-beamlines, for example, can operate at a central optical frequency of ${1.55}{eV}$ with spectral width of $\\Delta = {0.05}{eV}$ , and pulse length of $\\tau = {100}{fs}$ .",
"We consider the scenario in which both beams are focused down to their respective wavelengths.",
"Frequency doubling and tripling is also possible, as is the beam splitting in two independent arms.",
"We consider the beams to be focused down to their respective photon wavelengths.",
"Finally, we consider, in the reconversion region, a magnetic field of strength ${10}{T}$ , although the limits on $g_{a\\gamma \\gamma }$ scale only as $B^{-1/2}$ .",
"Due to the pulsed nature of the axion conversion, the magnetic field can be pulsed as well.",
"Such high fields require superconducting magnets.",
"The design of laser target chambers sets a lower bound to the distance between the interaction region and the detector of a few meters.",
"To ensure an aperture of at least ${0.03}{rad}$ we assume a magnetic field of ${30}{cm}$ in each dimension to evaluate equation REF .",
"Note that the length does not play a significant role as long as it is larger than the other spatial dimensions as when $qL\\gg 1$ , the $\\sin ^2(qL/2)$ in (REF ) oscillates quickly and the laser spectral width effectively averages over the oscillations.",
"In this case the length drops out of the probability.",
"Figure: Exclusion plot for axion parameter space.",
"The light blue region shows existing bounds from the OSQAR experiment ; the orange region is excluded by PVLAS ; the dashed blue line depicts CAST constraints ; the lower horizontal dashed line comes form stellar cooling lifetimes and the upper from solar Bragg diffraction experiments .",
"The dotted black and dark blue lines correspond to our proposal performed at ω=1.55eV\\omega ={1.55}{eV} and the first harmonic, respectively, with ELI parameters and no substructure coherence.",
"The black line shows the possible bounds using standing wave coherence and the red line indicates the same parameters but 1 day of shots per angular step instead of a single shot.",
"The region above the line is excluded in each case.",
"The QCD axion region indicates particular theoretical predictions for where the axion might be, given dark matter abundances .Figure REF shows the axion exclusion limits we expect to achieve with the proposed experiment.",
"The bounds are calculated assuming a single shot for each appropriately spaced angular step size.",
"The whole experiment entails around 30 shots.",
"We see that our technique is able to access regions of parameter space that are not already excluded by existing LSW experiments.",
"The lines in Figure REF must be interpreted as the upper bound on $g_{a\\gamma \\gamma }$ and, for given laser parameters, they are constructed by changing the scattering angle as discussed earlier so that we constrain an order of magnitude in $m_a$ .",
"The scalar exclusion plot gives very similar bounds.",
"The two upper lines in Figure REF are obtained without any coherent enhancement.",
"While PVLAS provides slightly better bounds, however, as discussed above, these suffer from irreducible Standard Model background limitations.",
"Our proposed searches, on the other hand, have no such background.",
"When scattering from coherent laser substructures is implemented, our bounds on the axion-photon coupling are competitive with CAST in the $1-5$ eV mass range, and model independent.",
"Astrophysical bounds, as is the case for all laboratory experiments in this mass range, provide more stringent bounds, again with the caveat that they are model dependent.",
"Despite the unfavourable scaling of the bounds with the number of shots, the red curve shows that the QCD band can be reached comfortably with 1 day run time per step size.",
"Considering the progress in laser technology over the past decades [36], it is not inconceivable that laser intensities would grow further over the coming years to allow new parameter space to be probed.",
"Given the cubic scaling with photon number for coherent scattering, these advances can achieve a significant improvement of the bounds presented here, and more so compared to other LSW experiment.",
"The mass range can also be extended by using XFELs, for which a higher degree of frequency tuning is possible.",
"This would allow for easier scanning of parameter space.",
"The experimental parameters we consider here do not maximise the coherence factor.",
"In fact, the reconversion efficiency can be improved if the magnetic field were spatially varying, such that its Fourier transform components match the momentum transfer in the axion-photon transition.",
"Moreover, the refractive index of the reconversion chamber could also be varied to reduce the difference in the axion mass and the effective photon mass, which would decrease the required momentum transfer.",
"We have not yet included these effects in our analysis, and as such our results can be considered a lower bound on the performance of this technique at high-intensity laser facilities.",
"We would like to thank Prof Subir Sarkar, Rami Shelbaya and especially Charlotte Palmer for many useful discussions.",
"We thank Prof Willy Fishler for comments on an early version of the paper.",
"The research leading to these results has received funding from AWE plc.",
"British Crown Copyright 2019/AWE."
]
] | 2001.03392 |
[
[
"Nine New Repeating Fast Radio Burst Sources from CHIME/FRB"
],
[
"Abstract We report on the discovery and analysis of bursts from nine new repeating fast radio burst (FRB) sources found using the Canadian Hydrogen Intensity Mapping Experiment (CHIME) telescope.",
"These sources span a dispersion measure (DM) range of 195 to 1380 pc cm$^{-3}$.",
"We detect two bursts from three of the new sources, three bursts from four of the new sources, four bursts from one new source, and five bursts from one new source.",
"We determine sky coordinates of all sources with uncertainties of $\\sim$10$^\\prime$.",
"We detect Faraday rotation measures for two sources, with values $-20(1)$ and $-499.8(7)$ rad m$^{-2}$, that are substantially lower than the RM derived from bursts emitted by FRB 121102.",
"We find that the DM distribution of our events, combined with the nine other repeaters discovered by CHIME/FRB, is indistinguishable from that of thus far non-repeating CHIME/FRB events.",
"However, as previously reported, the burst widths appear statistically significantly larger than the thus far non-repeating CHIME/FRB events, further supporting the notion of inherently different emission mechanisms and/or local environments.",
"These results are consistent with previous work, though are now derived from 18 repeating sources discovered by CHIME/FRB during its first year of operation.",
"We identify candidate galaxies that may contain FRB 190303.J1353+48 (DM = 222.4 pc cm$^{-3}$)."
],
[
"Introduction",
"FRBs are an enigmatic class of radio transients that exhibit millisecond durations, cosmological distances and large energy output [20].",
"Such extreme characteristics have resulted in a diverse and evolving landscape of physical models that differ in progenitor and emission-mechanism types, with possible interpretations ranging from compact objects to cosmic strings [50].",
"Among the most intriguing FRB sources are those that emit multiple bursts, as first seen in FRB 121102 [60], [56].",
"Repeating sources necessitate non-cataclysmic models.",
"Recently proposed models typically include young, highly magnetised neutron stars that likely interact with their environments, such as supernova remnants [32], [43], [8], [41] or massive black holes [64], or bursting due to internal magnetar instabilities affecting the magnetosphere [33].",
"No model yet fully explains all existing, albeit limited, observational data.",
"Therefore, discoveries of new repeating FRBs are important to constrain available models.",
"Interferometric follow-up observations of bursts can be performed to obtain precise sub-arcsecond positions, which can then be observed with optical telescopes to identify host galaxies and their redshifts.",
"The localization of FRB 121102 through direct imaging of repeat bursts [12], [35] pinpointed the source to a star-forming region in a dwarf galaxy at redshift $z = 0.193$ [61], [6], enabling multi-wavelength studies of the source environment [57].",
"Recent localizations of single-burst FRBs show that FRBs can also reside in more massive galaxies with a wide range of specific star formation rates [5], [53], [52], [34].",
"Additional localizations and subsequent multi-wavelength observations will provide a unique data set of host classifications, redshifts, assessments of source environments and other information.",
"The CHIME telescope and its FRB search backend [13] first detected 13 low-frequency bursts [14], and discovered the second repeating FRB source from this initial sample [15].",
"CHIME/FRB recently published a detection of FRB 121102 in the CHIME band [28], as well as eight new repeating sources whose bursts generally show complex morphological features and, as an ensemble, possess larger widths than those of thus far non-repeating FRBs [16].",
"The latter observation serves as significant evidence of different emission mechanisms between repeating sources and apparent single-burst FRB sources, and/or common environments surrounding repeating sources, though a larger sample of both FRB types will be helpful for verifying this apparent trend.",
"Moreover, the observed Faraday rotation measure (RM) for the CHIME/FRB repeater FRB 180916.J0158+65 ($-114.6 \\pm 0.6$ rad m$^{-2}$ ; Paper I) is considerably smaller than the RM$\\sim 10^5$ rad m$^{-2}$ measured for FRB 121102 [42]; moreover, FRB J180916.J0518+65 has no persistent radio counterpart [34] while FRB 121102 is coincident with a persistent source [35].",
"These facts suggest that dramatically different magneto-ionic environments can contain FRB sources.",
"In this work, we report the discovery and subsequent analyses of bursts emitted from nine new repeating FRB sources.",
"In Section , we highlight the observations taken with the CHIME telescope that enabled offline analyses of total-intensity and polarization spectra.",
"In Section , we describe the analysis used for quantifying per-burst and sample properties.",
"In Section , we discuss the results obtained from analyses of burst morphology, sample properties, baseband data and multi-wavelength counterparts.",
"In Section , we summarise our findings."
],
[
"Observations",
"All detections presented here were made during a period of telescope commissioning between 28 August 2018 and 30 September 2019 using the CHIME/FRB system.",
"As described by [13], the CHIME/FRB backend continuously receives a total-intensity, polarization-summed time series generated by a FX correlator with 0.98304 ms cadence for 16,384 frequency channels across the 400–800 MHz band.",
"The 128-node FRB system employs real-time radio frequency interference (RFI) mitigation and a modified tree de-dispersion algorithm.",
"Candidate signals with integrated S/N values greater than a configurable threshold are immediately forwarded to a post-detection pipelineDocumentation for the post-detection part of the CHIME/FRB realtime pipeline can be found at https://chimefrb.github.io/frb-l2l3/ for real-time classification to i) identify and ignore RFI-induced signals, ii) check for coincidence with known Galactic sources and iii) compare with predicted Galactic contributions to DM.",
"Signals are classified as extragalactic (i.e., FRBs) if they are not associated with any known Galactic sources, and their measured DMs exceed independent estimates of the maximum values predicted by available Galactic DM models [17], [63]; we do not account for DM contributions from electronic content in the Galactic halo.",
"We deemed a group of bursts to originate from the same repeating source if their measured position and DM values lied close to one another, such that the differences in these quantities satisfied the following criteria given estimated uncertainties: $\\Delta $ DM $<$ 1 pc cm$^{-3}$ ; $\\Delta $ R.A.$\\cos $ (Dec.) $<$ 1 deg.",
"; and $\\Delta $ Dec. $<$ 1 deg.",
"These thresholds were chosen based on a statistical analysis of chance coincidence in the presence of a large FRB sample; see Appendix for a discussion of simulations that motivate these bounds for CHIME/FRB, given that $\\sim $ 700 FRBs have been detected with the CHIME/FRB system during the aforementioned time period.Analysis of the $\\sim $ 700 FRBs found by CHIME/FRB is ongoing and will be published in a forthcoming catalog.",
"The thresholds used in this work are more stringent than those used in Paper I, with the DM threshold being increased by an order of magnitude.",
"The CHIME/FRB real-time pipeline automatically records 60-s segments of total-intensity data to disk at the resolution used by the FRB instrument for detection, for all burst events deemed astrophysical and extragalactic.",
"For sufficiently bright signals, the CHIME/FRB system also records buffered telescope baseband data — complex voltages measured with 4-bit precision (each for real and imaginary parts) by all 1,024 dual-polarization feeds at 2.56-$\\mu $ s cadence across 1,024 frequency channels — which allows for offline polarization detection, position verification, and high-time-resolution studies of burst morphology.",
"For this work, we analyzed polarization properties and verified localizations using available baseband data; we deferred morphological studies with baseband data for future study as analysing $\\mu $ s-level structure is beyond the scope of this work.",
"Since Paper I, we lowered the S/N thresholds for recording intensity and baseband data to disk as confidence in the real-time detection system grew.",
"For repeating sources, total-intensity spectra of bursts presented here are drawn from a wider S/N distribution, going down to a real-time detection S/N of 8 instead of 9.",
"A larger number of bursts were detected with 9 $<$ S/N $<$ 10 than in Paper I due to improved system sensitivity and classification algorithms during the telescope commissioning period.",
"For baseband recording, we used a threshold S/N of 9.",
"Table: Properties of Nine New CHIME/FRB Repeating Sources"
],
[
"Analysis & Results",
"We detected nine new repeating FRB sources with the CHIME/FRB instrument, using the criteria described in Section .",
"Source properties are summarized in Table REF , while individual burst properties are listed in Table REF ."
],
[
"Source localization",
"Burst localization was carried out following the methods described in Paper I.",
"For all sources, we used a model of the CHIME primary beam and S/N estimates from all beams that detected bursts in order to obtain burst-averaged sky positions using a $\\chi ^2$ -grid method.",
"While the methods remain identical to those used in Paper I, we have updated our underlying beam model to correct for an effective 0.071$^\\circ $ rotation of the telescope relative to true North (counter clockwise as viewed from the CHIME meridian) that was identified after publication of Paper I.This amount of rotation corresponds to a distance offset between the apparent and true N-S directions of $\\sim $ 5 cm at the North end of each cylinder We chose not to correct positions reported in Paper I in this work since their quoted uncertainties make the corrected positions statistically equivalent.",
"Our best estimates of source positions are provided in Table REF , and graphical depictions of the localization regions are shown in Figure REF .",
"We were not able to acquire total-intensity or baseband data for one burst from Source 3, since data acquisition was temporarily disabled around the time of this burst due to commissioning-related upgrades of the CHIME/FRB system.",
"However, since this burst satisfied detection-S/N and classification thresholds for FRB signals, we nonetheless recorded the metadata produced by the realtime detection pipeline.",
"Since these data contain the quantities needed for burst-averaged localization refinement, we used the metadata for this burst in generating $\\chi ^2$ grids and confidence intervals for the sky localization of Source 3.",
"Figure: NO_CAPTION"
],
[
"Exposure Determination",
"On-sky exposure (referenced to 600 MHz) across the FWHM region of the total-intensity beams synthesised for the CHIME/FRB system was determined in a manner similar to that used in Paper I.",
"In summary, we estimated the exposure for a grid of sky positions within the 90% confidence uncertainty region for each source (see Figure REF ).",
"We then calculated the weighted average and standard deviation for the exposure over all these positions, with the weights equal to the sky-position probability maps shown in Figure REF .",
"Exposure for each source during the interval from 28 August 2018 to 30 September 2019 is reported in Table REF and plotted in Appendix (see Figure REF ).",
"For high declination sources ($\\delta > +70^\\circ $ ), which transit across the primary beam of the telescope twice each day, we chose to report the exposure for the lower transit separately since the beam response is different for the two transits."
],
[
"Determination of Burst Fluence and Peak Flux Density",
"Dynamic spectra were calibrated and burst fluences and peak fluxes were determined as described in Paper I.",
"In summary, we used transit observations of steady sources with known spectral properties to obtain flux conversion factors as a function of frequency in the vicinity of each burst.",
"These calibration spectra were then applied to the total-intensity burst data to achieve calibrated dynamic spectra in flux units corrected for sensitivity variations across our band due to the telescope primary beam.",
"Fluences were calculated by integrating the extent of each burst in the band-averaged time series (binned at the full $0.98304\\,\\text{ms}$ resolution of total-intensity data) while peak fluxes were taken to be the highest value within the same extent.",
"Due to the narrow-band nature of the bursts in our sample, averaging the signal over the entire bandwidth also averages noise into our fluence and flux values.",
"However, we chose to quote burst fluence and fluxes from a the same frequency range for consistency.",
"If there were multiple sub-bursts in a given burst, then peak flux and fluence values were obtained for each component.",
"To simplify these calculations, we assumed that all bursts were detected along the meridian of the primary beam so that our fluences and fluxes represent lower bounds.",
"Uncertainties were estimated by taking into account beam and time variations in system sensitivity using steady calibrator sources as in Paper I.",
"For all sources except Source 4, we used steady sources within $5^{\\circ }$ of declination for calibration.",
"Since there are no calibration sources within $5^{\\circ }$ in declination from Source 4, we assumed that our beam is North-South symmetric and used a source within $1^{\\circ }$ of declination on the opposite side of zenith."
],
[
"Fluence Completeness Determination",
"We determined fluence completeness, the per-source threshold fluence for which all bursts above it are expected to be detected by the CHIME/FRB instrument, in a manner similar to that used in Paper I, with some minor modifications to how sensitivity variation is characterised.",
"Previously, for circumpolar sources observed twice a day, fluence thresholds were simulated for the transit where bursts were detected, then calibrators with similar declination were used to get a band-averaged sensitivity scaling for the transit without detections.",
"We caution that the per-frequency scaling between transits is highly structured, so the observed attenuation for a given source will depend strongly on its spectrum.",
"In this work, we extended the simulations to cover both transits, using our beam model to compute relative sensitivities.",
"This approach included a description of the synthesised beams in addition to the degradation in the primary beam sensitivity, which is significant for sources that pass between synthesised beams in one transit while crossing beam centers in the other transit.",
"In order to characterise the inter-day sensitivity variation of the CHIME/FRB system, we used the method described in [28].",
"The method involves estimating the daily variation in RMS noise at the location of each source by analysing distributions of S/N values of pulsars within 5$^\\circ $ of the source declination.",
"However, observations with the CHIME/FRB system through 2019 June suggest that small variations in RMS noise can have a significant effect on the FRB detection rate.",
"Since these variations cannot be adequately characterised by the small number of pulsars detectable in a 5$^\\circ $ declination range, we used pulsars which are robustly detected by the CHIME/FRB system within its entire observable declination range (with Dec. $> -11^\\circ $ ) in this work.",
"The estimate of the daily variation in RMS noise is obtained by averaging measurements from pulsars detected on each sidereal day for which the telescope was operating with the same gain calibration.",
"This approach is in contrast to the method used in [28], which combined measurements from pulsars detected on the same UTC day.",
"Our modification makes the measurement of the relative RMS noise more sensitive to changes in system sensitivity due to varying gain-calibration strategies.",
"Typical daily variations are at the 20% level."
],
[
"Characterisation of Burst Morphology",
"Many bursts from repeating FRB sources exhibit complex morphology, comprised of multiple sub-bursts that usually drift down in frequency as time progresses [27].",
"Finding the optimal DM necessitates methods beyond the typically employed S/N-optimisation, and here we used the same method as in Paper I — maximising burst structure by calculating the phase coherence of emission in all frequency channels with the DM_phase packagehttps://github.com/danielemichilli/DM_phase (Seymour et al.",
"in prep.)",
"over a range of trial DMs.",
"The sub-burst alignment after dedispersion was verified by eye and Figures REF and REF show dynamic spectra for all bursts, dedispersed to the best estimate of that burst's DM, as listed in Table REF .",
"We used the same modeling procedure discussed in Paper I for estimating widths, arrival times and scattering timescales from our calibrated total-intensity dynamic spectra, which we summarize here.",
"In all fits, we preserved the raw time and frequency resolution of the CHIME/FRB total-intensity data and fitted two-dimensional models of Gaussian temporal profiles and either Gaussian or weighted power-law spectral shapes.",
"For bursts with simple morphology i.e., with no significant multiple components), we fitted a single-burst spectrum and allowed the DM to float freely during each fit in order to determine robust uncertainties for all other burst-specific parameters.",
"In these cases, the DMs estimated from direct modeling of the dynamic spectra were statistically consistent with the values estimated from the structure-optimisation algorithm shown in Table REF .",
"For bursts with spectro-temporal structure typical of repeating sources, we fitted several components and held the DM fixed to the structure-optimisation value in Table REF in order to ensure separation of sub-burst components and the determination of robust width estimates.",
"For all bursts, we also fitted single-tail scattering profiles to dynamic spectra that arise from frequency-dependent, multi-path propagation of the signal through small-scale electron density inhomogeneities in the ISM.",
"We fitted for scattering by applying a temporal pulse broadening function [40] and assuming that the scattering timescales depends on frequency as $f^{-4}$ .",
"As done in previous works, we selected the superior model for each burst – one that explicitly fits for the scattering timescale or one that does not fit for scattering effects – based on the best-fit $\\chi ^2$ statistic [14].",
"Using this model selection procedure, we consider all widths reported in Table REF to approximate the intrinsic burst widths.",
"Significant estimates of the scattering timescale are presented with uncertainties in Table REF ; for non-detections of scattering timescales and widths, which we defined to be consistent with 0 ms at 3$\\sigma $ confidence, we present upper limits of the 2$\\sigma $ confidence interval.",
"As noted in Paper I, repeater morphology likely introduces bias into fits of scattering timescales due to faint, extended emission being detected as part of a scattering tail; we therefore urge caution when interpreting scattering timescales shown in Table REF .",
"As in Paper I, we determined burst drift rates using an autocorrelation analysis and present results for bursts where the drift rate is constrained in Table REF .",
"We calculated each linear drift rate by fitting a two-dimensional Gaussian profile to the two-dimensional auto-correlation.",
"We used a Monte Carlo method to obtain robust confidence intervals, de-dispersing our bursts to 100 DM values drawn from the DM uncertainty distribution and fitting a linear drift rate for each of 100 random noise realisations per DM value.",
"We found five bursts with significant drift rates (see Table REF ).",
"Table: Individual Burst Properties from Nine New CHIME/FRB Repeaters.",
"a ^aFigure: NO_CAPTIONFigure: NO_CAPTION"
],
[
"Baseband Detections",
"The CHIME/FRB baseband system [13] was triggered for one burst each from Sources 2, 4, 5, and 6.",
"We used the baseband data for these events to verify the localization regions obtained from the per-beam S/N analysis described in Section REF ; the results and details of future position refinement with baseband data will be presented elsewhere (Michilli et al., in prep.).",
"Once available, this analysis will reduce the localization precision to $\\sim \\frac{6.7}{S/N}$ arcmin [13].",
"Visual inspection of the baseband dynamic spectra confirmed successful capture of all four bursts by the baseband system.",
"All baseband events were incoherently dedispersed to the S/N-optimising DM after coherently dedispersing each of the 1,024 baseband channels to a nearby fiducial value, in order to mitigate intra-channel smearing.",
"All four baseband events were manually processed through a polarization analysis pipeline.",
"The pipeline, described in an upcoming paper (Mckinven et al., in prep.",
"), searches for an RM detection using the RM-tools packagehttps://github.com/CIRADA-Tools/RM that implements two independent methods: RM-synthesis [11], [10] and Stokes QU-fitting [47].",
"Using these algorithms, we detected moderate RM values for Sources 2 and 6.",
"Table REF summarises the main polarization products for sources with an RM detection.",
"RMs derived from RM-synthesis ($\\mathrm {RM_{FDF}}$ ) and Stokes QU-fitting ($\\mathrm {RM_{QUfit}}$ ) are reported, and show small but significant differences.",
"$\\mathrm {RM_{FDF}}$ is determined from the peak of the Faraday dispersion function (FDF), while $\\mathrm {RM_{QUfit}}$ is calculated from a parametric fit under a thin-screen model of Faraday rotation.",
"Uncertainties in $\\mathrm {RM_{FDF}}$ were determined in a manner consistent with previous analysis of a different burst presented in Paper I, while those for $\\mathrm {RM_{QUfit}}$ were determined from its marginal probability density output by the Nested Sampling routine.",
"The differences in estimates of $\\mathrm {RM_{FDF}}$ and $\\mathrm {RM_{QUfit}}$ are a more valid measure of the systematic RM measurement uncertainty than each method's formal measurement error.",
"The discrepant RMs are also a reflection of small differences in the measurement procedure of the two methods.",
"In particular, the Stokes Q,U fitting of RM-tools implements a Nested Sampling algorithm [59] to find the best-fitting parameters.",
"This method allows the partially degenerate parameters, RM and the de-rotated polarization angle ($\\chi _0$ ), to be simultaneously fit.",
"Figures REF and REF summarise the general properties of the polarized signal for Sources 2 and 6, respectively: dynamic spectra for Stokes I, Q, U and V parameters where data have been downsampled to $\\sim $ 1.56-MHz and $\\sim $ 0.33-ms resolution; FDFs that are cleaned of instrumental response introduced by limited bandwidth coverage [26]; and Stokes Q,U spectra normalised by the total linear polarization at each frequency.",
"We note that mixing of polarized signal between different Stokes parameters was not corrected for and appears to affect sources to varying degrees.",
"In particular, Source 6 (see Fig.",
"REF ) displays significant leakage of signal of Stokes U into Stokes V, as evidenced by a $\\lambda ^2$ modulation consistent with Faraday rotation.",
"This leakage makes the uncorrected linear polarized fraction (L/I, with $\\mathrm {L = \\sqrt{Q^2 + U^2}}$ ) for Source 6 a lower bound, which we estimate to be L/I $\\ge $ 0.2, and the circular polarized component highly uncertain.",
"Leakage appears to be subdominant for Source 2, as its polarized fraction is consistent with unity.",
"Corrections of these instrumental effects are a work in progress and, although important for accurate linear and circular polarized fractions, should not substantially change the RM values reported here.",
"Figure: NO_CAPTIONTable: Observed and Galactic-foreground RM values for CHIME/FRB repeaters.Figure: NO_CAPTIONFigure: NO_CAPTION"
],
[
"Discussion",
"The systematic monitoring of a large sky fraction by CHIME/FRB has enabled the discovery of nine new repeating FRB sources, and a total of 18 repeating sources when including the results from [15] and Paper I.",
"The current sample of known repeating FRB sources now consists of 20 members, with FRBs 121102 [56] and 171019 [30], [48] discovered using other observatories.",
"Besides offering opportunities for interferometric localization and multi-wavelength follow up [34], this number is large enough to enable studies of distributions of some source properties, although caution is needed as the CHIME/FRB pipeline has detection biases that remain to be quantified."
],
[
"Burst DMs and Morphologies",
"We compared the distribution of DMs of the 18 CHIME/FRB repeaters with that of the 12 published thus far non-repeating sources found by CHIME/FRB, and found no statistically significant difference using either a Kolmogorov-Smirnov or Anderson-Darling test.",
"Both distributions are subject to similar (though not necessarily identical) detection biases.",
"This fact is suggestive that both their distribution in space and their associated local environments do not differ strongly, unless the differences cancel one another, which seems unlikely.",
"Further comparison with additional CHIME/FRB sources that have not yet repeated (currently under analysis) will also be of interest.",
"Paper I noted a statistically significant difference in repeater and thus far non-repeater burst widths.",
"We repeated this analysis for this work with the new nine repeating sources included, using the $2\\sigma $ confidence upper limits where there are no significant measurements of the pulse width.",
"As discussed in Section REF , two best-fit models were obtained for each burst in the CHIME/FRB sample – one that directly models the effect of scattering and another that ignores scattering – and the model that yielded a better goodness of fit statistic was chosen as the superior model.",
"Therefore, all widths were obtained in a similar manner and likely reflect the intrinsic burst widths when accounting for effects from scattering and multiple burst components that is typical in repeating-FRB activity.",
"Additionally, we included in the comparison only bursts with a detection S/N $>10$ , which was the threshold for saving intensity data to disk at the time of detection of the non-repeating sources we considered.",
"This criterion ensures that the two samples have an identical selection function, but omits Source 3 from this analysis since all bursts from that source had S/N $<10$ .",
"We compare temporal widths of different Gaussian components of the bursts, including several measurements for any burst with multiple components in the samples used for the comparison.",
"These width distributions are shown in Figure REF .",
"Using both the Kolmogorov-Smirnov and Anderson-Darling tests, we found that the two samples are not drawn from the same distribution with $\\sim 5\\sigma $ and $\\sim 4\\sigma $ significance, respectively.",
"When combining width measurements of different bursts (or their components) from each repeating source using inverse-variance weighting, in order to perform this comparison strictly between FRB sources, we also found that the difference persists and supports the notion of possibly different emission mechanisms between repeating and thus far non-repeating FRBs.",
"Alternatively, significant variation in pulse widths could also be caused by differences in the properties of the circumburst media between repeating and thus far non-repeating FRBs.",
"However, this is a less likely possibility considering that the two DM distributions do not have any statistically significant differences.",
"As a population, the thus far non-repeaters reported in [14] show significant scattering, with half having scattering times of $\\stackrel{>}{_{\\sim }}1$ ms at 600 MHz.",
"By contrast, only six of the eighteen published CHIME/FRB repeating sources have statistically significant scattering measurements in the 400–800 MHz band whereas the remaining twelve sources have reported upper limits ([28], [15], Paper I).",
"However, as we described in Section REF and in Paper I, the complex morphology of repeating FRB sources makes their accuracy less transparent.",
"We nonetheless use scattering estimates for CHIME/FRB repeating FRBs at face value in the following statistical analysis.",
"Here we assumed that the scattering properties of the intervening medium do not change over the period of several months between detection and used the strictest constraint on scattering time listed in Table REF as the measured value for each source.",
"However, several of the reported upper limits are much larger than the measured scattering times for the thus far non-repeaters.",
"This is expected given the above-mentioned width disparity which makes it harder to detect scattering at the $\\sim $ 1-ms level in the repeating bursts.",
"The lack of measured scattering timescales for several of the repeating sources makes it difficult to directly compare their scattering properties with those of the thus far non-repeaters.",
"Instead, we compared the two distributions with the methodology used in Paper I which sets the reported upper limit on the scattering time for each source to be the $1\\sigma $ confidence interval of a normal probability density distribution.",
"A cumulative distribution function for a mixture distribution of the repeater scattering timescales was generated by adding the probability density distributions for all sources.",
"This cumulative distribution function was then compared with measured scattering times for the non-repeaters using a Kolmogorov-Smirnov test.",
"We found that we cannot rule out the possibility of the two samples being drawn from the same parent distribution.",
"We will repeat this analysis once a larger sample of repeater bursts with statistically significant measurements of scattering timescales is obtained.",
"Figure: NO_CAPTIONPaper I noted a statistically marginal, negative correlation between peak FRB flux and width based on the repeating FRB sample published therein.",
"We repeated this analysis, combining measurements presented in Table REF of the present work to the sample published in Paper I.",
"We found no evidence for correlation between peak fluxes and widths from this enlarged sample.",
"However, as in Paper I, we note that various sources of selection bias have not yet been quantified, and future analyses of such correlations may still be worthwhile.",
"The 190213 detection of Source 3 at $\\sim 1$ ms time resolution consists of two sub-bursts, separated by $\\sim 19$ ms without an apparent “bridge“ in emission in time or frequency – similar to the 181019 detection of Source 1 in Paper I.",
"Whether a detection like this constitutes one burst comprised of two sub-bursts or two separate bursts remains to be seen; the interpretation is likely model-dependent.",
"In Table REF , we reported new measurements of sub-burst linear drift rates.",
"However, in contrast to Paper I, where almost half of bursts showed visible drift, only five of the bursts discussed in this work have significant drift rates.",
"The difference is most likely due to more bursts in the present sample having lower S/N than in Paper I, thus making significant detection of drifting sub-bursts more difficult.",
"The measured linear drift rates appear to be drawn from a similar distribution as in Paper I, and the notion that repeating sources of FRBs exhibit drifting sub-bursts with linear drift rates of order few to tens of MHz/ms in the 400–800 MHz band remains valid.",
"The apparent differences in drift rate uncertainties reported here and in Paper I are due to i.)",
"a non-linear relation between the measured rotation $\\theta $ of a two-dimensional Gaussian and the drift rate df/dt $= 1/\\tan (-\\theta )$ , leading to measurements of higher drift rates being more uncertain and ii.)",
"the drift rate uncertainties were determined after marginalising over DM, with the DM uncertainties of the bursts for which we measure drift rate here all being only 0.2–0.3 pc cm$^{-3}$ .",
"As CHIME/FRB total-intensity data have relatively coarse time resolution, it is possible that apparently simple bursts would show substructure and measurable drift if examined at higher time resolution.",
"This circumstance has been seen in bursts from FRB 121102 [27].",
"In addition, the emission mechanism and/or local environments likely introduce burst-to-burst fluctuations to the observed morphology.",
"As with the sources in Paper I, we found that all bursts in this present sample of repeating sources possess small emission bandwidths of 100–200 MHz, similar to behavior seen in FRB 121102 [23], whereas a large fraction of non-repeating FRBs span the full CHIME band.",
"Given the large sample of repeat bursts obtained with CHIME/FRB, detected over a wide range of beam positions, it is likely that these small emission bandwidths reflect real, intrinsic differences in emission mechanisms and/or local environments between repeating and non-repeating FRB sources.",
"However, a robust statistical analysis of spectral features requires a sample of thus far non-repeating FRBs larger than that considered in Paper I."
],
[
"Implications of RM Values",
"Using CHIME/FRB baseband data, we measured RM values from two new repeating sources.",
"These results are summarised in Table REF , along with with a previous RM detection from a different CHIME/FRB repeater (Paper I; see Table REF ).",
"RM values reported are those derived from the Stokes QU-fitting method, which were found to give marginally higher linear polarization fractions after de-rotation.",
"We did not apply any redshift corrections to the values presented in Table REF as their redshifts are not currently known, and the RM contributions from local environments are not constrained.",
"We also computed the RM contributions from the Milky Way foreground ($\\mathrm {RM_{MW}}$ ), which are provided in Table REF .",
"We used two methods to determine the Galactic contribution to RM: i) the constructed RM foreground map of [45], previously applied in Paper I; and ii) the first Faraday moment of diffuse, polarized emission obtained from the Global Magneto-Ionic Medium Survey (GMIMS) [19].",
"The first moment is the mean of the Faraday spectrum, weighted by polarized intensity.",
"Along most high-latitude sightlines the Faraday spectrum is simple, and the first moment is equivalent to the peak Faraday depth.",
"Departures from this scenario arise when polarized emission is Faraday rotated by different amounts.",
"Such departures are commonly observed in diffuse, polarized emission from the Galactic foreground, where its extended nature leads to variable levels of Faraday rotation as a function of line-of-site distance.",
"Cases such as this display complex structure of the polarized emission in Faraday depth and can even appear as multiple peaks, preventing an accurate $\\mathrm {RM_{MW}}$ value from being inferred.",
"This does not appear to be a problem here, with all repeater directions yielding relatively simple Faraday spectra, as is also the case for FRB 121102 [42].",
"Significant differences can be seen between certain $\\mathrm {RM_{MW}}$ values when using the two methods.",
"Notably, FRB180916's position near the Galactic midplane likely complicates the interpretation of the first Faraday moment from GMIMS.",
"This discrepancy occurs because sightlines at lower latitudes probe a greater extent of the Galactic foreground emission.",
"In low-latitude cases such as this, the first Faraday moment of the emission is unlikely to be an accurate proxy for $\\mathrm {RM_{MW}}$ .",
"In fact, even idealized scenarios with minimal path length through the foreground and no magnetic field reversals can still produce systematic bias between the first Faraday moment of diffuse, polarized emission and the true $\\mathrm {RM_{MW}}$ value [46].",
"A larger sample of extragalactic polarized sources, including FRBs, will be helpful in identifying well behaved regions of the sky where diffuse, polarized emission can be leveraged to better subtract the foreground RM contribution.",
"Regardless of the method used for subtracting the RM contribution of the Galactic foreground, a considerable gap ($\\mathrm {>10^5 \\textrm { rad }m^{-2}}$ ) exists between RM values obtained for repeating sources observed by CHIME/FRBwith those associated with bursts from FRB121102.",
"It is possible that depolarization arising from intrachannel Faraday rotation greatly reduces sensitivity to polarized signal with extreme RMs, effectively preventing detection.",
"At the native channelization of CHIME baseband data, sensitivity to polarized signal drops significantly for RMs beyond $\\sim 10^3 \\textrm { rad }m^{-2}$ , with the exact value depending on the specifics of the burst spectrum.",
"However, even after accounting for this possibility, repeater RMs discovered with CHIME appear rather modest compared to the effective range over which RM detections are possible.",
"At the very least, these results demonstrate that RM is not a perfect discriminant between repeating and (apparently) non-repeating sources and further highlight the distinct nature of FRB 121102.",
"In addition to RM, polarization fraction (both linear and circular) and the polarization position angle (PA) across the burst phase offer additional diagnostic information for informing FRB emission models (see Paper I for details).",
"The flat PA curves for Sources 2 and 6 shown in Figure REF are consistent with those of FRB 180916.J0158+65 (Paper I) and FRB 121102 [60], [42], [22], [27]; yet this does not appear to be a common feature of all FRB sources with FRB 110523 displaying evidence for PA variation across the burst duration [39].",
"The significance of this burst being from a thus far non-repeating source remains to be seen and encourages future polarized FRB observations.",
"The nearly $\\sim 100\\%$ linear fraction polarization of Source 2 and the much lower $>20\\%$ of Source 6 are largely consistent with the level of heterogeneity expected from the published sample.",
"However, we urge caution when interpreting the fractional polarizations of Source 6 as significant instrumental effects lead to substantial mixing of the Stokes parameters, yielding misleading linear and circular polarization fractions.",
"However, these polarized instrumental effects are highly dependent on position in the primary beam of the telescope and for Source 2 appear to be sub-dominant.",
"Nonetheless, analysis techniques for correcting the differential polarized response across the primary beam are currently in development and will be reported elsewhere.",
"The combined polarization information of these new repeating FRB sources will likely motivate revision of existing FRB emission models, many of which use polarization properties of FRB 121102 as key information in their initial construction.",
"An example of such a case is the young magnetar model of [37], which posits a dense, ionized nebula around a central engine giving rise to the high RM observed in FRB 121102.",
"A variant of this model [41] has since been developed and found to be consistent with burst properties reported for the initial batch of CHIME repeaters (see Paper I).",
"Moreover, this model allows for a range of RM values, dependent on the age of the magnetar and the specifics of its formation channel [36].",
"According to this scenario, the RM should decay monotonically with time [49] giving rise to a trend of younger, more active sources with higher RMs and older, less active sources with lower RMs.",
"The fact that CHIME seems to preferentially observe repeaters with relatively low RM values is at odds with this prediction but can possibly be explained by invoking a concomitant evolution of the peak flux frequency as a function of age, such that CHIME/FRB tends to detect older repeating sources [29], [41].",
"The detection of only one burst from FRB121102 in the CHIME band supports this assertion, but a much larger sample of repeater RMs over a wide range of bandwidths will be needed to make any robust claims.",
"Alternatively, it is also possible that FRB 121102's large RM is not from an associated nebula but from highly magnetized environment commonly found in dense, star-forming regions and/or around massive accreting black holes [42], as in the case of the Galactic-center magnetar [18].",
"In this framework, the circumstance of FRB 121102 residing in such environments can be due purely to chance, or is perhaps a reflection of the precursor's preference for these environments.",
"In either case, the observed RM is independent of the central FRB engine driving emission and therefore not directly correlated with properties relating to its emission.",
"This interpretation can be tested in the future with a much larger RM sample combined with interferometric localization and multi-wavelength observations capable of probing the local environment.",
"In addition, multi-epoch observations of newly discovered repeaters may provide evidence for secular evolution of the RM similar to that seen for FRB 121102 [42], [22].",
"The existence of such a trend would be difficult to reconcile in this framework and would more strongly support emission models that explicitly link evolution of local environment with the FRB engine itself."
],
[
"Repetition Rates",
"We calculated the repetition rates of these repeaters and compared them to the limits on single bursts observed by the Australian Square Kilometre Array Pathfinder [58].",
"Note that a repeat burst was found from FRB 171019, one of the sources from [58], in follow up observations at 820 MHz with the Green Bank Telescope [30].",
"In order to compare the repetition rates across different fluence limits, we scale the observed rates (based on the detections and exposure in Table REF ) by $S^{1.5}$ where $S$ is the fluence sensitivity (also specified in Table REF ) and the exponent of 1.5 reflects an assumed Euclidean spatial distrbution of sources.",
"We calculated the rates for upper and lower transits separately for sources that are circumpolar in the CHIME/FRB field of view.",
"Figure REF shows the observed and scaled rates of repetition for repeaters from Paper I, this work, and from [30] as well as scaled upper limits on repetition rates for the single bursts reported by [58].",
"We found that most of the repetition rates for repeaters are at or lower than the 1-$\\sigma $ upper limits from the ASKAP observations.",
"The scaled repetition rate of Source 5 is marginally higher than most of the upper limits from ASKAP observations.",
"Figure: Repetition rates of repeaters and upper limits for single bursts from ASKAP .",
"Observed rates are denoted as black circles, while rates scaled by fluence sensitivity are shown as red triangles.",
"Repeaters from Paper I are labelled “P1\".",
"Repeaters from Table are labelled as “Sx\".",
"The repetition rate for FRB 171019 is calculated from the GBT 820 MHz observations .",
"The rates (red triangles) and upper limits are scaled to a fluence limit of 1 Jy–ms using a scaling of S 1.5 S^{1.5} where SS is the sensitivity of the search.",
"The “U\" and “L\" suffixes for CHIME/FRB-detected repeaters denote the rates calculated from number of detections, exposure and sensitivity in the upper and lower transits, respectively, for circumpolar sources.",
"1-σ\\sigma error bars are shown assuming a Poisson distribution."
],
[
"Multi-wavelength Follow-up",
"We checked for any catalogued ionized regions [3], [24] or star-forming regions [4], [54] within the Milky Way galaxy that are coincident with the localization areas of the 9 new repeaters presented in Table REF , and found none.",
"We also estimated the maximum redshift ($z_{\\rm max}$ ) limit for all FRBs to identify plausible galaxy candidates.",
"To estimate the Milky Way (MW) contribution to the observed DMs, we considered the smaller of NE2001 [17] and YMW16 models' [63] predicted Galactic DM values, and added a MW halo contribution of 50 pc cm$^{-3}$ [51].",
"We then subtracted this value from the observed DMs to estimate the extragalactic DM contribution.",
"These excess DMs were converted to z$_{\\rm max}$ by using the DM-redshift relation: DM$_{\\text{ex}}$ $\\approx 900z$ [65].",
"These redshift estimates are approximate upper limits as we did not account for the DM of the host galaxy.",
"Using these $z_{\\rm max}$ values, we searched various catalogues of nearby galaxy clusters [1], [9], [62], [25] and found none within the localization regions of the FRBs.",
"As our repeating FRBs have poor localization and large estimated $z_{\\rm max}$ , the chance coincidence probability of finding even a massive star-forming galaxy is large.",
"This circumstance remains true for Source 4 despite possessing the lowest DM excess among our repeating FRB sample, DM$_{\\rm ex} \\approx {115} \\textrm { pc cm}^{-3}$ .",
"However, the largest RM excess in our sample, $|{\\rm RM}_{\\rm ex}|$ $\\approx $ 490 rad m$^{-2}$ for Source 6, likely suggests considerable host DM contribution and, hence, a nearby host galaxy.",
"This fact prompted us to look for plausible host galaxy candidates around Source 6, which has the second lowest extragalactic DM among Sources 1-9 (DM$_{\\rm ex}\\approx $ 140 pc cm$^{-3}$ ).",
"We estimated z$_{\\rm max} \\approx $ 0.16 for Source 6 and found a pair of face-on star-forming merging galaxies, SDSS J135159.17+480729.0 and SDSS J135159.87+480714.2, at spectroscopic redshift = 0.064 [2]; these galaxies possess the lowest $z_{\\rm max}$ among the catalogued galaxies within the localization region of Source 6, and either one can easily account for the observed excess DM.",
"Using the luminosity function of massive galaxies from [21], we estimate the density of galaxies with M$_{B} < -$ 20 to be $\\sim $ 0.0015 Mpc$^{-3}$ .",
"By assuming a pair merger fraction = 0.01 [7], we estimated the probability of finding a pair of massive merging galaxies by chance to be $< 5\\%$ within the Source 6 containment region for $z_{\\rm max} =$ 0.16.",
"Therefore, we consider these galaxies to be interesting candidates for the host of Source 6."
],
[
"Conclusions",
"We have reported on the discovery of nine new repeating FRB sources from CHIME/FRB.",
"Multiple bursts from these nine sources were collected during a $\\sim $ 1 yr period of telescope operation, and display complex spectro-temporal behavior similar to previously reported repeating FRB sources [27].",
"We found that the statistical properties of the data set presented in this work confirm the findings of Paper I: the DM distributions of repeating and thus far non-repeating CHIME/FRB sources are indistinguishable, while the distributions of temporal widths between the same populations are statistically different at the $4\\sigma $ level, with the repeating-source population producing larger widths.",
"As first discussed in Paper I, this observation likely indicates intrinsic differences between emission mechanisms and/or environments local to the sources that produce repeating and thus far non-repeating bursts.",
"Future studies of these properties from the $\\sim $ 700 bursts observed by CHIME/FRB will place stronger constraints on these statistical differences and are forthcoming.",
"Analysis of baseband data acquired for two of these sources yielded significant estimates of Faraday rotation, with RM = $-$ 20(1) rad m$^{-2}$ for Source 2 and RM = $-$ 499.8(7) rad m$^{-2}$ for Source 6.",
"The large RM difference between bursts associated with FRB 121102 and those reported here further indicates a uniqueness in the local magnetized environment of FRB 121102.",
"Furthermore, the relatively modest RMs discovered with CHIME/FRB do not yet indicate clear differences with the non-repeating sample.",
"This suggests that RM is perhaps a poor discriminant for repeating and thus far non-repeating FRB sources, but any robust statistical claim requires a larger sample.",
"When comparing to expected Galactic contributions, the low excess DM and large excess RM for Source 6 are suggestive a potentially nearby host galaxy with possible star formation; we searched for candidates and identified a pair of merging galaxies at redshift $z = 0.064$ as a tentative candidate host galaxy.",
"A more precise localization of Source 6 is needed to confirm or exclude this tentative hypothesis.",
"To date, CHIME/FRB has discovered a total of 18 repeating FRB sources [15] out the 20 currently known, with two other repeating sources found using the Arecibo Observatory [60] and ASKAP [30].",
"Each source presents an opportunity for arcsecond localization with large telescope arrays and, once achieved, multi-wavelength studies of the host galaxy and any associated intra-galactic environment.",
"Indeed, interferometric follow-up of several CHIME/FRB repeating sources are underway with the realfast backend at the VLA [31] and the European VLBI Network [35]; the EVN recently localized FRB 180916.J0158+65, discovered by CHIME/FRB (Paper I), with mas precision to a star-forming region in a nearby massive spiral galaxy [34].",
"We encourage additional community involvement, both in localizing these sources and targeting localized sources for constraining emission and activity across the electromagnetic spectrum.",
"We thank the Dominion Radio Astrophysical Observatory, operated by the National Research Council Canada, for gracious hospitality and useful expertise.",
"The CHIME/FRB Project is funded by a grant from the Canada Foundation for Innovation 2015 Innovation Fund (Project 33213), as well as by the Provinces of British Columbia and Québec, and by the Dunlap Institute for Astronomy and Astrophysics at the University of Toronto.",
"Additional support was provided by the Canadian Institute for Advanced Research (CIFAR), McGill University and the McGill Space Institute via the Trottier Family Foundation, and the University of British Columbia.",
"The Dunlap Institute is funded by an endowment established by the David Dunlap family and the University of Toronto.",
"Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.",
"The National Radio Astronomy Observatory is a facility of the National Science Foundation (NSF) operated under cooperative agreement by Associated Universities, Inc. M. B. is supported by a Doctoral Research Award from the Fonds de Recherche du Québec: Nature et technologies (FQRNT).",
"P. C. is supported by an FRQNT Doctoral Research Award.",
"V. M. K. holds the Lorne Trottier Chair in Astrophysics and Cosmology and a Canada Research Chair, receives support from a Discovery Grant of the Natural Sciences and Engineering Research Council of Canada (NSERC) and Herzberg Award, as well as from an R. Howard Webster Foundation Fellowship from CIFAR, and from the FRQNT Centre de Recherche en Astrophysique du Québec.",
"D. M. is a Banting fellow.",
"Z. P. is supported by a Schulich Graduate Fellowship.",
"M. D. is supported by a Killam Fellowship and receives support from an NSERC Discovery Grant, CIFAR, and from the FRQNT Centre de Recherche en Astrophysique du Québec.",
"B. M. G. acknowledges the support of NSERC through grant RGPIN-2015-05948, and of the Canada Research Chairs program.",
"SMR is a CIFAR Fellow and is supported by the NSF Physics Frontiers Center award 1430284.",
"P. S. is a Dunlap Fellow and an NSERC Postdoctoral Fellow.",
"FRB work at UBC is supported by an NSERC Discovery Grant and by CIFAR.",
"The baseband voltage system is funded in part by a John R. Evans Leaders Fund award, from the Canadian Foundation for Innovation, to I.H.S."
],
[
"Chance Coincidence Probabilities with large numbers of FRBs",
"Once a large sample of FRBs has been detected, the probability of identifying two bursts with similar DM and sky location can become non-negligible.",
"In such circumstances there are two questions that must be answered: Given $N$ FRBs detected by a survey, what is the probability that any two FRBs will lie within the same phase-space bin of dimension ($\\Delta \\mathrm {DM},\\,\\Delta \\mathrm {R.A.}\\cos (\\mathrm {Dec.}),\\,\\Delta \\mathrm {Dec.}$ )?",
"We refer to this as the “global probability.\"",
"If we detect two FRBs in the same ($\\Delta \\mathrm {DM},\\,\\Delta \\mathrm {R.A.}\\cos (\\mathrm {Dec.}),\\,\\Delta \\mathrm {Dec.}$ ) bin during our survey, what is the probability that they are physically unrelated?",
"We refer to this as the “individual probability.\"",
"Here we answer both questions for all CHIME/FRB repeaters found to date (i.e., from this work and from Paper I), given that CHIME/FRB has detected $\\sim 700$ FRBs during the observing period mentioned in Section .",
"A detailed analysis of the $\\sim 700$ FRBs is underway and will be published elsewhere.",
"Here we limit our discussion to the declinations and DM distribution of the FRBs in the CHIME/FRB survey; we marginalised over the R.A. dimension for reasons discussed below."
],
[
"Global probability",
"The first question is mathematically equivalent to the non-uniform birthday problem, i.e., what is the chance that two people in a group of $N$ share the same birthday.",
"The uniform case, in which each birthday (or each $\\Delta \\mathrm {DM},\\,\\Delta \\mathrm {R.A.}\\cos (\\mathrm {Dec.}),\\,\\Delta \\mathrm {Dec.}$ bin) are equally likely, is straightforward to solve.",
"The solution for the non-uniform case has been described in terms of recursion relations for a coincidence of two among $N$ objects [38], and for a general case of $m$ among $N$ objects [55].",
"However, for the case of CHIME/FRB, the parameter space is large enough that the recursion formulae and their approximations are not computationally tractable.",
"Instead, we simulated the current results of the CHIME/FRB survey using Monte Carlo methods in order to understand how often we misidentify repeaters with a given criteria.",
"Following the method in Appendix A of Paper I we estimated the detection probability distribution function from the $\\sim $ 700 FRBs found by the CHIME/FRB survey.",
"This calculation incorporated variations in exposure times, sensitivity at different declinations and DMs.",
"We assumed that the probability of detection is independent of R.A. and local time since the survey has been operating for over a year and any variation in R.A. would have been averaged out.",
"The probability distribution was smoothed by a Gaussian kernel with a 350 $\\mathrm {pc\\,cm^{-3}}$ DM scale and 10$^{\\circ }$ angular scale in declination.",
"Figure: Histogram of the MC trials as a function of the number of detected chance coincidences.",
"Both trials had the same localization criteria, ΔR.A.cos( Dec .",
")<1deg,Δ Dec .<1deg\\Delta \\mathrm {R.A.}\\cos (\\mathrm {Dec.})<1\\deg ,\\,\\Delta \\mathrm {Dec.}<1\\deg .",
"The left panel displays a DM criterion of Δ DM <10 pc cm -3 \\Delta \\mathrm {DM} < 10\\,\\mathrm {pc\\,cm^{-3}}, and the right panel displays a DM criterion of Δ DM <1 pc cm -3 \\Delta \\mathrm {DM} < 1\\,\\mathrm {pc\\,cm^{-3}}.",
"A stricter DM cutoff allows for fewer chance coincidence detections when sampling 700 independent FRBs.We sampled 700 independent FRBs from this probability distribution for 10$^4$ trials and determined how many FRBs would be counted as repeaters with the following criteria: I) $\\Delta \\mathrm {DM} < 10\\,\\mathrm {pc\\,cm^{-3}},\\,\\Delta \\mathrm {RA}\\cos (\\mathrm {Dec.})<1\\deg ,\\,\\Delta \\mathrm {Dec.}<1\\deg $ and II) $\\Delta \\mathrm {DM} < 1\\,\\mathrm {pc\\,cm^{-3}},\\,\\Delta \\mathrm {R.A.}\\cos (\\mathrm {Dec.})<1\\deg ,\\,\\Delta \\mathrm {Dec.}<1\\deg $ .",
"Figure REF displays a histogram of the trials as a function of number of detected chance coincidences for the two criteria.",
"For the first criterion, $\\sim $ 40% of the trials yield at least one set of independent FRBs falsely identified as coming from the same repeating source.",
"With the second criterion, the probability is $\\sim $ 4%.",
"We then determined the selection criteria to use, such that our repeaters have a chance coincidence probability ($p_{\\rm CC}$ ) of $\\sim $ 1 in 100 simulations.",
"We found: i) given a localization uncertainty of $\\sim $ 20', the DM tolerance has to be $\\Delta \\mathrm {DM} < 2.0\\,\\mathrm {pc\\,cm^{-3}},$ and ii) given a DM tolerance of $10.0\\,\\mathrm {pc\\,cm^{-3}}$ , the spatial localization has to be $\\Delta \\mathrm {R.A.}\\cos (\\mathrm {Dec.})<0.1\\deg ,\\,\\Delta \\mathrm {Dec.}<0.1\\deg $ .",
"These criteria were found before looking at the properties of the repeating FRB candidates.",
"We note, however, that the probability distribution was smoothed in both declination and DM, and assumed to be uniform in R.A. Any clustering or non-uniformity at scales smaller than the smoothing scales would increase the $p_{\\rm CC}$ .",
"Additionally, the results of our simulations imply that all future repeating CHIME/FRB candidates must have extremely spatially well-localized bursts to keep our chance coincidence probabilities low.",
"A source with minimal spatial uncertainties would also allow for more DM evolution across the bursts, which has been seen in FRB 121102 at the $\\sim $ 1% level [27], [28]."
],
[
"Individual Probabilities",
"The above simulations do not identify which bursts might be misidentified as repeaters.",
"If $p$ is the probability of detecting a burst in the ($\\Delta \\mathrm {DM},\\,\\Delta \\mathrm {R.A.}\\cos (\\mathrm {Dec.}),\\,\\Delta \\mathrm {Dec.}$ ) neighborhood of a known burst, the probability that one of the $n$ subsequent bursts will be in the same bin by coincidence is $1-(1-p)^n$ .",
"Since these repeaters are identified from a population of $N\\approx 700$ FRBs, we use a trials factor of 700 to calculate the final coincidence probabilities.",
"Here, we considered only the repeaters with two detected bursts, since the chance coincidence probabilities for three or more bursts occurring within the same phase-space bin are negligible for our current sample size.",
"We started with the probability distribution above, normalised over the full sky and DM range.",
"Table REF specifies the candidate repeaters with two bursts and the range of $\\Delta \\mathrm {DM},\\,\\Delta \\mathrm {R.A.}\\cos (\\mathrm {Dec.}),\\,\\mathrm {and}\\,\\Delta \\mathrm {Dec.}$ for the detected bursts.",
"For Sources 1 and 2, the trials-adjusted chance coincidence probability ($Np_{\\rm CC}$ ) remains below the 0.1% level, indicating robust association of repeat bursts despite the large CHIME/FRB sample.",
"However, $Np_{\\rm CC}$ = 10% for Source 8, which makes this association statistically less significant.",
"Future detections of bursts from Source 8 will confirm its status as a repeater.",
"Table: Chance Coincidence Probabilities for Repeating FRB Candidates with Two Bursts"
],
[
"Exposure Estimation",
"The timeline of the exposure of the CHIME/FRB system to each of the sources presented in this work is plotted in Figure REF .",
"The exposure is calculated for the transit of each source across the FWHM region of the synthesized beams at 600 MHz and includes transits in the interval from 2018 August 28 to 2019 September 30.",
"We excised transits from the reported exposure for which the RMS noise (shown in Figure REF ) was different by more than one standard deviation from the mean RMS noise in the above-mentioned interval.",
"The fraction of excised transits averaged about 5% for each source.",
"The reported exposures only include intervals in each transit for which the detection pipeline was fully operational.",
"The reduction in daily exposure for Sources 1 and 6 for a period of several months can thus be attributed to the failure of the computing node designated to process data for one of the four synthesised beams through which these sources transit.",
"Sources 3 and 4 have declinations greater than +70$^\\circ $ thereby allowing both upper and lower transits to be observable with the CHIME/FRB system.",
"However, there were no bursts detected during the lower transit of either of these sources.",
"Source 3 is particularly interesting since a significant fraction of its 90% confidence localization region is located between the FWHM regions of two synthesized beams during the upper transit (see Figure 1).",
"Therefore, for this source, all allowed sky locations which transit between the two beams have zero exposure.",
"This circumstance results in an average exposure for Source 3 over the entire positional uncertainty region of $55 \\pm 52$ hours, despite its high declination.",
"Figure: Timeline of CHIME/FRB's daily exposure to the new repeating FRB sources for upper and lower transits, if observable.",
"Days on which a burst was detected are indicated by solid lines.",
"The errors on the exposure are due to uncertainties in the source positions.",
"The increase in exposure time from its typical value for some of the days is due to the occurrence of two transits in the same solar day caused by the length of a solar and a sidereal day being slightly different.",
"The RMS noise is estimated using pulsars detected by CHIME/FRB for each sidereal day for which the telescope was operating with the same gain calibration.",
"The RMS noise for each pulsar is measured relative to the median over all days the pulsar was detected with the weighted average of measurements from several pulsars (the number of which is denoted by the marker colors) plotted here.Figure: Timeline of CHIME/FRB's daily exposure to the new repeating FRB sources.",
"(cont.)"
]
] | 2001.03595 |
[
[
"Long-Range Phonon Spin Transport in Ferromagnet-Nonmagnetic Insulator\n Heterostructures"
],
[
"Abstract We investigate phonon spin transport in an insulating ferromagnet-nonmagnet-ferromagnet heterostructure.",
"We show that the magnetoelastic interaction between the spins and the phonons leads to nonlocal spin transfer between the magnets.",
"This transfer is mediated by a local phonon spin current and accompanied by a phonon spin accumulation.",
"The spin conductance depends nontrivially on the system size, and decays over millimeter length scales for realistic material parameters, far exceeding the decay lengths of magnonic spin currents."
],
[
"Long-Range Phonon Spin Transport in Ferromagnet$-$ Nonmagnetic Insulator Heterostructures Andreas Rückriegel Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands Rembert A. Duine Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Department of Applied Physics, Eindhoven University of Technology, P.O.",
"Box 513, 5600 MB Eindhoven, The Netherlands We investigate phonon spin transport in an insulating ferromagnet$-$ nonmagnet$-$ ferromagnet heterostructure.",
"We show that the magnetoelastic interaction between the spins and the phonons leads to nonlocal spin transfer between the magnets.",
"This transfer is mediated by a local phonon spin current and accompanied by a phonon spin accumulation.",
"The spin conductance depends nontrivially on the system size, and decays over millimeter length scales for realistic material parameters, far exceeding the decay lengths of magnonic spin currents.",
"One of the main goals of the field of spintronics is achieving long-range spin transport through electrical insulators [1], [2].",
"Up to now, the main focus of this research is on magnetic insulators, in which the spin is carried by spin waves (or magnons) that are the elementary excitations of the magnetic order parameter.",
"However, it was shown recently that the magnetization dynamics in a ferromagnet can even inject a spin current into an adjacent nonmagnetic insulator [3], analogous to the spin pumping at the interface of a ferromagnet and a normal metal [4].",
"In this case, the spin is carried by transverse acoustic phonons with circular polarization [3], [5], [6], [7], [8].",
"A long-range exchange coupling that was observed in a ferromagnet-semiconductor hybrid structure was similarly interpreted in terms of spin transfer by circularly polarized phonons [9].",
"This raises the possibility of using phonon currents to transfer spin in (non-)magnetic insulators.",
"Indeed, An et al.",
"have found that phonons in nonmagnetic gadolinium-gallium garnet (GGG) mediate a coherent coupling between two yttrium-iron garnet (YIG) films that are half a millimeter apart [10].",
"In their experiment, An et al.",
"coherently excited the ferromagnetic resonance (FMR) of a YIG film with a microwave field, which affected the coherent FMR dynamics of a second YIG film separated from the first by a nonmagnetic GGG spacer.",
"This coupling is interpreted in terms of a phonon spin current, with a propagation length that surpasses the analogous magnon propagation length by several orders of magnitude because of the low acoustic damping in these materials.",
"In view of possible spintronics applications, it would be desirable to drive the (phonon) spin current electrically instead of via a microwave field, e.g.",
"by exciting incoherent magnons via an electronic spin accumulation in a metallic lead [4], [1], [2].",
"In this work, we calculate the incoherent spin transport in an insulating ferromagnet$-$ nonmagnet$-$ ferromagnet heterostructure that is driven by a difference in magnon spin accumulation between the magnets, as depicted in Fig.",
"REF .",
"We assume that the magnets are attached to metallic leads with fixed electronic spin accumulations $\\mu _{L/R}$ that act as chemical potentials for the magnons [11].",
"The metallic contacts are furthermore assumed to be small enough that they do not influence the phonon modes in the heterostructure.",
"We find that the magnetoelastic interactions between magnons and phonons lead to finite phonon spin accumulations in all three layers, and show that a local phonon spin current mediates nonlocal spin transfer between the two ferromagnets over macroscopic distances.",
"Figure: Heterostructure consisting of a nonmagnetic insulator of length LLsandwiched between two identical magnetic insulators of length dd.The left and right spins interact via the phonons in the nonmagnetic insulatorthat may support a finite spin current between the magnets.The magnon spin accumulations in the magnets are parametrized with magnon chemical potentials μ L/R \\mu _{L/R}.These chemical potentials are generated by metallic leads (not shown) that are attached to the magnets.We consider a heterostructure consisting of a nonmagnetic insulator of length $L$ sandwiched between two identical magnetic insulators of length $d$ , see Fig.",
"REF .",
"We also assume that the magnetizations in both magnets are parallel to each other and to the phonon propagation direction, which maximizes the magnetoelastic phonon pumping [3].",
"In this setup, the spin of the left and right magnets interacts with the elastic lattice displacement field $ \\mathbf {u}(\\mathbf {r}) $ via the magnetoelastic Hamiltonian [12] $ {\\cal H}_{\\rm me} = \\frac{1}{s^2} \\sum _{X=L,R} \\int _V \\textrm {d}^3r \\sum _{\\alpha \\beta } B_{\\alpha \\beta }s_{X,\\alpha }(\\mathbf {r}) s_{X,\\beta }(\\mathbf {r}) \\epsilon _{\\alpha \\beta }(\\mathbf {r}) ,$ where $\\alpha $ and $\\beta $ run over the three spatial components $x$ , $y$ , $z$ ; $s$ and $V$ are the saturation spin density and volume of both magnets, $\\mathbf {s}_{L/R}(\\mathbf {r})$ is the local spin density, and $\\epsilon _{\\alpha \\beta }(\\mathbf {r}) = \\frac{1}{2} \\left[ \\partial u_\\beta (\\mathbf {r}) / \\partial r_\\alpha + \\partial u_\\alpha (\\mathbf {r}) / \\partial r_\\beta \\right]$ is the linearized strain tensor.",
"$ B_{\\alpha \\beta } = \\delta _{\\alpha \\beta } B_\\parallel + (1-\\delta _{\\alpha \\beta }) B_\\bot $ are the magnetoelastic constants of an isotropic system.",
"Focusing on the uniform macrospin modes of both magnets, we write $ \\mathbf {s}_{L/R}(\\mathbf {r}) \\simeq \\sqrt{ \\frac{s}{2V} } ( \\mathbf {e}_+ \\psi _{L/R}^\\dagger + \\mathbf {e}_- \\psi _{L/R} ) + \\mathbf {e}_z s $ , where $\\mathbf {e}_\\pm = \\mathbf {e}_x \\pm i \\mathbf {e}_y$ are circularly polarized transverse basis vectors, and $\\psi _{L/R}$ and $\\psi _{L/R}^\\dagger $ are destruction and creation operators for the macrospin magnon modes that satisfy the bosonic commutation relations $[ \\psi _X, \\psi _{X^{\\prime }}^\\dagger ] = \\delta _{X,X^{\\prime }}$ .",
"In this case the magnetoelastic Hamiltonian (REF ) reduces to ${\\cal H}_{\\rm me} =&\\frac{ B_\\bot }{ \\sqrt{2sd} } \\biggl \\lbrace \\psi _L^\\dagger \\mathbf {e}_+ \\cdot \\left[ \\mathbf {u}(z=0) - \\mathbf {u}(z=-d) \\right]\\nonumber \\\\&+ \\psi _R^\\dagger \\mathbf {e}_+ \\cdot \\left[ \\mathbf {u}(z=L+d) - \\mathbf {u}(z=L) \\right]+ {\\rm H.c.} \\biggr \\rbrace .$ Here, $ \\mathbf {u}(z) = \\sqrt{\\frac{d}{V}} \\int \\textrm {d}x\\int \\textrm {d}y \\mathbf {u}(\\mathbf {r}) $ is the normalized average of the displacement field on the device cross section, which we assumed to be large compared to the thickness $d$ of the magnets.",
"Note that the magnetoelastic coupling (REF ) only enters via boundary conditions, and that the macrospin magnons only couple to the circularly polarized transverse phonon fields $u_\\pm (z) = u_\\mp ^\\dagger (z) \\equiv \\mathbf {e}_\\pm \\cdot \\mathbf {u}(z)$ .",
"These phonons actually carry the internal angular momentum, or phonon spin [3], [5], [6], [7], [8], $ L_z = \\int \\textrm {d}z\\, l_z(z) $ , with the phonon spin density $l_z(z)&= \\mathbf {e}_z \\cdot \\left\\langle \\mathbf {u}(z) \\times \\rho (z) \\partial _t \\mathbf {u}(z) \\right\\rangle \\\\&= \\rho (z)\\, \\textrm {Im} \\left\\langle u_+^\\dagger (z) \\partial _t u_+(z) \\right\\rangle ,$ where $\\rho (z)$ is the local mass density.",
"We can thus interpret the magnetoelastic Hamiltonian (REF ) in terms of spin transfer between the magnetic and elastic subsystems.",
"To investigate the spin transport in the heterostructure depicted in Fig.",
"REF , we employ semiclassical stochastic differential equations, which is, however, equivalent to a fully quantum nonequilibrium Green's function approach in the linear regime [13].",
"For an elastically isotropic medium, we therefore consider the equations of motion $\\partial _t u_+(z,t)&= \\rho ^{-1}(z) \\pi _+(z,t) , \\\\[.2cm]\\partial _t \\pi _+(z,t)&= \\partial _z \\left[ \\mu (z) \\partial _z u_+(z,t) \\right] - 2 \\eta (z) \\pi _+(z,t)\\nonumber \\\\[.1cm]&- \\frac{ 2B_\\bot }{ \\sqrt{2sd} }\\psi _L(t) \\left[ \\delta (z) - \\delta (z+d) \\right]\\nonumber \\\\&- \\frac{ 2B_\\bot }{ \\sqrt{2sd} }\\psi _R(t) \\left[ \\delta (z-L-d) - \\delta (z-L) \\right] ,$ where $\\pi _+(z,t)$ is the momentum density conjugate to $u_+(z,t)$ , and $\\rho (z)$ , $\\mu (z)$ , and $\\eta (z)$ are the local mass density, shear modulus and elastic damping constant.",
"For $-d<z<0$ and $L<z<L+d$ , they are given by $\\rho (z) = \\rho $ , $\\mu (z) = \\rho c_\\bot ^2$ , and $\\eta (z) = \\eta $ , whereas for $0<z<L$ they are $\\rho (z) = \\tilde{\\rho }$ , $\\mu (z) = \\tilde{\\rho } \\tilde{c}_\\bot ^2$ , and $\\eta (z) = \\tilde{\\eta }$ , with the transverse sound velocities $c_\\bot $ and $\\tilde{c}_\\bot $ in the magnetic and nonmagnetic insulators.",
"The macrospin magnon modes are governed by $& ( 1+i\\alpha ^G +i\\alpha ^{\\rm sp} ) i\\partial _t \\psi _L(t) =\\left( \\omega _{\\textrm {FM}} + i\\alpha ^{\\rm sp} \\frac{\\mu _L}{\\hbar } \\right) \\psi _L(t)\\nonumber \\\\&- h_L^G(t) - h_L^{\\rm sp}(t)+ \\frac{ B_\\bot }{ \\hbar \\sqrt{2sd} } \\left[ u_+(0,t) - u_+(-d,t) \\right],\\\\& ( 1+i\\alpha ^G +i\\alpha ^{\\rm sp} ) i\\partial _t \\psi _R(t) =\\left( \\omega _{\\textrm {FM}} + i\\alpha ^{\\rm sp} \\frac{\\mu _R}{\\hbar } \\right) \\psi _R(t)\\nonumber \\\\&- h_R^G(t) - h_R^{\\rm sp}(t)+ \\frac{ B_\\bot }{ \\hbar \\sqrt{2sd} } \\left[ u_+(L+d,t) - u_+(L,t) \\right] .$ Here, $\\omega _{\\rm FM}$ is the FMR frequency, $\\alpha ^G$ the Gilbert damping constant, and $\\alpha ^{\\rm sp} = g^{\\uparrow \\mathrel {{-1mu}}\\downarrow }/4\\pi s d$ the Gilbert damping enhancement due to spin pumping to the metallic leads, with the spin-mixing conductance $g^{\\uparrow \\mathrel {{-1mu}}\\downarrow }$ [14].",
"Furthermore, $ h_{L/R}^{G/{\\rm sp}}(t) = (1/2\\pi )\\int \\textrm {d}\\omega e^{-i\\omega t} h_{L/R}^{G/{\\rm sp}}(\\omega ) $ are noise fields that satisfy the quantum fluctuation-dissipation theorems [13] $\\left\\langle h_{L/R}^G(\\omega ) h_{L/R}^{G*}(\\omega ^{\\prime }) \\right\\rangle =&2\\pi \\delta (\\omega -\\omega ^{\\prime }) \\alpha ^G \\omega \\coth \\left( \\frac{ \\hbar \\omega }{ 2 k_B T } \\right) ,\\\\\\left\\langle h_{L/R}^{\\rm sp}(\\omega ) h_{L/R}^{{\\rm sp}*}(\\omega ^{\\prime }) \\right\\rangle =&2\\pi \\delta (\\omega -\\omega ^{\\prime })\\frac{\\alpha ^{\\rm sp}}{\\hbar } \\left( \\hbar \\omega - \\mu _{L/R} \\right)\\nonumber \\\\&\\times \\coth \\left( \\frac{ \\hbar \\omega - \\mu _{L/R} }{ 2 k_B T } \\right) ,$ with the magnon temperature $T$ and the left and right magnon chemical potentials $\\mu _{L/R}$ .",
"We do not consider a similar noise field for the phonons or temperature gradients because we focus on genuine long-range interaction between the two macrospin modes.",
"This interaction is mediated by the phonons that are driven by the magnon distributions.",
"In contrast, thermal phonons only add to the damping of the two macrospin modes when there are local temperature gradients across the magnet$-$ nonmagnet interfaces.",
"The elastic equations of motion (REF ) are supplemented by elastic continuity boundary conditions at the interfaces: $u_+ ( 0^+,t ) =& u_+ ( 0^-,t ) , \\\\u_+ ( L^+,t ) =& u_+ ( L^-,t ) .$ At the interfaces and boundaries, we furthermore impose momentum conservation, yielding $\\mu (0^+) u_+^{\\prime }(0^+,t) - \\mu (0^-) u_+^{\\prime }(0^-,t) - \\frac{ 2B_\\bot }{ \\sqrt{2sd} } \\psi _L(t) =& 0 ,\\\\\\mu (L^+) u_+^{\\prime }(L^+,t) - \\mu (L^-) u_+^{\\prime }(L^-,t) + \\frac{ 2B_\\bot }{ \\sqrt{2sd} } \\psi _R(t) =& 0 ,\\\\\\mu (-d^+) u_+^{\\prime }(-d^+,t) + \\frac{ 2B_\\bot }{ \\sqrt{2sd} } \\psi _L(t) =& 0 ,\\\\- \\mu (L+d^-) u_+^{\\prime }(L+d^-,t) - \\frac{ 2B_\\bot }{ \\sqrt{2sd} } \\psi _R(t) =& 0 ,$ where $u_+^{\\prime }(z,t)=\\partial _z u_+(z,t)$ .",
"In a stationary state, we may write $ u_+(z,t) = (1/2\\pi )\\int \\textrm {d}\\omega e^{-i\\omega t} u_+(z,\\omega ) $ and $ \\psi _{L/R}(t) = (1/2\\pi )\\int \\textrm {d}\\omega e^{-i\\omega t} \\psi _{L/R}(\\omega ) $ ; then the solutions of the coupled magnetoelastic equations of motion (REF ) and (REF ) are given by $u_+(z,\\omega ) ={\\left\\lbrace \\begin{array}{ll}A(\\omega ) e^{ i k(\\omega ) ( z + d) } + B(\\omega ) e^{ - i k(\\omega ) z } , & -d < z < 0 , \\\\C(\\omega ) e^{ i \\tilde{k}(\\omega ) z } + D(\\omega ) e^{ i \\tilde{k}(\\omega ) ( L - z ) } , & 0 < z < L , \\\\E(\\omega ) e^{ i k(\\omega ) ( z - L ) } + F(\\omega ) e^{ i k(\\omega ) ( L + d - z ) } , & L < z < L + d ,\\end{array}\\right.",
"}$ and $\\psi _L(\\omega ) =&g_L(\\omega ) \\left\\lbrace - h_L^G(\\omega ) - h_L^{\\rm sp}(\\omega ) + \\frac{ B_\\bot }{ \\hbar \\sqrt{2sd} }\\left( e^{ i k(\\omega ) d } - 1 \\right) \\left[ A(\\omega ) - B(\\omega ) \\right] \\right\\rbrace ,\\\\\\psi _R(\\omega ) =&g_R(\\omega ) \\left\\lbrace - h_R^G(\\omega ) - h_R^{\\rm sp}(\\omega ) + \\frac{ B_\\bot }{ \\hbar \\sqrt{2sd} }\\left( e^{ i k(\\omega ) d } - 1 \\right) \\left[ E(\\omega ) - F(\\omega ) \\right] \\right\\rbrace ,$ with the phonon wave vectors $ c_\\bot k(\\omega ) = \\sqrt{ \\omega ^2 + 2 i \\eta \\omega } $ and $ \\tilde{c}_\\bot \\tilde{k}(\\omega ) = \\sqrt{ \\omega ^2 + 2 i \\tilde{\\eta } \\omega } $ , and the noninteracting magnon Green's functions $g_{L/R}(\\omega ) = \\frac{1}{ \\omega - \\omega _{\\textrm {FM}} + i \\alpha ^G \\omega + i \\alpha ^{\\rm sp} \\left( \\omega - \\mu _{L/R} / \\hbar \\right) } .$ The elastic amplitudes $A(\\omega ),\\ldots ,F(\\omega )$ are determined by the boundary conditions (REF ) and (REF ); as the explicit, analytical expressions for these amplitudes are rather involved, we omit them here.",
"Equipped with the full, analytical solution of the coupled magnetoelastic dynamics (REF ) and (REF ), we proceed to calculate various observables characterizing the spin transport through the heterostructure.",
"Throughout the remainder of this Letter, we use parameters of YIG for the magnets and GGG for the nonmagnet [10], [12], [15], i.e., $s = 7.5/\\textrm {nm}^{3}$ , $\\rho = 5170\\, \\textrm {kg}/\\textrm {m}^{3}$ , $c_\\bot = 3843\\, \\textrm {m}/\\textrm {s}$ , $B_\\bot = 6.96\\times 10^{5}\\, \\textrm {J}/\\textrm {m}^{3}$ , and $ \\alpha ^G = 9 \\times 10^{-5} $ , as well as $\\tilde{\\rho } = 7080\\, \\textrm {kg}/\\textrm {m}^{3}$ and $\\tilde{c}_\\bot = 3530\\, \\textrm {m}/\\textrm {s}$ .",
"The metallic leads are taken to be platinum, so that $ g^{\\uparrow \\mathrel {{-1mu}}\\downarrow } = 5\\, {\\rm nm}^{-2}$ [16].",
"Furthermore, we assume room temperature, $ T = 300\\, {\\rm K}$ , and fix the FMR frequency to $\\omega _{\\rm FM}/ 2\\pi = 5.49\\, \\textrm {GHz} $ .",
"For these parameters, the wavelengths of phonons in the magnets and the nonmagnet are $\\lambda = 2\\pi c_\\bot / \\omega _{\\rm FM} = 700\\, \\textrm {nm}$ and $\\tilde{\\lambda } = 2\\pi \\tilde{c}_\\bot / \\omega _{\\rm FM} = 643\\, \\textrm {nm}$ respectively.",
"Lastly, the phonon damping is taken to be $ \\eta / \\omega _{\\rm FM} = \\tilde{\\eta } / \\omega _{\\rm FM} = 6 \\times 10^{-5} $ , corresponding to the damping rate measured in Ref.",
"An2019 for a heterostructure of the same materials at room temperature.",
"The spin of the left and right macrospin is given by $ S_{R/L,z}(t) = \\hbar \\left[ sV + \\frac{1}{2} - \\left\\langle | \\psi _{R/L}(t) |^2 \\right\\rangle \\right] $ .",
"To obtain the spin current from the left to the right magnet that is detected in the right lead, we consider the spin lost by the right macrospin to this lead: $\\left( \\partial _t S_{R,z} \\right)^{\\rm sp} =&2 \\alpha ^{\\rm sp} \\textrm {Re}\\left\\langle \\psi _R^*(t) \\left[ i\\hbar \\partial _t - \\mu _R \\right] \\psi _R(t)\\right\\rangle \\\\=&- I_{R} - I_{L\\rightarrow R} .$ Here, $I_R$ is a local contribution, i.e., it only depends on the noise distribution of the right magnet itself.",
"On the other hand, $I_{L\\rightarrow R}$ is a genuine nonlocal spin current from the left to the right magnet that is mediated by the phonons; it is explicitly given by $I_{L\\rightarrow R} =&\\int \\frac{\\textrm {d}\\omega }{2\\pi } {\\cal T}_{L\\rightarrow R}(\\omega )\\nonumber \\\\&\\times \\left[f_B\\left( \\frac{ \\hbar \\omega - \\mu _R }{ k_B T } \\right) -f_B\\left( \\frac{ \\hbar \\omega - \\mu _L }{ k_B T } \\right)\\right] ,$ with the Bose function $f_B(x) = 1/\\left( e^x - 1 \\right)$ and the transmission function ${\\cal T}_{L\\rightarrow R}(\\omega ) =&2 \\frac{ \\left(\\alpha ^{\\rm sp}\\right)^2 B_\\bot ^2 }{ \\hbar ^3 s d }\\left( \\hbar \\omega - \\mu _R\\right) \\left( \\hbar \\omega - \\mu _L\\right)\\nonumber \\\\&\\times \\left| g_R(\\omega )\\left( e^{ i k(\\omega ) d } - 1 \\right)\\frac{ \\partial \\left[ E(\\omega ) - F(\\omega ) \\right] }{ \\partial h_L^{\\rm sp}(\\omega ) }\\right|^2 .$ For sufficiently small biasing, $\\mu _{L/R} \\ll \\hbar \\omega _{\\rm FM}$ , we may further linearize the Bose functions.",
"Then the spin current (REF ) reduces to $ I_{L\\rightarrow R} = \\sigma \\left( \\mu _R - \\mu _L \\right) $ , with the nonlocal spin conductance $ \\sigma = \\frac{1}{4 k_B T} \\int \\frac{\\textrm {d}\\omega }{2\\pi }\\frac{ {\\cal T}_{L\\rightarrow R}(\\omega ) }{ \\sinh ^2\\left( \\frac{ \\hbar \\omega }{ 2 k_B T } \\right) } .$ An intensity plot of this conductance is shown in Fig.",
"REF as function of the lengths $d$ and $L$ of the magnetic and nonmagnetic insulators.",
"Figure: Nonlocal spin conductance () as a function ofthe lengths dd and LL of the magnetic and nonmagnetic insulators,for the parameters stated in the main text.The lengths are given in units of half wavelengthsof phonons at the ferromagnetic resonance frequency in the respective material.The conductance exhibits pronounced minima for $ d = 2 n \\times \\lambda /2 $ , where $n=0,1,2,\\ldots $ The reason for this is that the macrospin exerts forces of equal magnitude but opposite direction on the elastic field at the interfaces, see Eqs.",
"(REF ).",
"Thus the excitation of phonons is favored when the length of the magnet is close to an odd number of phonon half-wavelengths and suppressed when it is close to an even number.",
"As function of the length $L$ of the nonmagnetic insulator the conductance also shows a modulation, with local maxima along the lines $ L /( \\tilde{\\lambda }/2 ) + 2 d /( \\lambda /2 ) = m$ , where $m = 1, 2, \\ldots $ This corresponds to standing waves for the whole heterostructure.",
"There is a slight deviation of the maxima from these straight lines because the hybridization of magnon and phonon modes leads to an anticrossing that shifts the phonon frequency away from the FMR frequency.",
"The decay of the conductance for increasing size $d$ of the magnets is explained by the decay of both the macrospin-phonon coupling [see Eq.",
"(REF )], and the Gilbert-damping enhancement $\\alpha ^{\\rm sp}\\propto 1/d$ that couples the magnets to the leads, as a function of the size of the magnets.",
"The behavior of the conductance for large nonmagnetic insulators is displayed in Fig.",
"REF , on a logarithmic scale.",
"Figure: Decay of the spin conductance () as a function of the length LLof the nonmagnetic insulator in units of half-wavelengths,for d=2λ/7=200 nm d=2\\lambda /7 = 200\\,{\\rm nm}and the parameters stated in the main text.Solid lines denote the envelope of the conductance curve;in the shaded area the conductance oscillates rapidly between the minimal and maximal values given by the envelope,see also Fig.",
".The dashed line is an exponential fit with decay length≈2650×λ ˜/2≈0.85 mm \\approx 2650 \\times \\tilde{\\lambda }/2 \\approx 0.85\\,{\\rm mm}.Similar to magnon conductances in magnetic insulators [1], [11], the conductance exhibits a power law decay for small $L$ , and eventually decays exponentially for large $L$ .",
"However, note that the crossover occurs at $L \\approx 1000 \\times \\tilde{\\lambda }/2 \\approx 0.3\\,{\\rm mm}$ for our parameters, while the characteristic decay length in the exponential regime is $ 0.85\\,{\\rm mm}$ .",
"Both of these length scales are almost two orders of magnitude larger than the analogous length scales of magnon spin currents in YIG [1], [11].",
"Another difference to magnons is that because of the constructive interference for standing waves in the heterostructure (see Fig.",
"REF ), the phonon conductance rapidly oscillates as a function of distance in the power-law decay regime.",
"To further substantiate our claim of long-range phonon spin transport, we note that away from the interfaces, the phonon spin density (REF ) satisfies a continuity equation: $\\partial _t l_z(z,t) + \\partial _z j_z(z,t)&= - 2 \\eta (z) l_z(z,t) ,$ where $ j_z(z,t) = - \\textrm {Im} \\left\\langle u_+^*(z,t) \\partial _z \\left[ \\mu (z) u_+(z,t) \\right] \\right\\rangle $ is the local phonon spin current density.",
"Both the phonon spin and spin current densities are shown in Fig.",
"REF .",
"Because of the spin-transfer via the magnetoelastic interaction there is a phonon spin accumulation in all three layers, see Fig.",
"REF (a).",
"For the same reason the phonon spin current shown in Fig.",
"REF (b) is finite even in the absence of biasing, $\\mu _L=0=\\mu _R$ .",
"However, this current is symmetric around zero and consequently does not lead to a net spin transfer between the magnets, in contrast to the biased setup with $\\mu _L-\\mu _R\\ne 0$ .",
"Note also that for the parameters shown in Fig.",
"REF that correspond to a standing wave, i.e., a maximum of the conductance, the phonon spin density is of the same order as the spin density of the macrospin magnons in the magnets.",
"For a minimum of the conductance or in the exponentially decaying long-range regime on the other hand, the phonon spin density is generally at least two orders of magnitude smaller than the magnon spin.",
"Figure: (a) Phonon spin and (b) phonon spin current densities for μ L =0=μ R \\mu _L=0=\\mu _R (solid lines),and μ L =0.01×ℏω FM \\mu _L=0.01 \\times \\hbar \\omega _{\\rm FM} and μ R =0\\mu _R=0 (dashed lines)for the parameters stated in the main text.The system size is set to d=2λ/7=200 nm d=2\\lambda /7=200\\,{\\rm nm} and L=3λ ˜/7=275.6 nm L= 3\\tilde{\\lambda }/7=275.6\\,{\\rm nm}.The shaded region denotes the magnets.The small jumps in l z l_z at the interfaces are due to the different mass densities of the magnetsand the nonmagnet.On the other hand,the jumps in j z j_z are not only caused by the change of shear modulusbut also by the magnetoelastic coupling, see Eqs.",
"()For comparison, the macrospin magnon spin densities in both magnets are ≈-600×2ℏ/λ\\approx -600\\times 2\\hbar /\\lambda .Lastly, one can show that the total spin $J_z=L_z+S_{L,z}+S_{R,z}$ satisfies the equation of motion $\\partial _t J_z(t)=&-2 \\int \\textrm {d}z \\eta (z) l_z(z,t)\\nonumber \\\\&- 2\\hbar \\sum _{X=L,R} \\textrm {Im}\\left\\langle \\psi _X^*(t) \\left[ \\alpha ^G \\partial _t \\psi _X(t) - h_X^G(t) \\right] \\right\\rangle \\nonumber \\\\&- 2\\hbar \\sum _{X=L,R} \\textrm {Im}\\left\\langle \\psi _X^*(t) \\left[ \\alpha ^{\\rm sp} \\partial _t \\psi _X(t) - h_X^{\\rm sp}(t) \\right] \\right\\rangle \\nonumber \\\\&- 2 \\alpha ^{\\rm sp} \\sum _{X=L/R} \\mu _X \\langle \\left| \\psi _X(t) \\right|^2 \\rangle .$ Therefore the total spin is conserved in the absence of dissipation ($\\eta =0=\\alpha ^{G/{\\rm sp}}$ ), and the damping and noise terms model the loss of spin to the environment consisting of nonuniform magnons, electronic leads, thermal phonons and the rigid-body dynamics of the lattice.",
"In consequence, the nonlocal spin current (REF ) between the two magnets must be mediated by the local phonon spin current density (REF ).",
"Utilizing the phonon degree of freedom in (non-)magnetic insulators provides a novel route for long-range spin transport.",
"We have shown that there is a finite phonon spin accumulation as well as a finite phonon spin current in an insulating magnet$-$ nonmagnet$-$ magnet heterostructure driven by the magnon distributions in the magnets.",
"If those magnon distributions are not in equilibrium with each other, there is a net spin current mediated by the phonons.",
"For realistic material parameters, we have found that this nonlocal spin current decays over millimeter length scales that are significantly larger than the decay lengths of magnonic spin currents in magnetic insulators.",
"A direct comparison of the magnitude of the phonon and magnon spin currents is less straightforward; however, since the magnon spin current is carried by a continuum of thermal magnons while the phonon spin current is driven by the single FMR mode, we expect this phonon spin current to be small compared to the magnon spin current at room temperature.",
"Experimentally, the phonon spin current is detectable electrically via the inverse spin Hall effect in the metallic leads [1], [11], [4].",
"The predicted phonon spin accumulation should be observable with Brillouin light scattering [7].",
"Because the spin transfer from the magnons to the phonons depends on a coherent magnon-phonon interconversion process at the interfaces, the nonlocal spin transport is particularly sensitive to the length of the magnets, and to a lesser extent also to the length of the nonmagnet.",
"In particular, spin transport is almost completely prohibited when the length of the magnets corresponds to an integer multiple of the phonon wavelength at the ferromagnetic resonance frequency.",
"This makes it possible to switch between a spin-conducting and a spin-nonconducting state by changing the ferromagnetic resonance frequency of the magnets, e.g.",
"via an external magnetic field.",
"While we have shown that long-range spin transport via acoustic phonons is possible, additional research is required to understand the effect of the phonon spin and angular momentum conservation in spin Seebeck experiments [17], and to understand the relaxation of the phonon spin beyond phenomenological models.",
"We acknowledge useful discussions with Simon Streib and Gerrit E. W. Bauer.",
"This work is supported by the European Research Council via Consolidator Grant No.",
"725509 SPINBEYOND.",
"R.D.",
"is a member of the D-ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).",
"This research was supported in part by the National Science Foundation under Grant No.",
"NSF PHY-1748958."
]
] | 2001.03462 |
[
[
"Optimal Sensor Position for a Computer Mouse"
],
[
"Abstract Computer mice have their displacement sensors in various locations (center, front, and rear).",
"However, there has been little research into the effects of sensor position or on engineering approaches to exploit it.",
"This paper first discusses the mechanisms via which sensor position affects mouse movement and reports the results from a study of a pointing task in which the sensor position was systematically varied.",
"Placing the sensor in the center turned out to be the best compromise: improvements over front and rear were in the 11--14% range for throughput and 20--23% for path deviation.",
"However, users varied in their personal optima.",
"Accordingly, variable-sensor-position mice are then presented, with a demonstration that high accuracy can be achieved with two static optical sensors.",
"A virtual sensor model is described that allows software-side repositioning of the sensor.",
"Individual-specific calibration should yield an added 4% improvement in throughput over the default center position."
],
[
"Introduction",
"In the wake of Douglas Engelbart's Mother of All Demos, the computer mouse has grown into one of the most engineered input devices [20].",
"Besides proprietary work, there has been published research on almost every factor imaginable: the shape of the device [17], [19], [23], [25], [26], [30], [34], [39], [41], its weight [16], the control-to-display (CD) gain function [13], [14], [50], the resolution of the displacement sensor [6], [42], [14], [44], and many more.",
"Yet it is fascinating to observe that for one factor, the placement of the sensor in the bottom of the device, the proper examination has been overlooked in academic research.",
"This factor turns out to have a strong but exploitable effect on users' pointing performance.",
"Figure: When a user operates a mouse, the mouse moves and rotates, and the resulting trajectory is affected by the position of the displacement sensor.When it is closer to the front, the increased radius of the arc results in the larger horizontal cursor displacement.Vertical cursor displacement stays the same irrespective of sensor position.To understand why sensor position matters, we need to revisit a misconception about mouse movement.",
"A mouse is a transducer that converts a planar movement into a two-dimensional (2D) cursor displacement.",
"We might thus readily believe that there is a one-to-one mapping between the physical and cursor movement, or at least that the mapping is modulated by the CD gain function.",
"That is not, in fact, the full picture.",
"The position of the displacement sensor has a covert effect on cursor movement.",
"The planar motion of a mouse, when held in hand, is produced as a combination of two-dimensional translation and one-dimensional rotation.",
"Though users may perceive translational movement of the cursor, joint rotation adds inadvertent rotational motion of the mouse body.",
"The wrist joint rotates about 15–25 degrees (radial–ulnar deviation) during mouse pointing [11], [16], [29], [48].",
"Also, the shoulder joint rotates, producing rotation around the elbow joint (mediolateral deviation).",
"Firstly, the sensor ignores rotation in motion, which distorts the cursor trajectory [32].",
"More interestingly, when the displacement sensor is placed further from or closer to the pivot of rotation, different paths result, as shown in Figure REF .",
"The differences can be large.",
"Our study revealed that a sensor positioned at the front produced almost twice the horizontal cursor displacement than one at the rear did (this is discussed further on).",
"The discussion surrounding sensor position is riddled with unproven hypotheses and misconceptions.",
"There is only a single academic paper on the topic, from 1989: Verplank and Oliver compared the task completion time in a maze-tracing task ($N=5$ ) with three mice prototypes, varying their shape and the position of ball sensors [47].",
"They concluded that a sensor position toward the front is favorable.",
"The designers of the Microsoft Mouse took this finding to indicate “a dramatic (performance) advantage in moving the ball to the front of the mouse” [38], explaining the advantage as being due to “increasing the apparent moment of inertia as the mouse is pivoted from [the] elbow or heel of the hand” [47].",
"However, this conclusion is potentially flawed as well as outdated.",
"Their study confounded the position of the sensor with the shape of the mouse, which we now know affects performance.",
"Moreover, no statistical analysis was reported.",
"Secondly, they used a low-CPI ball mouse of that era.",
"A modern optical mouse sensor operates like a camera with high speed but low resolution.",
"It computes translational displacement by calculating the cross-correlation between successive images.",
"Importantly, resolution, measured in Counts per Inch, or CPI, has improved dramatically.",
"While the mice of the 1980s have 100–400 CPI, an off-the-shelf mouse in 2019 exceeds 800 CPI, and a high-grade sensor can easily reach above 10K CPI, which is far beyond the human limit [10].",
"Within internet communities, two different hypotheses exist [2], [3], [4], [5].",
"According to the first, a sensor beneath the index finger performs best, because it reacts more swiftly (due to larger radius), and experience can be transferred from precise pen and touch manipulations.",
"The second hypothesis holds that the center is best, because an object can be more easily manipulated, supposedly, when force is exerted at its center of gravity.",
"In any case, both hypotheses have not yet been verified.",
"Manufacturers have taken very different positions on the topic, as Figure REF attests.",
"The center position seems to dominate, but the front and back also are represented.",
"To the best of our knowledge, this is the first paper to revisit the topic with an eye on the modern mouse.",
"Besides informing manufacturers and end-users about the matter, we see two other, more scientific, motivations.",
"Firstly, it is necessary to return to empirical research into the effect of sensor position with a study that is adequately reported upon and adheres to today's methodological standards.",
"Secondly, the optical mouse allows designing a device that can change the position of the sensor from the software side.",
"We present constructions for such a device, which we exploited for systematically finding the most suitable position for an individual user.",
"The paper is organized as follows.",
"After discussing relevant literature, we introduce the construction of two variable-sensor-position mice, one based on a sliding structure for the mechanical part, and another using a virtual sensor model with two sensors that uses a signal fusion technique.",
"The virtual sensor allows changing the sensor position via software.",
"Using our system, we estimate the effect of sensor position on user performance in pointing.",
"Finally, we offer two calibration methods for sensor position optimization.",
"The two takeaways for practitioners are 1) that placing the sensor in the center is the best compromise for regular users and 2) that a variable-sensor-position mouse can further improve the pointing throughput from that with a fixed center position.",
"Figure: Commercial mice (shown to scale) exhibit a wide distribution of displacement sensor designs.",
"Note that Ninox Astrum is equipped with a slider for a variable sensor position."
],
[
"Related Work",
"The engineering and ergonomics of the computer mice have been a topic of HCI research for decades.",
"We consider three core device-related factors here: shape, weight, and transfer function.",
"Reliable results have been obtained that link changes in these factors to ergonomics and movement performance.",
"Additionally, we review previous attempts of integrating two sensors on a mouse, which we employed to implement the variable-sensor-position mouse."
],
[
"Effect of shape",
"Shape has been the most intensively studied factor.",
"Continuous use of a mouse, especially in dragging, is known to increase carpal tunnel pressure [30] and potentially lead to carpal tunnel syndrome.",
"Slanted or vertical designs, also known as an ergonomic mouse, are known to reduce pronation of the wrist and muscle use but produce adverse effects on pointing performance in general [17], [39], [23], [41].",
"Some research has shown that a vertical (“ergonomic”) shape can indeed be beneficial ergonomically [25], and performance approaches the levels of a regular mouse design with practice [26].",
"Other attempts, such as a pen-like design [19] or alternative button designs [34], have been tried in efforts to minimize fatigue levels with mouse use.",
"Besides the slant angle, Isokoski and Raisamo [28] studied the effect of mouse size, interface (USB or PS/2), and sensor technology (ball or optical).",
"They found no practical performance differences among these.",
"However, it was later discovered that the size of a regular mouse is not suitable for children [27].",
"Hand size must be considered in choosing a comfortable mouse.",
"Shape-changing mouse designs have been proposed also, such as the Inflatable mouse [31] and the Adaptive mouse [46].",
"While we look at sensor position, not shape, we share the goal of adapting this factor to the user."
],
[
"Effect of weight",
"A mouse weighs from about 52 g (e.g., Cooler Master MM710) to over 150 g (e.g., Logitech G602 with battery).",
"This is another important specification for a choice of mouse and, similarly, interacts with biomechanical considerations.",
"Some models provide extra ballast so that a user can tune the weight distribution.",
"However, we could find only a few academic studies on this topic.",
"Chen et al.",
"[16] found that mouse weight affects the range of motion (ROM) of the wrist in a rapid targeting task.",
"A V-shaped relationship, centered at 130 g, was found between weight and ROM.",
"Excessively light or heavy mice tend to induce ulnar-side shifts over time, which results in rapid radial-side corrective motion.",
"Cabeças [12] found that a heavy mouse may increase friction between the mouse and the pad, thereby necessitating more muscle effort.",
"Still, no rigorous performance study of the effect of weight on performance has been conducted, so the relationship between weight and performance remains unresolved."
],
[
"Effect of transfer function",
"The CD gain function, also known as the transfer function, maps between the mouse motion and the cursor movement.",
"The goal has been to find the “best” transfer function – i.e., one that maximizes throughput [22] in pointing.",
"When a constant gain function is used, the ratio between the resolution of the mouse and screen has a single gain value.",
"An appropriate gain value is a level that does not cause clutching (or rely on imprecise limbs for control) and involves the correct quantization approach [14].",
"If the gain is set too low, the mouse cursor moves so slowly that a one-stroke mouse movement is not enough to reach the target; therefore, clutching arises.",
"Limb precision imposes a motor control limit, which is estimated to be around 700–1400 units per inch in the case of a mouse [10].",
"Quantization problems occur when the sensor resolution is not high enough to address one unit on the display.",
"Commonly, modern mice are in the 800–1000 CPI range, and display resolution is 72–100 PPI (pixels per inch), though some high-density displays (e.g., the Apple Retina Display) may reach 300+ PPI.",
"A proper gain should take both the device and display resolution into consideration.",
"Non-constant gain functions, often called pointer acceleration, have been investigated for solving the clutching and the quantization problem at the same time.",
"The general idea is to increase the CD gain in accordance with the speed of motion.",
"Thus, a cursor can jump toward a distant target with high-speed motion and address a small target at low speed.",
"Each operating system (MacOS, Windows, Linux, etc.)",
"provides a uniquely shaped gain function [13].",
"Beyond those default presets, a recent work named AutoGain [33] introduced an adaptive method for personalizing gain function."
],
[
"Mouse with two sensors",
"Having two sensors on a mouse allows capturing the final missing piece in 2D motion: rotation.",
"As stated before, conventional mouse sensors ignore rotation: the ball-and-wheel mechanism cannot capture it by principle, and the optical sensors ignore it by design (in fact, detecting rotation on a single optical sensor is possible [40]).",
"MacKenzie et al.",
"proposed the earliest prototype mouse with two balls in 1997 [37] to support three degree-of-freedom (3DoF) interaction, e.g., translate and rotate an object at the same time.",
"A series of studies followed [8], [7], [21], [24], [32], [40].",
"However, 3DoF manipulation was better supported by a wheel [8].",
"There is evidence that a major mouse company, Logitech, tested a two-sensor mouse for 3DoF support [1], but no mouse was successfully commercialized so far [36].",
"One inspiring work from Lee and Bang [32] exploited the two sensors to compensate for coordinate disturbance from the kinematic rotation of the human arm.",
"The goal of their work was not to introduce a new DoF, but to support better performance in a regular drawing task.",
"In this paper, the purpose of two-sensor implementation is only to simulate a physical sensor on an arbitrary location.",
"Similar simulation models were introduced previously [8], [40], but they have not been verified.",
"We validate the model theoretically and empirically.",
"The virtual sensor implementation allows better mechanical stability from no moving part, and opens up a new possibility of the software-side optimization procedure."
],
[
"Summary of prior work",
"While, to the best of our knowledge, the effect of sensor position has not been studied after the original study by Verplank and Oliver on the Microsoft Mouse [47], other findings – on the effects of joint rotation – hint at the possibility of an empirical effect.",
"Studies of computer ergonomics have revealed that the rotation of a mouse is connected with rotation in wrist and arm posture.",
"Large radial and ulnar deviations in the elbow and the wrist are common, with wrist deviation ranging from 10 to 25 degrees [11], [48], [29].",
"Those values are large, given that the range of motion in radial–ulnar deviation is about 30 to 40 degrees [15], [16].",
"In the extreme case, if accompanied by elbow rotation, a mouse could rotate by up to 60 degrees during common usage.",
"These observations about rotation provide an incentive for studying the effect of sensor position."
],
[
"A Variable-Sensor-Position Mouse Device",
"We demonstrate two constructions of variable-sensor-position mice.",
"An overview is given in Figure REF .",
"The physical sensor (Approach 1) allows manually sliding the sensor along a rail, in similarity to the Ninox Astrum in Figure REF but with a wider range.",
"The virtual sensor (Approach 2) is able to emulate any sensor position on the vertical axis, and the position of the sensor can be changed in software.",
"Below, we present the principles of emulating a sensor and our empirical testing of the accuracy achieved.",
"Figure: Constructions of variable-sensor-position mice.",
"(a) Physical, in which the displacement sensor is on a sliding rail.",
"(b) Virtual, with two sensors to compute a virtual sensor at any location in between.",
"We label the extreme front position as 0% and the rearmost position as 100%.",
"In the Physical device, the motion sensor can be placed between 20% and 80% position, in 10% increments."
],
[
"Approach 1: Physical sensor positioning",
"Mouse device: The dimensions of the shell are $117 \\times 62 \\times 38$ mm while the shape matches that of a popular commercial mouse (Logitech G102/G203; our implementation does not have a scroll wheel).",
"Our shells were 3D printed, including buttons and mounts for switches and sensors.",
"We designed the sensor mounts to align the optical center of the sensors with the centerline of the device.",
"We used a PixArt PMW3360 optical 2D displacement sensor, a model frequently used in high-end gaming mice.",
"The sensor reads incremental horizontal ($dX$ ) and vertical ($dY$ ) displacement.",
"We used two Omron D2F-F microswitches, mounted under the left and right buttons.",
"A SparkFun Pro Micro board interfaced with the sensor module via SPI protocol.",
"The board sampled the sensor data at 500 Hz.",
"The sensor resolution was set to the maximum value (12,000 CPI) internally.",
"Users can select their preferred resolution (UserCPI), and the raw readings from the sensor are down-scaled to that value for USB reports.",
"The microprocessor (Microchip ATmega32u4) supports native USB communication.",
"Two USB endpoints were implemented: an HID (Human Interface Device) mouse, and a CDC (Communication Device Class) serial for logging.",
"Sliding rail: The sensor module was attached to a rail at the bottom of the shell.",
"The rail offers seven mounting positions at 7.2 mm intervals between 20% and 80% positions (marked by arrows in the figure).",
"The sensor can be manually moved.",
"Tracking: In each sample, a timestamp, two button states, two raw sensor values ($dX$ , $dY$ ), and their down-scaled values ($mX = k \\times dX$ , $mY = k \\times dY$ ) are logged through CDC serial, where $k$ is a CPI multiplier calculated as $k=(\\texttt {UserCPI} / 12,000)$ .",
"Separately, an HID report packet consists of the down-scaled values ($mX$ , $mY$ ) and the two-button states, which are sent to the host computer and controlled the system cursor.",
"Because it only allows integer values, truncation occurs at the decimal point, and the remainder is added to the next sample."
],
[
"Approach 2: A virtual sensor",
"All properties of the Virtual device (Figure REF , right) are identical to those in Approach 1 except that there are two fixed sensors instead of one sensor on a sliding rail.",
"The front and the rear sensor measure displacement separately, denoted as ($dX_{front}$ , $dY_{front}$ ) and ($dX_{rear}$ , $dY_{rear}$ ), which allows emulating a virtual sensor position ($mX$ , $mY$ ) by means of Equation REF : $\\begin{aligned}mX &= k\\lbrace (1 - p) dX_{front} + p \\cdot dX_{rear}\\rbrace \\\\mY &= k\\frac{dY_{front} + dY_{rear}}{2}\\end{aligned}$ The $p$ value, denoting the emulated position of the sensor, can be set within the range 0–100% with 1% resolution.",
"In each sample, a timestamp, two button states, four raw sensor values ($dX_{front}$ , $dY_{front}$ , $dX_{rear}$ , $dY_{rear}$ ), and the down-scaled virtual sensor values ($mX$ , $mY$ ) are logged.",
"An HID mouse report is generated as in Approach 1.",
"In following, we show Equation REF forms a virtual sensor equivalent to the physical sensor.",
"Virtual and physical sensors at position $p$ yield identical measurements.",
"Figure: Illustration of a virtual sensor during motion.",
"We assume physical displacement sensors (in gray) in the front and rear part of the device.",
"A simulated virtual sensor (the red, black-outlined dot) is located between the two sensors.rr is the distance between physical sensors.pp is the position of the virtual sensor range between 0–100%.See the text for the calculations.Intuition: Let us look at what happens between two samples in a moving device.",
"Figure REF shows device locations in $(t-1)^{th}$ and $t^{th}$ samples.",
"Let us set a reference point at the rear sensor.",
"Now, its movement can be broken down into a translation vector $\\vec{T}$ and a rotation of $\\theta $ (unit: rad).",
"With only translation movements considered, the same displacement vector $\\vec{T}$ is read from both sensors.",
"Next, during the rotation process, the rear sensor does not undergo any translation movement, so it does not measure any displacement.",
"However, the front sensor is translated further by the arc length ($r\\theta $ ) of the rotation around the rear sensor.",
"Because the added translation is tangential with respect to the front sensor, the $r\\theta $ value is added to the horizontal displacement for only the front sensor reading.",
"Similarly, the imaginary virtual sensor first translates by $\\vec{T}$ , and the rotation adds $(1-p)r\\theta $ to its measurement of horizontal displacement.",
"The sensor readings are modeled by Equation REF .",
"Amount of rotation $\\theta $ and the simulated virtual sensor data can be derived from the two sensors as in Equation REF and Equation REF .",
"$\\begin{aligned}\\texttt {Rear} &= (dX_{rear}, dY_{rear}) = \\vec{T}\\\\\\texttt {Front} &= (dX_{front}, dY_{front})= (dX_{rear} + r\\theta , dY_{rear}) \\\\\\texttt {Virtual} &= (dX_{rear} + (1-p)r\\theta , dY_{rear})\\end{aligned}$ $\\begin{aligned}dX_{front} - dX_{rear} &= (dX_{rear}+r\\theta ) - (dX_{rear})= r\\theta \\\\\\therefore \\;\\; \\theta &= \\frac{dX_{front} - dX_{rear}}{r}\\end{aligned}$ $\\begin{aligned}\\texttt {Virtual} &= (dX_{rear} + (1-p)r\\theta , dY_{rear}) \\\\&= (dX_{rear} + (1-p)r\\frac{dX_{front} - dX_{rear}}{r}, dY_{rear}) \\\\&= ((1-p)dX_{front} + p \\cdot dX_{rear}, dY_{rear})\\end{aligned}$ In addition to Equation REF , we replace $dY_{rear}$ with $\\frac{dY_{front}+dY_{rear}}{2}$ although they are expected to be same.",
"By averaging them, we compensate a possible deviation between sensors.",
"Finally, we multiply a CPI multiplier $k$ to both the $X$ and $Y$ values for down-sampling, which yields Equation REF ."
],
[
"Accuracy measurements with a robot",
"The virtual sensor model assumes ideal sensors that only measure translational movement.",
"However, a real sensor may react to rotational movement as well.",
"We wanted to test the virtual sensor model empirically, using an instrumented mouse.",
"We assessed the accuracy of the Virtual device against a ground-truth device (Physical) by comparing trajectories produced by a high-precision robotic arm.",
"Method: The two mice were modified to have DIN ISO 9409-1-A50 mounts on top of their cover.",
"The center of the mount, the axis of rotation, was placed at the $p=50\\%$ position.",
"Each device was attached to a FRANKA EMIKAhttps://www.franka.de Panda robotic arm, which has a reported accuracy of $\\pm $ .1 mm.",
"It was set to move in $\\infty $ shape with and without rotation.",
"See the top row in Figure REF for the two programmed motions employed.",
"Planned path lengths of the both were 700 mm.",
"When rotation was applied, the mouse rotated by -20$$ at the leftmost position and +40$$ at the rightmost position.",
"As the robotic arm moved the device, a host computer collected raw sensor logs: ($dX$ , $dY$ ) data for the seven mounting positions (20%–80%) with the Physical device; ($dX_{front}$ , $dY_{front}$ ) and ($dX_{rear}$ , $dY_{rear}$ ) data from the 0% and 100% positions with Virtual.",
"We simulated the cursor movements for the 20%–80% positions from the Virtual device by using Equation REF with $k=1$ .",
"We repeated the execution three times.",
"The repeated measurements produced virtually identical data, so we picked only the last measurement for further analysis.",
"Results: We compared Virtual and Physical in all sensor-position conditions between 20%–80%.",
"See Figure REF and Table REF for the results.",
"We note that the data from Physical exhibited slightly more deviation, because of an angular and positional fixture error at each mounting position.",
"Looking at the translation-only motion (see Figure REF , left column) revealed trajectories that overlap almost perfectly.",
"The measured average trajectory length was 709 mm, which was only marginally different from the planned length of 700 mm.",
"When rotation is added (see Figure REF , right column), the cursor movements are distorted [32].",
"Moreover, changing the sensor placement altered the cursor trajectory systematically.",
"As the sensor moved from the front to the rear (toward 100%), the cursor moved less in the x direction.",
"As a result, the sensor at 20% traveled 6.9% longer compared to the sensor at 80%.",
"We can also confirm the simulated cursor motions overlapped the real one with tiny discrepancy (<1%).",
"Figure: The virtual-sensor method was assessed by comparing trajectories against a corresponding physical-sensor mouse.",
"A robotic arm (top) performed ∞\\infty -shaped trajectories (top row, 1–2–3–4–5–6–7–8–9–4–A–B–1) without (left column) and with rotation (right column).The virtual sensor (bottom row) generated nearly identical cursor trajectories relative to the physical sensor (middle row).",
"Note: Units are kilocount, where 1 kilocount ≈\\approx 2.12 mm with the 12,000 CPI resolution sensor.Table: Results of the accuracy measurement experiment.",
"The values are calculated from the detected movements from the sensors."
],
[
"Human-subject observation",
"Even though the robot experiment exhibited an effect of device rotation on cursor trajectory, we wanted to also check the effect with humans subjects.",
"We conducted a study of mouse usage pattern while playing a simple game.",
"Method: Six participants (age Mean 30.2, SD 5.9; 5 male, 1 female) played the game shown in Figure REF , using the virtual sensor mouse.",
"The virtual sensor was fixed to 50% position (set $p=0.5$ in Equation REF ).",
"In the game, targets are spawned in random positions, each persisting for 3.5 seconds.",
"Users were requested to click all the targets before they disappear.",
"The initial spawning rate was set to 2 targets/s, and the speed gradually increased or decreased in response to the performance, targeting 92% success rate.",
"During a five-minute practice session, each participant was asked to set the sensor resolution to his or her preferred value.",
"Three-minute test session was followed and samples were logged.",
"We asked participants to play as quickly and accurately as possible.",
"The average spawning speed reached by the participants in that session ($\\approx $ click rate) ranged from 1.2 to 2.2 targets/s.",
"Figure: The aimed-movement game AimBooster (in Auto-balanced mode with the default targets/second setting, http://aimbooster.com) was used to assess the sensor models empirically.",
"The goal of the game is to click targets that spawn in random positions.Table: Linear regression analysis between (dX front dX_{front}, dX rear dX_{rear}) and (dY front dY_{front}, dY rear dY_{rear}) pairs for each participant.",
"<1 slopes in dXdX mean that the front sensor reads larger values than the rear sensor values, while the dYdY values are almost identical (slope ≈\\approx 1.0, R 2 R^2=.99).",
"The intercepts are near zero (unit: count), meaning in-place rotation almost never happened.Results: Linear regression analyses were performed between $dX_{front}$ –$dX_{rear}$ and $dY_{front}$ –$dY_{rear}$ pairs.",
"Table REF shows that the choice of sensor position has an apparent effect on the displacement sensed.",
"All participants were very similar in the patterns exhibited, whatever their choice of CPI.",
"The rear sensor reads almost half the horizontal displacement ($dX$ ) read by the front sensor.",
"For vertical displacement ($dY$ ), the sensors' readings were nearly identical.",
"Low $R^2$ in $dX$ indicates users rotated the mouse inconsistently during gameplay, which implies the complex nature of the sensor position problem.",
"With human subjects, the front sensor read much larger $dX$ , up to 192% of the rear sensor.",
"For example, an $\\infty $ shape was hand-drawn with a single motion (see Figure REF ), and the front sensor drew about twice wide shape compared to the rear sensor.",
"In the robot experiment, this expansion was less pronounced (see Figure REF ).",
"These results indicate the effect of sensor position is a practical problem, even beyond the assumption we made in the robot experiment.",
"The issue is more apparent when performing a directional movement: e.g., when a user aims a target at 45 direction, the cursor may move towards a lower or higher angle depending on different sensor placements.",
"See Figure REF for an illustration of this situation.",
"Figure: Hand-drawn ∞\\infty shapes on different sensor positions.",
"One subject (P4) drew the shape once.",
"The front and rear data are raw data, and the center data is virtually synthesized using Equation .We wanted to rigorously measure the effect of sensor position in a controlled pointing task.",
"The previously published study, comparing three sensor positions, with three different mouse shapes, was from 1989, and five of the 12 participants had never used a mouse [47].",
"Our experiment adheres closely to the ISO 9241-411:2012 (previously ISO 9241-9:2000) standard for evaluation of input devices with minor differences: (1) the calculation of the effective index of difficulty $ID_e$ , and (2) the treatment of outliers [51].",
"The test is used to evaluate pointing performance in different movement directions.",
"To understand the systematic effect of sensor position, we varied it in the full range of available positions in 20% increments (=14.4 mm), plus the center (50%) which is the most common choice for commercial mouse devices (see Figure REF )."
],
[
"Method",
"We used a within-subject design.",
"Sensor position is an independent variable, having seven levels: 0%, 20%, 40%, 50%, 60%, 80%, and 100%.",
"Pointing throughput and path deviation are dependant variables.",
"Participants: We recruited hobby gamers, who represent users who are potentially interested in optimizing mouse performance.",
"We recruited 14 participants who regularly play games using a mouse (age Mean 27.8, SD 5.6; nine males and five females).",
"All participants had normal or corrected-to-normal vision and were right-handed.",
"They used a mouse daily: 1.6 h (SD 0.8) for gaming and 3.5 h (SD 1.8) for general tasks.",
"Their hand sizes (measured from the tip of the middle finger to the midcarpal joint) ranged from 15.5 cm to 20 cm (Mean 17.3, SD 1.4).",
"Their regular mice had sensors around the center, with some variations around 40%–60% positions.",
"No devices featured an extreme sensor design, such as in the Apple Magic Mouse.",
"All participants signed an informed consent form.",
"Each was given a movie ticket equivalent to 14 EUR for participating.",
"Apparatus: The virtual-sensor mouse was used.",
"UserCPI was fixed at 800, a common value used in practice.",
"A constant gain function was used, which translates 1 count = 1 pixel.",
"Besides adding realism, this decreases the possibility of our results being biased from the choice of transfer functions.",
"An experiment program was implemented in Processinghttps://processing.org/ and ran in full-screen mode.",
"We used a high-speed gaming monitor (BenQ XL2546, 24-inch, 1920$\\times $ 1080 px, 92 pixels per inch, 240 frames per second).",
"A desktop computer (Intel Core i7 8700 3.2 GHz, 32 GB RAM, NVIDIA GeForce RTX 2080) drove the experiment program.",
"Task: We adopted the multi-direction tapping test from ISO 9241-411:2012, with 15 round targets in a circular arrangement (see Figure REF ).",
"Distance ($D$ ) is defined as the diameter of the outer circle, and target width ($W$ ) is defined as the diameter of each target.",
"At the beginning, the first target, 1 (see Figure REF , Right), is highlighted in green.",
"Clicking on it starts a session, and immediately the next target, 2, gets highlighted.",
"The participant moves the cursor (1$\\rightarrow $2) and clicks it.",
"This is defined as one trial.",
"If the click is outside the target (a mistake), it is marked in red and no correction is allowed.",
"After all the targets are visited, clicking the first target again (15$\\rightarrow $1) completes the session.",
"Accordingly, one session strictly consists of 15 trials and includes 16 mouse clicks.",
"Figure: The multi-directional tapping test was used to measure the effect of sensor position.Left: Each trial consists of cursor movement (a) from the previous target (b) to the current target (c) and a button click.",
"The current target is highlighted in green.",
"If the previous attempt yielded a miss, the previous target turns red.",
"Right: Definitions of DD and WW, and the order of the targets.We used $D$ = {300, 900 px} $\\times $ $W$ = {20, 50, 120 px}Physical sizes: $D$ ={83, 248 mm}, $W$ ={5.5, 14, 33 mm} = 6 combinations, designed to distribute the index of difficulty ($ID$ ) over the range 1.8–5.5 bits.",
"Each combination was repeated three times; therefore, there were $6\\times 3=18$ sessions in one Position block.",
"In total, 7 (blocks) $\\times $ 18 (sessions) $\\times $ 15 (trials) = 1,890 trials were collected per participant.",
"The order of the sensor positions (blocks) was counterbalanced across participants via a balanced Latin Square design [49].",
"Within each block, the $D$ and $W$ combinations were presented in random order.",
"Procedure: After a participant arrived, an experimenter explained the purpose of the experiment: “We are testing the pointing performance with different mouse settings.” We did not mention that the mouse sensor position would be the setting changed.",
"The task was demonstrated, and participants practiced it with a default mouse setting (800 CPI, sensor position 50%) for 10 sessions with random $D$ and $W$ .",
"During the practice, they adjusted the position of the desk, chair, mouse, and monitor to be comfortable.",
"The experimenter configured the sensor position before each block.",
"To adapt initially to the sensor configuration presented, participants played the AimBooster game (see Figure REF ) for at least three minutes.",
"They were encouraged to strive for >2.5 targets/s (<400 ms between clicks, 500+ clicks in total).",
"We enforced this training to ensure that users adapted to the new condition.",
"After the adaptation period, one block of the task was completed: 2 ($D$ ) $\\times $ 3 ($W$ ) $\\times $ 3 (instances) = 18 sessions, in random order.",
"The participants could rest whenever they wished between sessions.",
"If the success rate within a block fell below 90%, the block had to be redone.",
"A one-minute rest period was enforced between blocks.",
"Finally, the experimenter collected participants' demographics and took pictures of their dominant hand holding the mouse.",
"The full procedure took about one hour.",
"Data preprocessing: Each trial produced a log entry with the previous target, the current target, and the timestamped cursor trajectory.",
"In all, 26,460 trials were carried out.",
"We found and screened out 39 outliers, defined by these criteria: (1) the movement distance being less than $D/2$ [51]; (2) the start point or endpoint being more than $W\\times 2$ from the desired position [51]; or (3) movement time exceeding $\\log _2(D/W+1)$ , which means an extremely slow trial with <1 bit/s throughput."
],
[
"Results",
"The grand success rate was maintained at 94.7% ($SD$ =1.5%, minimum 91.9%) [35].",
"For the following statistical analyses, the significance level $\\alpha $ was set to .05 unless otherwise stated.",
"The error bands in the graphs are 95% confidence intervals (CI).",
"Statistical tests were carried out with SPSS version 18 and Python SciPy module version 1.2.0.",
"Throughput: Fitts' throughput (TP) is a performance metric for both speed and accuracy, calculated in terms of information transfer in the pointing task [35].",
"For post-hoc treatments for accuracy, bivariate endpoint deviation $SD_{x,y}$ [51] was calculated for each session to estimate effective width $W_e = 4.133 \\times SD_{x,y}$ .",
"Similarly, effective distance $D_e$ was calculated by averaging the movement distances within a session [45].",
"The effective index of difficulty of a session was calculated as $ID_e = \\log _2(D_e/W_e+1)$ [45].",
"Figure: For most users in our study, the sensor position exhibited a significant effect.",
"The P1–P14 graphs are TPs of each participant, and the Averaged graph shows the mean TPs averaged overall.The horizontal bars indicate significant differences.",
"Error bands are 95% CI.The solid red marks show the best sensor position found for each participant.",
"The outlined marks are for positions in the best homogeneous subset if it has multiple items.",
"The numbers at the top of each subplot are the order of blocks, which were counterbalanced across participants.One data point ($ID_e$ , $MT$ ) was produced from a session of 15 trials, where $MT$ is the averaged movement time.",
"The throughput for a session $k$ was calculated as $\\textit {TP}_k=ID_{e_k}/MT_k$ , and for a block, we followed the mean-of-means approach to calculate $\\textit {TP}=\\frac{1}{N}\\sum _{i=1}^{N}\\textit {TP}_i$ , where $N=18$ sessions in a block.",
"The observed TPs ranged from 4.0 to 7.0 bits/s (Mean 5.5, $SD$ .6).",
"We performed a regression analysis for each block [45].",
"The Fitts' law model was well fitted (Mean $R^2=.91$ , $SD$ .04), with near-zero intercepts (Mean .016 s, $SD$ .057) as expected.",
"Figure REF illustrates TPs of each participant (P1–P14) for the range of tested sensor positions.",
"We compared the positions within a participant by using Friedman's chi-square test, followed by a post-hoc Tukey's HSD test.",
"The Friedman test revealed a significant effect of sensor position on TP within each individual except P4 ($\\chi ^2=5.92$ , $p=.43$ ).",
"This means that every user except one showed a strong effect of the sensor position.",
"Using the post-hoc comparison results, we calculated the best subset of conditions and picked a single best position for each individual (see the caption of Figure REF ).",
"The last plot (Averaged) is the mean TP over all the participants.",
"Repeated measure analysis of variance (RM-ANOVA) revealed a significant effect of sensor position on TP ($F_{(3.51, 45.6)}=20.83$ , $p<.001$ , $\\eta ^2_p=.616$ with Greenhouse-Geisser correction).",
"The TP values followed a quadratic distribution ($p<.001$ ).",
"Bonferroni post-hoc analyses showed the extremal (0% and 100%) sensor positions to be worse than all others, but no significant difference was found in the positions between 20% and 80%.",
"Compare to the 50% position (TP=5.77), 100% position was worse by 14.0% (TP=4.97), and 0% was worse by 10.7% (TP=5.16).",
"We further analyzed the effect of sensor position for each individual.",
"We performed two-way ANOVA on the TPs for all the sessions, grouped by Participant and sensor Position.",
"Because the observations are not independent, we focused on effect size instead of on statistical significance (all $p$ s < .001).",
"The partial eta-squared ($\\eta ^2_p$ ) values [18] for the factors were $0.462$ for Participant, $0.241$ for Position, and $0.165$ for Participant$\\times $Position.",
"This can be interpreted as follows: (1) individual-to-individual differences are very large (0.462), and (2) there is a large effect of sensor position (0.241).",
"Also, (3) a large interaction effect is present (0.165), suggesting that people differ in their optimal sensor positions.",
"To estimate the advantage of having a personalized optimal sensor position relative to the global optimum (center), we calculated the performance difference between the personal best and the 50% condition.",
"The averaged TP for the best positions was 6.034 bits/s, and the averaged TP at the 50% position was 5.798 bits/s.",
"This is roughly a 4.1% throughput advantage (0.236 bits/s).",
"Path deviation: To measure path deviation, we calculated mean absolute error (MAE).",
"The error was defined as a distance between the committed cursor path and the ideal path (Figure REF ).",
"MAE for a path was calculated as ${\\frac{1}{N}\\sum _{i=1}^{N}|\\textit {Error}_i|}$ where $N$ is all the points in the cursor trajectory and $\\textit {Error}_i$ is the error at point $i$ .",
"A high path deviation implies a user was struggling to direct the cursor toward the target because the cursor movement diverged from the user's intentions.",
"Figure: Path deviation, measured with mean absolute error (MAEMAE) between cursor path and the ideal path, is used to measure variability in repeated acts of pointing.",
"The ideal path is marked as a straight line connecting the origin and the target.Figure: Sensor position affects path deviation in pointing: (A) MAE plotted for various sensor positions, the horizontal bars indicate a significant difference between positions; (B) linear regression showing a significant (p<.001) negative association (r=-.69) between TP and MAE.Figure REF A shows the MAE values ranged 15.5–31.1 pixels (Mean 19.8, $SD$ 3.1) for different sensor positions.",
"RM-ANOVA revealed a significant effect of sensor position on MAE ($F_{(3.13, 40.6)} = 19.02 $ , $ p <.001 $ , $ \\eta ^2_p = .594 $ with Greenhouse-Geisser correction).",
"The distribution of MAE had a quadratic form ($p<.001$ ).",
"Bonferroni post-hoc analyses showed higher values for the extremal positions (0% and 100%) in general; and 50% and 60% formed the best homogeneous subset.",
"Compare to the 50% position (MAE=18.0), 100% position was worse by 19.9% (MAE=21.6), and 0% was worse by 22.5% (MAE=22.1).",
"This result is highly consistent with those of the TP analysis.",
"In addition, we performed a linear regression analysis between TP and MAE, which revealed a strong negative correlation (see Figure REF B).",
"This suggests that the performance drop may be attributable to greater path deviation.",
"To conclude, when the sensor is placed in a suboptimal position, cursor trajectories vary more and are less under the user's control (see Figure REF ).",
"As an alternative metric for measuring path deviation, we also calculated Root-Mean-Square Error (RMSE).",
"The RMSE values ranged 18.7–40.3 pixels (Mean 24.0, $SD$ 4.0).",
"RM-ANOVA revealed a significant effect of sensor position on RMSE ($F_{(2.9, 37.5)} = 16.96 $ , $ p <.001 $ , $ \\eta ^2_p = .566 $ with Greenhouse-Geisser correction).",
"The distribution of RMSE had a quadratic form ($p<.001$ ), centered at the 50% position.",
"Bonferroni post-hoc analyses showed higher errors in the extremal positions (0% and 100%) than all others; however, no significant difference was found for the 20%–80% positions.",
"Linear regression analysis exhibited a strong negative correlation ($r=-.70$ ) between TP and RMSE.",
"The same conclusion is drawn as in the MAE analysis."
],
[
"Discussion",
"In all the TP, MAE, and RMSE measurements, the data followed a quadratic distribution centered at the 50% position.",
"This suggests an advantage for the center position within our study setting.",
"The results obtained fly in the face of the strongly held belief in “a dramatic advantage of the front sensor” [38], but aligned with the “center of gravity” hypothesis rather more.",
"The sensor in the foremost position was almost as bad as the sensor in the rearmost location.",
"Figure: An example of high and low path deviations exhibited by P14 in three sessions (DD=900, WW=20, the optimal (60%) and the extremal sensor positions).",
"At the optimal position, the paths tend to be more straight and less deviates toward the target."
],
[
"Calibration and Software-Side Tuning",
"The virtual-sensor mouse permits software-side setting of the sensor position.",
"This can be exploited for improving the position to suit the individual or task.",
"We present two procedures and the related softwarehttps://userinterfaces.aalto.fi/mouse_sensor_position/, which take slightly different approaches.",
"A single-subject demonstration of the two procedures is provided in Figure REF .",
"Figure: Sensor position can be individualized using a calibration procedure or in-application optimization.This figure demonstrates single-subject data on both.",
"(Left) Calibration results in the multi-directional tapping test.",
"(Right) Bayesian human-in-the-loop optimization using AimBooster.",
"Both results converged to p=40%p=40\\% as the optimum.The calibration task: The virtual-sensor mouse can be calibrated for an individual in a one-shot manner employing the pointing task.",
"The entire calibration session takes about one hour, to guarantee enough repetitions for observing subtler differences between conditions.",
"We release software for carrying out the calibration with the virtual-sensor mouse.",
"The calibration procedure follows the experimental method of the pointing study and is organized thus: The software selects a sensor position randomly from the set ${20, 40, ..., 80\\%}$ , then asks the user to perform the multi-directional tapping task for four minutes.",
"Extreme positions (0% and 100%) are removed since, according to our study, they are unlikely to offer the best settings.",
"Rest is enforced before moving on to the next position.",
"Confidence intervals are computed post hoc for the throughput values, and path deviations obtained, like that reported in the experiment.",
"To choose the optimal sensor position for the user, we look at the highest-achieving conditions, and we choose the median $p$ if there is more than one within the confidence bounds.",
"The procedure can be repeated for increased confidence in the optimal position, or test for the other sensor resolution.",
"In-task optimization: In addition to the controlled calibration procedure, the virtual-sensor device can be exploited for human-in-the-loop optimization.",
"The position of the sensor can be optimized in any application wherein (1) a user aims toward a stationary target and selects by a click, (2) it requires repetitive target selections, and (3) clicks and cursor movement, including timestamps, can be logged.",
"Most of the point-and-click games, for example, Whac-A-Mole, AimBooster, etc., are subject to be optimized.",
"An objective function can be formulated on the basis of the close correlation between throughput and path deviation, MAE or RMSE.",
"We defined and used path deviation rate $PDR=\\frac{\\textrm {MAE}}{|Path|}$ or $\\frac{\\textrm {RMSE}}{|Path|}$ , where $|Path|$ is the length of the ideal path.",
"With this, we could relax the condition of prior knowledge about the target in throughput measurement.",
"Divide by the path length will further compensate for the effect of varying movement amplitude.",
"The ideal path is assumed to be a straight line between the start point (the beginning of the largest amplitude sub-movement [33]) and the clicked point.",
"To facilitate deployment, we offer a Bayesian optimizer script in Python for sensor-position optimization [43].",
"It uses a sparse Gaussian Process (GP) as the proxy model to avoid choking on the high volume of data [9].",
"The proxy model maps sensor position $p$ to PDR, which it tries to minimize.",
"We recommend using expected improvement (EI) as the acquisition function; which exploits our finding that most users' personal optima are somewhere near the center, and that the form of the p-to-deviation relationship is quadratic.",
"The optimizer changes $p$ of the virtual mouse, and PDR is measured during a task.",
"It takes about one minute for one sample to update the GP model.",
"Hence, it should be called not mid-game but, for example, during breaks."
],
[
"Summary and Discussion",
"With this paper, we have presented two variable-sensor-position mouse constructions and shown that an accurate virtual mouse model can be achieved via a fusion of two sensors.",
"We described a user study in which we systematically manipulated the position of the sensor, ascertaining that position does have a reliable effect on pointing performance.",
"Further, we showed that individuals differ in their optimal sensor position and, hence, calibrating the sensor yields additional benefits.",
"While we did not study the causal mechanisms underlying this effect, we hypothesize that they are attributable to musculoskeletal differences.",
"We believe that even small differences in rotation angles, posture, and limb lengths would carry over to this effect.",
"Alternatively, it may be due to the cognitive mechanism by which people abstract their hand (and a device) into a point.",
"For example, the user's internal model may prefer the center of the mouse as its reference point when planning pointing movements.",
"This mechanism may also explain the path deviation being minimal at the center sensor position during the pointing process.",
"To the best of our knowledge, sensor placement has never been examined systematically before.",
"Our work opens up opportunities to understand differences among gamers, one aspect of which may be trivially attributable to the mouse designs chosen.",
"While HCI research has focused on pointer-facilitation techniques for translational motion, biomechanical factors that affect rotational motion, such as coordinate disturbance and joint rotation, need more attention.",
"We foresee the data and hardware constructions that we release facilitating such efforts.",
"We can identify several opportunities to build on this work.",
"Our study was limited to a pointing task, and it will be important to reproduce these effects in other tasks, such as tracing or drawing.",
"We believe that the effect will persist and may even be amplified in fast motions in which users rely more on ballistic motions.",
"Also, our experimental task was relatively short.",
"The effect should be replicated in a longitudinal study, with longer training times.",
"In addition, while our study considered only vertical changes in sensor position, the horizontal location too may affect the motion.",
"This should be studied further.",
"We used the familiar Logitech design only, and it would be important to assess whether the position effect interacts with the shape and weight of the device or the CD gain functions.",
"We also expect the virtual sensor and the optimization techniques will be able to minimize the sense of incompatibility when using two or more mice by one person, for example, at work and at home.",
"The experiment had certain limitations.",
"Firstly, potential interaction between shape, sensor resolution, and sensor position was not examined.",
"The preferred sensor position may change at lower or higher CPI values or with different forms, although we find this unlikely in light of the lack of an effect in the displacement study.",
"Secondly, the participants may have had a bias in favor of the center position, produced by daily use.",
"To mitigate this, we provided a short but extensive (500+ clicks) training for every block before actual testing.",
"One evidence could be P13 who exhibited optimal position at 80% (see on Figure REF ).",
"At least for P13, the training was enough to overcome the existing bias from the centered sensor of the everyday mouse.",
"However, the effectiveness of the training, in general, was not evaluated.",
"Repeating the study with differently biased people (e.g., everyday mice with front sensors) could be an immediate follow-up investigation.",
"Lastly, the calibration was demonstrated only with a single user.",
"The result suggests preliminary evidence that the method will likely work.",
"Large-scale deployment and examination will strengthen the validity of the calibration process."
],
[
"Conclusion",
"Our work demonstrates that the choice of displacement sensor position affects pointing performance.",
"While the study result indicates that a central position is the best compromise, performance can be further improved with a virtual-sensor device that allows personalizing the position.",
"Our data suggests that an additional improvement of 4% is achievable with the personalized optimization.",
"This may seem small, but for top-competitors such as pro gamers, even a small improvement could contribute.",
"The virtual sensor is inexpensive to implement and would increase the cost of a high-end mouse only marginally.",
"We expect the findings to be of interest to HCI researchers, engineers, gamers, computer graphic designers, and anyone else engaged in performance-oriented mouse use."
],
[
"Acknowledgment and Open Science",
"This work has been funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant no.",
"637991), and Korea Creative Content Agency (award no.",
"R2019020010).",
"We thank Marko Repo for his great assistance in carrying out the experiments, and Francesco Verdoja and Jens Lundell for their help in the robotic arm experiment.",
"All data and code generated in the study are released at https://userinterfaces.aalto.fi/mouse_sensor_position/.",
"The page reports 3D models of the mouse apparatus, circuit schematics, firmware, all raw and processed data reported here, as well as the scripts for data analysis and the calibration and optimization procedures."
]
] | 2001.03352 |
[
[
"Degenerate band edges in periodic quantum graphs"
],
[
"Abstract Edges of bands of continuous spectrum of periodic structures arise as maxima and minima of the dispersion relation of their Floquet--Bloch transform.",
"It is often assumed that the extrema generating the band edges are non-degenerate.",
"This paper constructs a family of examples of $\\mathbb{Z}^3$-periodic quantum graphs where the non-degeneracy assumption fails: the maximum of the first band is achieved along an algebraic curve of co-dimension 2.",
"The example is robust with respect to perturbations of edge lengths, vertex conditions and edge potentials.",
"The simple idea behind the construction allows generalizations to more complicated graphs and lattice dimensions.",
"The curves along which extrema are achieved have a natural interpretation as moduli spaces of planar polygons."
],
[
"Introduction",
"Periodic media play a prominent role in many fields including mathematical physics and material sciences.",
"A classical instance is the study of crystals, one of the most stable form of all solids that can be found throughout nature.",
"In a perfectly ordered crystal, the atoms are placed in a periodic order and this order is responsible for many properties particular to this material.",
"On the mathematical level, the stationary Schrödinger operator $-\\Delta +V$ with a periodic potential $V$ is used to describe the one-electron model of solid state physics ; here $V$ represents the field created by the lattice of ions in the crystal.",
"The resulting differential operator with periodic coefficients has been studied intensively in mathematics and physics literature for almost a century.",
"A standard technique in spectral analysis of periodic operators is called the Floquet-Bloch theory (see e.g., Kbook, Ksurvey).",
"This technique is applicable not only to the above model example of periodic Schrödinger operators on Euclidean space, but also to a wide variety of elliptic periodic equations on manifolds and branching structures (graphs).",
"Periodic elliptic operators of mathematical physics as well as their periodic elliptic counterparts on manifolds and quantum graphs do share an important feature of their spectra: the so-called band-gap structure (see e.g., BK,KOS,Kbook,Ksurvey).",
"Namely, the spectrum of a periodic elliptic operator can be represented in a natural way as the union of finite closed intervals, called spectral bands, and sometimes they may leave open intervals between them, called spectral gaps.",
"An endpoint of a spectral gap is called a gap edge.",
"For each spectral band, there is also a corresponding band function whose image is exactly that spectral band.",
"The set consisting of all graphs of band functions is called the dispersion relation.",
"The analytical and geometrical properties of dispersion relations encode significant information about the spectral features of the operator.These features are also called “threshold effects” whenever they depend only on the infinitesimal structure (e.g., a finite number of Taylor coefficients) of the dispersion relation at the spectral edges.",
"Hence studying structural properties of the dispersion relation may reveal interesting results for periodic differential operators.",
"A well-known and widely believed conjecture in physics literature says that generically (with respect to perturbations of the coefficients of the operator) the extrema are attained by a single band of the dispersion relation, are isolated, and have non-degenerate Hessian.",
"The non-degeneracy of extrema at the edges of the spectrum is often assumed to establish many important results such as finding asymptotics of Green's functions of a periodic elliptic operator near and at its gap edge Kha, KKR, KucRai, homogenization Bir, BirSus2000, BirSus2004, or counting dimensions of spaces of solutions with polynomial growth KP1, KP2, just to name a few.",
"In the continuous situation, the generic simplicity of spectral gap edges was obtained in .",
"The well-known result in established the validity of the full conjecture for the bottom of the spectrum of a periodic Schrödinger operator in Euclidean spaces, however the full conjecture still remains unproven for internal edges.",
"It is worth mentioning that in the two dimensional situation, a “variable period\" version of the non-degeneracy conjecture was found in and the isolated nature of extrema for a wide class of $\\mathbb {Z}^2$ -periodic elliptic operators was recently established in .",
"In the discrete graph situation, the statement of the conjecture fails for periodic Schrödinger operators on a diatomic lattice (see ).",
"However, in the example of there are only 2 free parameters to perturb the operator with and therefore the degeneracy may be attributed to the paucity of available perturbations.",
"To investigate this question further, considered a wider class of $\\mathbb {Z}^2$ -periodic discrete graphs and it was found that the set of parameters of vertex and edge weights for which the dispersion relation of the discrete Laplace-Beltrami operator has a degenerate extremum is a semi-algebraic subset of co-dimension 1 in the space of all parameters.",
"These examples show that the non-degeneracy of gap edges is a delicate issue even in the discrete setting.",
"In this paper, we propose two examples of periodic metric (or “quantum”) graphs whose Schrödinger operator dispersion relation has a degenerate band edge.",
"Remarkably, this band edge remains degenerate under a continuum of perturbations: one may vary edge lengths, vertex coupling constants and the edge potentials.",
"Our examples can be considered quantum-graph versions of the counterexample in , and they clearly show that the main reason for the degeneracy is not the small number of perturbation degrees of freedom, but rather the drastic effect a suitably chosen rank-1 perturbation has on the topology of the graph."
],
[
"The main result",
"We now introduce the quantum graph of our main theorem and formulate the result.",
"The description of principal notions used in the main theorem, such as quantum graphs, covers and periodicity, and the Floquet–Bloch transform, are deferred to Sections REF , and REF correspondingly.",
"Expanded versions of these descriptions are available in several sources, such as BK,Berincol17,Sunada,Ksurvey.",
"We will in fact describe two variants of our graph, $X_1$ and $X_2$ ; the main theorem will apply equally to both.",
"We start by describing one layer of the graph, which looks like planar hexagonal lattice shown in Figure REF .",
"It has vertices of two types, type $A$ and type $B$ denoted by red filled and blue empty circles correspondingly.",
"The graph $X_1$ will have $\\delta $ -type conditions at vertices $A$ and $B$ , with real coupling constants $\\gamma _A$ and $\\gamma _B$ , $\\gamma _A \\ne \\gamma _B$ .",
"The graph $X_2$ will have only Neumann–Kirchhoff (NK) conditions but the vertices of type $A$ are decorated by attaching a “tail”, i.e.",
"an edge leading to a vertex of degree one, shown as a smaller black circle in Figure REF (right).",
"Either version is a $\\mathbb {Z}^2$ -periodic graph in $\\mathbb {R}^2$ and its period lattice is generated by the two brown dashed vectors.",
"The edges of the same color (parallel edges) are related by $\\mathbb {Z}^2$ -shifts.",
"They are assumed to have the same length and to have the same potential (if any) placed on them.",
"Figure: Two layers of graphs X 1 X_1 (left) and X 2 X_2 (right) respectively.",
"These layers are ℤ 2 \\mathbb {Z}^2-periodic with respect to theBravais lattice generated by the two brown dashed.",
"The only difference between these two layers is the extra black tails added in the right layer.Figure: The graph X 1 X_1 is generated by stacking together infinitelymany copies of the layer graph along the height axis.",
"A layer isconnected to the next layer by certain green edges.",
"To get the graph X 2 X_2, one just simply adds black tails at the red filled vertices of X 1 X_1.The $\\mathbb {Z}^3$ -periodic graphs $X_1$ and $X_2$ are obtained by stacking the corresponding layers infinitely many times in both directions of the height axis, see Figure REF .",
"The layers are connected in a periodic fashion by edges (shown in green) between vertices of type B in a lower level and vertices of type A in the upper level.",
"Roughly speaking, one may think of the result as an infinite sheeted cover of the layers in Figure REF .",
"In particular, $X_1$ is a 3-dimensional topological diamond lattice, see .",
"In Figure REF we sketch a choice of the fundamental domain of the graph $X_1$ with respect to the $\\mathbb {Z}^3$ -periodic lattice.",
"Figure: A fundamental domain for the graph X 1 X_1.",
"Here the three gray vertices are not included in the fundamental domain.",
"The graph X 1 X_1 can be obtained by shifting this fundamental domain along the three dashed directions, which are its periods.The graphs $X_1$ and $X_2$ we defined above are actually the maximal abelian covers of finite graphs (see e.g., Baez, Sunada for more details on maximal abelian covers of graphs).",
"Taking the quotient of $X$ with respect to the periodic lattice we obtain the respective graphs in Figure REF .",
"The graph $\\Gamma _1 = X_1 / \\mathbb {Z}^3$ has two vertices, $A$ and $B$ , which are the images of the vertices of type $A$ and $B$ in $X_1$ under the canonical covering map from $X_1$ to $\\Gamma _1$ .",
"The four edges of $\\Gamma _1$ are the images of the sets of parallel edges in $X_1$ .",
"The graph $\\Gamma _2 = X_2 / \\mathbb {Z}^3$ has three vertices and five edges.",
"For either graph $\\Gamma $ , the first integral homology group is $H_1(\\Gamma ,\\mathbb {Z})\\cong \\mathbb {Z}^3$ .",
"We will be using notation $X$ when a statement applies equally to both $X_1$ and $X_2$ ; similarly we use $\\Gamma $ to refer to both graphs $\\Gamma _1$ and $\\Gamma _2$ .",
"Figure: The graphΓ 1 \\Gamma _1 (left) and the graph Γ 2 \\Gamma _2 (right).",
"In both graphs, the vertices AA, BB correspond to red-filled and blue-empty type vertices in XX, while the vertex CC corresponds to the decorated vertices in the black tails in X 2 X_2.",
"Here e 0 e_0 is the line CACA and e j e_j,1≤j≤41 \\le j \\le 4 are the corresponding edges between the twovertices AA and BB.The graphs $X$ are metric graphs: each edge $e$ in $X$ is identified with the interval $[0,\\ell (e)]$ , where $\\ell (e)$ is the length of the edge $e$ .",
"The lengths of edges related by a periodic shift (i.e.",
"belonging to the same $\\mathbb {Z}^3$ -equivalence class or having the same color) are the same.",
"We denote by $\\ell _j$ , $j\\in \\lbrace 1,\\ldots ,4\\rbrace $ the distinct lengths of edges in the graph $X_1$ ; the graph $X_2$ has an additional length — the length of the tail — which we denote by $\\ell _0$ .",
"This metric information on $X$ can be viewed as a pull-back of the metric on $\\Gamma $ via the covering map $\\pi : X \\rightarrow \\Gamma $ .",
"Notice that unlike the periodic realization of graphene and its multi-layer variants, we do not assume that each hexagon in the layer graph is regular, i.e.",
"the lengths of edges with distinct colors may be different.",
"On the edges of the graph $X$ we consider the Laplacian $-\\Delta _X=-\\frac{d^2}{dx^2}$ or, more generally, the Schrödinger operator $-\\Delta _X + q_e(x)$ with piecewise continuous potential $q_e(x)$ .",
"The potential is assumed to be the same on the edges of the same equivalence class (color), taking into account the edge's orientation.",
"This ensures the potential is $\\mathbb {Z}^3$ -periodic like the rest of the graph; we do not impose any other symmetry conditions on $q_e$ .",
"Regularity of the potential also plays no role in our examples, the same results can be extended to $L^1$ potentials with minor modifications.",
"At every vertex of the graph $X_2$ , we impose the standard Neumann-Kirchhoff boundary condition; we impose $\\delta $ -type conditions with distinct coupling constants $\\gamma _A$ and $\\gamma _B$ (one of them may be zero) on the corresponding vertices of the graph $X_1$ .",
"For the precise definition of vertex conditions, the reader is referred to Section REF .",
"The graphs $X$ are non-compact, $\\mathbb {Z}^3$ -periodic quantum graphs.",
"According to the Floquet-Bloch theory, the spectrum of the operator $-\\Delta _X$ is the union of the ranges of the band functions $\\lambda _j = \\lambda _j(k)$ , $j \\ge 1$ , where the quasimomentum ${\\mathbf {k}}$ ranges over the torus $\\mathbb {T}^3 :=\\left(\\mathbb {R}/2\\pi \\mathbb {Z}\\right)^3=(-\\pi ,\\pi ]^3$ and $\\lambda _1({\\mathbf {k}}) \\le \\lambda _2({\\mathbf {k}}) \\le \\cdots \\qquad \\mbox{for any } {\\mathbf {k}}\\in \\mathbb {T}^3.$ Now we state our main result.",
"Theorem 2.1 The spectrum of the operator $-\\Delta _X$ has an open gap between the first and the second band functions, i.e.",
"$\\max _{\\mathbf {k}}\\lambda _1({\\mathbf {k}})<\\min _{\\mathbf {k}}\\lambda _2({\\mathbf {k}}).$ If the lengths $\\ell _j$ ($1 \\le j\\le 4$ ) are approximately equal, then there exists a non-trivial one-dimensional algebraic curve $\\mu $ in $\\mathbb {T}^3$ such that $\\lambda _1$ attains its maximum value on $\\mu $ .",
"Consequently, there exists a degenerate band edge in the spectrum of $-\\Delta _X$ .",
"The degenerate band edge in the spectrum is persistent under a small perturbation of edge lengths, vertex coupling constants or edge potentials.",
"Theorem REF will be proved in Section after reviewing relevant definitions and tools in Section .",
"It will become clear during the proof that the phenomenon described in the Theorem is very robust.",
"Informally speaking, the extremum responsible for a band edge is frequently degenerate for any graph where removing a single vertex (but not the edges incident to it) reduces the rank of the fundamental group by 3 or more.",
"In particular, the condition on the edge lengths in part (b) of the Theorem serves only to insure a degenerate band edge particularly for the first band.",
"For almost all choices of edge lengths one can show that a finite proportion of bands will have degenerate edges.This is a consequence of Barra–Gaspard ergodicity of quantum graphs: informally, what happens once for one choice of lengths will happen with finite frequency for almost all choices of lengths.",
"For more precise statements, see BarGasjsp00,BerWintams10,BanBerprl13,ExnTurjpa17 The decorations introduced at vertices $A$ and $B$ ($\\delta $ -type conditions in $X_1$ and the tail edge in $X_2$ ) serve to break symmetry in the periodic graph and thus create a band gap.",
"If the symmetry is not broken, one would expect the bands to touch along the curve $\\mu $ ; see for related results.",
"Finally, the topology of the degeneracy submanifold $\\mu $ may be non-trivial in the higher-dimensional analogues of our example.",
"We touch upon it in in Section ."
],
[
"Quantum graphs and vertex conditions",
"In this section we recall some notations and basic notions of quantum graphs; for more details the reader is encouraged to consult , .",
"Consider a graph $\\mathcal {G}=(\\mathcal {V},\\mathcal {E})$ where $\\mathcal {V}$ and $\\mathcal {E}$ are the sets of vertices and edges of $\\mathcal {G}$ , respectively.",
"For each vertex $v\\in \\mathcal {V}$ , let $\\mathcal {E}_v$ be the set of edges $e$ incident to the vertex $v$ .",
"The degree $d_v$ of the vertex $v$ is the cardinality of the set $\\mathcal {E}_v$ .",
"The graph $\\mathcal {G}$ is a metric graph if each edge $e$ of the graph is give a length, $\\ell _e$ and can thus be identified with the interval $[0,\\ell _e]$ .",
"A function $f$ on the graph $\\mathcal {G}$ is henceforth a collection of functions $\\lbrace f_e\\rbrace _{e\\in \\mathcal {E}}$ , each defined on the corresponding interval.",
"Let us denote by $L^2(\\mathcal {G})$ (correspondingly $H^2(\\mathcal {G})$ ) the space of functions on the graph $\\mathcal {G}$ such that on each edge $e$ in $\\mathcal {E}$ , $f_e$ belongs to $L^2(e)$ (corresp.",
"$H^2(e)$ ) and, moreover, $\\sum _{e \\in \\mathcal {E}} \\Vert f\\Vert ^2_{L^2(e)}<\\infty \\qquad \\left(\\mbox{corresp.\\ }\\sum _{e \\in \\mathcal {E}} \\Vert f\\Vert ^2_{H^2(e)}<\\infty \\right).$ $\\mathcal {G}$ is called a quantum graph if it is a metric graph equipped with a self-adjoint differential operator $\\mathcal {H}$ of the Schrödinger type acting in $L^2(\\mathcal {G})$ .",
"We will take $\\mathcal {H}$ to act as $-\\Delta _{\\mathcal {G}}+q_e(x)$ on the edge $e$ , where $q_e$ are assumed to be piecewise continuous.",
"The domain of the operator will be the Sobolev space $H^2(\\mathcal {G})$ further restricted by a set of vertex conditions which involve the values of $f_e(v)$ and the derivatives $\\frac{df_e}{dx}(v)$ calculated at the vertices.",
"We list some commonly used vertex conditions below.",
"Dirichlet condition at a vertex $v \\in \\mathcal {V}$ requires that the function $f$ vanishes at the vertex, $f(v) = 0.$ This is an example of a decoupling condition.",
"Namely, if the Dirichlet condition is imposed at a vertex of degree $d>1$ , it is equivalent to disconnecting the edges incident to the vertex and imposing Dirichlet conditions at the resulting $d$ vertices of degree 1.",
"$\\delta $ -type condition at a vertex $v\\in \\mathcal {V}$ requires the function to be continuous at $v$ in addition to the condition $\\sum _{e \\in \\mathcal {E}_v}\\frac{df_e}{dx}(v) = \\gamma _v f(v),\\qquad \\gamma _v\\in \\mathbb {R},$ where $\\frac{df_e}{dx}(v)$ is the derivative of the function $f_e$ taken in the direction into the edge.",
"We note that the value $f(v)$ is well-defined because of the assumed continuity.",
"The real parameter $\\gamma _v$ is called the vertex coupling constant.",
"The special case of the $\\delta $ -type condition with $\\gamma _v=0$ is the Neumann-Kirchhoff (NK) or “standard” condition.",
"The Dirichlet condition defined above can be naturally interpreted as $\\gamma _v=+\\infty $ .",
"quasi-NK or magnetic condition at a vertex $v\\in \\mathcal {V}$ : Assume that the degree of the vertex $v$ is $d_v$ , $\\mathcal {E}_v=\\lbrace 1, \\ldots , d_v\\rbrace $ and we are given $d_v$ unit complex scalars $z_1, \\ldots , z_{d_v} \\in \\mathbb {S}^1$ .",
"We impose the following two conditions: ${\\left\\lbrace \\begin{array}{ll}&z_1 f_1(v) = z_2 f_2(v) = \\ldots = z_{d_v} f_{d_v}(v) \\\\&\\sum _{j=1}^{d_v}z_j \\frac{df_j}{dx}(v) = 0,\\end{array}\\right.",
"}$ Of course, the NK condition is a special case of (REF ) when all $z_j$ are equal.",
"If every vertex of the graph $\\mathcal {G}$ is equipped with one of the above conditions, the operator $\\mathcal {H}$ is self-adjoint (see *Theorem 1.4.4 and references therein).",
"The last set of conditions allow one to introduce magnetic field on the graph without modifying the operator (see and also , for more recent appearances).",
"They also arise as a result of Floquet–Bloch reduction reviewed in the next section."
],
[
"Floquet-Bloch reduction",
"Let us now return to our periodic graph $X$ .",
"Recall that the $\\delta $ -type conditions are imposed at all vertices of $X$ and hence the operator $-\\Delta _X$ is self-adjoint.",
"A standard Floquet-Bloch reduction (see e.g., BK, Kbook, Ksurvey) allows us to reduce the consideration of the spectrum of $-\\Delta _X$ to a family of spectral problems on a compact quantum graph (a fundamental domain).",
"More precisely, denote by $g_1, g_2, g_3$ some choice of generators of the shift lattice $\\mathbb {Z}^3$ .",
"For each ${\\mathbf {k}}=(k_1, k_2, k_3) \\in (-\\pi ,\\pi ]^3 =: \\mathbb {T}^3$ , let $-\\Delta _X^{({\\mathbf {k}})}$ be the Laplacian that acts on the domain consisting of functions $u \\in H^2_{loc}(X)$ that satisfy the $\\delta $ -type conditions at vertices along with the following Floquet conditions, $u_{g_1e}(x) = e^{ik_1}u_e(x),\\quad u_{g_2e}(x) = e^{ik_2}u_e(x),\\quad u_{g_3e}(x) = e^{ik_3}u_e(x),$ for all $x \\in X$ and $n=(n_1, n_2, n_3) \\in \\mathbb {Z}^3$ .",
"Then $-\\Delta _X$ is the direct integral of $-\\Delta _X^{({\\mathbf {k}})}$ and therefore, $\\sigma (-\\Delta _X) = \\bigcup _{{\\mathbf {k}}\\in \\mathbb {T}^3}\\sigma (-\\Delta _X^{({\\mathbf {k}})}).$ The operator $-\\Delta _X^{({\\mathbf {k}})}$ has discrete spectrum $\\sigma (-\\Delta _X^{({\\mathbf {k}})})=\\lbrace \\lambda _j({\\mathbf {k}})\\rbrace _{j=1}^\\infty $ where we assume that $\\lambda _j$ is increasing in $j$ , see (REF ).",
"The dispersion relation of the operator $-\\Delta _X$ is the multivalued function ${\\mathbf {k}}\\mapsto \\lbrace \\lambda _j({\\mathbf {k}})\\rbrace $ and the spectrum of $-\\Delta _X$ is the range of the dispersion relation for quasimomentum ${\\mathbf {k}}$ in $\\mathbb {T}^3$ .",
"Hence, it suffices to focus on solving the eigenvalue problems $-\\Delta _X^{({\\mathbf {k}})} u =\\lambda u$ where $\\lambda \\in \\mathbb {R}$ for $u$ in the domain of $-\\Delta _X^{({\\mathbf {k}})}$ .",
"This problem is unitarily equivalent to the eigenvalue problem on the compact graph $\\Gamma $ , $-\\frac{d^2}{dx^2}u=\\lambda u, \\quad \\lambda \\in \\mathbb {R},$ where $u$ satisfies the respective vertex conditions at the vertices $A$ and $C$ and the quasi-NK conditions at the vertex $B$ : ${\\left\\lbrace \\begin{array}{ll}&e^{ik_1}u_1(B)=e^{ik_2}u_2(B)=e^{ik_3}u_3(B)=u_4(B) \\\\&e^{ik_1}u_1^{\\prime }(B)+e^{ik_2}u_2^{\\prime }(B)+e^{ik_3}u_3^{\\prime }(B)+u_4^{\\prime }(B)=\\gamma _B u(B),\\end{array}\\right.",
"}$ where $\\gamma _B$ is taken to be 0 for the graph $\\Gamma _2$ and $u_j$ are the restrictions of the function $u$ to the edges $e_j$ .",
"We will use the notation $\\Gamma ^{{\\mathbf {k}}}_1$ and $\\Gamma ^{{\\mathbf {k}}}_2$ (or $\\Gamma ^{{\\mathbf {k}}}$ if the distinction between the two graphs is irrelevant) to denote the eigenvalue problem with condition (REF ) at the vertex $B$ .",
"From now on, we shall emphasize the vertex conditions pictorially by replacing the names of the vertices by their corresponding boundary conditions, see Fig.",
"REF .",
"We will use $\\gamma _A$ , $\\mathbf {NK}$ , $\\mathbf {D}$ and $\\mathbf {Q_{k,\\gamma _B}}$ to indicate the $\\delta $ -type, Neumann–Kirchhoff, Dirichlet and quasi-NK vertex conditions respectively.",
"We will also occasionally use this convention in the text, e.g., the vertex $B$ in the above graph $\\Gamma ^{{\\mathbf {k}}}$ will be mentioned as the $\\mathbf {Q_k}$ -vertex.",
"Finally, we will use the symbol $\\lambda _j(\\Gamma ^{{\\mathbf {k}}})$ for the $j^{th}$ -eigenvalue of the quantum graph $\\Gamma ^{{\\mathbf {k}}}$ .",
"In particular, we have $\\sigma (-\\Delta _X) = \\bigcup _{j \\ge 1, k \\in \\mathbb {T}^3}\\left\\lbrace \\lambda _j(\\Gamma ^{{\\mathbf {k}}})\\right\\rbrace .$ Figure: The quantum graphs Γ 1 𝐤 \\Gamma ^{{\\mathbf {k}}}_1 (left) and Γ 2 𝐤 \\Gamma ^{{\\mathbf {k}}}_2 (right) and their vertexconditions.",
"In the figures, the types of the boundary conditions are bold letters while the labels of the vertices are regular letters."
],
[
"Eigenvalue comparison under some surgery transformations",
"In this section we list some eigenvalue comparison results that will be useful to prove the existence of a gap in the dispersion relation in Theorem REF (a).",
"The following interlacing inequality is often useful when variation of a coupling constant is used to interpolate between different $\\delta $ -type conditions and also the Dirichlet condition (which is interpreted as the $\\delta $ -type condition with coupling $+\\infty $ ).",
"Theorem 3.1 (A special case of *Theorem 3.4) If the graph $\\widehat{G}$ is obtained from $G$ by changing the coefficient of the $\\delta $ -type condition at a single vertex $v$ from $\\gamma _v$ to $\\widehat{\\gamma }_v \\in (\\gamma _v, \\infty ]$ .",
"Then their eigenvalues satisfy the interlacing inequalities $\\lambda _{k}(G) \\le \\lambda _{k}(\\widehat{G})\\le \\lambda _{k+1}(G) \\le \\lambda _{k+1}(\\widehat{G}),\\qquad k \\ge 1.$ If a given value $\\Lambda $ has multiplicities $m$ and $\\widetilde{m}$ in the spectra of $G$ and $\\widehat{G}$ respectively, then the $\\Lambda $ -eigenspaces of $G$ and $\\widehat{G}$ intersect along a subspace of dimension $\\min (m, \\widetilde{m})$ .",
"Note that by (REF ), $\\widetilde{m}$ must be equal to $m-1$ , $m$ or $m+1$ .",
"For simplicity, from now on, if the graph $G_1$ is obtained from $G_2$ by changing the $\\delta $ -type conditions to Dirichlet conditions at a single vertex, we will say that $G_1$ is a rank one Dirichlet perturbation of the graph $G_2$.",
"We now consider the effect on the eigenvalue of the enlargement of a graph, which is realized by attaching a subgraph at a designated vertex.",
"The following theorem is quoted in the narrowest form that is sufficient for our needs.",
"Theorem 3.2 (A special case of *Theorem 3.10) Suppose that $\\widehat{G}$ is formed from graphs $G$ and $H$ by identifying or “gluing” two Neumann–Kirchhoff vertices $v_0\\in G$ and $w_0\\in H$ .",
"If $\\lambda _1(H)<\\lambda _1(G)$ and the eigenvalue $\\lambda _1(G)$ has an eigenfunction which does not vanish at $v_0$ then $\\lambda _1(\\widehat{G})<\\lambda _1(G)$ ."
],
[
"Topology of moduli spaces of polygons",
"Given $n$ positive real numbers $\\lbrace a_j\\rbrace $ one can ask what is the topology of the space of all planar polygons whose side lengths are $\\lbrace a_j\\rbrace $ .",
"Two polygons are identified if they can be mapped into each other by a composition of rotation and translation.",
"The resulting spaces may not be smooth and their full classification is surprisingly rich, see and references therein.",
"These spaces make an appearance in our question as the degenerate curves on which the dispersion relation has an extremum.",
"Figure: Quadrangle corresponding toequation ().For our example we will only require the following simple lemma (which follows from the results of ) addressing the topology of the set of quadrangles with given four edge lengths, see Figure REF .",
"Lemma 3.3 The curve $\\mu $ of solutions ${\\mathbf {k}}=(k_1,k_2,k_3)\\in \\mathbb {T}^3$ of $\\sum _{1 \\le j \\le 3} e^{ik_j}a_j+a_4=0$ is an algebraic curve of co-dimension 2 if and only if $a_m < \\sum _{j\\ne m} a_j$ for every $m=1,\\ldots ,4$ .",
"If there is an $m$ with the inequality reversed, the set of solutions $\\mu $ is empty.",
"If there is an $m$ with inequality turning into equality, the set of solutions is a single point.",
"The topology of $\\mu $ in this particular case has been described, for example, in *Sec 12.",
"The curve is smooth unless there is a linear combination of $\\lbrace a_j\\rbrace $ with coefficients $\\pm 1$ that is equal to zero.",
"If the curve $\\mu $ is smooth it is either a circle or a disjoint union of two circles.",
"The non-generic cases when $\\mu $ is not a smooth manifold are of the following types: two circles intersecting at a point, two circles intersecting at two points and three circles with one intersection among each pair.",
"Figure: The set of roots of () for twochoices of {a j }\\lbrace a_j\\rbrace ; two views of the same plot are shown.",
"Theranges are adjusted to k 1 ∈(0,2π]k_1 \\in (0,2\\pi ] andk 2 ,k 3 ∈(-π,π]k_2,k_3 \\in (-\\pi ,\\pi ] for a smoother plot.",
"Straight red linescorrespond to a j =1a_j=1 for all jj; Black stars (appear as a thickfuzzy line) are produced using a 1 =1.1a_1=1.1, a 2 =0.95a_2=0.95, a 3 =0.9a_3=0.9and a 4 =1a_4=1.The latter case arises when all $a_j$ are equal.",
"It is shown in red solid line in Figure REF .",
"Note that the plot is on a torus, therefore each pair of parallel lines is actually a single line forming a circle.",
"A smooth curve $\\mu $ for a generic choice of $a_j\\approx 1$ is also shown."
],
[
"Proof of the main result",
"In this section, we present the details of the proof of Theorem REF .",
"Without loss of generality, for the graph $\\Gamma _1$ we will make the assumption $\\gamma _A < \\gamma _B.$ Starting with the graph $\\Gamma _1$ , we introduce two of its modifications.",
"The graph $\\Gamma _1^A$ is obtained by changing the condition at the vertex $A$ to Dirichlet; the graph $\\Gamma _1^B$ is obtained similarly by placing a Dirichlet condition at the vertex $B$ .",
"Remembering that a Dirichlet condition is decoupling, we can picture the result as shown in Fig.",
"REF .",
"Figure: The two “star” graphs Γ 1 A \\Gamma ^A_1 (left)and Γ 1 B \\Gamma ^B_1 (right) after disconnecting the corresponding Dirichlet vertices (𝐃)(\\textbf {D}).By placing Dirichlet conditions at vertices $A$ or $B$ of the graph $\\Gamma _2$ , we analogously construct the two graphs $\\Gamma _2^A$ and $\\Gamma _2^B$ .",
"We remark that the graph $\\Gamma _2^A$ has two connected components, see Fig.",
"REF and Fig.",
"REF .",
"Using the tools introduced in Section REF we establish the following comparison result, which compares the first eigenvalue $\\Gamma _j^A$ with the first eigenvalue of $\\Gamma _j^B$ , where $j$ is either 1 or 2.",
"Figure: The quantum graphs Γ 2 A,1 \\Gamma _2^{A,1} (left) and Γ 2 A,2 \\Gamma _2^{A,2}after disconnecting from the Dirichlet vertex of their union Γ 2 A \\Gamma _2^A.Figure: The quantum graph Γ 2 B \\Gamma _2^B.Lemma 4.1 The first eigenvalue of $\\Gamma ^B$ is always strictly less than than the first eigenvalue of $\\Gamma ^A$ , $\\lambda _1(\\Gamma ^B)<\\lambda _1(\\Gamma ^A)$ The graphs $\\Gamma _1^B$ and $\\Gamma _1^A$ differ only in the coefficient of the $\\delta $ -type condition at the vertex of degree 4 (we are in the situation of pure Laplacian, with no potential).",
"Since the coefficient of $\\Gamma _1^B$ (which is $\\gamma _A$ ) is smaller than the coefficient of $\\Gamma _1^A$ , see equation (REF ), we immediately get from Theorem REF that $\\lambda _1(\\Gamma _1^B) \\le \\lambda _1(\\Gamma _1^A)$ .",
"The case of equality is excluded because the ground state must be non-zero on the vertex of degree 4 which means it cannot satisfy $\\delta $ -type conditions with two different constants (hence it cannot be a common eigenfunction).",
"For the graph $\\Gamma _2$ we establish two inequalities, $\\lambda _1(\\Gamma _2^B) < \\lambda _1(\\Gamma _2^{A,1})$ and $\\lambda _1(\\Gamma _2^B) < \\lambda _1(\\Gamma _2^{A,2})$ .",
"The first follows by changing the condition at vertex $A$ of the graph $\\Gamma _2^B$ from NK to Dirichlet: the eigenvalue strictly increases (since the eigenfunction of $\\Gamma _2^B$ is non-zero at $A$ ) and the graph decouples into several disjoint parts one of which coincides with $\\Gamma _2^{A,1}$ .",
"To prove the second inequality, we start with $\\lambda _1(\\Gamma _2^{A,2})>0$ whose eigenfunction does not vanish on vertex $B$ , and attach to $B$ a Neumann interval of length $\\ell _0$ whose first eigenvalue is $0<\\lambda _1(\\Gamma _2^{A,2})$ .",
"The strict inequality follows from Theorem REF .",
"In our terminology, the graphs $\\Gamma ^A$ and $\\Gamma ^B$ are the rank one Dirichlet perturbations of the corresponding graph $\\Gamma $ .",
"The next important observation is that they are also, in fact, the rank one Dirichlet perturbations of the corresponding graph $\\Gamma ^{\\mathbf {k}}$ for any ${\\mathbf {k}}$ .",
"Lemma 4.2 The rank one Dirichlet perturbation of the graph $\\Gamma ^{\\mathbf {k}}$ at the vertex $A$ (corresp.",
"$B$ ) is unitarily equivalent to $\\Gamma ^A$ (corresp.",
"$\\Gamma ^B$ ) for any ${\\mathbf {k}}\\in \\mathbb {T}^3$ .",
"Since the Dirichlet perturbation is decoupling, the resulting graphs have no cycles and therefore any quasi-momenta can be removed by a gauge transform, see *Thm 2.6.1.",
"To put it another way, replacing the vertex condition (REF ) at $B$ with Dirichlet removes all dependence on the quasi-momenta ${\\mathbf {k}}$ .",
"Similarly, the quasi-NK conditions could be equivalently imposed at the vertex $A$ , where replacing them with Dirichlet also removes all dependence on ${\\mathbf {k}}$ .",
"Lemma 4.3 The first eigenvalue $\\lambda _1$ of $-\\Delta $ on $\\Gamma ^B$ is simple.",
"If $\\ell _1 = \\ell _2 = \\ell _3 = \\ell _4$ , the eigenfunction corresponding to $\\lambda _1$ is identical on these four edges, $\\phi _1 \\equiv \\phi _2 \\equiv \\phi _3 \\equiv \\phi _4$ , and non-zero except at $B$ .",
"The proof is identical for $\\Gamma ^B_1$ and $\\Gamma ^B_2$ .",
"Simplicity of the eigenvalue follows from general variational principles (or can be deduced from the secular equation for the corresponding graphs, see also the proof of Proposition REF below).",
"The first eigenfunction is known to be positive, except where a Dirichlet condition is enforced, for a large family of vertex conditions .",
"Symmetry can be deduced by, for example, restricting $-\\Delta $ to the symmetric subspace of the operator's domain , observing that the first eigenfunction of the restricted operator is positive and concluding that it corresponds to a positive eigenfunction of the full operator and therefore must be the ground state.",
"Since by Lemma REF $\\Gamma ^A$ and $\\Gamma ^B$ are obtained by a rank-1 Dirichlet perturbation from the quantum graph $\\Gamma ^{\\mathbf {k}}$ for any ${\\mathbf {k}}$ , Theorem REF yields the inequalities $\\lambda _1(\\Gamma ^{\\mathbf {k}}) \\le \\lambda _1(\\Gamma ^B)\\le \\lambda _2(\\Gamma ^{\\mathbf {k}}),$ and $\\lambda _1(\\Gamma ^{\\mathbf {k}}) \\le \\lambda _1(\\Gamma ^A)\\le \\lambda _2(\\Gamma ^{\\mathbf {k}}),$ which hold of all ${\\mathbf {k}}\\in \\mathbb {T}^3$ .",
"Adding the result of Lemma REF , we get $\\lambda _1(\\Gamma ^{\\mathbf {k}}) \\le \\lambda _1(\\Gamma ^B)< \\lambda _1(\\Gamma ^A) \\le \\lambda _2(\\Gamma ^{\\mathbf {k}}),$ obtaining part (a) of Theorem REF .",
"We will now show that the first inequality in (REF ) turns into equality $\\lambda _1(\\Gamma ^{\\mathbf {k}})=\\lambda _1(\\Gamma ^B)$ for $k$ in a one-dimensional curve $\\gamma $ in $\\mathbb {T}^3$ .",
"Let $\\varphi $ be the $\\lambda _1(\\Gamma ^B)$ -eigenfunction of $\\Gamma ^B$ .",
"By Theorem REF , equality (REF ) holds if and only if $\\varphi $ is also an eigenfunction of $\\Gamma ^{\\mathbf {k}}$ .",
"We denote by $\\varphi _j$ the restriction of $\\varphi $ on $e_j$ for $0\\le j\\le 4$ .",
"Obviously, $\\varphi $ satisfies the first condition in (REF ) at the vertex $B$ .",
"Therefore, equality (REF ) holds if and only if ${\\mathbf {k}}=(k_1,k_2,k_3) \\in [-\\pi ,\\pi )^3$ is such that $\\sum _{1 \\le j \\le 3} e^{ik_j}\\varphi _j^{\\prime }(B)+\\varphi _4^{\\prime }(B)=0.$ By Lemma REF , the set of solutions of (REF ) is a non-trivial algebraic curve of co-dimension 2 if $2\\max _{j} \\left|\\varphi _j^{\\prime }(B)\\right| <\\sum _{1 \\le j\\le 4} \\left|\\varphi _j^{\\prime }(B)\\right|.$ If the lengths $\\ell _j$ ($1 \\le j\\le 4$ ) are approximately equal then (by eigenfunction continuity and Lemma REF ) all $\\left|\\varphi _j^{\\prime }(B)\\right|$ are approximately equal and condition (REF ) is satisfied.",
"This completes the proof of part (b).",
"Finally, the robustness of the degenerate gap edge under a small perturbation of edge lengths or edge potentials follows directly from continuity of eigenvalue and eigenfunction data BerKucincol12,KucZhajmp19 and the fact that conditions for the degenerate gap edge are inequalities (REF ) and (REF ).",
"With a little extra effort we can provide a quantitative condition on the lengths $\\ell _j$ to ensure the validity of the quadrangle inequalities (REF ) whenever all of the derivatives $\\varphi _1^{\\prime }(B),\\ldots ,\\varphi _4^{\\prime }(B)$ are not zero.",
"Proposition 4.4 Let $\\rho _0$ be the unique solution in $(2,3)$ to the equation $\\rho ^2-\\frac{\\rho ^3}{3}=\\frac{\\pi ^2}{24},$ and assume further that $\\min \\left\\lbrace \\left(\\rho _0\\cdot \\min _{1 \\le j\\le 4}\\ell _j\\right), \\ell _0\\right\\rbrace \\ge \\max _{1 \\le j\\le 4}\\ell _j$ Then $2\\cdot |\\varphi _j^{\\prime }(B)|<|\\varphi _1^{\\prime }(B)|+|\\varphi _2^{\\prime }(B)|+|\\varphi _3^{\\prime }(B)|+|\\varphi _4^{\\prime }(B)|$ for each $1\\le j\\le 4$ .",
"As a consequence, the same conclusion in part (b) of Theorem REF holds.",
"Without loss of generality, assume that $\\ell _4 \\ge \\ell _3 \\ge \\ell _2 \\ge \\ell _1$ .",
"On the edge $e_j$ where $1 \\le j\\le 4$ , we write $\\varphi _j(x)=\\alpha _j \\sin (\\beta x)$ , where $0 \\le x \\le \\ell _j$ , $\\alpha _j\\in \\mathbb {R}$ and $\\beta =\\left(\\lambda _1(\\Gamma ^B)\\right)^{1/2}$ .",
"Here we identify the vertex $B$ as $x=0$ on each edge $e_j$ .",
"Observe that $0<\\beta =\\lambda _1(\\Gamma ^B)^{1/2} \\le \\min _{1\\le j \\le 4}\\left\\lbrace \\frac{\\pi }{2\\ell _0},\\frac{\\pi }{\\ell _j}\\right\\rbrace =\\frac{\\pi }{2\\ell _0}$ This implies that $\\beta \\ell _j \\in (0,\\pi /2]$ for each $j$ .",
"So $\\min _{1 \\le j \\le 4}|\\sin (\\beta \\ell _j)|=\\sin (\\beta \\ell _1)$ .",
"Moreover, from the fact that $\\varphi _j(\\ell _j) \\ne 0$ and the continuity of $\\varphi $ at the vertex $A$ , we have $\\beta ^{-1}\\varphi _j^{\\prime }(B)=\\alpha _j = \\alpha _4 \\cdot \\frac{\\sin (\\beta \\ell _4)}{\\sin (\\beta \\ell _j)}$ Therefore, it is enough to show $\\sin (\\beta \\ell _1) \\cdot \\sum _{i=2}^4 \\frac{1}{\\sin (\\beta \\ell _i)}>1$ Put $\\displaystyle \\rho :=\\frac{\\ell _4}{\\ell _1} \\in [1,\\rho _0]$ then we get $1-\\frac{\\pi ^2}{24\\rho ^2}>\\frac{\\rho }{3}$ From (REF ), $\\beta \\ell _1<\\frac{\\pi \\ell _1}{2 \\ell _0}\\le \\frac{\\pi }{2\\rho }$ and so it implies $1 - \\frac{(\\beta \\ell _1)^2}{6} >\\frac{\\ell _4}{3\\ell _1}$ Since $\\sin (\\beta \\ell _1) \\ge \\beta \\ell _1 - (\\beta \\ell _1)^3/6$ and $\\sin (\\beta \\ell _j) \\le (\\beta \\ell _4)$ , (REF ) follows from (REF )."
],
[
"Discussion",
"Our Theorem REF provides a quantum graph counterexample to the mentioned conjecture at the beginning of the paper, about the genericity of non-degenerate spectral edges in spectra of $\\mathbb {Z}^d$ -periodic quantum graphs, where $d>2$ .",
"Note that this construction can also be modified to provide an example of a $\\mathbb {Z}^d$ -discrete graph whose dispersion relation of the discrete Laplacian operator contains a degenerate band edge.",
"Indeed, let $\\Gamma _d$ be the graph with two vertices such that there are exactly $d+1$ - edges between them and therefore, its maximal abelian covering $X_d$ is a $d$ -dimensional topological diamond.",
"One can write down explicitly the dispersion relation of$X_d$ and then proceed a similar calculation as in to derive the degeneracy of the extrema of the band functions.",
"Our construction of the graphs $\\Gamma $ required that the dimension of the dual torus of quasimomenta $k$ be of dimension at least three.",
"The same method and proof will still work if we increase the number of edges connecting the two vertices $A$ and $B$ (at least four edges).",
"In fact, the entire mechanism of the proof is extremely robust: two rank-one perturbations that reduce the number of cycles by 3 or more help create the gaps between conductivity bands, while a continuum of solutions to an equation similar to (REF ) will make the band edge degenerate.",
"The degeneracy curve thus still has a natural interpretation as the set of possible $(n-1)$ -tuples of angles in a planar $n$ -gon with the given edge lengths; here $n$ is the number of cycles broken by the rank one perturbation.",
"For $n>3$ the topology of such objects becomes increasingly complicated.",
"Their homology groups were studied by many authors, see and references therein.",
"A heuristic reason for the degeneracy may be put forward using the classical idea of Wigner and von Neumann: a family of complex Hermitian matrices depending on 3 parameters is expected to have isolated point degeneracies (where a pair of eigenvalues meet).",
"This is what the eigenvalues want to do here, but there are hard bounds (REF ) and (REF ) from the rank one perturbations, so the eigenvalues instead accumulate at the bound."
],
[
"Acknowledgments",
"The work of the first author was partially supported by NSF DMS–1815075 grant and the work of the second author was partially supported by the AMS–Simons Travel grant.",
"Both authors express their gratitude to Peter Kuchment for introducing them to this exciting topic and to Lior Alon and Ram Band for many deep discussions.",
"We thank an anonymous referee for several improving suggestions.",
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]
] | 2001.03566 |
[
[
"The two-sorted algebraic theory of states, and the universal states of\n MV-algebras"
],
[
"Abstract We introduce a two-sorted algebraic theory whose models are states of MV-algebras and, to within a categorical equivalence that extends Mundici's well-known one, states of Abelian lattice-groups with (strong order) unit.",
"We discuss free states, and their relation to the universal state of an~MV-algebra.",
"We clarify the relationship of such universal states with the theory of affine representations of lattice-groups.",
"Main result: The universal state of any locally finite MV-algebra---in particular, of any Boolean algebra---has semisimple codomain."
],
[
"Introduction",
"We are concerned in this paper with the algebraic theory of states of MV-algebras and Abelian lattice-groups with a (strong order) unit, here called “unital Abelian $\\ell $ -groups”.",
"Usually, states are defined as normalised positive real-valued linear functionals on Riesz spaces with unit or, more generally, normalised positive group homomorphisms to $\\operatorname{\\mathbb {R}}$ of unital Abelian $\\ell $ -groups [10].",
"Part of their importance stems from the long-recognised fact that unital Abelian lattice-groups and their states provide an abstraction of bounded real random variables and of expected-value operators, respectively.",
"To illustrate, the collection $\\operatorname{{M}_{\\rm b}}{(X,{A})}$ of bounded ${A}$ -measurable functions from $X$ to $\\operatorname{\\mathbb {R}}$ is an Abelian $\\ell $ -group under pointwise addition and order; and the function $X\\rightarrow \\operatorname{\\mathbb {R}}$ constantly equal to 1 is a unit of this $\\ell $ -group, due to our boundedness assumption.",
"If $\\mu $ is a probability measure on ${A}$ , then the expected-value operator $\\int _X-\\,{\\rm d}\\mu \\colon \\operatorname{{M}_{\\rm b}}{(X,{A})}\\rightarrow \\operatorname{\\mathbb {R}}$ is a state of the unital Abelian $\\ell $ -group $\\operatorname{{M}_{\\rm b}}{(X,{A})}$ .",
"In this paper we allow states to take values in any unital Abelian $\\ell $ -group, and not just in the real numbers; thus, if $G$ and $H$ are such groups, a state of $G$ with values in $H$ is a positive group homomorphism from $G$ to $H$ that carries the unit of $G$ to the unit of $H$ .",
"We shall see that this level of generality allows us to investigate universal constructions that, while ubiquitous in algebra, are not available if one insists that states be real-valued.",
"We shall be interested, specifically, in the existence of a “most general state”, or universal state, of a given unital Abelian $\\ell $ -group.",
"We will prove that such a universal state indeed always exists, as a consequence of the standard fact that free algebras exist in algebraic categories.",
"One obstruction to this plan is that the category of unital Abelian $\\ell $ -groups and their unit-preserving homomorphisms is not algebraic with respect to its underlying-set functor.",
"This is because the characteristic property of the unit 1, that its multiples $n{1+\\cdots +1}$ ($n$ times) should eventually exceed any given element, is not even definable in first-order logic, by a standard compactness argument.",
"Nonetheless, a well-known result of Mundici [20] tells us that the category of unital Abelian $\\ell $ -groups is equivalent to the algebraic category of MV-algebras [6] and their homomorphisms.",
"We shall prove a version of Mundici's result for states, rather than just for unital homomorphisms, that will allow us to carry our programme out to within an equivalence of categories.",
"To begin with, the theory of (real-valued) states has already been adapted to MV-algebras.",
"The original reference for this is [22], where MV-algebraic states were introduced with the motivation of modelling the notion of “average truth degree\" in many-valued logic.",
"For a primer on states of MV-algebras see [24], [8].",
"In line with what is discussed above for lattice-groups, in this paper we consider states between any two MV-algebras; see Definition REF below.",
"Basic facts about the functor $\\Gamma $ that features in Mundici's equivalence are recalled in Section .",
"In Theorem REF we extend Mundici's equivalence to one between ${\\sf MV_{st}}$ (the category whose objects are MV-algebras and whose morphisms are states) and ${{{\\sf A_1}}}$ (the category whose objects are unital Abelian $\\ell $ -groups and whose morphisms are states).",
"In Section states are treated as two-sorted algebras, cf.",
"Definition REF .",
"The elementary and yet key Proposition REF about their equational presentation is proved.",
"The proposition enables us to identify the category of states ${\\sf ES}$ with the category of models in ${\\sf Set}$ of a finitely axiomatised two-sorted equational theory.",
"There is a corresponding, non-algebraic category ${\\sf S}$ of states between unital Abelian $\\ell $ -groups.",
"Theorem REF then shows that the categories ${\\sf ES}$ and ${\\sf S}$ are equivalent.",
"Section deals with free objects (free states) in ${\\sf ES}$ , and with universal states (see below for precise definitions).",
"We describe the free object generated by a two-sorted set by universal states and binary coproducts in the category of MV-algebras (Theorem REF ).",
"This constitutes our first main result.",
"The second part of our paper is devoted to the issue of representing universal states explicitly, insofar as this is possible.",
"For this, in Section we show how Choquet's theory of affine representations [10] relates to the construction of a universal state.",
"Specifically, Proposition REF says that the codomain of a universal state is Archimedean (or semisimple, in the case of MV-algebras) precisely when it coincides with the extended form of the affine representation that we introduce.",
"Thus, universal states with a semisimple codomain admit of a satisfactory concrete description through affine representations.",
"Our second main result, proved in Section , is that the codomain of the universal state of any locally finite MV-algebra (in particular, of any Boolean algebra) is semisimple.",
"The proof uses the duality between finitely presented MV-algebras and the category of compact rational polyhedra with piecewise linear maps with integer coefficients as morphisms.",
"We assume familiarity with MV-algebras and unital Abelian $\\ell $ -groups; see [6], [24] and [4], [10] for background information.",
"We often adopt the standard practice in algebra of omitting underlying-set functors, if clarity is not impaired.",
"We shall also omit parentheses in application of functors and functions, writing e.g.",
"$FI$ in place of $F(I)$ , when this improves readability.",
"We assume $\\operatorname{\\mathbb {N}}\\lbrace 1,2,\\ldots \\rbrace $ ."
],
[
"Mundici's equivalence, for states",
"Let ${{\\sf A_1^\\ell }}$ be the category that has unital Abelian $\\ell $ -groups as objects and unital $\\ell $ -homomorphisms as morphisms, and let ${\\sf MV}$ be the category of MV-algebras and their homomorphisms.",
"In [20], Mundici established a categorical equivalence between ${{\\sf A_1^\\ell }}$ and ${\\sf MV}$ which we recall here, without proofs.",
"To each unital $\\ell $ -group $G$ we associate its unit interval $\\Gamma (G,1)\\lbrace a\\in G\\mid 0\\leqslant a \\leqslant 1\\rbrace ,$ equipped with the operations $a\\oplus b & (a+b)\\wedge 1, \\\\\\lnot a & 1-a.$ Then $(\\Gamma (G,1),\\oplus ,\\lnot ,0)$ is an MV-algebra [6].",
"We will also make use of binary operations $\\odot $ and $\\ominus $ on $\\Gamma (G,1)$ defined by $a\\odot b & \\lnot (\\lnot a \\oplus \\lnot b),\\\\a\\ominus b & a \\odot \\lnot b.$ The operations $+,\\oplus $ , and $\\odot $ in $\\Gamma (G,1)$ are related as follows (see [6]): $a+b=(a\\oplus b) + (a\\odot b), \\qquad a,b\\in \\Gamma (G,1).$ Since every unital $\\ell $ -homomorphism $f\\colon G\\rightarrow H$ restricts to a homomorphism of MV-algebras $\\Gamma (G,1)\\rightarrow \\Gamma (H,1)$ , we obtain a functor $\\Gamma \\colon {{\\sf A_1^\\ell }}\\longrightarrow {\\sf MV}.$ If the unit 1 is understood, we write $\\Gamma G$ in place of $\\Gamma (G,1)$ .",
"In order to describe a functor in the other direction, the notion of good sequence was introduced in [20] (see also [6]).",
"For $M$ an MV-algebra, we say that $\\mathbf {a}(a_{i})_{i\\in \\operatorname{\\mathbb {N}}} \\in M^{\\operatorname{\\mathbb {N}}}$ is a good sequence in $M$ if $a_{i}\\oplus a_{i+1}=a_{i}$ for each $i\\in \\operatorname{\\mathbb {N}}$ , and there is $n_{0}\\in \\operatorname{\\mathbb {N}}$ such that $a_{n}=0$ for all $n\\geqslant n_{0}$ .",
"We shall write $(a_{1},\\dots ,a_{k})$ in place of $(a_{1},\\dots ,a_{k},0,0,\\dots )$ ; in particular, $(a)$ is short for $(a,0,0,\\dots )$ , given $a\\in M$ .",
"Addition of good sequences $(a_{i})_{i\\in \\operatorname{\\mathbb {N}}}$ and $(b_{i})_{i\\in \\operatorname{\\mathbb {N}}}$ is defined by $(a_{i})_{i\\in \\operatorname{\\mathbb {N}}} + (b_{i})_{i\\in \\operatorname{\\mathbb {N}}} (a_{i}\\oplus (a_{i-1}\\odot b_{1 }) \\oplus \\dots \\oplus (a_{1}\\odot b_{i-1})\\oplus b_{i})_{i\\in \\operatorname{\\mathbb {N}}}.$ The set of all good sequences in $M$ equipped with $+$ becomes a commutative monoid $A_{M}\\subseteq M^{\\operatorname{\\mathbb {N}}}$ with neutral element $(0)$ .",
"By general algebra, the full inclusion of the category of Abelian groups into that of commutative monoids has a left adjoint; write $\\eta _{A_M}\\colon A_M\\rightarrow \\Xi M$ for the component at $A_M$ of the unit of this adjunction.",
"The monoid $A_{M}$ can be shown to be cancellative, so the monoid homomorphism $\\eta _{A_M}$ is injective.",
"To describe the elements of $\\Xi M$ explicitly, let us say two ordered pairs of good sequences $(\\mathbf {a},\\mathbf {b})$ and $(\\mathbf {a}^{\\prime },\\mathbf {b}^{\\prime })$ are equivalent if $\\mathbf {a}+\\mathbf {b}^{\\prime }=\\mathbf {a}^{\\prime }+ \\mathbf {b},$ and let us write $[\\mathbf {a},\\mathbf {b}]$ for the equivalence class of $(\\mathbf {a},\\mathbf {b})$ .",
"Then $\\Xi M$ is defined as the set of all equivalence classes of the form $[\\mathbf {a},\\mathbf {b}]$ equipped with the addition $[\\mathbf {a},\\mathbf {b}] + [\\mathbf {c},\\mathbf {d}] [\\mathbf {a}+\\mathbf {c},\\mathbf {b}+\\mathbf {d}],$ with the neutral element $[(0),(0)]$ , and with the unary inverse operation $-[\\mathbf {a},\\mathbf {b}][\\mathbf {b},\\mathbf {a}].$ Moreover, the monoid $A_M$ is lattice-ordered by the restriction of the product order of $M^{\\operatorname{\\mathbb {N}}}$ , and this lattice order on $A_{M}$ extends (in the obvious sense, through the injection $\\eta _{A_M}$ ) to exactly one translation-invariant lattice order on $\\Xi M$ .",
"Thus, $\\Xi M$ is an Abelian $\\ell $ -group with unit $[(1),(0)]$ .",
"Lemma 2.1 For any MV-algebra $M$ , and any unital Abelian $\\ell $ -group $G$ , the following hold.",
"The function $\\varphi _M \\colon M \\rightarrow \\Gamma \\Xi M$ given by $\\varphi _M(a)[(a),(0)],\\quad a\\in M,$ is an isomorphism of MV-algebras.",
"The lattice-ordered monoids $G^{+}\\lbrace a\\in G\\mid a\\geqslant 0\\rbrace $ and $A_{\\Gamma G}$ are isomorphic through the function $g\\colon G^{+}\\rightarrow A_{\\Gamma G}$ that sends $a\\in G^{+}$ to the unique good sequence $g(a)(a_{1},\\dots ,a_{n})$ of elements $a_{i}\\in \\Gamma G$ such that $a=a_{1}+\\cdots + a_{n}$ .",
"The function $\\varepsilon _G\\colon G\\rightarrow \\Xi \\Gamma G$ defined by $\\varepsilon _G (a) [g(a^+),g(a^-)], \\quad a\\in G,$ is an isomorphism of unital $\\ell $ -groups, where $g$ is as in item (2) above, and, as usual, $a^{+}a\\vee 0$ , $a^{-}-a \\vee 0$ .",
"See Theorem 2.4.5, Lemma 7.1.5, and Corollary 7.1.6 in [6].",
"A homomorphism $h\\colon M\\rightarrow N$ of MV-algebras lifts to a function $h^{*}\\colon A_{M}\\rightarrow A_{N}$ upon setting $h^{*}((a_{i})_{i\\in \\operatorname{\\mathbb {N}}})(h(a_{i}))_{i\\in \\operatorname{\\mathbb {N}}}$ .",
"Then Lemma REF and the universal construction of $\\Xi M$ from $A_M$ entail that $h^{*}$ has exactly one extension to a unital $\\ell $ -homomorphism $\\Xi h\\colon \\Xi M\\rightarrow \\Xi N$ .",
"We thereby obtain a functor $\\Xi \\colon {\\sf MV}\\longrightarrow {{\\sf A_1^\\ell }}.$ Theorem 2.1 (Mundici's equivalence) The functors $\\Gamma \\colon {{\\sf A_1^\\ell }}\\rightarrow {\\sf MV}$ and $\\Xi \\colon {\\sf MV}\\rightarrow {{\\sf A_1^\\ell }}$ form an equivalence of categories.",
"Remark 2.1 In what follows we shall often tacitly identify $M$ with $\\Gamma \\Xi M\\subseteq \\Xi M$ , and thus speak of functions defined on $M$ having an extension to $\\Xi M$ , etc.",
"In the rest of this section we lift the equivalence of Theorem REF from homomorphisms to states.",
"Definition 2.1 (States) For MV-algebras $M$ and $N$ , a function $s\\colon M\\rightarrow N$ is a state (of $M$ with values in $N$) if $s(1)=1$ , and for each $a,b\\in M$ with $a \\odot b=0$ the equality $s(a\\oplus b) =s(a)+s(b)$ holds, where $+$ is interpreted in $\\Xi N$ .",
"For unital Abelian $\\ell $ -groups $G$ and $H$ , a function $s\\colon G\\rightarrow H$ is a state (of $G$ with values in $H$) if $s(1)=1$ , and $s$ is a group homomorphism that is positive, i.e., for each $g\\in G^+$ we have $s(g)\\in H^+$ .",
"We write ${\\sf MV_{st}}$ for the category whose objects are MV-algebras and whose morphisms are states, and ${{{\\sf A_1}}}$ for the category whose objects are unital Abelian $\\ell $ -groups and whose morphisms are states.",
"States of MV-algebras can be defined in several equivalent ways.",
"For instance, a function $s\\colon M\\rightarrow N$ satisfying $s(1)=1$ is a state if, and only if, for each $a,b\\in M$ with $a \\odot b=0$ , the equalities $s(a\\oplus b) =s(a)\\oplus s(b)$ and $s(a)\\odot s(b)=0$ hold—see [15].",
"In Section we will give an equational characterisation of states (Proposition REF ).",
"Remarks 2.1 (1) It is elementary that requiring the group homomorphism $s\\colon G\\rightarrow H$ to be positive is equivalent to asking that it be order-preserving.",
"(2) States are a classical notion in the theory of partially ordered Abelian groups (see e.g.",
"[10]), where they are most often assumed to have codomain $\\operatorname{\\mathbb {R}}$ .",
"For emphasis, we refer to the latter states as real-valued.",
"A significant example is the Lebesgue integral $\\int _{[0,1]}- \\,{\\rm d}\\lambda $ over the unital Abelian $\\ell $ -group of all continuous functions $[0,1]\\rightarrow \\operatorname{\\mathbb {R}}$ .",
"Let us stress that states do not necessarily preserve infima and suprema and, therefore, in general they are not morphisms in ${{\\sf A_1^\\ell }}$ .",
"(3) States of MV-algebras also are a well-studied notion (see e.g.",
"[24], [8]), and they are usually assumed to have codomain $[0,1]\\subseteq \\operatorname{\\mathbb {R}}$ .",
"We refer to the latter states as real-valued.",
"Analogously to the previous item, states of MV-algebras are not morphisms in ${\\sf MV}$ , in general.",
"Lemma 2.2 Let $M$ and $N$ be MV-algebras, and $G$ and $H$ be unital Abelian $\\ell $ -groups.",
"Any state of unital Abelian $\\ell $ -groups $s\\colon G\\rightarrow H$ restricts to a function $\\Gamma (s)\\colon \\Gamma G\\rightarrow \\Gamma H$ that is a state of MV-algebras.",
"Any state of MV-algebras $s\\colon M\\rightarrow N$ has exactly one extension to a state of unital Abelian $\\ell $ -groups $\\Xi (s)\\colon \\Xi M\\rightarrow \\Xi N$ .",
"To prove the first item, observe that $0\\leqslant a \\leqslant 1$ in $G$ entails $0\\leqslant s(a)\\leqslant 1$ in $H$ , because $s$ preserves the order; further, $s(1)=1$ by definition.",
"If $0\\leqslant a_1,a_2\\leqslant 1$ in $G$ , and $a_1\\odot a_2 =0$ in the MV-algebra $\\Gamma G$ , then (REF ) yields $a_1\\oplus a_2=a_2+a_2$ .",
"For the second item, let us regard $s$ as a function $\\Gamma \\Xi M\\rightarrow \\Gamma \\Xi N$ .",
"Then $s$ is easily seen to be order-preserving, [24].",
"For $a,b\\in \\Gamma \\Xi M$ we have $s(a+b)=s(a)+s(b)$ as soon as $a+b$ belongs to $\\Gamma \\Xi M$ .",
"Indeed, the latter happens precisely when $a\\odot b=0$ by (REF ), and then $s(a+b)=s(a\\oplus b)=s(a)+s(b)$ by the definition of state.",
"Now existence and uniqueness of $\\Xi (s)$ is granted by the general extension result [12].",
"(The hypotheses of the cited proposition require $G$ to be directed, and to have the Riesz interpolation property; it is classical and elementary that any $\\ell $ -group satisfies these properties.)",
"In light of Lemma REF we consider the functors $\\Gamma \\colon {{{\\sf A_1}}}\\longrightarrow {\\sf MV_{st}}$ and $\\Xi \\colon {\\sf MV_{st}}\\longrightarrow {{{\\sf A_1}}}$ that extend the homonymous ones in (REF ) and (REF ), respectively.",
"Theorem 2.2 (Mundici's equivalence, for states) The functors $\\Gamma \\colon {{{\\sf A_1}}}\\rightarrow {\\sf MV_{st}}$ and $\\Xi \\colon {\\sf MV_{st}}\\longrightarrow {{{\\sf A_1}}}$ form an equivalence of categories.",
"The isomorphism $\\varphi _M \\colon M \\rightarrow \\Gamma \\Xi M$ from Lemma REF (1) is natural: if $s\\colon M\\rightarrow N$ is a state of MV-algebras, the naturality square $\\begin{tikzcd}M [rr, \"\\varphi _M\"] [d, \"s\"^{\\prime }] & {} & \\Gamma \\Xi M [d, \"\\Gamma \\Xi (s)\"]\\\\N [rr, \"\\varphi _N\"^{\\prime }] & {} & \\Gamma \\Xi N\\end{tikzcd}$ commutes by the definitions of $\\Xi $ and $\\Gamma $ .",
"The isomorphism $\\varepsilon _{G}\\colon G\\rightarrow \\Xi \\Gamma G$ is also natural: if $s\\colon G\\rightarrow H$ is a state of unital Abelian $\\ell $ -groups, the naturality square $\\begin{tikzcd}G [rr, \"\\varepsilon _G\"] [d, \"s\"^{\\prime }] & {} & \\Xi \\Gamma G [d, \"\\Xi \\Gamma (s)\"]\\\\H [rr, \"\\varepsilon _H\"^{\\prime }] & {} & \\Xi \\Gamma H\\end{tikzcd}$ commutes.",
"Indeed, $\\Xi \\Gamma (s)\\varepsilon _G$ and $\\varepsilon _Hs$ are states $G\\rightarrow \\Xi \\Gamma H$ , and it follows from direct inspection of the definitions involved that they agree on the unit interval of $G$ ; but then they agree on the whole of $G$ , by (1) and (2) in Lemma REF ."
],
[
"The two-sorted variety of states",
"For multi-sorted universal algebra see the pioneering [13], [5], and the textbook reference [1].",
"We are concerned with the two-sorted case only.",
"Unlike [5], and like [1], we allow arbitrary multi-sorted sets as carriers of algebras—no non-emptyness requirement is enforced.",
"The difference is immaterial for the present paper, because each of our two sorts has constants.",
"We recall that the product category ${\\sf Set^2}{\\sf Set}\\times {\\sf Set}$ of two-sorted sets and two-sorted functions has as objects the ordered pairs $(A,B)$ of sets, and as morphisms $f\\colon (A_{1},B_{1}) \\longrightarrow (A_{2},B_2)$ the ordered pairs $f(f_{1},f_{2})$ of functions $f_{1}& \\colon A_{1} \\rightarrow A_{2},\\\\f_{2}&\\colon B_{1} \\rightarrow B_{2}.$ Composition of morphisms and identity morphisms are defined componentwise.",
"We consider two sorts $\\mathcal {R}$ and $\\mathcal {E}$ of random variables and degrees of expectation, respectively, and operations as follows.",
"(T1) Operations $\\oplus \\colon \\mathcal {R}^2\\rightarrow \\mathcal {R}$ , $\\lnot \\colon \\mathcal {R}\\rightarrow \\mathcal {R}$ , and $0\\colon \\mathcal {R}^{\\emptyset }\\rightarrow \\mathcal {R}$ .",
"Thus, these are operations of arities 2, 1, and 0, respectively, in the sort of random variables.",
"(T2) Operations $\\oplus \\colon \\mathcal {E}^2\\rightarrow \\mathcal {E}$ , $\\lnot \\colon \\mathcal {E}\\rightarrow \\mathcal {E}$ , and $0\\colon \\mathcal {E}^{\\emptyset }\\rightarrow \\mathcal {E}$ .",
"Thus, these are operations of arities 2, 1, and 0, respectively, in the sort of expectation degrees.",
"They are purposefully denoted by the same symbols as their counterparts in the sort $\\mathcal {R}$ .",
"(T3) One operation $s\\colon \\mathcal {R}\\rightarrow \\mathcal {E}$ from the sort of random variables to that of degrees of expectation.",
"Items (T1)–(T3) define a two-sorted (similarity) type.",
"For the sake of clarity, let us spell out that a two-sorted function $(m,n)\\colon (M_{1},N_{1})\\rightarrow (M_{2},N_{2})$ is a homomorphism between algebras of this two-sorted type precisely when $m\\colon M_{1}\\rightarrow M_{2}$ and $n\\colon N_{1}\\rightarrow N_{2}$ are homomorphisms in the type of $\\mathcal {R}$ and $\\mathcal {E}$ , respectively, and moreover the square $\\begin{tikzcd}M_{1} [rr, \"s\"] [d, \"m\"^{\\prime }] & {} & N_{1} [d, \"n\"]\\\\M_{2} [rr, \"s\"^{\\prime }] & {} & N_{2}\\end{tikzcd}$ commutes.",
"Definition 3.1 (States as two-sorted algebras) A state is an algebra $(M,N)$ of the two-sorted type (T1)–(T3) such that the following equational conditions hold.",
"(S1) $(M,\\oplus ,\\lnot ,0)$ is an MV-algebra.",
"(S2) $(N,\\oplus ,\\lnot ,0)$ is an MV-algebra.",
"(S3) For every $a,b\\in M$ , $s\\colon M\\rightarrow N$ satisfies $s(a\\oplus b)=s(a) \\oplus s(b\\wedge \\lnot a)$ , $s(\\lnot a)=\\lnot s(a)$ , and $s(1)=1$ .",
"Remark 3.1 The presented axiomatisation (S3) is originally inspired by that of internal states [9].",
"It was used already in [16] in case of states whose domains are Boolean algebras; see also [15].",
"Let us emphasise that in the above and throughout we denote a two-sorted algebra simply by its underlying two-sorted set $(M,N)$ .",
"All operations—$\\oplus $ , $s$ , and so forth—are tacitly understood.",
"This is in keeping with standard usage in algebra.",
"Item (S3) in Definition REF amounts to requiring that $s\\colon M\\rightarrow N$ be a state: Proposition 3.1 Let $s\\colon M \\rightarrow N$ be a function between MV-algebras $M$ and $N$ .",
"The following are equivalent.",
"The function $s$ is a state.",
"The function $s$ satisfies (S3) in Definition REF : $s(a\\oplus b)=s(a) \\oplus s(b\\wedge \\lnot a)$ , $s(\\lnot a)=\\lnot s(a)$ , $s(1)=1$ .",
"We will need the following equations, valid in all MV-algebras: $b\\wedge \\lnot a=b\\ominus (a\\odot b)$ , $a\\oplus b=a\\oplus (b\\wedge \\lnot a)$ , $a\\odot (b\\wedge \\lnot a)=0$ .",
"In detail, REF holds by the definition of $\\wedge $ , since $b\\wedge \\lnot a =b\\odot (\\lnot b \\oplus \\lnot a)=b\\odot \\lnot (a\\odot b)=b\\ominus (a\\odot b).$ The operation $\\oplus $ distributes over $\\wedge $ by [6], so $a\\oplus (b\\wedge \\lnot a) = (a\\oplus b) \\wedge (a\\oplus \\lnot a)=a\\oplus b,$ which proves REF .",
"Finally, for REF , $a\\odot (b\\wedge \\lnot a)=a\\odot (b\\ominus (a\\odot b))=(a\\odot b) \\odot \\lnot (a\\odot b)=0.$ Assume $s$ is a state.",
"Then REF holds by definition.",
"Further, REF , REF , and the definition of state yield $s(a\\oplus b)=s(a\\oplus (b\\wedge \\lnot a))=s(a) + s(b\\wedge \\lnot a).$ Then $s(a) + s(b\\wedge \\lnot a)\\in N$ , and so the $+$ above agrees in fact with $\\oplus $ by (REF ).",
"Therefore, REF holds.",
"Finally, since $a\\odot \\lnot a =0$ , we can write $1=s(a\\oplus \\lnot a)=s(a) + s(\\lnot a),$ which proves REF because $1-s(a)$ equals $\\lnot s(a)$ in $\\Xi N$ .",
"Thus, $s$ satisfies REF –REF .",
"Conversely, assume $s\\colon M \\rightarrow N$ has properties REF –REF .",
"We first prove that $s$ is order-preserving.",
"Assume $a\\leqslant b$ , or equivalently, by definition, $b=a\\oplus (b\\ominus a)$ .",
"Then REF yields $s(b)=s(a\\oplus (b\\ominus a))=s(a)\\oplus s((b\\ominus a) \\wedge \\lnot a) \\geqslant s(a),$ where the last inequality follows from monotonicity of $\\oplus $ in each coordinate [6].",
"Let $a\\odot b=0$ .",
"Then REF gives $b\\wedge \\lnot a=b$ and, by REF , $s(a\\oplus b)=s(a)\\oplus s(b).$ It remains to check that $s(a)\\odot s(b)=0$ , which implies $s(a)\\oplus s(b)=s(a)+s(b)$ as is to be shown.",
"Employing REF , REF , and the assumption $a\\odot b=0$ , we get $s(\\lnot a)\\oplus s(\\lnot b \\wedge a)= s(\\lnot a \\oplus \\lnot b)=s(\\lnot (a\\odot b))=s(\\lnot 0)=1.$ Since $s$ is order-preserving, and since $\\oplus $ is monotone in each coordinate by [6], we infer $s(\\lnot a) \\oplus s(\\lnot b)\\geqslant s(\\lnot a)\\oplus s(\\lnot b \\wedge a)=1$ .",
"Then by REF this gives $s(a)\\odot s(b)=\\lnot (s(\\lnot a) \\oplus s(\\lnot b))=0,$ which completes the proof.",
"We consider the category ${\\sf ES}$ (for “Equational States”) of states in the sense of Definition REF , and their homomorphisms.",
"By its very definition, ${\\sf ES}$ is the category of models in ${\\sf Set}$ of a (finitely axiomatised) two-sorted equational theory.",
"Thus, ${\\sf ES}$ is a two-sorted variety—i.e., it is closed under homomorphic images, subalgebras, and products inside the category of all algebras of the type (T1)–(T3)—by the easy implication in Birkhoff's Variety Theorem.",
"(See [2] for details on Birkhoff's Theorem in the multi-sorted setting.)",
"Let us recall that “homomorphic images” here are the codomains of those homomorphisms that are surjective in each sort, and that these are exactly the regular epimorphisms [1].",
"We further consider the category ${\\sf S}$ of states whose objects are all states $G\\rightarrow H$ of unital Abelian $\\ell $ -groups $G$ with values in any unital Abelian $\\ell $ -groups $H$ , and whose morphisms are pairs of unital $\\ell $ -homomorphisms $G_1\\rightarrow G_2$ and $H_1\\rightarrow H_2$ making the obvious square commute.",
"Thus, the objects of ${\\sf S}$ are exactly the arrows $G\\rightarrow H$ in ${{{\\sf A_1}}}$ .",
"We define a functor $\\Gamma ^2\\colon {\\sf S}\\longrightarrow {\\sf ES}$ by setting $\\Gamma ^2(G\\rightarrow H)(\\Gamma G,\\Gamma H)$ , for $G\\rightarrow H$ an object of ${\\sf S}$ , where the operation $s\\colon \\Gamma G\\rightarrow \\Gamma H$ of the two-sorted algebra $(\\Gamma G,\\Gamma H)$ is defined as the restriction of the given state $G\\rightarrow H$ (cf.",
"item (1) in Lemma REF ).",
"Concerning arrows, given the morphism $(g,h)\\colon (G_1\\rightarrow H_1)\\rightarrow (G_2\\rightarrow H_2)$ in ${\\sf S}$ —i.e., given the pair of unital $\\ell $ -homomorphisms $g\\colon G_1\\rightarrow G_2$ and $h\\colon H_1\\rightarrow H_2$ that form the required commutative square with the states $G_i\\rightarrow H_i$ , $i=1,2$ —we set $\\Gamma ^2(g,h)(\\Gamma g,\\Gamma h)$ , which is evidently a morphism in ${\\sf ES}$ .",
"We also define a functor $\\Xi ^2\\colon {\\sf ES}\\longrightarrow {\\sf S}$ by setting $\\Xi ^2(M,N)$ to be the state $\\Xi s\\colon \\Xi M \\rightarrow \\Xi N$ (cf.",
"item (2) in Lemma REF ), where $s\\colon M\\rightarrow N$ is the two-sorted operation of $(M,N)$ .",
"For a morphism $(m,n)$ from $(M_1,N_1)$ to $(M_2,N_2)$ , we let $\\Xi ^2(m,n)(\\Xi m,\\Xi n)$ ; this is a morphism in ${\\sf S}$ because $\\Xi $ is a functor.",
"Theorem 3.1 (Equational characterisation of states) The functors $\\Gamma ^2\\colon {\\sf S}\\rightarrow {\\sf ES}$ and $\\Xi ^2\\colon {\\sf ES}\\rightarrow {\\sf S}$ form an equivalence of categories.",
"The morphism $(\\varphi _{M},\\varphi _{N})\\colon (M,N) \\rightarrow \\Gamma ^2\\Xi ^2(M,N)$ is an isomorphism in ${\\sf ES}$ , where the components of $(\\varphi _{M},\\varphi _{N})$ are as in Lemma REF and thus are isomorphisms; naturality is verified componentwise, and thus follows at once from Theorem REF .",
"Similarly, for every state $G\\rightarrow H$ in ${{{\\sf A_1}}}$ , $(\\varepsilon _{G},\\varepsilon _{H})\\colon (G\\rightarrow H)\\rightarrow \\Xi ^2\\Gamma ^2(G,H)$ is an isomorphism, where $\\varepsilon _{G}$ and $\\varepsilon _{H}$ are as in Lemma REF , and naturality reduces to naturality in each sort, which holds by Theorem REF ."
],
[
"Free and universal states",
"For a set $I$ , let us write $FI$ for the free MV-algebra generated by $I$ , and $\\iota _I\\colon I\\rightarrow F I$ for the “inclusion of free generators”, i.e., for the component at $I$ of the unit of the free/underlying-set adjunction $F\\dashv |-|$ .",
"By general algebraic considerations, the functor $|-|_2\\colon {\\sf ES}\\longrightarrow {\\sf Set^2}$ that takes a state of MV-algebras to its carrier two-sorted set has a left adjoint $F^2\\colon {\\sf Set^2}\\longrightarrow {\\sf ES}.$ The existence of this left adjoint has nothing to do with MV-algebras specifically, and is entailed by the existence of free algebras in varieties of multi-sorted algebras.",
"The original reference for this latter result seems to be [13], with a generalisation established in [5].",
"For a two-sorted set $S$ , write $\\eta _ S\\colon S \\rightarrow F^2S$ for the component at $S$ of the unit of the adjunction $F^2\\dashv |-|_2$ .",
"Then $\\eta _S$ is characterised as the essentially unique two-sorted function $S \\rightarrow F^2S$ such that, for any two-sorted function $f\\colon S \\rightarrow (M,N)$ with a state in the codomain, there is exactly one morphism $h\\colon F^2S\\rightarrow (M,N)$ in ${\\sf ES}$ making the diagram $\\begin{tikzcd}S [r, \"\\eta _S\"] [dr, \"f\"^{\\prime }] & F^2S [d, dashed, \"h\"]\\\\& (M,N)\\end{tikzcd}$ commute.",
"In algebraic parlance, $F^2S$ is a state freely generated by the two-sorted set $S$ ; it is evidently unique to within a unique isomorphism, and therefore any state satisfying the preceding universal property will be called the free state generated by $S$.",
"Lemma 4.1 Consider the two-sorted set $(\\emptyset ,S_2)$ , where $S_2$ is any set.",
"Writing $\\mathbb {2}$ for the initial MV-algebra (=two-element Boolean algebra), $(\\mathbb {2} ,FS_{2})$ is the free state in ${\\sf ES}$ generated by $(\\emptyset ,S_2)$ , where the operation $s$ is the unique homomorphism $\\mathbb {2}\\rightarrow FS_2$ .",
"Given a two-sorted function $(f_!,f_2)\\colon (\\emptyset ,S_2)\\rightarrow (M,N)$ , where $f_!\\colon \\emptyset \\rightarrow M$ is the unique function from $\\emptyset $ to $M$ , write $h_!\\colon \\mathbb {2}\\rightarrow M$ for the unique homomorphism, and $h_2\\colon FS_2\\rightarrow N$ for the unique homomorphism such that $f_2=h_2\\iota _{S_2}$ , where $\\iota _{S_2}$ is as in (REF ).",
"Further, consider the component-wise inclusion $\\iota \\colon (\\emptyset ,S_2)\\subseteq (\\mathbb {2},FS_2)$ .",
"Then $(h_!,h_2)\\colon (\\mathbb {2},FS_{2})\\rightarrow (M,N)$ is a morphism in ${\\sf ES}$ , because $\\mathbb {2}$ is initial in ${\\sf MV_{st}}$ .",
"Also, $(f_!,f_2)=(h_!,h_2)\\iota $ , and $(h_!,h_2)$ is clearly unique with this property.",
"If $M$ is any MV-algebra, a state $\\upsilon _M\\colon M\\longrightarrow \\Upsilon M$ is said to be universal (for $M$ ) if for each state $s\\colon M\\rightarrow N$ there is exactly one homomorphism of MV-algebras $h\\colon \\Upsilon M\\rightarrow N$ satisfying $h\\upsilon _M=s$ .",
"Universal states, when they exist, are evidently unique to within a unique isomorphism.",
"Any state satisfying the preceding universal property will therefore be called the universal state (of $M$).",
"Lemma 4.2 Consider the two-sorted set $(S_1,\\emptyset )$ , where $S_1$ is any set.",
"Then the free state $F^2(S_1,\\emptyset )$ generated by $(S_1,\\emptyset )$ is (isomorphic in ${\\sf ES}$ to) $(FS_{1},\\Upsilon F S_1),$ where the operation $s$ is the universal state $\\upsilon _{FS_1}$ of $FS_1$ .",
"Let us display the components of $F^2(S_1,\\emptyset )$ as the pair of MV-algebras $(A,B)$ .",
"If $M$ is any MV-algebra and $\\mathbb {1}$ is the terminal (=one-element) MV-algebra, we consider the object $(M,\\mathbb {1})$ of ${\\sf ES}$ whose operation $s$ is the unique homomorphism $M\\rightarrow \\mathbb {1}$ .",
"Given any function $f\\colon S_1\\rightarrow M$ , the unique two-sorted function $(f,!",
")\\colon (S_1,\\emptyset )\\rightarrow (M,\\mathbb {1})$ whose first component is $f$ (and whose second component, necessarily, is the only possible function $!\\colon \\emptyset \\rightarrow \\mathbb {1}$ ) has exactly one extension to a homomorphism $(h_1,!",
")\\colon (A,B)\\rightarrow (M,\\mathbb {1})$ .",
"The second component of this homomorphism is the only possible one $!\\colon B\\rightarrow \\mathbb {1}$ .",
"Then $h_1\\colon A\\rightarrow M$ is a homomorphism that extends $f$ , and it is the unique such: if $h_1^{\\prime }\\colon A\\rightarrow M$ extends $f$ then $(h_1^{\\prime },!",
")\\colon (A,B)\\rightarrow (M,\\mathbb {1})$ extends $(f,!",
")$ , and thus $h_1^{\\prime }=h_1$ by the uniqueness of $(h_1,!",
")$ .",
"This shows that $A$ is $FS_1$ .",
"In the rest of this proof we write $FS_1$ and drop $A$ .",
"Let us next consider the operation $s\\colon FS_1\\rightarrow B$ of the object $(FS_1,B)$ , with the intent of proving that the state $s$ is universal for $FS_1$ .",
"Consider any state $t\\colon FS_1\\rightarrow N$ , and the corresponding object $(FS_1,N)$ of ${\\sf ES}$ .",
"Writing $\\iota \\colon S_1\\rightarrow FS_1$ for the insertion of free generators, we consider the two-sorted function $(\\iota ,!",
")\\colon (S_1,\\emptyset )\\rightarrow (FS_1,N)$ .",
"The universal property of $(FS_1,B)$ yields exactly one homomorphism $(1_{FS_1},h_2)\\colon (FS_1,B)\\rightarrow (FS_1,N)$ extending $(\\iota ,!",
")$ , the first component of which is, necessarily, the identity $1_{FS_1}\\colon FS_1\\rightarrow FS_1$ .",
"The second component $h_2$ therefore satisfies $h_2s=t$ .",
"Further, $h_2$ is the only homomorphism with this property.",
"Indeed, if $h^{\\prime }_2\\colon B \\rightarrow N$ satisfies $h^{\\prime }_2s=t$ , then $(1_{FS_1},h^{\\prime }_2)\\colon (FS_1,B)\\rightarrow (FS_1,N)$ extends $(\\iota ,!",
")$ and so $h^{\\prime }_2=h_2$ by the universal property of $(FS_1,B)$ .",
"In conclusion, $s\\colon FS_1\\rightarrow B$ is the universal state $\\upsilon _{F S_1}$ of $FS_1$ , and therefore $B$ is $\\Upsilon F S_1$ .",
"This completes the proof.",
"Lemma REF establishes the existence of the universal state of a free MV-algebra.",
"For the general case, we take a quotient.",
"If $f\\colon A\\rightarrow B$ is a function, we write $\\operatorname{Ker}{f}\\lbrace (x,y)\\in A^2\\mid fx=fy\\rbrace $ .",
"This notation is opposed to $\\ker {f}f^{-1}\\lbrace 0\\rbrace $ , which we also use later on in the paper and which makes sense when $A$ and $B$ are either MV-algebras or lattice-groups.",
"Corollary 4.1 Every MV-algebra has a universal state.",
"Let $M$ be any MV-algebra, write $FM$ for the free MV-algebra generated by the set $M$ , and let $q_1\\colon FM\\rightarrow M$ be the unique, automatically surjective homomorphism extending the identity function $M\\rightarrow M$ .",
"By Lemma REF , $F^2(M,\\emptyset )$ is $(FM,\\Upsilon F M)$ , where the operation $s$ is the universal state $\\upsilon _{FM}$ of $FM$ .",
"In the rest of this proof we write $\\upsilon $ , tout court.",
"Set $\\theta _1=\\operatorname{Ker}{q_1}$ and $\\theta _2=\\langle \\lbrace (\\upsilon x, \\upsilon y) \\mid (x,y)\\in \\theta _1\\rbrace \\rangle $ , where $\\langle - \\rangle $ denotes the congruence generated by $-$ on $\\Upsilon F M$ .",
"Then $(\\theta _1,\\theta _2)$ is evidently a congruence on $(FM,\\Upsilon F M)$ , and it is the congruence generated by the two-sorted relation $(\\theta _1,\\emptyset )$ —indeed, any congruence on $(FM,\\Upsilon F M)$ that contains $(\\theta _1,\\emptyset )$ must contain $(\\theta _1,\\lbrace (\\upsilon x, \\upsilon y) \\mid (x,y)\\in \\theta _1\\rbrace )$ , by the compatibility with $\\upsilon $ , and thus must contain $(\\theta _1,\\theta _2)$ .",
"We consider the quotient state $(M,\\frac{\\Upsilon FM}{\\theta _2})$ of $(FM,\\Upsilon FM)$ modulo the congruence $(\\theta _1,\\theta _2)$ , with operation $M\\rightarrow \\frac{\\Upsilon FM}{\\theta _2}$ denoted $s$ , $\\begin{tikzcd}[row sep=large]FM [r, \"\\upsilon \"] [d, \"q_1\"^{\\prime }]& \\Upsilon FM[d, \"q_2\"] [dd, \"k\", bend left, dashed]\\\\M [d, \"t\"^{\\prime }] [r, \"s\"] & \\frac{\\Upsilon FM}{\\theta _2} [d, \"h\", dashed]\\\\N [r, \"1_N\"^{\\prime }]& N\\end{tikzcd}$ and we verify that $s$ is the universal state of $M$ .",
"(We write $q_2\\colon \\Upsilon FM\\rightarrow \\frac{\\Upsilon FM}{\\theta _2}$ for the natural quotient map.)",
"For this, let $t\\colon M \\rightarrow N$ be any state.",
"The composite $tq_1\\colon FM\\rightarrow N$ is a state, too.",
"Since $\\upsilon $ is universal for $FM$ , there is exactly one homomorphism $k\\colon \\Upsilon F M\\rightarrow N$ such that $k\\upsilon =tq_1$ .",
"Let us show $\\operatorname{Ker}{q_2}\\subseteq \\operatorname{Ker}{k}$ .",
"Since $\\operatorname{Ker}{q_2}$ is $\\theta _2$ , it suffices to show that the inclusion holds for $\\lbrace (\\upsilon x, \\upsilon y) \\mid (x,y)\\in \\theta _1\\rbrace $ , because the latter is a generating set of $\\theta _2$ .",
"Given $(\\upsilon x, \\upsilon y)$ with $(x,y)\\in \\theta _1$ , from $q_1x=q_1y$ we obtain $tq_1x=tq_1y$ , and thus $k\\upsilon x = k\\upsilon y$ , that is $(\\upsilon x,\\upsilon y)\\in \\operatorname{Ker}{k}$ , as we intended to show.",
"By the universal property of quotients (applied to MV-algebras), there is exactly one homomorphism $h\\colon \\frac{\\Upsilon FM}{\\theta _2} \\rightarrow N$ with $hq_2=k$ .",
"This homomorphism satisfies $hs=t$ .",
"Indeed, from $hq_2=k$ and $k \\upsilon = tq_1$ we obtain $hq_2\\upsilon =tq_1$ , and therefore $hsq_1=tq_1$ ; but $q_1$ is epic, whence $hs=t$ .",
"Finally, if $h^{\\prime }$ is any homomorphism that satisfies $h^{\\prime }s=t$ , then $h^{\\prime }sq_1=tq_1$ and so $h^{\\prime }q_2\\upsilon =tq_1$ ; applying the uniqueness property of $k$ , we infer $h^{\\prime }q_2=k$ , and applying that of $h$ we conclude $h=h^{\\prime }$ .",
"This completes the proof that $s$ is universal for $M$ .",
"Corollary REF entails at once by elementary category theory (see e.g.",
"[17]) that the faithful, non-full inclusion functor $|-|\\colon {\\sf MV}\\longrightarrow {\\sf MV_{st}}$ has a left adjoint $\\Upsilon \\colon {\\sf MV_{st}}\\longrightarrow {\\sf MV}\\,.$ Thus, $\\Upsilon M$ is the MV-algebra freely generated by $|M|$ .",
"Remark 4.1 The faithful, non-full inclusion of ${{\\sf A_1^\\ell }}$ into the category of unital partially ordered Abelian groups (with morphisms the unital order-preserving group homomorphisms) is proved to have a left adjoint in [4]; the argument there is for the non-unital case, but is easily adapted.",
"Thus, the unital Abelian $\\ell $ -group freely generated by any unital partially ordered Abelian group exists.",
"As a special case of this, one has that the unital Abelian $\\ell $ -group freely generated by a unital partially ordered Abelian group that happens to be lattice-ordered exists.",
"Upon applying the results in Section to translate into the language of ordered groups, Corollary REF provides an alternative proof of this result that is streamlined by the use of two-sorted algebraic theories.",
"For clarity, we mention that Bigard, Keimel, and Wolfenstein in [4] distinguish between “universal” and free $\\ell $ -groups: the former are in fact what we call “free”, as is now standard; the latter have the further property that the universal arrow is an order embedding.",
"Our own usage of “universal” for the components of the unit of the adjunction $\\Upsilon \\dashv |-|$ is meant as mere emphasis, in view of the probabilistic meaning of the construction.",
"General free algebras in ${\\sf ES}$ reduce to universal states and coproducts in ${\\sf MV}$ ; see [21] for the latter.",
"We also say “sum” for “coproduct”.",
"We write $+$ to denote binary sums in ${\\sf MV}$ .",
"If $A\\rightarrow A+B \\leftarrow B$ is a coproduct, we call the two arrows the coproduct injections (with no implication about their injectivity as functions), and we often denote them ${\\rm in}_1$ and ${\\rm in}_2$ , respectively.",
"Theorem 4.1 For any two-sorted set $(S_{1},S_{2})$ , the state $(F S_{1},\\Upsilon F S_1 +F S_{2} )$ in ${\\sf ES}$ with operation $s$ equal to ${\\rm in}_{1} \\upsilon _{FS_1}$ , where $\\upsilon _{FS_1}$ is the universal state of $FS_1$ , and ${\\rm in}_{1}\\colon \\Upsilon FS_1\\rightarrow \\Upsilon F S_1 + F S_{2}$ is the first coproduct injection, is the free state generated by $S(S_1,S_2)$ , the component of the unit at $S$ being $\\eta _S(\\iota _{S_1}, {\\rm in_2}\\iota _{S_2} )$ with $\\iota _{S_i}$ as in (REF ), $i=1,2$ , and ${\\rm in_2}\\colon FS_2\\rightarrow \\Upsilon F S_1 +F S_{2}$ the second coproduct injection.",
"We consider a two-sorted function $(f_1,f_2)\\colon (S_1,S_2)\\rightarrow (M,N)$ .",
"With reference to the diagram below, $\\begin{tikzcd}S_{1} [dr, \"f_1\"^{\\prime }] [r, \"\\iota _{S_1}\"] & F S_{1} [dr] [d, \"h_1\"] [r, \"\\upsilon _{FS_1}\"] & \\Upsilon F S_{1} [d] [r, \"{\\rm in}_1\"] & \\Upsilon F S_1 +F S_{2} [dl, \"h_2\"^{\\prime }] \\\\& M [r] & N & [l] F S_{2} [u, \"{\\rm in}_2\"^{\\prime }] \\\\& & & S_2 [u, \"\\iota _{S_2}\"^{\\prime }] [ul, \"f_2\"]\\end{tikzcd}$ we have: Exactly one homomorphism $h_1\\colon FS_1\\rightarrow M$ making the upper left triangle commute, by the freeness of $FS_1$ ; the state $FS_1\\rightarrow N$ given by the composition $FS_1\\rightarrow M\\rightarrow N$ ; exactly one homomorphism $\\Upsilon FS_1\\rightarrow N$ making at its immediate left commute; exactly one homomorphism $FS_2\\rightarrow N$ making the lower right triangle commute, by the freeness of $FS_2$ ; and therefore, by the universal property of coproducts, there is exactly one homomorphism $h_2\\colon \\Upsilon FS_1+ FS_2\\rightarrow N$ such that precomposing it with the coproduct injections ${\\rm in}_1$ and ${\\rm in_2}$ yields $\\Upsilon FS_1\\rightarrow N$ and $FS_2\\rightarrow N$ , respectively.",
"By construction, the pair $(h_1,h_2)$ is a morphism in ${\\sf ES}$ that satisfies $(f_1,f_2)=(h_1,h_2)(\\iota _{S_1}, {\\rm in_2}\\iota _{S_2} )$ .",
"That it is the unique such follows readily from the diagram above."
],
[
"Universal states, and Choquet's affine representation",
"We recall basic facts about the theory of affine representations.",
"For background information and references see [10].",
"We formulate the results in the language of ordered groups, as is traditional; they may be translated for MV-algebras via the equivalence in Theorem REF .",
"For a unital Abelian $\\ell $ -group $G$ , set $\\operatorname{St}{G}\\left\\lbrace s\\colon G\\rightarrow \\operatorname{\\mathbb {R}}\\mid s \\text{ is a real-valued state}\\right\\rbrace \\subseteq \\operatorname{\\mathbb {R}}^G\\,.$ Equip $\\operatorname{\\mathbb {R}}$ with its Euclidean topology, $\\operatorname{\\mathbb {R}}^G$ with the product topology, and $\\operatorname{St}{G}$ with the subspace topology.",
"Then $\\operatorname{St}{G}$ , the state space of $G$, is a compact Hausdorff space, which is moreover a convex set in the vector space $\\operatorname{\\mathbb {R}}^G$ .",
"For every $a\\in G$ we consider the function $\\hat{a}\\colon \\operatorname{St}{G}&\\longrightarrow \\operatorname{\\mathbb {R}}\\\\\\nonumber s &\\longmapsto s(a).$ The function $\\hat{a}$ is continuous and affine; we write $\\operatorname{{\\rm A}}{(\\operatorname{St}{G})}$ for the set of all continuous affine maps $\\operatorname{St}{G}\\rightarrow \\operatorname{\\mathbb {R}}$ .",
"Then [10] says that $\\operatorname{{\\rm A}}{(\\operatorname{St}{G})}$ is a unital Abelian $\\ell $ -group under pointwise addition and order, with unit the function constantly equal to 1.",
"The map $e_G\\colon G&\\xrightarrow{}\\operatorname{{\\rm A}}{(\\operatorname{St}{G})}\\\\ \\nonumber a&{\\ \\ \\ \\ } \\hat{a},$ induced by (REF ) is a state $G\\rightarrow \\operatorname{{\\rm A}}{(\\operatorname{St}G)}$ .",
"It is an isomorphism onto its range precisely when $G$ is Archimedean [10].",
"Recall that $G$ is Archimedean if for $a,b\\in G$ , $na\\leqslant b$ for all positive integers $n$ implies $a\\leqslant 0$ .",
"In any case, even when $G$ fails to be Archimedean, we call (REF ) the affine representation of $G$ .",
"For any compact Hausdorff space $X$ , write $$ for the unital Abelian $\\ell $ -group of continuous real-valued functions on $X$ , operations being defined pointwise; the function constantly equal to 1 is the unit.",
"For every unital Abelian $\\ell $ -group $G$ , the inclusion $\\operatorname{{\\rm A}}{(\\operatorname{St}{G})}\\subseteq $ preserves the unit, the group structure, and the partial order.",
"In general, however, it fails to preserve the lattice structure.",
"Let us therefore consider the sublattice-subgroup $\\widehat{G}$ of $$ generated by the image of $G$ under the affine representation map $e_G$ in (REF ).",
"Modifying the codomain of the affine representation accordingly, but retaining the same notation, we obtain the state $e_G\\colon G&\\xrightarrow{}\\widehat{G}\\\\ \\nonumber a&{\\ \\ \\ \\ } \\hat{a},$ which we call the extended affine representation of $G$ .",
"Reformulating the notion of universal state of MV-algebras, a state $\\upsilon _G\\colon G\\longrightarrow \\Upsilon G$ of the unital Abelian $\\ell $ -groups $G$ will be called universal (for $G$ ) if for each state $s\\colon G\\rightarrow H$ there is exactly one unital $\\ell $ -homomorphism $h\\colon \\Upsilon G\\rightarrow H$ satisfying $h\\upsilon _G=s$ .",
"We will relate the extended affine representation (REF ) of $G$ with the codomain $\\Upsilon G$ of a universal state.",
"In light of the universal property of $v_G$ there is exactly one comparison unital $\\ell $ -homomorphism $q_G\\colon \\Upsilon G\\longrightarrow \\widehat{G}$ that satisfies $q_G v_G=e_G\\,.$ Proposition 5.1 Let $G$ be a unital Abelian $\\ell $ -group.",
"The homomorphism $q_G$ in (REF ) is an isomorphism if, and only if, $\\Upsilon G$ is Archimedean.",
"First observe that $q_G$ is always surjective.",
"Indeed, $v_G[G]$ and $e_G[G]$ generate $\\Upsilon G$ and $\\widehat{G}$ as $\\ell $ -groups, respectively.",
"Therefore, by (REF ), the map $q_G$ throws a generating set of $\\Upsilon G$ onto a generating set of $\\widehat{G}$ ; hence it is surjective.",
"Further, recall that the radical ideal of a unital Abelian $\\ell $ -group $H$ is defined as $\\operatorname{Rad}{H}\\bigcap \\left\\lbrace \\ker {h}\\mid h\\colon H\\rightarrow \\operatorname{\\mathbb {R}}\\text{ is a homomorphism}\\right\\rbrace ,$ and that $H$ is Archimedean if, and only if, $\\operatorname{Rad}{H}=\\lbrace 0\\rbrace $ .",
"It now suffices to prove that $\\ker {q_G}=\\operatorname{Rad}{\\Upsilon G}$ .",
"Since $\\widehat{G}$ is Archimedean by construction, and unital $\\ell $ -homomorphisms such as $q_G$ preserve radical ideals, the inclusion $\\operatorname{Rad}{\\Upsilon G}\\subseteq \\ker {q_G}$ is clear.",
"For the converse inclusion, suppose by contraposition that $x\\notin \\operatorname{Rad}{\\Upsilon G}$ , with the intent of showing $q_Gx\\ne 0$ .",
"By the hypothesis there is a unital $\\ell $ -homomorphism $h\\colon \\Upsilon G\\rightarrow \\operatorname{\\mathbb {R}}$ such that $hx\\ne 0$ .",
"Thus we have a real-valued state $sh v_G\\colon G\\longrightarrow \\operatorname{\\mathbb {R}}.$ Evaluation of elements of $\\widehat{G}$ at $s$ produces a homomorphism ${\\rm ev}_{s}\\colon \\widehat{G}&\\longrightarrow \\operatorname{\\mathbb {R}}\\\\f&\\longmapsto fs\\in \\operatorname{\\mathbb {R}}$ such that ${\\rm ev}_{s} e_G= h v_G.$ To see that (REF ) holds, pick $a\\in G$ and compute: $({\\rm ev}_s e_G)(a) = {\\rm ev}_s(e_G(a))=(e_G(a))(s)=s(a)$ , where the last equality is given by the definition (REF ).",
"Since $s= h v_G$ , (REF ) holds.",
"From (REF –REF ) we deduce ${\\rm ev}_{s} q_G v_G = h v_G\\,,$ which by the universal property of $v_G$ entails ${\\rm ev}_s q_G=h.$ Since $hx\\ne 0$ by hypothesis, from (REF ) we infer $q_Gx\\ne 0$ , as was to be shown.",
"Remark 5.1 The question of when the Abelian $\\ell $ -group freely generated by an Archimedean partially ordered Abelian group is itself Archimedean has long had an important place in the theory of ordered groups.",
"Bernau [3] gave an example of an Archimedean partially ordered Abelian group, which moreover happens to be lattice-ordered, such that the Abelian $\\ell $ -group it freely generates fails to be Archimedean.",
"Further, Bernau obtained in [3] a necessary and sufficient condition for the Abelian $\\ell $ -group freely generated by a partially ordered Abelian group to be Archimedean; the condition, which we do not reproduce here due to its length, is known as the uniform Archimedean property of a partially ordered group.",
"Via the results in Section , Bernau's example entails that the universal state of a semisimple MV-algebra may have a non-semisimple codomain.",
"However, we shall see in the next section that this cannot happen when the semisimple MV-algebra in question is locally finite."
],
[
"The universal state of a locally finite MV-algebra",
"An MV-algebra is locally finite if each of its finitely generated subalgebras is finite.",
"By general algebraic considerations, any algebraic structure is the direct union of its finitely generated subalgebras [14].",
"Thus, every locally finite MV-algebra is the direct union of its finite subalgebras.",
"We are going to prove: Theorem 6.1 Let $\\upsilon _M\\colon M\\rightarrow \\Upsilon M$ be the universal state of a locally finite MV-algebra $M$ (cf.",
"Corollary REF ).",
"Then $\\Upsilon M$ is semisimple.",
"The proof of this theorem will require a number of lemmas.",
"For each integer $i\\geqslant 1$ , let $M_i\\lbrace 0,\\frac{1}{i},\\dots , \\frac{i-1}{i},1\\rbrace $ be the finite totally ordered MV-algebra of cardinality $i+1$ .",
"By [6], a finite MV-algebra $A$ is isomorphic to a product $M_{k_1}\\times \\dots \\times M_{k_n}$ , for uniquely determined integers $k_1,\\dots ,k_n \\geqslant 1$ , $n\\geqslant 0$ .",
"(When $n=0$ , $A$ is the terminal MV-algebra $\\lbrace 0=1\\rbrace $ .)",
"For the rest of this section we use $A$ to denote such a finite MV-algebra.",
"Lemma 6.1 Let $A=M_{k_1}\\times \\dots \\times M_{k_n}$ be a finite MV-algebra and $N$ be any MV-algebra.",
"Write $\\lbrace a_1,\\ldots ,a_n\\rbrace $ for the atoms of $A$ .",
"Any state $s\\colon A\\rightarrow N$ is uniquely determined by its values at the atoms of $A$ .",
"A function $s\\colon \\lbrace a_1,\\dots ,a_n\\rbrace \\rightarrow N$ has an extension to a state $A\\rightarrow N$ if, and only if, $k_1s(a_1) + \\dots + k_ns(a_n)=1$ , and in that case the extension is unique.",
"The unital Abelian $\\ell $ -group $\\Xi A$ is the simplicial group $\\operatorname{\\mathbb {Z}}^n$ with unit the element $(k_1,\\ldots ,k_n)$ , and the atoms $a_i$ are the standard basis elements of $\\operatorname{\\mathbb {Z}}^n$ ; from this item 1 follows at once.",
"As for item 2, the left-to-right implication follows directly from the definition of state.",
"Conversely, assume $k_1s(a_1) + \\dots + k_ns(a_n)=1$ .",
"For any $a\\in \\Gamma \\Xi A\\subseteq \\Xi A= \\operatorname{\\mathbb {Z}}^n$ , we can write $a=\\sum _{i=1}^nc_ia_i$ for uniquely determined integers $c_i$ .",
"Setting $s^{\\prime }(a)\\sum _{i=1}^n c_is(a_i)$ , one verifies that $s^{\\prime }$ is a state, and then, by item 1, $s^{\\prime }$ is the unique state extending $s$ .",
"Set $\\mathbf {k}(k_1,\\ldots ,k_n)$ , for short.",
"If $Fn$ is the MV-algebra freely generated by the set $\\lbrace x_1,\\ldots ,x_n\\rbrace $ , there is a finitely generated (hence principal, by [6]) ideal $U(\\mathbf {k})$ of $Fn$ determined by the partition-of-unity relation $\\sum _{i=1}^nk_ix_i=1,$ where addition is interpreted in the unital Abelian $\\ell $ -group $\\Xi Fn$ .",
"In more detail, there is a term $\\sigma (x_1,\\ldots ,x_n)$ in the language of MV-algebras such that, for any MV-algebra $M$ , and for any elements $b_1,\\ldots ,b_n\\in M$ , $\\sigma (b_1,\\dots ,b_n)=0$ holds in $M$ if, and only if, $\\sum _{i=1}^nk_ib_i=1$ holds in $\\Xi M$ .",
"Thus, $U(\\mathbf {k})$ is the ideal of $Fn$ generated by $\\sigma (x_1,\\ldots ,x_n)$ , when the latter is regarded as an element of $Fn$ in the usual way.",
"For an explicit computation of $\\sigma $ the interested reader can consult [23].",
"We write $S_{\\mathbf {k}}$ for the quotient algebra $F n/U(\\mathbf {k})$ .",
"By Lemma REF , the function $a_i\\longmapsto x_i,\\qquad i=1,\\ldots ,n,$ is extended by exactly one state $\\upsilon _A\\colon A \\longrightarrow S_{\\mathbf {k}},$ where $A=M_{k_1}\\times \\dots \\times M_{k_n}$ .",
"Lemma 6.2 The state (REF ) is the universal state of the MV-algebra $A=M_{k_1}\\times \\dots \\times M_{k_n}$ .",
"If $s\\colon A \\rightarrow N$ is any state, and $\\lbrace a_1,\\ldots ,a_n\\rbrace $ is the set of atoms of $A$ , consider the assignment $x_i\\mapsto s(a_i)$ , $i=1,\\ldots , n$ .",
"We have $\\sum _{i=1}^n k_is(a_i)=1$ in $N$ , because $s$ is a state.",
"Then, by the definition of $S_{\\mathbf {k}}$ and the universal property of homomorphic images in varieties, this assignment has exactly one extension to a homomorphism $h\\colon S_{\\mathbf {k}}\\rightarrow N$ .",
"For each $a_i\\in A$ we have $(h\\upsilon _A)a_i=s(a_i)$ , which entails $h\\upsilon _A=s$ because $\\lbrace a_1,\\ldots , a_n\\rbrace $ is a $\\operatorname{\\mathbb {Z}}$ -module basis of $\\Xi A=\\operatorname{\\mathbb {Z}}^n$ .",
"Such $h$ is unique, because if $h^{\\prime }\\colon S_{\\mathbf {k}}\\rightarrow N$ satisfies $h^{\\prime }\\upsilon _A=s$ then $h$ and $h^{\\prime }$ must agree on the generating set $\\lbrace x_1,\\ldots ,x_n\\rbrace $ of $S_{\\mathbf {k}}$ : indeed, $h^{\\prime }\\upsilon _A=h\\upsilon _A$ entails $h^{\\prime }x_i=(h^{\\prime }\\upsilon _A)a_i=(h\\upsilon _A)a_i=hx_i$ , $i=1,\\ldots ,n$ .",
"For the proof of the last lemma we need, Lemma REF below, we shall apply geometric techniques.",
"We use the integer $d\\geqslant 0$ to denote the dimension of the real vector space $\\operatorname{\\mathbb {R}}^d$ .",
"A function $f \\colon \\operatorname{\\mathbb {R}}^d \\rightarrow \\operatorname{\\mathbb {R}}$ is PL (for piecewise linear) if it is continuous with respect to the Euclidean topology on $\\operatorname{\\mathbb {R}}^d$ and $\\operatorname{\\mathbb {R}}$ , and there is a finite set of affine functions $l_1,\\ldots ,l_u\\colon \\operatorname{\\mathbb {R}}^d\\rightarrow \\operatorname{\\mathbb {R}}$ such that for each $x \\in \\operatorname{\\mathbb {R}}^d$ one has $f(x)=l_i(x)$ for some choice of $i=1,\\ldots ,u$ .",
"(We note in passing that the terminology “piecewise linear” is traditional, even though “piecewise affine” would be, strictly speaking, more appropriate.)",
"If moreover, each $l_i\\colon \\operatorname{\\mathbb {R}}^d\\rightarrow \\operatorname{\\mathbb {R}}$ can be chosen so as to restrict to a function $\\operatorname{\\mathbb {Z}}^d\\rightarrow \\operatorname{\\mathbb {Z}}$ , then $f$ is a $\\operatorname{\\mathbb {Z}}$ -map.",
"(This terminology comes from [24].)",
"In coordinates, this is equivalent to asking that $l_i$ can be written as a linear polynomial with integer coefficients.",
"For an integer $d^{\\prime }\\geqslant 0$ , a function $\\lambda =(\\lambda _1,\\ldots ,\\lambda _{d^{\\prime }})\\colon \\operatorname{\\mathbb {R}}^d \\rightarrow \\operatorname{\\mathbb {R}}^{d^{\\prime }}$ is a PL map (respectively, a $\\operatorname{\\mathbb {Z}}$ -map) if each one of its scalar components $\\lambda _j\\colon \\operatorname{\\mathbb {R}}^d\\rightarrow \\operatorname{\\mathbb {R}}$ is a PL function ($\\operatorname{\\mathbb {Z}}$ -map).",
"One defines PL maps ($\\operatorname{\\mathbb {Z}}$ -maps) $A \\rightarrow B$ for arbitrary subsets $A\\subseteq \\operatorname{\\mathbb {R}}^d$ , $B\\subseteq \\operatorname{\\mathbb {R}}^{d^{\\prime }}$ as the restriction and co-restriction of PL maps ($\\operatorname{\\mathbb {Z}}$ -maps).",
"A convex combination of a finite set of vectors $v_1,\\ldots ,v_u \\in \\operatorname{\\mathbb {R}}^d$ is any vector of the form $\\lambda _1v_1+\\cdots +\\lambda _uv_u$ , for non-negative real numbers $\\lambda _i\\geqslant 0$ satisfying $\\sum _{i=1}^u\\lambda _i=1$ .",
"If $S\\subseteq \\operatorname{\\mathbb {R}}^d$ is any subset, we let $\\operatorname{conv}{S}$ denote the convex hull of $S$ , i.e.",
"the collection of all convex combinations of finite sets of vectors $v_1,\\ldots ,v_u \\in S$ .",
"A polytope is any subset of $\\operatorname{\\mathbb {R}}^d$ of the form $\\operatorname{conv}{S}$ , for some finite $S\\subseteq \\operatorname{\\mathbb {R}}^d$ , and a (compact) polyhedron is a union of finitely many polytopes in $\\operatorname{\\mathbb {R}}^d$ .",
"A polytope is rational if it may be written in the form $\\operatorname{conv}{S}$ for some finite set $S\\subseteq \\operatorname{\\mathbb {Q}}^d\\subseteq \\operatorname{\\mathbb {R}}^d$ of vectors with rational coordinates.",
"Similarly, a polyhedron is rational if it may be written as a union of finitely many rational polytopes.",
"It is clear that the composition of $\\operatorname{\\mathbb {Z}}$ -maps is again a $\\operatorname{\\mathbb {Z}}$ -map.",
"Rational polyhedra and $\\operatorname{\\mathbb {Z}}$ -maps thus form a category, which we denote ${\\sf PL}_{\\mathbb {Z}}$ .",
"This is a non-full subcategory of the classical compact polyhedral category ${\\sf PL}$ whose objects are polyhedra and whose morphisms are PL maps.",
"Remark 6.1 The full subcategory of ${\\sf PL}_{\\mathbb {Z}}$ whose objects are rational polyhedra lying in unit cubes $[0,1]^d$ , as $d$ ranges over all non-negative integers, is equivalent to ${\\sf PL}_{\\mathbb {Z}}$ , see [19].",
"An analogous remark applies to ${\\sf PL}$ .",
"We shall make use of these facts whenever convenient, without further warning.",
"Given a subset $S\\subseteq \\operatorname{\\mathbb {R}}^d$ , we write ${\\operatorname{\\nabla _{\\mathbb {Z}}}}S$ for the set of all $\\operatorname{\\mathbb {Z}}$ -maps $S \\rightarrow [0,1]$ .",
"Then ${\\operatorname{\\nabla _{\\mathbb {Z}}}}S$ inherits from $[0,1]$ the structure of an MV-algebra, upon defining operations pointwise; in other words, ${\\operatorname{\\nabla _{\\mathbb {Z}}}}S$ is a subalgebra of the MV-algebra $$ of all continuous functions $S\\rightarrow [0,1]$ .",
"It can be proved that if $X\\subseteq \\operatorname{\\mathbb {R}}^d$ is a rational polyhedron then ${\\operatorname{\\nabla _{\\mathbb {Z}}}}X$ is finitely presentable; see e.g.",
"[24], where the result is stated for $X\\subseteq [0,1]^d$ —the case $X\\subseteq \\operatorname{\\mathbb {R}}^d$ is then a consequence of Remark REF .",
"Following tradition, from now on we say `finitely presented' instead of `finitely presentable', even when the latter would be the proper expression.",
"By ${\\sf MV_{fp}}$ we denote the full subcategory of ${\\sf MV}$ whose objects are finitely presented MV-algebras.",
"Further, if $P\\subseteq \\operatorname{\\mathbb {R}}^d$ and $Q\\subseteq \\operatorname{\\mathbb {R}}^{d^{\\prime }}$ are rational polyhedra for some integers $d,d^{\\prime }\\geqslant 0$ , a $\\operatorname{\\mathbb {Z}}$ -map $\\lambda \\colon P \\rightarrow Q$ induces a function ${\\operatorname{\\nabla _{\\mathbb {Z}}}}\\lambda \\colon {\\operatorname{\\nabla _{\\mathbb {Z}}}}Q \\xrightarrow{} {\\operatorname{\\nabla _{\\mathbb {Z}}}}P$ given by $f \\in {\\operatorname{\\nabla _{\\mathbb {Z}}}}Q {\\ {\\operatorname{\\nabla _{\\mathbb {Z}}}}\\lambda \\ } f\\lambda \\in {\\operatorname{\\nabla _{\\mathbb {Z}}}}P.$ It can be shown that ${\\operatorname{\\nabla _{\\mathbb {Z}}}}\\lambda $ is a homomorphism of MV-algebras, see e.g.",
"[19].",
"We thereby obtain a functor $\\operatorname{\\nabla _{\\mathbb {Z}}}\\colon {\\sf PL}_{\\mathbb {Z}}\\xrightarrow{} {\\sf MV_{fp}^{\\rm op}}.$ To define a functor $\\operatorname{Max}\\colon {\\sf MV_{fp}^{\\rm op}}\\xrightarrow{} {\\sf PL}_{\\mathbb {Z}}$ in the other direction, let us assume that a specific finite presentation of an MV-algebra is given.",
"That is, we are given a finite set $R\\lbrace \\rho _i\\in Fn\\mid i=1,\\ldots ,m\\rbrace $ of elements of $Fn$ , the free MV-algebra generated by $n$ elements $x_1,\\ldots ,x_n$ , and we consider the finitely presented quotient $Fn/\\langle R \\rangle $ , where $\\langle R \\rangle $ denotes the ideal of $Fn$ generated by $R$ .",
"Since every finitely generated ideal of any MV-algebra is principal by [6], we may safely assume that $\\langle R \\rangle $ is generated by the single element $\\rho $ .",
"Writing $\\pi _i\\colon [0,1]^n\\rightarrow [0,1]$ for the $i^{\\rm th}$ projection function, the assignment $x_i\\mapsto \\pi _i$ , $i=1,\\ldots ,n$ , extends to exactly one homomorphism $Fn\\rightarrow \\operatorname{\\nabla _{\\mathbb {Z}}}[0,1]^n$ by the universal property of the free algebra, and this homomorphism is an isomorphism by [6].",
"Thus, ${\\operatorname{\\nabla _{\\mathbb {Z}}}}[0,1]^n$ is the MV-algebra freely generated by the projection functions.",
"Under this isomorphism, $\\rho $ corresponds to an element $\\rho ^{\\prime }$ of ${\\operatorname{\\nabla _{\\mathbb {Z}}}}[0,1]^n$ , and the assignment $x_i\\mapsto \\pi _i$ , $i=1,\\ldots ,n$ , extends to exactly one isomorphism $Fn/\\langle \\rho \\rangle \\rightarrow {\\operatorname{\\nabla _{\\mathbb {Z}}}}[0,1]^n/\\langle \\rho ^{\\prime }\\rangle $ .",
"Then we set $\\operatorname{Max}{Fn/\\langle \\rho \\rangle }\\rho ^{\\prime -1}\\lbrace 0\\rbrace $ .",
"Here, the zero set $\\rho ^{\\prime -1}\\lbrace 0\\rbrace $ of the $\\operatorname{\\mathbb {Z}}$ -map $\\rho ^{\\prime }$ is a rational polyhedron in $[0,1]^n$ .",
"When regarded as a topological space, by [6] the polyhedron $\\rho ^{\\prime -1}\\lbrace 0\\rbrace $ is homeomorphic to the maximal spectral space of the MV-algebra $Fn/\\langle \\rho \\rangle $ , whence the name of the functor (REF ).",
"In turn, the maximal ideals of any MV-algebra $M$ are in bijection with the kernels of the homomorphisms $M\\rightarrow [0,1]$ ([24]).",
"Therefore the points of $\\rho ^{\\prime -1}\\lbrace 0\\rbrace $ are in bijection with (the kernels of) the homomorphisms $Fn/\\langle \\rho \\rangle \\rightarrow [0,1]$ .",
"To define $\\operatorname{Max}$ on arrows, then, consider a homomorphism $h\\colon Fn/\\langle \\rho \\rangle \\rightarrow Fm/\\langle \\tau \\rangle $ .",
"Precomposing with $h$ , a morphism $Fm/\\langle \\tau \\rangle \\rightarrow [0,1]$ is sent to a morphism $Fn/\\langle \\rho \\rangle $ , so that we obtain a mapping $\\operatorname{Max}h\\colon \\tau ^{\\prime -1}\\lbrace 0\\rbrace \\rightarrow \\rho ^{\\prime -1}\\lbrace 0\\rbrace $ which can be proved to be a $\\operatorname{\\mathbb {Z}}$ -map.",
"This is the definition of the functor $\\operatorname{Max}$ for finitely presented MV-algebras; for the general case of a finitely presentable algebra, to complete the definition one chooses a finite presentation for each algebra.",
"Theorem 6.2 (Duality theorem for finitely presented MV-algebras) The functors $\\operatorname{\\nabla _{\\mathbb {Z}}}\\colon {\\sf PL}_{\\mathbb {Z}}\\xrightarrow{} {\\sf MV_{fp}^{\\rm op}}$ and $\\operatorname{Max}\\colon {\\sf MV_{fp}^{\\rm op}}\\xrightarrow{} {\\sf PL}_{\\mathbb {Z}}$ form an equivalence of categories.",
"Two different proofs of this result are given in [18] and [19].",
"Remark 6.2 We will apply Theorem REF in the sequel without explicit reference to its numbered statement, freely using such expressions as “the dual rational polyhedron $X$ of the finitely presented MV-algebra $A$ ”.",
"If $x \\in \\operatorname{\\mathbb {Q}}^d$ , there is a unique way to write out $x$ in coordinates as $x = \\left(\\frac{p_1}{q_1},\\ldots , \\frac{p_d}{q_d}\\right)$ with $p_i,q_i \\in \\operatorname{\\mathbb {Z}}\\ , \\ q_i > 0$ , $p_i$ and $q_i$ relatively prime for each $i=1,\\ldots , d$ .",
"The positive integer $\\operatorname{\\rm den}{x}{\\rm lcm}\\,{\\lbrace q_1,q_2,\\ldots ,q_d\\rbrace }$ is the denominator of $x$ .",
"For a non-negative integer $t$ , a $t$ -dimensional simplex (or just $t$ -simplex) in $\\operatorname{\\mathbb {R}}^d$ is the convex hull of $t+1$ affinely independent points in $\\operatorname{\\mathbb {R}}^d$ , called its vertices; the vertices of a simplex are uniquely determined.",
"A face of a simplex $\\sigma $ is the convex hull of a nonempty subset of the vertices of $\\sigma $ , and as such it is itself a simplex.",
"The relative interior of $\\sigma $ is the set of points expressible as convex combinations of its vertices with strictly positive coefficients.",
"A simplex is rational if its vertices are.",
"The following notion is fundamental to the arithmetic geometry of ${\\sf PL}_{\\mathbb {Z}}$ : A simplex $\\sigma \\subseteq \\operatorname{\\mathbb {R}}^d$ is regular if whenever $x$ is a rational point lying in the relative interior of some face $\\tau $ of $\\sigma $ with vertices $v_1,\\ldots ,v_l$ , then $\\operatorname{\\rm den}{x}\\geqslant \\sum _{i=1}^l\\operatorname{\\rm den}{v_i}$ .",
"See [24].",
"Regular simplices are also called unimodular in the literature.",
"Since the finite MV-algebra $A=M_{k_1}\\times \\cdots \\times M_{k_n}$ is finitely presented, it has a dual polyhedron, which by duality must be the sum of the duals of $M_{k_i}$ , $i=1,\\ldots ,{n}$ .",
"The dual of each $M_{k_i}$ is immediately seen to be a single rational point $p$ (in some $\\operatorname{\\mathbb {R}}^d$ ) such that $\\operatorname{\\rm den}{p}=k_i$ .",
"Thus, the dual of $A$ is any finite set of points $\\lbrace p_1,\\ldots ,p_n\\rbrace $ such that $\\operatorname{\\rm den}{p_i}=k_i$ , $i=1,\\ldots ,n$ .",
"A specific coordinatisation of such a rational polyhedron is obtained as follows.",
"Let $\\lbrace e_1,\\ldots , e_n\\rbrace $ be the standard basis vectors in $\\operatorname{\\mathbb {R}}^n$ .",
"Then $\\operatorname{\\rm den}{\\frac{e_i}{k_i}}=k_i$ .",
"Thus, $\\lbrace \\frac{e_1}{k_1},\\ldots , \\frac{e_n}{k_n}\\rbrace $ is the rational polyhedron dual to $A$ .",
"The convex hull $\\Delta _{\\mathbf {k}}\\operatorname{conv}{\\left\\lbrace \\frac{e_1}{k_1},\\ldots , \\frac{e_n}{k_n}\\right\\rbrace }$ is a regular simplex by direct inspection.",
"For each $i=1,\\ldots , n$ , we write $\\pi _i^{\\mathbf {k}}\\colon \\Delta _{\\mathbf {k}}\\rightarrow [0,1]$ for the restriction of the projection function $\\pi _i\\colon [0,1]^n\\rightarrow [0,1]$ .",
"Evidently, $\\pi _i^\\mathbf {k}\\in {\\operatorname{\\nabla _{\\mathbb {Z}}}}\\Delta _{\\mathbf {k}}$ .",
"Observe that $\\sum _{i=1}^n k_i\\pi _i^\\mathbf {k}=1.$ Lemma 6.3 For any $\\mathbf {k}=(k_1,\\ldots ,k_n)$ , the dual polyhedron of the finitely presented MV-algebra $S_{\\mathbf {k}}$ is $\\Delta _{\\mathbf {k}}$ .",
"In more detail, writing $x_i^\\mathbf {k}$ for the generators provided by the presentation of $S_{\\mathbf {k}}$ , the assignment $x_i^\\mathbf {k}\\mapsto \\pi _i^\\mathbf {k}$ , $i=1,\\ldots ,n$ , extends to exactly one homomorphism $S_{\\mathbf {k}}\\rightarrow {\\operatorname{\\nabla _{\\mathbb {Z}}}}\\Delta _{\\mathbf {k}}$ , and that homomorphism is an isomorphism.",
"Applying the definition of the functor $\\operatorname{Max}$ in (REF ), the polyhedron $\\operatorname{Max}S_{\\mathbf {k}}$ dual to $S_{\\mathbf {k}}$ is the solution set in $[0,1]^n$ of the equation $\\sum _{i=1}^n k_i\\pi _i=1$ that presents $S_{\\mathbf {k}}$ (or equivalently, it is the zero set of $\\sigma (\\pi _1,\\ldots ,\\pi _n)$ , where $\\sigma $ is as in the definition of $S_{\\mathbf {k}}$ ).",
"Computation confirms that this solution set is $\\Delta _{\\mathbf {k}}$ ; and the remaining part of the statement is a direct consequence of the duality Theorem REF .",
"Corollary 6.1 There is exactly one state $\\upsilon _A\\colon A \\longrightarrow {\\operatorname{\\nabla _{\\mathbb {Z}}}}\\Delta _{\\mathbf {k}}$ that extends the assignment $a_i\\longmapsto \\pi _i,\\qquad i=1,\\ldots ,n,$ and this is the universal state of the MV-algebra $A=M_{k_1}\\times \\cdots \\times M_{k_n}$ .",
"The existence and uniqueness of the extension follows from Lemma REF , upon recalling (REF ).",
"That the state in question is naturally isomorphic to $\\upsilon _A$ in Lemma REF follows at once from the isomorphism in Lemma REF .",
"We record a well-known elementary property of $\\operatorname{\\mathbb {Z}}$ -maps for which we could not locate a reference (however, see the related [24]).",
"Lemma 6.4 Let $X\\subseteq \\operatorname{\\mathbb {R}}^d$ and $Y\\subseteq \\operatorname{\\mathbb {R}}^{d^{\\prime }}$ be rational polyhedra, for integers $d,d^{\\prime } \\geqslant 0$ , and let $f\\colon X\\rightarrow Y$ be a $\\operatorname{\\mathbb {Z}}$ -map.",
"For each $x \\in X\\cap \\operatorname{\\mathbb {Q}}^d$ , the number $fx$ is rational and $\\operatorname{\\rm den}{fx}$ divides $\\operatorname{\\rm den}{x}$ ; in symbols, $\\operatorname{\\rm den}{fx}\\mid \\operatorname{\\rm den}{x}$ .",
"The $\\operatorname{\\mathbb {Z}}$ -map $f\\colon X \\rightarrow Y$ may be written in vectorial form as $(f_1, \\ldots , f_{d^{\\prime }})$ , with each $f_i\\colon X \\rightarrow \\operatorname{\\mathbb {R}}$ a $\\operatorname{\\mathbb {Z}}$ -map.",
"In the rest of this proof, assume all rational numbers are expressed in reduced form.",
"Pick a point $x=(\\frac{p_i}{q_i})\\in X\\cap \\operatorname{\\mathbb {Q}}^d$ , and regard $x$ as a column vector.",
"By definition, each $f_i$ agrees locally at the point $x$ with an affine function $\\operatorname{\\mathbb {R}}^d \\rightarrow \\operatorname{\\mathbb {R}}$ with integer coefficients.",
"Thus, there is a column vector $c=(c_i)\\in \\operatorname{\\mathbb {Z}}^{d^{\\prime }}$ together with a $(d^{\\prime }\\times d)$ matrix $M=(z_{ij})$ with integer entries such that $fx=Mx+c.$ This shows that $fx$ is rational if $x$ is.",
"Further, let us write $fx=(\\frac{a_i}{b_i})\\in \\operatorname{\\mathbb {Q}}^{d^{\\prime }}$ .",
"By (REF ) we have $\\frac{a_i}{b_i}=\\sum _{j=1}^{d}z_{ij}\\frac{p_j}{q_j} + c_i$ , so that $b_i \\mid \\operatorname{\\rm den}{x}$ , and therefore ${\\rm lcm}\\,\\lbrace b_i\\rbrace _{i=1}^{d^{\\prime }}=\\operatorname{\\rm den}{fx}\\mid \\operatorname{\\rm den}{x}$ .",
"We can now prove the promised lemma.",
"Lemma 6.5 If $h\\colon A\\rightarrow B$ is any injective homomorphism between finite MV-algebras, $\\Upsilon h\\colon \\Upsilon A \\rightarrow \\Upsilon B$ is injective.",
"Let $\\lbrace e_1/k_1,\\ldots ,e_n/k_n\\rbrace \\subseteq \\operatorname{\\mathbb {R}}^n$ be the dual finite rational polyhedron of $A$ , with $\\operatorname{\\rm den}{e_i/k_i}=k_i$ , $i=1,\\ldots ,n$ .",
"Similarly, let $\\lbrace e_1/t_1,\\ldots ,e_m/t_m\\rbrace \\subseteq \\operatorname{\\mathbb {R}}^m$ be the dual of $B$ , with $\\operatorname{\\rm den}{e_i/t_i}=t_i$ , $i=1,\\ldots ,m$ .",
"Let us write $h^*\\operatorname{Max}h\\colon \\lbrace e_1/t_1,\\ldots ,e_m/t_m\\rbrace \\rightarrow \\lbrace e_1/k_1,\\ldots ,e_n/k_n\\rbrace $ for the $\\operatorname{\\mathbb {Z}}$ -map dual to $h$ .",
"By Lemma REF we have $k_i\\mid t_j$ whenever $h^*(e_j/t_j)=e_i/k_i$ .",
"Moreover, the $\\operatorname{\\mathbb {Z}}$ -map $h^*$ has exactly one extension to an affine map $\\alpha \\colon \\Delta _{\\mathbf {t}}\\rightarrow \\Delta _{\\mathbf {k}}$ , where $\\mathbf {t}(t_i)_{i=1}^m$ , by the elementary properties of affine functions and simplices.",
"The computation in [24] (or else, which is essentially the same, direct computation of the matrix form of $\\alpha $ with respect to standard bases) confirms that $\\alpha $ is a $\\operatorname{\\mathbb {Z}}$ -map.",
"Since $h$ is injective, it is a monomorphism, and thus $h^*$ is an epimorphism by duality.",
"But then a straightforward verification shows that $h^*$ is a surjective function.",
"This entails at once that its affine extension $\\alpha $ is surjective, too.",
"For any two continuous functions $f,g\\colon \\Delta _{\\mathbf {k}}\\rightarrow \\operatorname{\\mathbb {R}}$ with $f\\ne g$ we then have $f\\alpha \\ne g\\alpha $ , by the surjectivity of $\\alpha $ .",
"In particular, this means that the homomorphism ${\\operatorname{\\nabla _{\\mathbb {Z}}}}\\alpha \\colon {\\operatorname{\\nabla _{\\mathbb {Z}}}}\\Delta _{\\mathbf {k}}\\longrightarrow {\\operatorname{\\nabla _{\\mathbb {Z}}}}\\Delta _{\\mathbf {t}},$ which by the definition of the functor $\\operatorname{\\nabla _{\\mathbb {Z}}}$ carries a $\\operatorname{\\mathbb {Z}}$ -map $f\\colon \\Delta _{\\mathbf {k}}\\rightarrow [0,1]$ to the $\\operatorname{\\mathbb {Z}}$ -map $f\\alpha \\colon \\Delta _{\\mathbf {t}}\\rightarrow [0,1]$ , is injective.",
"The proof is now completed by showing that $\\Upsilon h$ is ${\\operatorname{\\nabla _{\\mathbb {Z}}}}\\alpha $ to within a natural isomorphism.",
"That is, in light of Corollary REF , we need to check that the square $\\begin{tikzcd}A [d, \"\\upsilon _A\"^{\\prime }] [r, \"h\"] & B [d, \"\\upsilon _B\"]\\\\{\\operatorname{\\nabla _{\\mathbb {Z}}}}\\Delta _{\\mathbf {k}} [r, \"{\\operatorname{\\nabla _{\\mathbb {Z}}}\\alpha }\"^{\\prime }]& {\\operatorname{\\nabla _{\\mathbb {Z}}}}\\Delta _{\\mathbf {t}}\\end{tikzcd}$ commutes, which amounts to a straightforward application of the definitions which we leave to the reader.",
"Finally: Write $\\Sigma $ for the directed partially ordered set of all finite subalgebras of $M$ , so that $M=\\bigcup _{A\\in \\Sigma }A$ (direct union).",
"The functor $\\Upsilon $ in (REF ) is left adjoint and thus preserves colimits [17].",
"Moreover, by Lemma REF , each inclusion $A\\rightarrow B$ with $A,B\\in \\Sigma $ is sent by $\\Upsilon $ to an injective homomorphism.",
"Together with Corollary REF , this says that $\\Upsilon M$ is a directed colimit of semisimple MV-algebras with injective transition homomorphisms.",
"Since the radical ideal of semisimple MV-algebras is trivial, the radical of the directed colimit $\\Upsilon M$ is trivial, too, by [6].",
"Hence, $\\Upsilon M$ is semisimple.",
"Corollary 6.2 For any locally finite MV-algebra $M$ , the extended affine representation of $M$ (cf.",
"Section ) is the universal state of $M$ .",
"This happens, in particular, for all Boolean algebras.",
"Theorem REF and Proposition REF .",
"The last assertion holds because of the standard fact that finitely generated Boolean algebras are finite."
],
[
"Further research",
"The present paper is part of a nascent programme aimed at exploring universal constructions in probability theory.",
"While we refrain from sketching that programme here, we do point out some connections between the line of research pursued in this paper, and many-valued logic.",
"Fuzzy Probability Logic FP(Ł) over infinite-valued Łukasiewicz logic was developed by Hájek [11], and later extended by Cintula and Noguera to a two-tier modal logic aimed at modelling uncertainty [7].",
"The logic FP(Ł) formalises reasoning about properties of states, similarly to probabilistic logics designed for reasoning about probability.",
"The main feature of FP(Ł) is a two-level syntax.",
"Probability assessments are represented in the language by a unary modality ‘Probably’, which can be applied to Boolean formulæ only.",
"The class of MV-algebras and states provides a possible complete semantics for FP(Ł); see [7] for more details.",
"States as two-sorted algebras, as introduced here, may provide an equivalent multi-sorted algebraic semantics to the logic FP(Ł).",
"Details will be pursued in further research."
],
[
"Acknowledgments",
"Both authors are deeply grateful to an anonymous reviewer for his/her careful and competent reading of a previous version of this paper, which contained two significant oversights in the proof of Corollary REF .",
"The work of Tomáš Kroupa has been supported from the GAČR grant project GA17-04630S and from the project RCI (CZ.02.1.01/0.0/0.0/16_019/0000765)."
]
] | 2001.03533 |
[
[
"Demo: Light-Weight Programming Language for Blockchain"
],
[
"Abstract This demo abstract introduces a new light-weight programming language koa which is suitable for blockchain system design and implementation.",
"In this abstract, the basic features of koa are introduced including working system (with playground), architecture, and virtual machine operations.",
"Rum-time execution of software implemented by koa will be presented during the session."
],
[
"Introduction",
"Recently, blockchain technologies are getting a lot of attentions due to its decentralized scalable nature.",
"Among them, Bitcoin and Ethereum are major leading blockchain technologies.",
"The Bitcoin is a blockchain-based cryptocurrency system which was initiated by Nakamoto Satoshi in 2008 [1], [2].",
"The trading in Bitcoin conducts via peer-to-peer networking under the encryption/description with SHA-256.",
"All transactions are stored as a data structure called Merkle tree by every node participating in blockchain.",
"The Bitcoin solves the double-spending problem using Proof-of-Work (PoW) consensus algorithm for generating blocks.",
"The Ethereum is a distributed computing platform based on blockchain developed by Vitalik Buterin in 2015.",
"Similar to the Bitcoin, All transactions are stored by every node in Etheruem.",
"However, it can execute applications on the blockchain using Smart Contract as well as trading cryptocurrency.",
"Thus, Etheruem applies the blockchain to various areas such as SNS, e-mail, games, and etc.",
"In this paper, we propose a new programming language which is easy to understand and write a code that implements the blockchain system in a scalable way.",
"This language, called Koa, solves difficulties of understanding and analyzes the code what the script language of the Bitcoin has with high-level language style.",
"Since the script language is a sequence of instructions, it is difficult to understand the flow of codes at a glance.",
"In addition, it is tricky to implement complicated logics as instructions.",
"For example, when expressing a branch of conditional statement in script language, there is no distinction between the parts to compare a condition, to execute if-true, and to execute if-false.",
"Besides, it is not easy to figure out what each instruction is used for.",
"The verifying process is performed by sequentially running the unlocking and lokcing script.",
"It is difficult to understand what kind of actions to perform because this script code is not distinguishable by semantic units.",
"Furthermore, when scripts are implemented, there is a difficulty to program considering the status of stack.",
"If the function is defined in koa, it can be distinguishable by semantic units.",
"When verifying a value, it needs to define a new instruction or to assemble many instructions complicatedly.",
"On the other hand, Koa can implement this by combining supported operators.",
"Using Koa, it has better readability than script languages and is easier to write the code simply.",
"Meanwhile, solidity is used as a smart contract language for Ethereum.",
"If a transaction is executed and the state changes, the Ethereum uses gas.",
"Applying gas as a kind of transaction fees, Ethereum prevents infinite loop attack.",
"However, the gas makes solidity code more complicated and hinder usability.",
"Furthermore, the state of Ethereum makes the static analysis of solidity code difficult.",
"Since the result of executing function differs according to state, it is difficult to predict gas consumption.",
"However, because the Koa is a stateless language, it can be statically analyzed.",
"Since there are not any branches of code execution depending on state nor loop, static analysis for the code is easy.",
"In addition, it can prevent the infinite loop attack.",
"This allows the Koa to predict the costs associated with running the smart contract.",
"Therefore, the Koa which is high-level language helps programmers to implement the blockchain quickly and easily."
],
[
"System Overview",
"The structure of the Koa programming language proposed in as Fig.",
"REF and it is mainly divided into two parts.",
"First of all, Koa that performs lexing, parsing, compiling, and vm running steps sequentially, as shown in Fig.",
"REF .",
"In addition, Koa playground is a kind of IDE environment software platforms and its corresponding web IDE is as presented in Fig.",
"REF .",
"When users write and compile a source code, it generates a bytecode through lexing, parsing, and compiling steps.",
"If users deploy this bytecode, it would be registered in blockchain and users receive an address where the program is deployed.",
"When users conduct execution call to the address for the desired functions, they can get execution results.",
"Figure: koa AST."
],
[
"Implementation",
"The form of abstract syntax tree (AST) can be illustrated as Fig.",
"REF .",
"Koa parser uses Pratt Parsing which is recursive parsing which uses the parsing function for each token instead of grammar rules.",
"It makes the parser intuitive and easy to understand.",
"Furthermore, Koa receives tokens from the lexer using TokenBuffer.",
"This allows the parser to take responsibility for syntax analysis (parsing) without having tokens directly.",
"The bytecode generated by koa is passed to Virtual Machine (VM) and then executed.",
"The VM operations include Stack, Memory, and Heap.",
"First of all, the stack is used for storing the data and numerical operators.",
"The, the memory is used to store the data, e.g., variables.",
"Lastly, the heap is used to store the data which have dynamic sizes, e.g., string.",
"When users send a call for requesting a function, VM stores this call data in callfunc.",
"Then, it executes the bytecode with stack, memory, and heap.",
"When all functions are called by users, it returns the output and the program exits.",
"This research was supported by the National Research Foundation of Korea (2016R1C1B1015406, 2017R1A4A1015675); Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korea government (MSIT) (No.",
"2018-0-00170, Virtual Presence in Moving Objects through 5G); and IITP grant funded by the Korea government (MSIP) (No.",
"2017-0-00068, A Development of Driving Decision Engine for Autonomous Driving using Driving Experience Information).",
"J. Kim is the corresponding author of this paper."
]
] | 2001.03365 |
[
[
"Data-Dependence of Plateau Phenomenon in Learning with Neural Network\n --- Statistical Mechanical Analysis"
],
[
"Abstract The plateau phenomenon, wherein the loss value stops decreasing during the process of learning, has been reported by various researchers.",
"The phenomenon is actively inspected in the 1990s and found to be due to the fundamental hierarchical structure of neural network models.",
"Then the phenomenon has been thought as inevitable.",
"However, the phenomenon seldom occurs in the context of recent deep learning.",
"There is a gap between theory and reality.",
"In this paper, using statistical mechanical formulation, we clarified the relationship between the plateau phenomenon and the statistical property of the data learned.",
"It is shown that the data whose covariance has small and dispersed eigenvalues tend to make the plateau phenomenon inconspicuous."
],
[
"Plateau Phenomenon",
"Deep learning, and neural network as its essential component, has come to be applied to various fields.",
"However, these still remain unclear in various points theoretically.",
"The plateau phenomenon is one of them.",
"In the learning process of neural networks, their weight parameters are updated iteratively so that the loss decreases.",
"However, in some settings the loss does not decrease simply, but its decreasing speed slows down significantly partway through learning, and then it speeds up again after a long period of time.",
"This is called as “plateau phenomenon”.",
"Since 1990s, this phenomena have been reported to occur in various practical learning situations (see REF (a) and [8], [2]) .",
"As a fundamental cause of this phenomenon, it has been pointed out by a number of researchers that the intrinsic symmetry of neural network models brings singularity to the metric in the parameter space which then gives rise to special attractors whose regions of attraction have nonzero measure, called as Milnor attractor (defined by [6]; see also Figure 5 in [2] for a schematic diagram of the attractor)."
],
[
"Who moved the plateau phenomenon?",
"However, the plateau phenomenon seldom occurs in recent practical use of neural networks (see REF (b) for example).",
"In this research, we rethink the plateau phenomenon, and discuss which situations are likely to cause the phenomenon.",
"First we introduce the student-teacher model of two-layered networks as an ideal system.",
"Next, we reduce the learning dynamics of the student-teacher model to a small-dimensional order parameter system by using statistical mechanical formulation, under the assumption that the input dimension is sufficiently large.",
"Through analyzing the order parameter system, we can discuss how the macroscopic learning dynamics depends on the statistics of input data.",
"Our main contribution is the following: Under the statistical mechanical formulation of learning in the two-layered perceptron, we showed that macroscopic equations can be derived even when the statistical properties of the input are generalized.",
"In other words, we extended the result of [10] and [9].",
"By analyzing the macroscopic system we derived, we showed that the dynamics of learning depends only on the eigenvalue distribution of the covariance matrix of the input data.",
"We clarified the relationship between the input data statistics and plateau phenomenon.",
"In particular, it is shown that the data whose covariance matrix has small and disparsed eigenvalues tend to make the phenomenon inconspicuous, by numerically analyzing the macroscopic system."
],
[
"Related works",
"The statistical mechanical approach used in this research is firstly developed by [10].",
"The method reduces high-dimensional learning dynamics of nonlinear neural networks to low-dimensional system of order parameters.",
"They derived the macroscopic behavior of learning dynamics in two-layered soft-committee machine and by analyzing it they point out the existence of plateau phenomenon.",
"Nowadays the statistical mechanical method is applied to analyze recent techniques ([5], [15], [13] and [11]), and generalization performance in over-parameterized setting ([3]) and environment with conceptual drift ([12]).",
"However, it is unknown that how the property of input dataset itself can affect the learning dynamics, including plateaus.",
"Plateau phenomenon and singularity in loss landscape as its main cause have been studied by [2], [14], [1] and [4].",
"On the other hand, recent several works suggest that plateau and singularity can be mitigated in some settings.",
"[7] shows that skip connections eliminate the singularity.",
"Another work by [16] points out that output dimensionality affects the plateau phenomenon, in that multiple output units alleviate the plateau phenomenon.",
"However, the number of output elements does not fully determine the presence or absence of plateaus, nor does the use of skip connections.",
"The statistical property of data just can affect the learning dynamics dramatically; for example, see REF for learning curves with using different datasets and same network architecture.",
"We focus on what kind of statistical property of the data brings plateau phenomenon.",
"Figure: (a) Training loss curves when two-layer perceptron with 4-4-3 units and ReLU activation learns IRIS dataset.",
"(b) Training loss curve when two-layer perceptron with 784-20-10 units and ReLU activation learns MNIST dataset.For both (a) and (b), results of 100 trials with random initialization are overlaid.",
"Minibatch size of 10 and vanilla SGD (learning rate: 0.01) are used.Figure: Loss curves yielded by student-teacher learning with two-layer perceptron which has 2 hidden units, 1 output unit and sigmoid activation, and with (a) IRIS dataset, (b) MNIST dataset, (c) a dataset in ℝ 60000×784 \\mathbb {R}^{60000\\times 784} drawn from standard normal distribution, as input distribution p(ξ)p({\\xi }).In every subfigure, results for 20 trials with random initialization are overlaid.",
"Vanilla SGD (learning rate: (a)(b) 0.005, (c) 0.001) and minibatch size of 1 are used for all three settings."
],
[
"Student-Teacher Model",
"We consider a two-layer perceptron which has $N$ input units, $K$ hidden units and 1 output unit.",
"We denote the input to the network by ${\\xi } \\in \\mathbb {R}^N$ .",
"Then the output can be written as $s = \\sum _{i=1}^K w_i g({J}_i\\cdot {\\xi }) \\in \\mathbb {R}$ , where $g$ is an activation function.",
"We consider the situation that the network learns data generated by another network, called “teacher network”, which has fixed weights.",
"Specifically, we consider two-layer perceptron that outputs $t = \\sum _{n=1}^M v_n g({B}_n\\cdot {\\xi }) \\in \\mathbb {R}$ for input ${\\xi }$ as the teacher network.",
"The generated data $({\\xi }, t)$ is then fed to the student network stated above and learned by it in the on-line manner (see REF ).",
"We assume that the input ${\\xi }$ is drawn from some distribution $p({\\xi })$ every time independently.",
"We adopt vanilla stochastic gradient descent (SGD) algorithm for learning.",
"We assume the squared loss function $\\varepsilon = \\frac{1}{2} (s - t)^2$ , which is most commonly used for regression."
],
[
"Statistical Mechanical Formulation",
"In order to capture the learning dynamics of nonlinear neural networks described in the previous subsection macroscopically, we introduce the statistical mechanical formulation in this subsection.",
"Let $x_i := {J}_i\\cdot {\\xi }$ ($1 \\le i \\le K$ ) and $y_n := {B}_n\\cdot {\\xi }$ ($1 \\le n \\le M$ ).",
"Then $&\\quad (x_1, \\ldots , x_K, y_1, \\ldots , y_M)\\sim \\mathcal {N} \\left( 0, [{J}_1, \\ldots , {J}_K, {B}_1, \\ldots , {B}_M]^T \\Sigma [{J}_1, \\ldots , {J}_K, {B}_1, \\ldots , {B}_M]\\right)$ holds with $N \\rightarrow \\infty $ by generalized central limit theorem, provided that the input distribution $p({\\xi })$ has zero mean and finite covariance matrix $\\Sigma $ .",
"Next, let us introduce order parameters as following: $Q_{ij} := {J}_i^T\\Sigma {J}_j = \\langle x_ix_j \\rangle $ , $R_{in} := {J}_i^T\\Sigma {B}_n = \\langle x_iy_n \\rangle $ , $T_{nm} := {B}_n^T\\Sigma {B}_m = \\langle y_ny_m \\rangle $ and $D_{ij} := w_i w_j$ , $E_{in} := w_i v_n$ , $F_{nm} := v_n v_m$ .",
"Then $(x_1, \\ldots , x_K, y_1, \\ldots , y_M) \\sim \\mathcal {N}({0}, \\left(\\begin{array}{cc}{Q} & {R} \\\\{R^T} & {T}\\end{array}\\right)).$ The parameters $Q_{ij}$ , $R_{in}$ , $T_{nm}$ , $D_{ij}$ , $E_{in}$ , and $F_{nm}$ introduced above capture the state of the system macroscopically; therefore they are called as “order parameters.” The first three represent the state of the first layers of the two networks (student and teacher), and the latter three represent their second layers' state.",
"$Q$ describes the statistics of the student's first layer and $T$ represents that of the teacher's first layer.",
"$R$ is related to similarity between the student and teacher's first layer.",
"$D, E, F$ is the second layers' counterpart of $Q, R, T$ .",
"The values of $Q_{ij}$ , $R_{in}$ , $D_{ij}$ , and $E_{in}$ change during learning; their dynamics are what to be determined, from the dynamics of microscopic variables, i.e.",
"connection weights.",
"In contrast, $T_{nm}$ and $F_{nm}$ are constant during learning.",
"Figure: Overview of student-teacher model formulation."
],
[
"Higher-order order parameters",
"The important difference between our situation and that of [10] is the covariance matrix $\\Sigma $ of the input ${\\xi }$ is not necessarily equal to identity.",
"This makes the matter complicated, since higher-order terms $\\Sigma ^e$ ($e = 1, 2, \\ldots $ ) appear inevitably in the learning dynamics of order parameters.",
"In order to deal with these, here we define some higher-order version of order parameters.",
"Let us define higher-order order parameters $Q_{ij}^{(e)}$ , $R_{in}^{(e)}$ and $T_{nm}^{(e)}$ for $e = 0, 1, 2, \\ldots $ , as $Q_{ij}^{(e)} := {J}_i^T\\Sigma ^e{J}_j, \\quad R_{in}^{(e)} := {J}_i^T\\Sigma ^e{B}_n, \\quad \\text{and} \\quad T_{nm}^{(e)} := {B}_n^T\\Sigma ^e{B}_m.$ Note that they are identical to $Q_{ij}$ , $R_{in}$ and $T_{nm}$ in the case of $e = 1$ .",
"Also we define higher-order version of $x_i$ and $y_n$ , namely $x^{(e)}_i$ and $y^{(e)}_n$ , as $x^{(e)}_i := {\\xi }^T \\Sigma ^e {J}_i,y^{(e)}_n := {\\xi }^T \\Sigma ^e{B}_n.$ Note that $x^{(0)}_i = x_i$ and $y^{(0)}_n = y_n$ ."
],
[
"Derivation of dynamics of order parameters",
"At each iteration of on-line learning, weights of the student network ${J}_i$ and $w_i$ are updated with $\\begin{aligned}\\Delta {J}_i &= -\\frac{\\eta }{N} \\frac{d\\varepsilon }{d{J}_i} = \\frac{\\eta }{N} [({t}-{s})\\cdot {w}_i] g^{\\prime }(x_i) {\\xi }= \\frac{\\eta }{N} \\left[ \\left( \\sum _{n=1}^M {v}_n g(y_n) - \\sum _{j=1}^K {w}_j g(x_j) \\right) \\cdot {w}_i \\right] g^{\\prime }(x_i) {\\xi }, \\\\\\Delta {w}_i &= -\\frac{\\eta }{N} \\frac{d\\varepsilon }{d{w}_i} = \\frac{\\eta }{N} g(x_i) ({t}-{s})= \\frac{\\eta }{N} g(x_i) \\left(\\sum _{n=1}^M {v}_n g(y_n) - \\sum _{j=1}^K {w}_j g(x_j) \\right),\\end{aligned} $ in which we set the learning rate as $\\eta /N$ , so that our macroscopic system is $N$ -independent.",
"Then, the order parameters $Q_{ij}^{(e)}$ and $R_{in}^{(e)}$ ($e = 0, 1, 2, \\ldots $ ) are updated with $\\begin{aligned}\\Delta Q_{ij}^{(e)} &= ({J}_i + \\Delta {J}_i)^T \\Sigma ^e ({J}_j + \\Delta {J}_j) - {J}_i^T \\Sigma ^e {J}_j= {J}_i^T \\Sigma ^e \\Delta {J}_j + {J}_j^T \\Sigma ^e \\Delta {J}_i + \\Delta {J}_i^T \\Sigma ^e \\Delta {J}_j \\\\&= \\frac{\\eta }{N} \\left[ \\sum _{p=1}^M E_{ip} g^{\\prime }(x_i)x_j^{(e)}g(y_p) - \\sum _{p=1}^K D_{ip} g^{\\prime }(x_i)x_j^{(e)}g(x_p)\\right.",
"\\\\&\\qquad \\qquad + \\left.",
"\\sum _{p=1}^M E_{jp} g^{\\prime }(x_j)x_i^{(e)}g(y_p) - \\sum _{p=1}^K D_{jp} g^{\\prime }(x_j)x_i^{(e)}g(x_p) \\right] \\\\&+ \\frac{\\eta ^2}{N^2} {\\xi }^T \\Sigma ^e {\\xi } \\left[\\sum _{p, q}^{K, K} D_{ip}D_{jq} g^{\\prime }(x_i)g^{\\prime }(x_j)g(x_p)g(x_q) \\right.\\left.",
"+ \\sum _{p, q}^{M, M} E_{ip}E_{jq} g^{\\prime }(x_i)g^{\\prime }(x_j)g(y_p)g(y_q) \\right.",
"\\\\&\\qquad \\left.",
"- \\sum _{p, q}^{K, M} D_{ip}E_{jq} g^{\\prime }(x_i)g^{\\prime }(x_j)g(x_p)g(y_q) \\right.\\left.",
"- \\sum _{p, q}^{M, K} E_{ip}D_{jq} g^{\\prime }(x_i)g^{\\prime }(x_j)g(y_p)g(x_q) \\right], \\\\\\Delta R_{in}^{(e)} &= ({J}_i + \\Delta {J}_i)^T \\Sigma ^e {B}_n - {J}_i^T \\Sigma ^e {B}_n= \\Delta {J}_i^T \\Sigma ^e {B}_n \\\\&= \\frac{\\eta }{N} \\left[ \\sum _{p=1}^M E_{ip} g^{\\prime }(x_i)y_n^{(e)}g(y_p) - \\sum _{p=1}^K D_{ip} g^{\\prime }(x_i)y_n^{(e)}g(x_p) \\right].\\end{aligned}$ Since $&{\\xi }^T \\Sigma ^e {\\xi } \\approx N \\mu _{e+1} \\qquad \\text{where}\\quad \\mu _d := \\frac{1}{N} \\sum _{i=1}^N \\lambda _i^d, \\qquad \\lambda _1, \\ldots , \\lambda _N : \\text{eigenvalues of $\\Sigma $}$ and the right hand sides of the difference equations are $O(N^{-1})$ , we can replace these difference equations with differential ones with $N \\rightarrow \\infty $ , by taking the expectation over all input vectors ${\\xi }$ : $\\begin{aligned}\\frac{dQ_{ij}^{(e)}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_3(x_i, x^{(e)}_j, y_p) - \\sum _{p=1}^K D_{ip} I_3(x_i, x^{(e)}_j, x_p) \\right.",
"\\\\&\\qquad \\qquad \\left.",
"+\\sum _{p=1}^M E_{jp} I_3(x_j, x^{(e)}_i, y_p) - \\sum _{p=1}^K D_{jp} I_3(x_j, x^{(e)}_i, x_p) \\right] \\\\&+ \\eta ^2 \\mu _{e+1} \\left[\\sum _{p, q}^{K, K} D_{ip}D_{jq} I_4(x_i, x_j, x_p, x_q) \\right.\\left.",
"+ \\sum _{p, q}^{M, M} E_{ip}E_{jq} I_4(x_i, x_j, y_p, y_q) \\right.",
"\\\\&\\qquad \\left.",
"- \\sum _{p, q}^{K, M} D_{ip}E_{jq} I_4(x_i, x_j, x_p, y_q) \\right.\\left.",
"- \\sum _{p, q}^{M, K} E_{ip}D_{jq} I_4(x_i, x_j, y_p, x_q) \\right] , \\\\\\frac{dR_{in}^{(e)}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_3(x_i, y^{(e)}_n, y_p) - \\sum _{p=1}^K D_{ip} I_3(x_i, y^{(e)}_n, x_p) \\right]\\end{aligned}$ $\\begin{aligned}\\text{where} \\quad I_3(z_1, z_2, z_3) &:= \\langle g^{\\prime }(z_1)z_2g(z_3)\\rangle \\quad \\text{and}\\quad I_4(z_1, z_2, z_3, z_4) := \\langle g^{\\prime }(z_1)g^{\\prime }(z_2)g(z_3)g(z_4)\\rangle .\\end{aligned}$ In these equations, $\\tilde{\\alpha } := \\alpha /N$ represents time (normalized number of steps), and the brackets $\\langle \\cdot \\rangle $ represent the expectation when the input ${\\xi }$ follows the input distribution $p({\\xi })$ .",
"The differential equations for $D$ and $E$ are obtained in a similar way: $\\begin{aligned}\\frac{dD_{ij}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_2(x_j, y_p) - \\sum _{p=1}^K D_{ip} I_2(x_j, x_p) \\right.\\left.",
"+ \\sum _{p=1}^M E_{jp} I_2(x_i, y_p) - \\sum _{p=1}^K D_{jp} I_2(x_i, x_p) \\right], \\\\\\frac{dE_{in}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M F_{pn} I_2(x_i, y_p) - \\sum _{p=1}^K E_{pn} I_2(x_i, x_p) \\right]\\end{aligned}$ $\\begin{aligned}\\text{where} \\quad I_2(z_1, z_2) &:= \\langle g(z_1)g(z_2)\\rangle .\\end{aligned}$ These differential equations (REF ) and (REF ) govern the macroscopic dynamics of learning.",
"In addition, the generalization loss $\\varepsilon _g$ , the expectation of loss value $\\varepsilon ({\\xi }) = \\frac{1}{2} \\Vert {s} - {t}\\Vert ^2$ over all input vectors ${\\xi }$ , is represented as $\\begin{aligned}\\varepsilon _g &= \\langle \\frac{1}{2} \\Vert {s}-{t}\\Vert ^2 \\rangle = \\frac{1}{2}\\left[ \\sum _{p, q}^{M, M} F_{pq} I_2(y_p, y_q) + \\sum _{p, q}^{K, K} D_{pq} I_2(x_p, x_q) \\right.\\left.",
"- 2\\sum _{p, q}^{K, M} E_{pq} I_2(x_p, y_q) \\right].\\end{aligned}$"
],
[
"Expectation terms",
"Above we have determined the dynamics of order parameters as (REF ), (REF ) and (REF ).",
"However they have expectation terms $I_2(z_1, z_2)$ , $I_3(z_1, z_2^{(e)}, z_3)$ and $I_4(z_1, z_2, z_3, z_4)$ , where $z$ s are either $x_i$ or $y_n$ .",
"By studying what distribution ${z}$ follows, we can show that these expectation terms are dependent only on 1-st and $(e+1)$ -th order parameters, namely, $Q^{(1)}, R^{(1)}, T^{(1)}$ and $Q^{(e+1)}, R^{(e+1)}, T^{(e+1)}$ ; for example, $I_3(x_i, x_j^{(e)}, y_p) &= \\int dz_1dz_2dz_3 \\ g^{\\prime }(z_1)z_2g(z_3) \\ \\mathcal {N}({z} | {0}, \\left(\\begin{array}{ccc}{Q_{ii}^{(1)}} & {Q_{ij}^{(e+1)}} & {R_{ip}^{(1)}} \\\\{Q_{ij}^{(e+1)}} & {*} & {R_{jp}^{(e+1)}} \\\\{R_{ip}^{(1)}} & {R_{jp}^{(e+1)}} & {T_{pp}^{(1)}}\\end{array}\\right))$ holds, where $*$ does not influence the value of this expression (see Supplementary Material A.1 for more detailed discussion).",
"Thus, we see the `speed' of $e$ -th order parameters (i.e.",
"(REF ) and (REF )) only depends on 1-st and $(e+1)$ -th order parameters, and the generalization error $\\varepsilon _g$ (equation (REF )) only depends on 1-st order parameters.",
"Therefore, with denoting $(Q^{(e)}, R^{(e)}, T^{(e)})$ by $\\Omega ^{(e)}$ and $(D, E, F)$ by $\\chi $ , we can write $\\frac{d}{d\\tilde{\\alpha }}\\Omega ^{(e)} &= f^{(e)}(\\Omega ^{(1)}, \\Omega ^{(e+1)}, \\chi ), \\qquad \\frac{d}{d\\tilde{\\alpha }}\\chi = g(\\Omega ^{(1)}, \\chi ), \\quad \\text{and}\\quad \\varepsilon _g = h(\\Omega ^{(1)}, \\chi )$ with appropriate functions $f^{(e)}$ , $g$ and $h$ .",
"Additionally, a polynomial of $\\Sigma $ $&P(\\Sigma ) := \\prod _{i=1}^d (\\Sigma - \\lambda ^{\\prime }_i I_N) = \\sum _{e=0}^d c_e \\Sigma ^e \\qquad \\text{where}\\quad \\lambda ^{\\prime }_1, \\ldots , \\lambda ^{\\prime }_d \\quad \\text{are distinct eigenvalues of $\\Sigma $}$ equals to 0, thus we get $\\begin{aligned}\\Omega ^{(d)} = - \\sum _{e=0}^{d-1} c_e \\Omega ^{(e)}.\\end{aligned}$ Using this relation, we can reduce $\\Omega ^{(d)}$ to expressions which contain only $\\Omega ^{(0)}, \\ldots , \\Omega ^{(d-1)}$ , therefore we can get a closed differential equation system with $\\Omega ^{(0)}, \\ldots , \\Omega ^{(d-1)}$ and $\\chi $ .",
"In summary, our macroscopic system is closed with the following order parameters: $&\\text{Order variables}:\\quad Q_{ij}^{(0)}, Q_{ij}^{(1)}, \\ldots , Q_{ij}^{(d-1)},\\quad R_{in}^{(0)}, R_{in}^{(1)}, \\ldots , R_{in}^{(d-1)},\\quad D_{ij}, E_{in} \\\\&\\text{Order constants}:\\quad T_{nm}^{(0)}, T_{nm}^{(1)}, \\ldots , T_{nm}^{(d-1)},\\quad F_{nm}.\\qquad \\text{($d$: number of distinct eigenvalues of $\\Sigma $)}$ The order variables are governed by (REF ) and (REF ).",
"For the lengthy full expressions of our macroscopic system for specific cases, see Supplementary Material A.2."
],
[
"Dependency on input data covariance $\\Sigma $",
"The differential equation system we derived depends on $\\Sigma $ , through two ways; the coefficient $\\mu _{e+1}$ of $O(\\eta ^2)$ -term, and how ($d$ )-th order parameters are expanded with lower order parameters (as (REF )).",
"Specifically, the system only depends on the eigenvalue distribution of $\\Sigma $ ."
],
[
"Evaluation of expectation terms for specific activation functions",
"Expectation terms $I_2$ , $I_3$ and $I_4$ can be analytically determined for some activation functions $g$ , including sigmoid-like $g(x) = \\textrm {erf}(x/\\sqrt{2})$ (see [10]) and $g(x) = \\textrm {ReLU}(x)$ (see [15])."
],
[
"Analysis of numerical solutions of macroscopic differential equations",
"In this section, we analyze numerically the order parameter system, derived in the previous sectionWe executed all computations on a standard PC.. We assume that the second layers' weights of the student and the teacher, namely $w_i$ and $v_n$ , are fixed to 1 (i.e.",
"we consider the learning of soft-committee machine), and that $K$ and $M$ are equal to 2, for simplicity.",
"Here we think of sigmoid-like activation $g(x) = \\textrm {erf}(x/\\sqrt{2})$ ."
],
[
"Consistency between macroscopic system and microscopic system",
"First of all, we confirmed the consistency between the macroscopic system we derived and the original microscopic system.",
"That is, we computed the dynamics of the generalization loss $\\varepsilon _g$ in two ways: (i) by updating weights of the network with SGD (REF ) iteratively, and (ii) by solving numerically the differential equations (REF ) which govern the order parameters, and we confirmed that they accord with each other very well ( REF ).",
"Note that we set the initial values of order parameters in (ii) as values corresponding to initial weights used in (i).",
"For dependence of the learning trajectory on the initial condition, see Supplementary Material A.3.",
"Figure: Example dynamics of generalization error ε g \\varepsilon _g computed with (a) microscopic and (b) macroscopic system.",
"Network size: NN-2-1.",
"Learning rate: η=0.1\\eta =0.1.",
"Eigenvalues of Σ\\Sigma : λ 1 =0.4\\lambda _1 = 0.4 with multiplicity 0.5N0.5 N, λ 2 =1.2\\lambda _2 = 1.2 with multiplicity 0.3N0.3 N, and λ 3 =1.6\\lambda _3 = 1.6 with multiplicity 0.2N0.2 N. Black lines: dynamics of ε g \\varepsilon _g.",
"Blue lines: Q 11 ,Q 12 ,Q 22 Q_{11}, Q_{12}, Q_{22}.",
"Green lines: R 11 ,R 12 ,R 21 ,R 22 R_{11}, R_{12}, R_{21}, R_{22}."
],
[
"Case of scalar input covariance $\\Sigma = \\sigma I_N$",
"As the simplest case, here we consider the case that the convariance matrix $\\Sigma $ is proportional to unit matrix.",
"In this case, $\\Sigma $ has only one eigenvalue $\\lambda = \\mu _1$ of multiplicity $N$ , then our order parameter system contains only parameters whose order is 0 ($e = 0$ ).",
"For various values of $\\mu _1$ , we solved numerically the differential equations of order parameters (REF ) and plotted the time evolution of generalization loss $\\varepsilon _g$ ( REF (a)).",
"From these plots, we quantified the lengths and heights of the plateaus as following: we regarded the system is plateauing if the decreasing speed of log-loss is smaller than half of its terminal converging speed, and we defined the height of the plateau as the median of loss values during plateauing.",
"Quantified lengths and heights are plotted in REF (b)(c).",
"It indicates that the plateau length and height heavily depend on $\\mu _1$ , the input scale.",
"Specifically, as $\\mu _1$ decreases, the plateau rapidly becomes longer and lower.",
"Though smaller input data lead to longer plateaus, it also becomes lower and then inconspicuous.",
"This tendency is consistent with REF (a)(b), since IRIS dataset has large $\\mu _1$ ($\\approx 15.9$ ) and MNIST has small $\\mu _1$ ($\\approx 0.112$ ).",
"Considering this, the claim that the plateau phenomenon does not occur in learning of MNIST is controversy; this suggests the possibility that we are observing quite long and low plateaus.",
"Note that REF (b) shows that the speed of growing of plateau length is larger than $O(1/\\mu _1)$ .",
"This is contrast to the case of linear networks which have no activation; in that case, as $\\mu _1$ decreases the speed of learning gets exactly $1/\\mu _1$ -times larger.",
"In other words, this phenomenon is peculiar to nonlinear networks.",
"Figure: (a) Dynamics of generalization error ε g \\varepsilon _g when input variance Σ\\Sigma has only one eigenvalue λ=μ 1 \\lambda = \\mu _1 of multiplicity NN.",
"Plots with various values of μ 1 \\mu _1 are shown.",
"(b) Plateau length and (b) plateau height, quantified from (a)."
],
[
"Case of different input covariance $\\Sigma $ with fixed {{formula:ae6c2f6a-6f8e-42cf-9503-1f64a0730fee}}",
"In the previous subsection we inspected the dependence of the learning dynamics on the first moment $\\mu _1$ of the eigenvalues of the covariance matrix $\\Sigma $ .",
"In this subsection, we explored the dependence of the dynamics on the higher moments of eigenvalues, under fixed first moment $\\mu _1$ .",
"In this subsection, we consider the case in which the input covariance matrix $\\Sigma $ has two distinct nonzero eigenvalues, $\\lambda _1 = \\mu _1 - \\Delta \\lambda /2$ and $\\lambda _2 = \\mu _1 + \\Delta \\lambda /2$ , of the same multiplicity $N/2$ ( REF ).",
"With changing the control parameter $\\Delta \\lambda $ , we can get eigenvalue distributions with various values of second moment $\\mu _2 = \\langle \\lambda _i^2 \\rangle $ .",
"Figure: Eigenvalue distribution with fixed μ 1 \\mu _1 parameterized by Δλ\\Delta \\lambda , which yields various μ 2 \\mu _2.",
"REF (a) shows learning curves with various $\\mu _2$ while fixing $\\mu _1$ to 1.",
"From these curves, we quantified the lengths and heights of the plateaus, and plotted them in REF (b)(c).",
"These indicate that the length of the plateau shortens as $\\mu _2$ becomes large.",
"That is, the more the distribution of nonzero eigenvalues gets broaden, the more the plateau gets alleviated.",
"Figure: (a) Dynamics of generalization error ε g \\varepsilon _g when input variance Σ\\Sigma has two eigenvalues λ 1,2 =μ 1 ±Δλ/2\\lambda _{1,2} = \\mu _1 \\pm \\Delta \\lambda /2 of multiplicity N/2N/2.",
"Plots with various values of μ 2 \\mu _2 are shown.",
"(b) Plateau length and (c) plateau height, quantified from (a)."
],
[
"Conclusion",
"Under the statistical mechanical formulation of learning in the two-layered perceptron, we showed that macroscopic equations can be derived even when the statistical properties of the input are generalized.",
"We showed that the dynamics of learning depends only on the eigenvalue distribution of the covariance matrix of the input data.",
"By numerically analyzing the macroscopic system, it is shown that the statistics of input data dramatically affect the plateau phenomenon.",
"Through this work, we explored the gap between theory and reality; though the plateau phenomenon is theoretically predicted to occur by the general symmetrical structure of neural networks, it is seldom observed in practice.",
"However, more extensive researches are needed to fully understand the theory underlying the plateau phenomenon in practical cases."
],
[
"Acknowledgement",
"This work was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research(A) (No.",
"18H04106).",
"Data-Dependence of Plateau Phenomenon in Learning with Neural Network — Statistical Mechanical Analysis (Supplementary Material)"
],
[
"Properties of expectation term $I_2$ , {{formula:ce46152f-8f42-4bf1-b8f6-6a7276c317e7}} and {{formula:75714516-6b25-4732-af90-9316229692bd}}",
"The differential equations of learning dynamics (3) and (5) in the main text have expectation terms, $I_2(z_1, z_2), I_3(z_1, z_2, z_3)$ and $I_4(z_1, z_2, z_3, z_4)$ .",
"Since their $z$ s are either $x^{(e)}_i = {\\xi }^T \\Sigma ^e {J}_i$ or $y^{(e)}_n = {\\xi }^T \\Sigma ^e {B}_n$ , any tuple $(z_1, z_2, \\ldots )$ follows multivatiate normal distribution $\\mathcal {N}({z} | 0, \\langle {z}\\cdot {z}^T \\rangle )$ when $N \\rightarrow \\infty $ by generalized central limit theorem, provided that the input ${\\xi }$ has zero mean and finite covariance.",
"Thus the expectation terms only depend on the covariance matrix $\\langle {z}\\cdot {z}^T \\rangle $ , and their elements can be calculated as $\\langle x^{(e)}_i x^{(f)}_j \\rangle = Q_{ij}^{(e+f+1)}$ , $\\langle x^{(e)}_i y^{(f)}_n \\rangle = R_{in}^{(e+f+1)}$ and $\\langle y^{(e)}_n y^{(f)}_m \\rangle = T_{nm}^{(e+f+1)}$ .",
"For example, $I_2(x_i, y_p) &= \\int dz_1dz_2 \\ g(z_1)g(z_2) \\ \\mathcal {N}({z} | {0}, \\left(\\begin{array}{cc}{Q_{ii}^{(1)}} & {R_{ip}^{(1)}} \\\\{} & {T_{pp}^{(1)}}\\end{array}\\right)), \\\\I_3(x_i, x_j^{(e)}, y_p) &= \\int dz_1dz_2dz_3 \\ g^{\\prime }(z_1)z_2g(z_3) \\ \\mathcal {N}({z} | {0}, \\left(\\begin{array}{ccc}{Q_{ii}^{(1)}} & {Q_{ij}^{(e+1)}} & {R_{ip}^{(1)}} \\\\{} & {Q_{jj}^{(2e+1)}} & {R_{jp}^{(e+1)}} \\\\{} & {} & {T_{pp}^{(1)}}\\end{array}\\right)), \\\\I_4(x_i, x_j, y_p, y_q) &= \\int dz_1dz_2dz_3dz_4 \\ g(z_1)g(z_2)g(z_3)g(z_4) \\ \\mathcal {N}({z} | {0}, \\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {R^{(1)}_{ip}} & {R^{(1)}_{iq}} \\\\{} & {Q^{(1)}_{jj}} & {R^{(1)}_{jp}} & {R^{(1)}_{jq}} \\\\{} & {} & {T^{(1)}_{pp}} & {T^{(1)}_{pq}} \\\\{} & {} & {} & {T^{(1)}_{qq}}\\end{array}\\right)}).$ Note that all the covariance matrix is symmetric.",
"Their left-bottom sides are not shown for notational simplicity.",
"Substituting these for $I$ s shown in equations (3) and (5) in the main text, we see that the `speed' of $e$ -th order parameters can be dependent only on 1-st, $(e+1)$ -th, and $(2e+1)$ -th order parameters.",
"Here we prove the following proposition, in order to show that the `speed' of $e$ -th order parameters are not dependent on $(2e+1)$ -th order parameters.",
"Proposition.",
"The expectation term $I_3(z_1, z_2, z_3) := \\int dz_1dz_2dz_3 \\ g^{\\prime }(z_1)z_2g(z_3) \\ \\mathcal {N}({z} | {0}, C)$ does not depend on $C_{22}$ .",
"Proof.",
"Since $C$ is positive-semidefinite, we can write $C = VV^T$ for some squared matrix $V$ .",
"Thus, when ${\\xi } \\sim \\mathcal {N}(0, I_N)$ , $A{\\xi } \\sim \\mathcal {N}(0, C)$ holds.",
"Therefore, we can regard that $z_i (i = 1, 2, 3)$ is generated by $z_i = {v}_i^T {\\xi }$ where ${v}_i$ is $i$ -th row vector of $V$ and ${\\xi }$ follows the standard normal distribution.",
"We can write ${v}_2 = c_1 {v}_1 + c_3 {v}_3 + {v}^\\perp $ for some coefficient $c_1, c_3 \\in \\mathbb {R}$ and some vector ${v}^\\perp $ perpendicular to ${v}_1$ and ${v}_3$ .",
"Then $I_3$ is written as $I_3(z_1, z_2, z_3)= \\langle g^{\\prime }(z_1) z_2 g(z_3) \\rangle = c_1 \\langle g^{\\prime }(z_1) z_1 g(z_3) \\rangle + c_3 \\langle g^{\\prime }(z_1) z_3 g(z_3) \\rangle + \\langle g^{\\prime }(z_1) {v}^{\\perp T}{\\xi } g(z_3) \\rangle .$ Since ${\\xi } \\sim \\mathcal {N}(0, I_N)$ and ${v}^\\perp \\perp {v}_1, {v}_3$ hold, $(z_1, z_3)$ and ${v}^{\\perp T}{\\xi }$ is independent.",
"Therefore the third term in the right hand side of the equation above is $\\langle g^{\\prime }(z_1) {v}^{\\perp T}{\\xi } g(z_3) \\rangle = \\langle g^{\\prime }(z_1) g(z_3) \\rangle \\langle {v}^{\\perp T}{\\xi } \\rangle = 0.$ In addition, we can determine $c_1$ and $c_3$ by solving $C_{12}&= {v}_2^T{v}_1 = (c_1 {v}_1^T+ c_3 {v}_3^T + {v}^{\\perp T}){v}_1= c_1 C_{11} + c_3 C_{13} \\quad \\text{and} \\\\C_{23}&= {v}_2^T{v}_3 = (c_1 {v}_1^T+ c_3 {v}_3^T + {v}^{\\perp T}){v}_3= c_1 C_{13} + c_3 C_{33}.$ Together with these, we get $I_3(z_1, z_2, z_3) = \\frac{(C_{12}C_{33}-C_{13}C_{23}) \\ I_3(z_1, z_1, z_3)+ (C_{11}C_{23}-C_{12}C_{13}) \\ I_3(z_1, z_3, z_3)}{C_{11}C_{33}-C_{13}^2},$ which shows that $I_3$ is independent to $C_{22}$ .",
"$\\blacksquare $"
],
[
"Full expression of order parameter system",
"Here we describe the whole system of the order parameters, with specific eigenvalue distribution of $\\Sigma $ ."
],
[
"Case with $\\Sigma = \\sigma I_N$",
"In this case, the order parameters are $&\\text{Order variables}:\\qquad Q_{ij}^{(0)},\\qquad R_{in}^{(0)},\\qquad D_{ij}, E_{in} \\\\&\\text{Order constants}:\\qquad T_{nm}^{(0)},\\qquad F_{nm}.$ Note that $Q_{ij}^{(1)}$ is identical to $Q_{ij}^{(0)}$ .",
"This is same for $R$ and $T$ .",
"The order parameter system is described as following, with omitting ${}^{(0)}$ -s for notational simplicity: $\\begin{aligned}\\frac{dQ_{ij}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_3\\left(\\begin{array}{ccc}{Q_{ii}} & {Q_{ij}} & {R_{ip}} \\\\{} & {*} & {R_{jp}} \\\\{} & {} & {T_{pp}}\\end{array}\\right) - \\sum _{p=1}^K D_{ip} I_3\\left(\\begin{array}{ccc}{Q_{ii}} & {Q_{ij}} & {Q_{ip}} \\\\{} & {*} & {Q_{jp}} \\\\{} & {} & {Q_{pp}}\\end{array}\\right) \\right.",
"\\\\&+ \\left.",
"\\sum _{p=1}^M E_{jp} I_3\\left(\\begin{array}{ccc}{Q_{jj}} & {Q_{ji}} & {R_{jp}} \\\\{} & {*} & {R_{ip}} \\\\{} & {} & {T_{pp}}\\end{array}\\right) - \\sum _{p=1}^K D_{jp} I_3\\left(\\begin{array}{ccc}{Q_{jj}} & {Q_{ji}} & {Q_{jp}} \\\\{} & {*} & {Q_{ip}} \\\\{} & {} & {Q_{pp}}\\end{array}\\right) \\right] \\\\&+ \\eta ^2 \\left[\\sum _{p, q}^{K, K} D_{ip}D_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q_{ii}} & {Q_{ij}} & {Q_{ip}} & {Q_{iq}} \\\\{} & {Q_{jj}} & {Q_{jp}} & {Q_{jq}} \\\\{} & {} & {Q_{pp}} & {Q_{pq}} \\\\{} & {} & {} & {Q_{qq}}\\end{array}\\right)}+ \\sum _{p, q}^{M, M} E_{ip}E_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q_{ii}} & {Q_{ij}} & {R_{ip}} & {R_{iq}} \\\\{} & {Q_{jj}} & {R_{jp}} & {R_{jq}} \\\\{} & {} & {T_{pp}} & {T_{pq}} \\\\{} & {} & {} & {T_{qq}}\\end{array}\\right)} \\right.",
"\\\\&\\qquad \\left.",
"- \\sum _{p, q}^{K, M} D_{ip}E_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q_{ii}} & {Q_{ij}} & {Q_{ip}} & {R_{iq}} \\\\{} & {Q_{jj}} & {Q_{jp}} & {R_{jq}} \\\\{} & {} & {Q_{pp}} & {R_{pq}} \\\\{} & {} & {} & {T_{qq}}\\end{array}\\right)}- \\sum _{p, q}^{M, K} E_{ip}D_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q_{ii}} & {Q_{ij}} & {R_{ip}} & {Q_{iq}} \\\\{} & {Q_{jj}} & {R_{jp}} & {Q_{jq}} \\\\{} & {} & {T_{pp}} & {R_{pq}} \\\\{} & {} & {} & {Q_{qq}}\\end{array}\\right)} \\right] , \\\\\\frac{dR_{in}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_3\\left(\\begin{array}{ccc}{Q_{ii}} & {R_{in}} & {R_{ip}} \\\\{} & {*} & {T_{np}} \\\\{} & {} & {T_{pp}}\\end{array}\\right) - \\sum _{p=1}^K D_{ip} I_3\\left(\\begin{array}{ccc}{Q_{ii}} & {R_{in}} & {Q_{ip}} \\\\{} & {*} & {R_{pn}} \\\\{} & {} & {Q_{pp}}\\end{array}\\right) \\right]\\end{aligned}$ and $\\begin{aligned}\\frac{dD_{ij}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_2\\left(\\begin{array}{cc}{Q_{jj}} & {R_{jp}} \\\\{} & {T_{pp}}\\end{array}\\right) - \\sum _{p=1}^K D_{ip} I_2\\left(\\begin{array}{cc}{Q_{jj}} & {Q_{jp}} \\\\{} & {Q_{pp}}\\end{array}\\right) \\right.",
"\\\\&\\qquad \\left.",
"+ \\sum _{p=1}^M E_{jp} I_2\\left(\\begin{array}{cc}{Q_{ii}} & {R_{ip}} \\\\{} & {T_{pp}}\\end{array}\\right) - \\sum _{p=1}^K D_{jp} I_2\\left(\\begin{array}{cc}{Q_{ii}} & {Q_{ip}} \\\\{} & {Q_{pp}}\\end{array}\\right) \\right], \\\\\\frac{dE_{in}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M F_{pn} I_2\\left(\\begin{array}{cc}{Q_{ii}} & {R_{ip}} \\\\{} & {T_{pp}}\\end{array}\\right) - \\sum _{p=1}^K E_{pn} I_2\\left(\\begin{array}{cc}{Q_{jj}} & {R_{jp}} \\\\{} & {T_{pp}}\\end{array}\\right) \\right]\\end{aligned}$ , $\\begin{aligned}\\text{where} \\quad I_2(C) &= \\frac{2}{\\pi }\\arcsin \\frac{C_{12}}{\\sqrt{1+C_{11}}\\sqrt{1+C_{22}}}, \\\\I_3(C) &= \\frac{2}{\\pi }\\cdot \\frac{1}{\\sqrt{(1+C_{11})(1+C_{33})-C_{13}^2}}\\frac{C_{23}(1+C_{11})-C_{12}C_{13}}{1+C_{11}}, \\\\I_4(C) &= \\frac{4}{\\pi ^2}\\cdot \\frac{1}{\\sqrt{1+2C_{11}}}\\arcsin \\frac{(1+2C_{11})C_{23}-2C_{12}C_{13}}{\\sqrt{(1+2C_{11})(1+C_{22}) - 2 C_{12}^2}\\sqrt{(1+2C_{11})(1+C_{33}) - 2 C_{13}^2}}\\end{aligned}$ for $g(x) = \\textrm {erf}(x/\\sqrt{2})$ activation, as [10] showed."
],
[
"Case with $\\Sigma $ which has two distinct eigenvalues, {{formula:736312cf-85b7-4b23-8b20-5f0b270fb274}} of multiplicity {{formula:9e857a3d-9fd4-4ace-810d-7629ba031854}} and {{formula:eb174f51-773f-4eb1-b5fd-18d8d727bd20}} of multiplicity {{formula:694cbda1-f7bf-4c6a-8f25-25442df480ec}}",
"In this case, the order parameters are $&\\text{Order variables}:\\qquad Q_{ij}^{(0)}, Q_{ij}^{(1)},\\qquad R_{in}^{(0)}, R_{in}^{(1)},\\qquad D_{ij}, E_{in} \\\\&\\text{Order constants}:\\qquad T_{nm}^{(0)}, T_{nm}^{(1)},\\qquad F_{nm}.$ Since $\\Sigma ^2 - (\\lambda _1 + \\lambda _2)\\Sigma + \\lambda _1\\lambda _2 I_N = 0$ , the relation $Q^{(2)}_{ij} = (\\lambda _1 + \\lambda _2)Q^{(1)}_{ij} - \\lambda _1\\lambda _2 Q^{(0)}_{ij}$ holds.",
"This is same for $R$ and $T$ .",
"Then the order parameter system is described as following: $\\begin{aligned}\\frac{dQ^{(0)}_{ij}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {R^{(1)}_{ip}} \\\\{} & {*} & {R^{(1)}_{jp}} \\\\{} & {} & {T^{(1)}_{pp}}\\end{array}\\right) - \\sum _{p=1}^K D_{ip} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {Q^{(1)}_{ip}} \\\\{} & {*} & {Q^{(1)}_{jp}} \\\\{} & {} & {Q^{(1)}_{pp}}\\end{array}\\right) \\right.",
"\\\\&+ \\left.",
"\\sum _{p=1}^M E_{jp} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{jj}} & {Q^{(1)}_{ji}} & {R^{(1)}_{jp}} \\\\{} & {*} & {R^{(1)}_{ip}} \\\\{} & {} & {T^{(1)}_{pp}}\\end{array}\\right) - \\sum _{p=1}^K D_{jp} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{jj}} & {Q^{(1)}_{ji}} & {Q^{(1)}_{jp}} \\\\{} & {*} & {Q^{(1)}_{ip}} \\\\{} & {} & {Q^{(1)}_{pp}}\\end{array}\\right) \\right] \\\\&+ \\eta ^2 (r_1\\lambda _1 + r_2\\lambda _2) \\left[\\sum _{p, q}^{K, K} D_{ip}D_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {Q^{(1)}_{ip}} & {Q^{(1)}_{iq}} \\\\{} & {Q^{(1)}_{jj}} & {Q^{(1)}_{jp}} & {Q^{(1)}_{jq}} \\\\{} & {} & {Q^{(1)}_{pp}} & {Q^{(1)}_{pq}} \\\\{} & {} & {} & {Q^{(1)}_{qq}}\\end{array}\\right)}+ \\sum _{p, q}^{M, M} E_{ip}E_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {R^{(1)}_{ip}} & {R^{(1)}_{iq}} \\\\{} & {Q^{(1)}_{jj}} & {R^{(1)}_{jp}} & {R^{(1)}_{jq}} \\\\{} & {} & {T^{(1)}_{pp}} & {T^{(1)}_{pq}} \\\\{} & {} & {} & {T^{(1)}_{qq}}\\end{array}\\right)} \\right.",
"\\\\&\\qquad \\left.",
"- \\sum _{p, q}^{K, M} D_{ip}E_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {Q^{(1)}_{ip}} & {R^{(1)}_{iq}} \\\\{} & {Q^{(1)}_{jj}} & {Q^{(1)}_{jp}} & {R^{(1)}_{jq}} \\\\{} & {} & {Q^{(1)}_{pp}} & {R^{(1)}_{pq}} \\\\{} & {} & {} & {T^{(1)}_{qq}}\\end{array}\\right)}- \\sum _{p, q}^{M, K} E_{ip}D_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {R^{(1)}_{ip}} & {Q^{(1)}_{iq}} \\\\{} & {Q^{(1)}_{jj}} & {R^{(1)}_{jp}} & {Q^{(1)}_{jq}} \\\\{} & {} & {T^{(1)}_{pp}} & {R^{(1)}_{pq}} \\\\{} & {} & {} & {Q^{(1)}_{qq}}\\end{array}\\right)} \\right] , \\\\\\frac{dQ^{(1)}_{ij}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{ii}} & {(\\lambda _1 + \\lambda _2)Q^{(1)}_{ij} - \\lambda _1\\lambda _2 Q^{(0)}_{ij}} & {R^{(1)}_{ip}} \\\\{} & {*} & {(\\lambda _1 + \\lambda _2)R^{(1)}_{jp} - \\lambda _1\\lambda _2 R^{(0)}_{jp}} \\\\{} & {} & {T^{(1)}_{pp}}\\end{array}\\right) \\right.",
"\\\\&- \\left.",
"\\sum _{p=1}^K D_{ip} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{ii}} & {(\\lambda _1 + \\lambda _2)Q^{(1)}_{ij} - \\lambda _1\\lambda _2 Q^{(0)}_{ij}} & {Q^{(1)}_{ip}} \\\\{} & {*} & {(\\lambda _1 + \\lambda _2)Q^{(1)}_{jp} - \\lambda _1\\lambda _2 Q^{(0)}_{jp}} \\\\{} & {} & {Q^{(1)}_{pp}}\\end{array}\\right) \\right.",
"\\\\&+ \\left.",
"\\sum _{p=1}^M E_{jp} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{jj}} & {(\\lambda _1 + \\lambda _2)Q^{(1)}_{ji} - \\lambda _1\\lambda _2 Q^{(0)}_{ji}} & {R^{(1)}_{jp}} \\\\{} & {*} & {(\\lambda _1 + \\lambda _2)R^{(1)}_{ip} - \\lambda _1\\lambda _2 R^{(0)}_{ip}} \\\\{} & {} & {T^{(1)}_{pp}}\\end{array}\\right) \\right.",
"\\\\&- \\left.",
"\\sum _{p=1}^K D_{jp} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{jj}} & {(\\lambda _1 + \\lambda _2)Q^{(1)}_{ji} - \\lambda _1\\lambda _2 Q^{(0)}_{ji}} & {Q^{(1)}_{jp}} \\\\{} & {*} & {(\\lambda _1 + \\lambda _2)Q^{(1)}_{ip} - \\lambda _1\\lambda _2 Q^{(0)}_{ip}} \\\\{} & {} & {Q^{(1)}_{pp}}\\end{array}\\right) \\right] \\\\&+ \\eta ^2 (r_1\\lambda _1^2 + r_2\\lambda _2^2) \\left[\\sum _{p, q}^{K, K} D_{ip}D_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {Q^{(1)}_{ip}} & {Q^{(1)}_{iq}} \\\\{} & {Q^{(1)}_{jj}} & {Q^{(1)}_{jp}} & {Q^{(1)}_{jq}} \\\\{} & {} & {Q^{(1)}_{pp}} & {Q^{(1)}_{pq}} \\\\{} & {} & {} & {Q^{(1)}_{qq}}\\end{array}\\right)}+ \\sum _{p, q}^{M, M} E_{ip}E_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {R^{(1)}_{ip}} & {R^{(1)}_{iq}} \\\\{} & {Q^{(1)}_{jj}} & {R^{(1)}_{jp}} & {R^{(1)}_{jq}} \\\\{} & {} & {T^{(1)}_{pp}} & {T^{(1)}_{pq}} \\\\{} & {} & {} & {T^{(1)}_{qq}}\\end{array}\\right)} \\right.",
"\\\\&\\qquad \\left.",
"- \\sum _{p, q}^{K, M} D_{ip}E_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {Q^{(1)}_{ip}} & {R^{(1)}_{iq}} \\\\{} & {Q^{(1)}_{jj}} & {Q^{(1)}_{jp}} & {R^{(1)}_{jq}} \\\\{} & {} & {Q^{(1)}_{pp}} & {R^{(1)}_{pq}} \\\\{} & {} & {} & {T^{(1)}_{qq}}\\end{array}\\right)}- \\sum _{p, q}^{M, K} E_{ip}D_{jq} I_4\\scalebox {0.8}{\\left(\\begin{array}{cccc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ij}} & {R^{(1)}_{ip}} & {Q^{(1)}_{iq}} \\\\{} & {Q^{(1)}_{jj}} & {R^{(1)}_{jp}} & {Q^{(1)}_{jq}} \\\\{} & {} & {T^{(1)}_{pp}} & {R^{(1)}_{pq}} \\\\{} & {} & {} & {Q^{(1)}_{qq}}\\end{array}\\right)} \\right] , \\\\\\frac{dR^{(0)}_{in}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{ii}} & {R^{(1)}_{in}} & {R^{(1)}_{ip}} \\\\{} & {*} & {T^{(1)}_{np}} \\\\{} & {} & {T^{(1)}_{pp}}\\end{array}\\right) - \\sum _{p=1}^K D_{ip} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{ii}} & {R^{(1)}_{in}} & {Q^{(1)}_{ip}} \\\\{} & {*} & {R^{(1)}_{pn}} \\\\{} & {} & {Q^{(1)}_{pp}}\\end{array}\\right) \\right], \\\\\\frac{dR^{(1)}_{in}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{ii}} & {(\\lambda _1 + \\lambda _2)R^{(1)}_{in} - \\lambda _1\\lambda _2R^{(0)}_{in}} & {R^{(1)}_{ip}} \\\\{} & {*} & {(\\lambda _1 + \\lambda _2)T^{(1)}_{np} - \\lambda _1\\lambda _2T^{(0)}_{np}} \\\\{} & {} & {T^{(1)}_{pp}}\\end{array}\\right) \\right.",
"\\\\&- \\left.",
"\\sum _{p=1}^K D_{ip} I_3\\left(\\begin{array}{ccc}{Q^{(1)}_{ii}} & {(\\lambda _1 + \\lambda _2)R^{(1)}_{in} - \\lambda _1\\lambda _2R^{(0)}_{in}} & {Q^{(1)}_{ip}} \\\\{} & {*} & {(\\lambda _1 + \\lambda _2)R^{(1)}_{pn} - \\lambda _1\\lambda _2R^{(0)}_{pn}} \\\\{} & {} & {Q^{(1)}_{pp}}\\end{array}\\right) \\right]\\end{aligned}$ , and $\\begin{aligned}\\frac{dD_{ij}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M E_{ip} I_2\\left(\\begin{array}{cc}{Q^{(1)}_{jj}} & {R^{(1)}_{jp}} \\\\{} & {T^{(1)}_{pp}}\\end{array}\\right) - \\sum _{p=1}^K D_{ip} I_2\\left(\\begin{array}{cc}{Q^{(1)}_{jj}} & {Q^{(1)}_{jp}} \\\\{} & {Q^{(1)}_{pp}}\\end{array}\\right) \\right.",
"\\\\&\\qquad \\left.",
"+ \\sum _{p=1}^M E_{jp} I_2\\left(\\begin{array}{cc}{Q^{(1)}_{ii}} & {R^{(1)}_{ip}} \\\\{} & {T^{(1)}_{pp}}\\end{array}\\right) - \\sum _{p=1}^K D_{jp} I_2\\left(\\begin{array}{cc}{Q^{(1)}_{ii}} & {Q^{(1)}_{ip}} \\\\{} & {Q^{(1)}_{pp}}\\end{array}\\right) \\right], \\\\\\frac{dE_{in}}{d\\tilde{\\alpha }}&= \\eta \\left[ \\sum _{p=1}^M F_{pn} I_2\\left(\\begin{array}{cc}{Q^{(1)}_{ii}} & {R^{(1)}_{ip}} \\\\{} & {T^{(1)}_{pp}}\\end{array}\\right) - \\sum _{p=1}^K E_{pn} I_2\\left(\\begin{array}{cc}{Q^{(1)}_{jj}} & {R^{(1)}_{jp}} \\\\{} & {T^{(1)}_{pp}}\\end{array}\\right) \\right]\\end{aligned}$ ."
],
[
"Dependence of learning trajectory on initial conditions on macroscopic parameters",
"In the statistical mechanical formulation, by considering $N$ as large, the dynamics of the system is reduced to macroscopic differential equations with small ($N$ -independent) dimensions.",
"The macroscopic system we derived is deterministic in the sense that randomness brought by stochastic gradient descent is vanished.",
"However, note that the trajectory of the macroscopic state can vary in accordance with its initial condition.",
"REF shows this variability with shades.",
"How does the initial condition affect the learning trajectory?",
"Consider a typical initialization that the microscopic parameters ${J}_1$ , ${J}_2$ , ${B}_1$ and ${B}_2$ are initialized as $({J}_i)_k, ({B}_n)_k \\stackrel{\\textrm {i.i.d.",
"}}{\\sim } \\mathcal {N}(0, 1/N)$ .",
"Then the mean and variance of corresponding initial macroscopic parameters $Q$ , $R$ and $T$ are $\\mathbb {E}[Q_{ii}^{(e)}] &= \\mu _e, \\quad \\mathbb {V}[Q_{ii}^{(e)}] = \\frac{3 \\mu _{2e}}{N}, \\quad \\mathbb {E}[Q_{ij}^{(e)}] = 0, \\quad \\mathbb {V}[Q_{ij}^{(e)}] = \\frac{\\mu _{2e}}{N}, \\\\\\mathbb {E}[R_{in}^{(e)}] &= 0, \\quad \\mathbb {V}[R_{in}^{(e)}] = \\frac{\\mu _{2e}}{N}, \\\\\\mathbb {E}[T_{nn}^{(e)}] &= \\mu _e, \\quad \\mathbb {V}[T_{nn}^{(e)}] = \\frac{3 \\mu _{2e}}{N}, \\quad \\mathbb {E}[T_{nm}^{(e)}] = 0, \\quad \\mathbb {V}[T_{nm}^{(e)}] = \\frac{\\mu _{2e}}{N}$ With $N \\rightarrow \\infty $ , these probabilistic parameters converge to $(Q^{(e)}, R^{(e)}, T^{(e)}) = (\\mu _e I_K, 0, \\mu _e I_M)$ .",
"However, the solution trajectory starting from just $(\\mu _e I_K, 0, \\mu _e I_M)$ cannot break the weight symmetry at all.",
"To argue practical learning trajectory, we have to consider the initial value slightly off from that point.",
"How close the initial condition is to that point affects how long it takes to break the weight symmetry, that is, the plateau length.",
"This is why REF (b) with $N = 10^7$ exhibits plateau slightly longer than that of REF (a) with $N = 10^5$ ."
]
] | 2001.03371 |
[
[
"Correlance and Discordance: Computable Measures of Nonlocal Correlation"
],
[
"Abstract We present six new measures of nonlocal correlation for discrete multipartite quantum systems; correlance, statance, probablance, strong discordance, discordance, and diagonal discordance.",
"The correlance measures all nonlocal correlation (even bound entanglement), and is exactly computable for all pure and mixed states.",
"Statance and probablance are not yet computable, but motivate the strong discordance (for nonlocal correlation beyond that achievable by a strictly classical state), discordance (a measure of all nonlocal correlation in distinguishably quantum states), and diagonal discordance (for nonlocal correlation in diagonal states), all of which are exactly computable for all states.",
"We discuss types of correlation and notions of classicality, and compare correlance, strong discordance, and discordance to quantum discord.",
"We also define diagonal correlance to handle strictly classical probability distributions, providing a powerful tool with wide-ranging applications."
],
[
"Introduction",
"The strong nonlocal correlations achievable in quantum systems can cause novel physical effects that are impossible in classical systems, and have led to an intense worldwide search for applications of these properties, particularly in quantum computing [1], [2], quantum cryptography [3], [4], [5], and quantum communications [6], [7], [8], [9].",
"While entanglement has been the star of this show, other types of quantum nonlocal correlation exist that may also be useful.",
"Thus, there is a need to quantify such nonlocal correlation to asses which states are most useful for a given application.",
"One such measure is quantum discord, which we now briefly summarize.",
"In the case of bipartite mixed states $\\rho \\equiv \\rho ^{(1,2)}$ , where parenthetical superscripts denote labels of subsystems (modes) of a Hilbert space $\\mathcal {H}\\equiv \\mathcal {H}^{(1,2)}\\equiv \\mathcal {H}^{(1)}\\otimes \\mathcal {H}^{(2)}$ of dimension $n\\!\\equiv \\!\\dim (\\rho )\\!=\\!",
"n_{1}n_{2}$ , where $n_{m}$ is the dimension of mode $m$ , the quantum mutual information of $\\rho $ is $\\mathcal {I}(\\rho ) \\equiv S({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}) + S({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(2)}) - S(\\rho ),$ where ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}$ is the reduced state for mode $m$ (see [sec:App.A]App.sec:App.A), and $S(\\rho ) \\!\\equiv \\!",
"- \\text{tr}[\\rho \\log _{n} (\\rho )]$ is the von Neumann entropy [10].",
"It has been suggested that $\\mathcal {I}(\\rho )$ can be partitioned as $\\mathcal {I}(\\rho )=\\mathcal {C}(\\rho )+\\mathcal {Q}(\\rho ),$ where $\\mathcal {C}\\equiv \\mathcal {C}(\\rho )$ is a measure of distinctly classical nonlocal correlation, and $\\mathcal {Q}{\\hspace{-1.0pt}}\\equiv {\\hspace{-1.0pt}}\\mathcal {Q}(\\rho )$ is a measure of all quantum (nonclassical) correlations called quantum discord (see [sec:App.B]App.sec:App.B for a full definition of quantum discord) [11], [12], [13].",
"It has been shown that $\\mathcal {Q}$ is not necessarily only due to entanglement, nor is it only due to nonentanglement quantum correlations (so-called quantum nonlocality without entanglement) [14], [15], [16].",
"However, quantum discord is not the only way to quantify nonclassical correlations.",
"In this paper, we identify several mechanisms of nonlocal correlations (not necessarily all distinct), and propose six measures to quantify them.",
"The most general of these, correlance, is computable for all possible input states and provides a single measure that can detect all forms of nonlocal correlation, while discordance offers an alternative to quantum discord, yet is also computable for all states.",
"Here we define the condition of no nonlocal correlation, and three general mechanisms of nonlocal correlation.",
"[leftmargin=*,labelindent=4pt] 1.",
"Absence of Nonlocal Correlation: An $N$ -mode state $\\rho $ (mixed or pure) has no nonlocal correlation if and only if (iff) it can be decomposed in product form, $\\rho = \\mathop \\otimes \\limits _{m = 1}^N \\rho ^{(m)} = \\rho ^{(1)} \\otimes \\cdots \\otimes \\rho ^{(N)} ,$ where each $\\rho ^{(m)}$ is a possibly mixed state in mode $m$ .",
"See [sec:App.C]App.sec:App.C for details about purity.",
"2.",
"Entanglement Correlation: An $N$ -mode state $\\rho \\equiv \\rho ^{(1, \\ldots ,N)}$ is separable (fully $N$ -partite separable) iff it has a set of decomposition probabilities and pure decomposition states $\\lbrace p_{j},\\rho _{j}\\rbrace $ such that $\\rho = \\sum \\nolimits _j {p_j \\rho _j^{(1)} \\otimes \\cdots \\otimes \\rho _j^{(N)} },$ otherwise it is entangled (fully $N$ -partite entangled).",
"See [sec:App.D]App.sec:App.D for details and basic examples.",
"3.",
"Decomposition-State Correlation: A necessary condition for $\\rho $ to have product form is that it has a decomposition with pure decomposition states $\\lbrace \\rho _{j}\\rbrace $ such that, for each $j$ , the $\\rho _{j}$ are pure product states, $\\rho _{j}= \\mathop \\otimes \\limits _{m = 1}^N \\rho _j^{(m)} = \\rho _j^{(1)} \\otimes \\cdots \\otimes \\rho _j^{(N)} \\;\\;\\forall j,$ where each $\\rho _j^{(m)}$ also has mode independence, $\\rho _j^{(m)}= \\rho _{(j_1 , \\ldots ,j_N )}^{(m)} = \\rho _{j_m }^{(m)} \\;\\;\\forall m \\in 1, \\ldots ,N.$ If (eq.5–eq.6) are not satisfied, then $\\rho $ must have some nonlocal correlation, even if it is separable, and we say there is decomposition-state correlation.",
"See [sec:App.E]App.sec:App.E for an example illustrating mode independence.",
"4.",
"Probability Correlation: A necessary condition for $\\rho $ to have product form is that it has a decomposition with probabilities $\\lbrace p_{j}\\rbrace $ with $N$ factors, $\\begin{array}{*{20}l}{p_{j}} &\\!\\!",
"{=\\prod \\nolimits _{m = 1}^N {p_j^{(m)} } = p_j^{(1)} \\cdots p_j^{(N)} \\;\\;\\forall j,} \\\\\\end{array}$ and also with mode independence, $\\forall j$ , ${\\hspace{5.0pt}}\\begin{array}{*{20}l}{p_j^{(m)}{\\hspace{-2.0pt}}} &\\!\\!",
"{={\\hspace{-1.0pt}} p_{(j_1 , \\ldots ,j_N )}^{(m)} {\\hspace{-2.0pt}} ={\\hspace{-1.0pt}} p_{j_m }^{(m)};\\;\\;\\sum \\nolimits _{j_m}p_{j_m}^{(m)}{\\hspace{-2.0pt}}={\\hspace{-2.0pt}}1; \\;\\;\\forall m {\\hspace{-2.0pt}}\\in {\\hspace{-2.0pt}} 1 {\\scriptstyle ,\\ldots ,}{\\hspace{0.5pt}} N .}",
"\\\\\\end{array}{\\hspace{-5.0pt}}$ Any state with no decomposition-state-correlation-minimizing decomposition (see [sec:IV]Sec.sec:IV) able to fulfill (eq.7–eq.8) must have some nonlocal correlation (even if separable), a feature we call probability correlation.",
"Theorem1: Together, (eq.5–eq.8) form a set of necessary and sufficient conditions for $\\rho $ to have product form.",
"Proof: Convert between general $\\rho $ and (eq.3), as in [sec:App.F]App.sec:App.F.",
"The combinations of (eq.5–eq.8) yield the six families of nonlocal correlation in [tab:1]Table tab:1, visualized in [fig:1]Fig.fig:1.",
"These families are not mutually exclusive and represent only the most general mechanisms of nonlocal correlation; more specific mechanisms are merely subsets of these, such as entanglement which is a subset of Families 1 and 2.",
"Note that since $p_j {\\hspace{-1.0pt}} ={\\hspace{-1.0pt}} p_j ^{(1)} {\\hspace{-1.0pt}} \\cdots p_j ^{(N)} \\;\\forall j$ is always possible, that does not generate more families, as proved in [sec:App.G]App.sec:App.G.",
"Table: Six families of nonlocal correlation, meaning states that can be decomposed satisfying (eq.5–eq.8) in these various ways.",
"In all cases, j≡j 1 ,...,j N j\\equiv j_{1},\\ldots , j_{N}.",
"We abbreviate “family” as F, “product-form” as PF, and “mode-independent” as MI.",
"Figure: (color online) Depiction of the six families of nonlocal correlation from [tab:1]Table tab:1.",
"Not all set relations are written here; for example 5⊂15\\subset 1 and 6⊂46\\subset 4 are implied, etc.",
"Not shown are Quasi-Families 2|32|3, 2|52|5, and 4|54|5, which consist of states with different decompositions in different distinct families, but with no decomposition in their intersection; see [sec:App.G.3]App.sec:App.G.3 for details.",
"For multipartite generalizations, see [sec:VII]Sec.sec:VII."
],
[
"Correlance: A Measure of All Nonlocal Correlation",
"Given any $N$ -mode state $\\rho \\in \\mathcal {H}\\equiv \\mathcal {H}^{(1)}\\otimes \\cdots \\otimes \\mathcal {H}^{(N)}$ , a measure of all possible nonlocal $N$ -mode correlation (full $N$ -partite correlation) is the correlance, given by $\\mathcal {X}(\\rho ) \\equiv \\frac{{\\widetilde{\\mathcal {X}}(\\rho )}}{{\\mathcal {N}_{\\mathcal {X}} }},$ where the raw correlance (unnormalized correlance) is $\\widetilde{\\mathcal {X}}(\\rho ) \\equiv \\text{tr}[(\\rho - \\varsigma )^2 ],$ where we define the reduction product, $\\varsigma \\equiv \\varsigma (\\rho ) \\equiv \\mathop \\otimes \\limits _{m = 1}^N {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)} = {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)} \\otimes \\cdots \\otimes {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(N)},$ where ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}$ is the mode-$m$ reduction of $\\rho $ (see [sec:App.A]App.sec:App.A), and the normalization factor is $\\mathcal {N}_{\\mathcal {X}} \\equiv \\mathop {\\max }\\limits _{\\forall \\rho ^{\\prime } \\in \\mathcal {H}} [\\widetilde{\\mathcal {X}}(\\rho ^{\\prime })]= \\widetilde{\\mathcal {X}}(\\rho _{\\text{ME}} ) = \\widetilde{\\mathcal {X}}(\\rho _{\\text{ME}_{\\text{TGX}} } ),$ where $\\rho _{\\text{ME}}$ is any maximally entangled (ME) state in $\\mathcal {H}$ and $\\rho _{\\text{ME}_{\\text{TGX}}}$ is any maximally entangled true-generalized X (TGX) state where the TGX states, first defined in [17] are generalizations of the Bell states (see [sec:App.H]App.sec:App.H).",
"The 13-step algorithm of [18] makes ME TGX states methodically to compute $\\mathcal {N}_{\\mathcal {X}}$ , while [sec:II.B]Sec.sec:II.B gives exact forms.",
"Figure: (color online) Correlance of arbitrary two-qubit mixed states for each of the six families of nonlocal correlation from [tab:1]Table tab:1, 10 5 10^5 states each (colors not related to [fig:1]Fig.fig:1).",
"This gives a sense of the upper limits of each family, and shows that merely having mode independence in either probabilities or decomposition states alone is not enough to get independent modes; both must have mode independence to get product form and 𝒳=0\\mathcal {X}=0.",
"Families 1 and 2 are generally entangled, Families 3–5 are generally nonlocally correlated without entanglement, and Family 6 has no nonlocal correlation at all.",
"The correlance $\\mathcal {X}(\\rho )$ is 0 iff $\\rho $ is expressible in product form as $\\rho {\\hspace{-1.0pt}}={\\hspace{-1.0pt}}\\varsigma {\\hspace{-1.0pt}}={\\hspace{-1.0pt}}{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)} {\\hspace{-1.0pt}} \\otimes {\\hspace{-1.0pt}} \\cdots {\\hspace{-1.0pt}}\\otimes {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(N)}$ , meaning that all modes are uncorrelated.",
"Furthermore, $\\mathcal {X}(\\rho ){\\hspace{-2.0pt}}={\\hspace{-2.0pt}}1$ iff $\\rho $ is maximally entangled (meaning fully $N$ -partite entangled here; see [sec:VII]Sec.sec:VII for generalizations), meaning that the $N$ modes share the most nonlocal correlation possible.",
"[fig:2]Figure fig:2 explores the correlance for the families of [tab:1]Table tab:1.",
"Correlance $\\mathcal {X}$ measures all forms of nonlocal correlation; entanglement, decomposition-state correlation (includes entanglement), and probability correlation.",
"See [sec:II]Sec.sec:II for details, including proofs and numerical tests.",
"Here, we test the correlance to check its performance and give details on its normalization.",
"For proof that correlance is necessary and sufficient for measuring all nonlocal correlation, see [sec:App.I]App.sec:App.I."
],
[
"Normalization Tests of Correlance",
"[fig:3]Figure fig:3 shows the results of a necessary test of the normalization of the correlance $\\mathcal {X}$ , which is that if it is properly normalized, then we should not be able to find any states with a value of $\\mathcal {X}$ that exceeds 1.",
"The proof that $\\mathcal {X}$ is properly normalized is given in [sec:App.J]App.sec:App.J.",
"Figure: (color online) Normalization test of (eq.9), the correlance 𝒳\\mathcal {X} of arbitrary general quantum mixed states ρ\\rho for all discrete multipartite systems up to n=18n=18 levels, 30,00030,\\!000 states for each system.",
"This shows 540,000540,\\!000 consecutive examples that do not produce 𝒳\\mathcal {X} higher than 1, providing strong evidence that 𝒳\\mathcal {X} is not improperly normalized.",
"For proof that (eq.12) is the correct normalization factor, see [sec:App.J]App.sec:App.J."
],
[
"Exact Calculation of Correlance Normalization",
"The correlance normalization factor for all systems can be computed without using any ME states as $\\mathcal {N}_{\\mathcal {X}} = 1 - \\prod \\limits _{m = 1}^N {P_{\\text{MP}}^{(m)} (L_* )},$ as proved in [sec:App.K]App.sec:App.K, where $P_{\\text{MP}}^{(m)}(L_* )$ is the minimum physical reduction purity of mode $m$ given a pure maximally entangled parent state of $L_*$ levels of equal nonzero probability such that the combination of all $P_{\\text{MP}}^{(m)}(L_* )$ is minimized, the calculation of which is given in [sec:App.K]App.sec:App.K.",
"For systems where more than one mode has the largest mode size $n_{\\max }\\equiv \\max (\\mathbf {n})=\\max \\lbrace n_{1},\\ldots ,n_{N}\\rbrace $ , $\\mathcal {N}_{\\mathcal {X}} = 1 - \\frac{1}{n}; {\\hspace{10.0pt}}\\text{(multiple modes with $n_{\\max }$)},$ for systems such as $2\\times 3\\times 3$ or $N$ -qudit systems like $2\\times 2$ or $3\\times 3\\times 3$ .",
"However for systems with exactly one largest mode, like $2\\times 3$ or $2\\times 2\\times 3$ , we must use (eq.13) or (eq.12).",
"Table: Examples of normalization factors 𝒩 𝒳 \\mathcal {N}_{\\mathcal {X}} for the correlance of (eq.9) as computed by (eq.13), all of which were checked and found correct by using the 13-step algorithm 𝒜 13 \\mathcal {A}_{13} of to generate ME TGX states ρ ME TGX \\rho _{\\text{ME}_{\\text{TGX}}} for use in (eq.12)."
],
[
"Correlance Adapted for Diagonal-Only Input",
"If the input states $\\rho $ are definitely always diagonal, we can define the diagonal correlance as $\\mathcal {X}_D (\\rho ) \\equiv \\frac{{\\widetilde{\\mathcal {X}}(\\rho )}}{{\\mathcal {N}_{\\mathcal {X}_D } }},$ where $\\widetilde{\\mathcal {X}}(\\rho )$ is from (eq.10), but the normalization is $\\mathcal {N}_{\\mathcal {X}_D }\\equiv \\mathop {\\max }\\limits _{\\lbrace \\rho _D ^{\\prime }\\rbrace } [\\widetilde{\\mathcal {X}}(\\rho _D ^{\\prime })] = \\widetilde{\\mathcal {X}}(\\rho _{D_{\\text{max}}} ),$ where $\\lbrace \\rho _D\\rbrace $ is the set of all diagonal states, and $\\rho _{D_{\\text{max}}} $ is any diagonal state that maximizes $\\widetilde{\\mathcal {X}}(\\rho )$ .",
"Surprisingly, the states $\\rho _{D_{\\text{max}}} $ that maximize $\\mathcal {X}_D$ are not merely maximally dephased maximally entangled states in general.",
"Instead, one simple example that maximizes $\\mathcal {X}_D$ in all systems is $\\rho _{D_{\\max } } = \\frac{1}{2}(|1\\rangle \\langle 1| + |n\\rangle \\langle n|),$ yielding the normalization factor, valid for all systems, $\\mathcal {N}_{\\mathcal {X}_D }=\\widetilde{\\mathcal {X}}(\\rho _{D_{\\text{max}}} )=\\frac{1}{2}-\\frac{1}{2^{N}}.$ [fig:4]Figure fig:4 gives a necessary test showing strong evidence that (eq.18) is not wrong, while [sec:App.L]App.sec:App.L gives a sketched proof that (eq.17) is a valid $\\rho _{D_{\\max } }$ , and uses it to derive (eq.18).",
"$\\mathcal {X}_D$ can be used for strictly classical probability distributions, for quantum states with no coherence, or for diagonal states exhibiting combinations of both classical and quantum features (see [sec:III]Sec.sec:III).",
"$\\mathcal {X}_D$ is useful because in strictly classical situations, entanglement is not possible, so ME states would not be a reasonable standard to use for normalization.",
"Figure: (color online) Normalization test of (eq.15), the diagonal correlance 𝒳 D \\mathcal {X}_D of arbitrary general diagonal mixed states ρ D \\rho _D for all discrete multipartite systems up to n=18n=18 levels, 30,00030,\\!000 states for each system.",
"This shows 540,000540,\\!000 consecutive examples that do not produce 𝒳 D \\mathcal {X}_D higher than 1, providing strong evidence that 𝒳 D \\mathcal {X}_D is not improperly normalized, where 𝒩 𝒳 D \\mathcal {N}_{\\mathcal {X}_D} of (eq.18) was used.",
"For a sketched proof that (eq.17) is a valid ρ D max \\rho _{D_{\\text{max}}} , and a derivation of 𝒩 𝒳 D \\mathcal {N}_{\\mathcal {X}_D } in (eq.18), see [sec:App.L]App.sec:App.L."
],
[
"Diagonal Correlance as a Measure of General Nonlocal Correlation of Strictly Classical Data",
"For $N$ classical random variables (RVs), meaning $N$ -tuples of variables $\\mathbf {x}\\equiv (x^{(1)},\\ldots ,x^{(N)})$ , where the data consists of $n_{S}$ samples of $N$ -tuple data points $X\\equiv \\lbrace \\mathbf {x}_j \\rbrace \\equiv \\lbrace (x^{(1)}_{j} , \\ldots ,x^{(N)}_{j}) \\rbrace |_{j = 1}^{n_S }$ , we can model the variables as a discrete system of $N$ modes, where mode $m$ represents RV $x^{(m)}$ .",
"The size of mode $m$ is $n_m$ (the number of discrete values $x^{(m)}$ can have or its number of histogram bins when viewed as a quantized continuous variable), so the total $N$ -mode system has size $n=n_{1}\\cdots n_{N}$ .",
"For simplicity, we will refer to the algorithm for constructing a density matrix $\\rho $ from strictly classical data $X$ as $\\mathcal {A}_{\\rho }\\equiv \\mathcal {A}_{\\rho }(\\mathbf {n},X)$ , and its output is $\\rho =\\mathcal {A}_{\\rho }(\\mathbf {n},X);$ [sec:App.M]App.sec:App.M shows how to implement $\\mathcal {A}_{\\rho }$ .",
"[fig:5]Figure fig:5 shows four examples of bivariate data treated by this method, and compares the traditional Pearson correlation coefficient ${r_{\\text{P}}}$ (see [sec:App.N]App.sec:App.N) [19], [20], [21] to the diagonal correlance $\\mathcal {X}_D$ .",
"As seen in [fig:5]Fig.fig:5, [fig:5]Fig.fig:5a is where RVs $x$ and $y$ are completely independent, and both $|{r_{\\text{P}}}|\\approx 0.01$ and $\\mathcal {X}_{D}\\approx 0.01$ , which is appropriately near zero for a finite sample of uncorrelated variables.",
"In [fig:5]Fig.fig:5b, the data has some nonlinear correlation, and $|{r_{\\text{P}}}|\\approx 0.03$ , while $\\mathcal {X}_{D}\\approx 0.24$ , showing that $\\mathcal {X}_{D}$ correctly detects the presence of nonlocal correlation, while $|{r_{\\text{P}}}|$ has trouble detecting it because $|{r_{\\text{P}}}|$ is merely a measure of linear correlation.",
"[fig:5]Figure fig:5c shows the linear focus of $|{r_{\\text{P}}}|$ since the linear data causes $|{r_{\\text{P}}}|\\approx 0.99$ , close to its maximum value of 1, while $\\mathcal {X}_{D}\\approx 0.57$ , showing that $\\mathcal {X}_{D}$ acknowledges this correlation but does not consider it to be maximal.",
"Figure: (color online) The left column of plots show four bivariate dimensionless data sets of 200 points for random variables (RVs) xx and yy, each quantized to four outcomes (not shown) before further calculation.",
"Each RV has a uniformly distributed random deviation of 0.050.05 about ideal functions: (a) (x,y)=(rand,rand)(x,y)=(\\text{rand},\\text{rand}), (b) (x,y)=(t,1 2[1+cos(2πt)])(x,y)=(t,\\frac{1}{2}[1+\\cos (2\\pi t)]), (c) (x,y)=(t,t)(x,y)=(t,t), and (d) (x,y)=(round[t],round[t])(x,y)=(\\text{round}[t],\\text{round}[t]), where rand\\text{rand} is a random number on [0,1][0,1], and t∈[0,1]t\\in [0,1].",
"The center column of plots are the density matrices ρ\\rho for the quantized 2-mode data in single-index form, and the right column of plots shows the magnitude of the Pearson correlation coefficient |r P ||{r_{\\text{P}}}| of [sec:App.N]App.sec:App.N and the diagonal correlance 𝒳 D \\mathcal {X}_D from (eq.15).Finally, [fig:5]Fig.fig:5d shows data that is indeed more correlated than linear since both $x$ and $y$ tend to have matching values while also producing a $\\rho $ that does not factor into a product state, giving $|{r_{\\text{P}}}|\\approx 1.00$ and $\\mathcal {X}_{D}\\approx 1.00$ , showing that $|{r_{\\text{P}}}|$ cannot distinguish the data of [fig:5]Fig.fig:5d from that of [fig:5]Fig.fig:5c, while $\\mathcal {X}_{D}$ correctly detects the difference and values the data of [fig:5]Fig.fig:5d as being maximally correlated for a diagonal state.",
"Important: States such as $\\rho $ in [fig:5]Fig.fig:5d or (eq.17) in general only have maximal diagonal correlance for the set of all diagonal states, and are not generally maximally correlated in the context of all quantum states, as maximally entangled states are.",
"Thus, if the data comes from a system with novel quantum correlations, application of $\\mathcal {X}_{D}$ could yield values greater than unity, even though this is guaranteed not to happen for strictly classical data.",
"Thus, the diagonal correlance $\\mathcal {X}_{D}$ provides a more sensitive measure of correlations in data with strictly diagonal density matrices than the Pearson correlation coefficient, and is not limited to two RVs."
],
[
"CLASSICAL VS. QUANTUM",
"Before we can develop other measures capable of distinguishing the types of correlation from [sec:I.A]Sec.sec:I.A, we must first confront the issue of classicality that is raised by the diagonal correlance.",
"Our main questions here are What does it mean for a state to be diagonal in the context of classicality?",
"and Is there a precise and unambiguous definition for classicality?",
"These questions and their answers are extremely relevant in any discussion on quantum correlations that includes quantum discord, since its definition requires that we identify the part of a state that represents “distinctly classical nonlocal correlation.” Therefore here, we will make a series of observations, and from these deduce some concrete answers to these questions that will then allow us to construct the desired measures."
],
[
"Diagonal States Are Not Necessarily Classical",
"Quantum superposition is an inherently nonclassical, distinctly quantum phenomenon.",
"The idea that an object's state could simultaneously have multiple exclusive outcomes at once is nonsense in any classical model.",
"For pure quantum states, superposition appears in density matrices as nonzero off-diagonal elements.",
"For mixed quantum states, having nonzero off-diagonals is called coherence.",
"However, as shown in App.H of [22], we can also have superposition without coherence.",
"For example, given a general two-qubit pure parent state expanded as $|\\psi \\rangle \\equiv a_1 |1\\rangle \\!",
"+\\!",
"a_2 |2\\rangle \\!+\\!",
"a_3 |3\\rangle \\!",
"+\\!",
"a_4 |4\\rangle $ , where $|1\\rangle \\equiv |1,1\\rangle $ , $|2\\rangle \\equiv |1,2\\rangle $ , $|3\\rangle \\equiv |2,1\\rangle $ , $|4\\rangle \\equiv |2,2\\rangle $ , if that state is the Bell state $|\\psi \\rangle =\\frac{1}{\\sqrt{2}}(|1\\rangle +|4\\rangle )$ , where $a_1 =\\!a_4 \\!=\\frac{1}{\\sqrt{2}}$ and $a_2 \\!=\\!a_3 \\!=\\!0$ , then its density matrix is $\\begin{array}{*{20}l}\\rho &\\!\\!",
"{ =\\!|\\psi \\rangle \\langle \\psi |\\!= \\!\\!\\left( {\\begin{array}{*{20}c}{a_1 a_1 ^* } & \\cdot & \\cdot & {a_1 a_4 ^* } \\\\\\cdot & \\cdot & \\cdot & \\cdot \\\\\\cdot & \\cdot & \\cdot & \\cdot \\\\{a_4 a_1 ^* } & \\cdot & \\cdot & {a_4 a_4 ^* } \\\\\\end{array}} \\right)\\!\\!}",
"&\\!\\!",
"{ = \\!\\!\\left( {\\begin{array}{*{20}c}{\\frac{1}{2}} & \\cdot & \\cdot & {\\frac{1}{2}} \\\\\\cdot & \\cdot & \\cdot & \\cdot \\\\\\cdot & \\cdot & \\cdot & \\cdot \\\\{\\frac{1}{2}} & \\cdot & \\cdot & {\\frac{1}{2}} \\\\\\end{array}} \\right)\\!,} \\\\\\end{array}$ and therefore its mode-1 reduction is diagonal, ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)} \\!=\\!\\left(\\!\\!",
"{\\begin{array}{*{20}c}{\\rho _{1,1} \\!+\\!",
"\\rho _{2,2} } &\\, {\\rho _{1,3} \\!+\\!",
"\\rho _{2,4} } \\\\{\\rho _{3,1} \\!+\\!",
"\\rho _{4,2} } &\\, {\\rho _{3,3} \\!+\\!",
"\\rho _{4,4} } \\\\\\end{array}}\\!\\!",
"\\right)\\!{\\hspace{-2.0pt}}={\\hspace{-2.0pt}}\\!\\left(\\!\\!",
"{\\begin{array}{*{20}c}{a_1 a_1^* } &\\!\\!",
"0 \\\\0 &\\!\\!",
"{a_4 a_4^* } \\\\\\end{array}}\\!\\!",
"\\right)\\!",
"= \\!\\left(\\!",
"{\\begin{array}{*{20}c}{\\frac{1}{2}} & \\cdot \\\\\\cdot & {\\frac{1}{2}} \\\\\\end{array}}\\!",
"\\right)\\!,$ but its probabilities are directly inherited from the wave-function overlaps of the superposition amplitudes of its Bell parent state as $p_1^{(1)} \\equiv {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{1,1}=\\rho _{1,1}=a_1 a_1^*$ and $p_2^{(1)} \\equiv {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{2,2}=\\rho _{4,4}=a_4 a_4^*$ , where $a_k \\equiv \\langle k|\\psi \\rangle $ are wave-function overlaps of basis states $|k\\rangle $ with the pure parent state $|\\psi \\rangle $ , which has matrix elements $\\rho _{j,k}=a_j a_k^*$ .",
"Thus, (eq.21) is an example of superposition without coherence.",
"In contrast, the probabilities of classical discrete states are estimators of which outcome to expect from a step function of pure computational basis states on average.",
"Therefore, in principle, there is a sample-time window which could reveal the true step-function behavior of any truly classical mixed state.",
"For diagonal quantum states such as (eq.21), no such time window exists, because the probabilities inherit the instantaneous nature of superposition from the wave-function overlaps of the pure parent state, as in (eq.21).",
"A popular misconception is that Fock states [23] (photon number states $|n\\rangle $ , not to be confused with our generic basis states used in the rest of this paper) are nonclassical.",
"This comes from the preconception that coherent states [24], [25] are somehow classical simply because they fulfill some necessary conditions for what we think the word “classical” should mean, such as having all quantum coherence functions equal 1 which only measures similarity to coherent states [24], [25], [26], [27] (see [sec:App.O]App.sec:App.O).",
"The problem of misdiagnosing coherent states as classical and then mistakenly using them as standards of classicality is really a language problem, as we explain next."
],
[
"The Need for a New Term: ",
"The main reason for confusion about classicality is that “classical” is already a colloquial word in everyday speech.",
"Similarly, “work” has an everyday meaning that has nothing to do with physics, but it also has a well-defined physics-meaning (though it is aptly named).",
"In physics, all models predating quantum theory were unofficially lumped into a category colloquially referred to as “classical,” where its use is just an adjective describing those models as part of the pre-quantum mindset.",
"However, the colloquial use of the word “classical” is often incorrectly interpreted as a formal physics definition, and this has led to some ridiculous conclusions, such as considering coherent states to be classical even though they have quantum superposition in the Fock basis.",
"Therefore, since improper use of the word “classical” is already ubiquitous, we need a new term with a clear, formal definition that lists the rules that a physical model must obey in a hypothetical world that is truly without quantum-mechanical properties.",
"Thus, we propose the term strictly classical as being the simplest, most transparent label for this idea that also lets existing literature have the word “classical” as either a colloquial adjective or a colloquial term improperly used as a formal physics term.",
"[fig:6]Figure fig:6 summarizes these terms, while [sec:III.C]Sec.sec:III.C presents a formal definition of strict classicality.",
"Figure: (color online) Informal and formal ideas of classicality."
],
[
"Definition of Strictly Classical States",
"Almost all (if not all) of the strange behavior of quantum mechanics comes from its distinctive feature of quantum superposition, the primary idea that caused so much reluctance to accept quantum mechanics, because it is at such odds with classical physics.",
"Therefore we use the absence of superposition as the core criterion for what constitutes strict classicality; Strict Classicality: Let a strictly classical system be a hypothetical physical system for which a necessary condition is that quantum superposition plays no role, is impossible, and cannot be created.",
"A system that is strictly classical can be said to have strict classicality.",
"While there may be other conditions we need to identify for strict classicality, the above definition immediately leads to the following rules; [leftmargin=*,labelindent=4pt] 1.",
"Strictly classical states must have no coherence (off-diagonals of the density matrix must be zero, so the density matrix is diagonal).",
"2.",
"Strictly classical states must have no diagonal superposition in any complete orthonormal pure product-state basis (no cases of superposition without coherence as described in [sec:III.A]Sec.sec:III.A and seen in (eq.21)).",
"3.",
"The actual instantaneous state of any strictly classical system must be a pure computational basis state.",
"4.",
"Mixed strictly classical states (having purity less than 1) must be averages of step functions of computational basis states, and the lack of purity comes entirely from lack of human knowledge of the instantaneous state (which is always a pure computational basis state).",
"5.",
"(Corollary to Rule 3) The only transformations possible for a discrete strictly classical system are permutation unitaries; no transformation can convert a strictly classical state to a state with quantum superposition.",
"Thus, strictly classical states are strictly classical in all reference frames, and reference frames themselves are defined only in terms of strictly classical states.",
"Thus, by Rule 2 we must be careful when using the word “classical.” A list of probabilities is not enough information to determine whether the system is strictly classical or quantum; we must also be told or find more information about the origins of the physical system itself.",
"A strictly classical situation often arises in the macroscopic world, where we happen to know that outcomes are already in step functions of particular basis states, and we simply sample them over certain longer time windows and build estimators of the probabilities yielding a diagonal statistical mixture.",
"In such cases, we can use the diagonal correlance $\\mathcal {X}_D$ exclusively, but our physical interpretation of “nonlocal correlations” must be due to distinctly nonquantum features, meaning they have nothing to do with superposition and its related effects such as entanglement or relative quantum phase.",
"If both superposition and classical probability exist, such as a step function of quantum states sampled with a long time window giving a mixture containing both wave-function overlap probabilities and ignorance-based probabilities, our interpretation of $\\mathcal {X}$ or $\\mathcal {X}_D$ must acknowledge that both probability mechanisms are part of the model.",
"[fig:7]Figure fig:7 depicts some types of hybrid quantum/classical states.",
"Figure: (color online) Depiction of density matrices of some types of physical systems (omitting quantum phase for simplicity).",
"The first three columns of plots show the instantaneous state as a step function in time, and the fourth column shows the density matrix as it would be constructed from ideal tomography after using measurement time windows that were much longer than the duration of each step (to show the effect of human-induced mixing).",
"The state in (eq.21) is a special case of scenario (d) where the constant mixed quantum state is diagonal.",
"In contrast, a strictly classical state can never be instantaneously mixed, as seen in (a).",
"In these examples, the instantaneous states are ontic, while the time-averaged states are epistemic , , , as defined and discussed in [sec:App.O]App.sec:App.O.At this point, a keen reader might object to nonlocal correlations in strictly classical systems.",
"However, the concept of state, which belongs to both classical and quantum physics, is inherently global and therefore nonlocal, so nonlocality is an inherent part of classical physics.",
"For example, even in a purely classical world, if you sneeze here on Earth, the state of the entire universe changes instantaneously.",
"Similarly, solutions to the classical heat equation propagate with infinite velocity because the state is global and therefore changes everywhere at once.",
"Thus, nonlocal correlations are perfectly permissible in mixed strictly classical states, because nonlocality is built-in to the concept of state.",
"Next, we define a few more specific measures which are not practical to calculate, but help to justify several fully computable measures similar to quantum discord."
],
[
"STATANCE AND PROBABLANCE",
"Here we propose two measures of nonlocal correlation, statance and probablance, that are unlikely to be practical to calculate in general, but which nevertheless give us sufficient insight into the mechanisms of nonlocal correlation to define three more measures, strong discordance, discordance, and diagaonal discordance, that are exactly computable for all mixed and pure states and may be preferable to quantum discord."
],
[
"Statance",
"The statance $\\hat{\\mathcal {S}}(\\rho )$ for any $N$ -mode quantum state $\\rho $ is $\\hat{\\mathcal {S}}(\\rho ) \\equiv \\frac{1}{{\\mathcal {N}_{\\mathcal {S}} }}\\mathop {\\min }\\limits _{\\lbrace U\\rbrace } [\\mathcal {S}(\\rho )],$ over all unitary matrices $U$ of $D\\in r,\\ldots , r^2$ levels where $r\\equiv \\text{rank}(\\rho )$ , $\\mathcal {N}_{\\mathcal {S}}$ normalizes $\\min _{\\lbrace U\\rbrace }[\\mathcal {S}(\\rho )]$ over all $\\rho $ , and $\\mathcal {S}$ is the unoptimized statance, $\\mathcal {S}(\\rho ) \\equiv {\\hspace{-26.0pt}} \\sum \\limits _{{\\hspace{26.0pt}}j_1 , \\ldots ,j_N =1,\\ldots ,1}^{D_{1},\\ldots ,D_{N}} {{\\hspace{-26.0pt}}{\\hspace{-2.0pt}}\\text{tr}[(\\rho _{(j_1 , \\ldots ,j_N )} - \\mu _{(j_1 , \\ldots ,j_N )} )^2 ]}{\\hspace{0.5pt}} ,$ where $D_{m}{\\hspace{-0.5pt}}\\in {\\hspace{-0.5pt}} 1,\\ldots ,n_{m}^2\\text{~s.t.~}{\\hspace{-1.5pt}}\\prod \\nolimits _{m=1}^{N}D_{m}{\\hspace{-0.5pt}}={\\hspace{-0.5pt}}D$ (see [sec:App.P]App.sec:App.P), and $\\mu _j \\equiv \\mu _{(j_1 , \\ldots ,j_N )}$ are $\\mu _j \\equiv {\\hspace{-1.0pt}} \\mathop \\otimes \\limits _{m = 1}^N {\\hspace{-1.0pt}}\\text{tr}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} {\\hspace{-2.0pt}}\\left({\\hspace{-1.5pt}} {\\frac{1}{{D_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} }}{\\hspace{-1.0pt}}{\\hspace{-26.0pt}}\\sum \\limits _{{\\hspace{26.0pt}}j_1^{\\lbrace m\\rbrace } , \\ldots ,j_N^{\\lbrace m\\rbrace } =1,\\ldots ,1}^{D_{1},\\ldots ,D_{N}} {{\\hspace{-26.0pt}}{\\hspace{-3.0pt}}\\delta _{j_m^{\\lbrace m\\rbrace } ,j_m } \\rho _{(j_1^{\\lbrace m\\rbrace } , \\ldots ,j_N^{\\lbrace m\\rbrace } )} } }{\\hspace{-1.5pt}} \\right){\\hspace{-2.0pt}} ,$ where $D_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}\\equiv \\frac{D}{D_{m}}$ , $\\text{tr}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}(A)$ is the partial trace of $A$ over all modes that are not mode $m$ (see [sec:App.A]App.sec:App.A), superscripts $\\lbrace m\\rbrace $ in $j^{\\lbrace m\\rbrace }\\equiv (j_1^{\\lbrace m\\rbrace } , \\ldots ,j_N^{\\lbrace m\\rbrace } )$ distinguish indices inside each factor of the tensor product, and the pure decomposition states $\\rho _j \\equiv \\rho _{(j_1 , \\ldots ,j_N )} $ are $\\rho _j \\equiv \\frac{1}{{p_j }}\\sum \\limits _{k,l = 1,1}^{r,r} {\\!",
"U_{j,k} U_{j,l}^* \\sqrt{\\lambda _k \\lambda _l } |e_k \\rangle \\langle e_l |},$ where $\\lbrace \\lambda _{k},|e_k \\rangle \\rbrace $ are the eigenvalues and eigenstates of $\\rho $ $\\text{s.t.~}\\lambda _{1}\\geqslant \\cdots \\geqslant \\lambda _{r}>0$ , and $ \\sum \\nolimits _{k = 1}^{r } { |U_{j,k} |^2 }\\ne 0\\;\\forall j$ as explained in [sec:App.P]App.sec:App.P, and $p_j \\equiv \\sum \\nolimits _{k = 1}^{r } {\\lambda _{k} |U_{j,k} |^2 }$ .",
"See [31] for a useful parameterization of unitary matrices.",
"Basically, $\\mu _j$ uses the $\\rho _j$ to form states [inside the partial trace in (eq.24)] constructed to have mode-independent (MI) probabilities (due here to the fact that they are all equal) so that the only way the tensor product of the mode-$m$ reductions of those states could also be MI would be if all the $\\rho _j$ were MI (see [sec:App.Q]App.sec:App.Q).",
"The statance $\\hat{\\mathcal {S}}(\\rho )=0$ iff $\\rho $ has a decomposition where all of the decomposition states $\\rho _j$ have MI product form as $\\rho _j = \\otimes _{m = 1}^N \\rho _{j_m }^{(m)} = $ $\\rho _{j_1 }^{(1)} \\otimes \\cdots \\otimes \\rho _{j_N }^{(N)}\\;\\;\\forall j$ , so $\\hat{\\mathcal {S}}(\\rho )=0$ is guaranteed in Families 5 and 6 of [tab:1]Table tab:1.",
"If $\\hat{\\mathcal {S}}(\\rho ){\\hspace{-1.0pt}}>{\\hspace{-1.0pt}}0$ , then $\\rho $ has decomposition-state correlation, meaning that it has no decomposition for which all $\\rho _j$ have both product form and mode independence, so its optimal decomposition has at least one $\\rho _j$ that violates either (eq.5) and (eq.6) or violates just (eq.6).",
"Thus, statance $\\hat{\\mathcal {S}}(\\rho )$ is a necessary and sufficient measure of decomposition-state correlation (see [sec:App.Q]App.sec:App.Q for proof, and [sec:App.R]App.sec:App.R for a special example)."
],
[
"Probablance",
"Our goal here is to define a measure of probability correlation, but there are several fine points to consider.",
"[leftmargin=*,labelindent=4pt] 1.",
"Recalling that Family 5 of [tab:1]Table tab:1, with defining form $\\rho =\\sum \\nolimits _j p_{j}\\rho _{j_{1}}^{(1)}\\otimes \\cdots \\otimes \\rho _{j_{N}}^{(N)}$ , has the ability to yield $\\mathcal {X}(\\rho )>0$ in [fig:2]Fig.fig:2, this shows that there are indeed some states whose only type of correlation is probability correlation (since those states were constructed with MI decomposition states so they have no decomposition-state correlation).",
"Moreover, that means that for Family 5 only, $\\mathcal {X}(\\rho )>0$ exclusively indicates probability correlation.",
"2.",
"Probability correlation cannot be judged by itself over all decompositions, because it is always possible to make a decomposition with equal probabilities which are therefore MI and have no probability correlation, as we will show in the next few paragraphs.",
"3.",
"The fact that Family 5 has MI decomposition states and yet can have $\\mathcal {X}(\\rho )>0$ means that to get the defining optimal form of that family with respect to probability correlation, we must first find decompositions that minimize the decomposition-state correlation, and then minimize over the probability sets of those decompositions to judge probability correlation.",
"Thus, for Family 5, since this restriction causes us to only consider decompositions with MI decomposition states when calculating the probabilities, then by Fine Point 1, if $\\mathcal {X}(\\rho )>0$ , we are guaranteed to be unable to find MI probabilities from that set of decompositions.",
"Keeping the above in mind, we define the probablance for any $N$ -mode quantum state $\\rho $ as $\\hat{\\mathcal {P}}(\\rho )\\equiv \\frac{1}{{\\mathcal {N}_\\mathcal {P}}}\\mathop {\\min }\\limits _{\\lbrace U^{\\prime }\\rbrace } [\\mathcal {P}(\\rho )];{\\hspace{6.0pt}}\\lbrace U^{\\prime }\\rbrace \\equiv \\text{arg}\\left({\\mathop {\\min }\\limits _{\\lbrace U\\rbrace } [\\mathcal {S}(\\rho )]}\\right),$ where $\\lbrace U^{\\prime }\\rbrace $ is the set of all unitaries that minimize $\\mathcal {S}(\\rho )$ from (eq.23), $\\mathcal {N}_\\mathcal {P}$ normalizes over all $\\rho $ , and $\\mathcal {P}(\\rho )$ is the unoptimized probablance, $\\mathcal {P}(\\rho ) \\equiv \\sum \\limits _{j=1}^{D} {\\left| {p_{j}^{\\prime } - q_{j}^{\\prime } } \\right|^2 } ,$ where $D$ and $j$ are as in [sec:IV.A]Sec.sec:IV.A, and $q_{j}^{\\prime }\\equiv q_{(j_1 , \\ldots ,j_N )}^{\\prime }$ is $\\begin{array}{*{20}l}{q_{(j_1 , \\ldots ,j_N )}^{\\prime }} &\\!\\!",
"{\\equiv \\prod \\limits _{m = 1}^N {\\varrho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{{\\prime }{\\hspace{1.0pt}}(m)}_{j_{m},j_{m}};{\\hspace{10.0pt}}\\varrho ^{\\prime }\\equiv \\sum \\limits _{l=1}^{D}p_{l}^{\\prime }|l\\rangle \\langle l|} \\\\{} &\\!\\!",
"{=\\prod \\limits _{m = 1}^N {{\\hspace{-3.0pt}}\\left( {\\rule {0pt}{20pt}{\\hspace{-1.0pt}}} \\right.",
"{\\hspace{-29.0pt}}\\sum \\limits _{{\\hspace{29.0pt}}\\rule {0pt}{8pt}j_1^{\\lbrace m\\rbrace } , \\ldots ,\\,j_N^{\\lbrace m\\rbrace } =1,\\ldots ,1}^{D_{1},\\ldots ,D_{N}} {{\\hspace{-29.0pt}}\\delta _{j_m^{\\lbrace m\\rbrace }{\\hspace{-1.0pt}} ,{\\hspace{1.0pt}}j_m } p_{(j_1^{\\lbrace m\\rbrace } , \\ldots ,j_N^{\\lbrace m\\rbrace } )}^{\\prime } } \\left.",
"{\\rule {0pt}{20pt}{\\hspace{-3.0pt}}} \\right)},} \\\\\\end{array}$ where $|l\\rangle \\equiv |l_{1}\\rangle \\otimes \\cdots \\otimes |l_{N}\\rangle $ are computational basis states in a Hilbert space of mode sizes $\\mathbf {n}^{\\prime }\\equiv (D_{1},\\ldots ,D_{N})$ , and $p_j^{\\prime }\\equiv p_{(j_1 , \\ldots ,j_N )}^{\\prime } \\equiv \\sum \\limits _{k= 1}^{r } {\\lambda _{k} |U^{\\prime }_{(j_1 , \\ldots ,j_N ),k} |^2 }{\\hspace{-1.0pt}} ,$ where $U^{\\prime }$ is defined in (eq.26), based on $U$ from [sec:IV.A]Sec.sec:IV.A.",
"To see how $\\hat{\\mathcal {P}}(\\rho )$ works, $\\varrho ^{\\prime }$ in (eq.28) creates a $D$ -level diagonal state that automatically has $D$ MI decomposition states so that the only way $\\varrho ^{\\prime }$ can have product form is if its diagonal elements are MI, and those are the probabilities of a given decomposition specified by $U^{\\prime }$ .",
"The quantity $q_j^{\\prime }$ then exploits this fact by making a product that equals $p_j^{\\prime }$ iff the set of $p_j^{\\prime }$ is MI (see [sec:App.S]App.sec:App.S for proof).",
"The reason we cannot simply ignore statance in this definition is that if we did ignore it, we could just choose $U$ to be an $r$ -dimensional Fourier matrix, and then (eq.29) would yield $p_j^{\\prime } =\\frac{1}{r}$ for $j\\in 1,\\ldots ,r$ (so $D=r$ ), and since any set of equal probabilities is automatically MI, that would yield $\\mathcal {P}(\\rho )=0$ for all $\\rho $ , which cannot be true for any valid measure of probability correlation, because the nonzero correlance of some states in Family 5 demonstrates that probability correlation exists.",
"Therefore, we must first find the particular $\\lbrace U^{\\prime }\\rbrace $ that minimize $\\mathcal {S}(\\rho )$ from (eq.23) and then find a particular $U^{\\prime }$ that minimizes $\\mathcal {P}(\\rho )$ to define the $p_j^{\\prime }$ for $\\hat{\\mathcal {P}}(\\rho )$ .",
"Probablance $\\hat{\\mathcal {P}}(\\rho )$ measures how far the decomposition probabilities are from having mode-independent factorizability for a decomposition that minimizes the decomposition-state correlation.",
"For a given $\\rho $ with decompositions $\\rho = \\sum \\nolimits _j {p_j } \\rho _j $ , we get $\\hat{\\mathcal {P}}(\\rho )=0$ iff there exists a decomposition-state-correlation-minimizing decomposition for which $p_j = \\prod \\nolimits _{m = 1}^N {p_{j_{m}}^{(m)} = p_{j_{1}}^{(1)} \\cdots p_{j_{N}}^{(N)} }\\;\\forall j$ , so $\\hat{\\mathcal {P}}(\\rho )=0$ is guaranteed for Families 2, 4, and 6 of [tab:1]Table tab:1.",
"If $\\hat{\\mathcal {P}}(\\rho )>0$ , then there is no decomposition-state-correlation-minimizing decomposition of $\\rho $ for which all $p_j$ have mode-independent product form.",
"Thus, $\\hat{\\mathcal {P}}(\\rho )$ is a valid measure of probability correlation as defined in (eq.7–eq.8) (see [sec:App.S]App.sec:App.S for proof, and [sec:App.T]App.sec:App.T for an example).",
"Caution: While it is tempting to call $\\hat{\\mathcal {P}}(\\rho )$ the “classical correlation,” that would be inaccurate because although strictly classical states must have no superposition and are therefore diagonal in the computational basis, there are quantum states of the same form with inherently different meaning, as explained in [sec:III]Sec.sec:III.",
"Therefore, the correlation measured by $\\hat{\\mathcal {P}}(\\rho )$ can either be classical, quantum, or a composite of both depending on the physical system described by $\\rho $ , so we just call this correlation the “probability correlation,” since that is the most transparent term, and makes no extra assumptions.",
"For strictly classical states, since they are step functions of computational basis states according to Rule 3 in [sec:III.C]Sec.sec:III.C, only decompositions for which all decomposition states have no superposition in the computational basis are allowed, so the only permissible decompositions have $D=r$ , $U$ is an $r$ -level permutation unitary, and $\\rho $ 's eigenstates are required to be the computational basis sates.",
"The reason $D=r$ is because $U$ is limited to a permutation unitary (because superposition is not allowed), which would lead to some $p_j =0$ if $D>r$ , and that would cause our measures to include $\\rho _j$ that do not contribute to $\\rho $ .",
"Thus, for strictly classical states, statance $\\hat{\\mathcal {S}}(\\rho )$ only considers the finite set of $r!$ permutation unitaries of dimension $r$ as $\\lbrace U\\rbrace $ and probablance $\\hat{\\mathcal {P}}(\\rho )$ only considers the $\\lbrace U^{\\prime }\\rbrace $ from that set that minimize statance."
],
[
"Observations about Statance and Probablance",
"Here, we use statance and probablance to make simple observations and do some special-case examples to lay the groundwork for a new measure that is comparable to quantum discord, but with several advantages, thereby showing statance and probablance to be worthwhile measures despite the difficulties in their calculation.",
"Theorem2: Entanglement is merely sufficient for causing $\\hat{\\mathcal {S}}(\\rho )>0$ , but is not necessary for that [i.e., states with $\\hat{\\mathcal {S}}(\\rho )>0$ do not all have entanglement, but all states with entanglement have $\\hat{\\mathcal {S}}(\\rho )>0$ ].",
"Proof: By definition, decomposition states of an entangled state cannot all achieve product form, so they also cannot achieve mode independence as in (eq.6), and therefore entangled states can never have $\\hat{\\mathcal {S}}(\\rho )=0$ , which proves that entanglement is sufficient to cause $\\hat{\\mathcal {S}}(\\rho )>0$ .",
"The fact that there exist separable states $\\rho = \\sum \\nolimits _j {(p_{j_{1}}^{(1)} \\cdots p_{j_{N}}^{(N)}) \\rho _j^{(1)} \\otimes \\cdots \\otimes \\rho _j^{(N)} } $ which are not entangled by definition, but lack any decomposition states with mode independence (so they have $\\hat{\\mathcal {S}}(\\rho )>0$ as in [fig:2]Fig.fig:2 for Family 4 [for which $\\mathcal {X}(\\rho )>0$ implies $\\hat{\\mathcal {S}}(\\rho )>0$ ]), proves that entanglement is not necessary to achieve $\\hat{\\mathcal {S}}(\\rho )>0$ .",
"(Thus, entanglement is merely a special type of decomposition-state correlation.)",
"Theorem3: Entanglement is the only kind of nonlocal correlation that pure states can have.",
"Proof: If $\\rho $ is pure, then $\\hat{\\mathcal {S}}(\\rho )$ would only measure entanglement because all pure states have no probability correlation (since its only decomposition is it itself with probability 1), and any decomposition-state correlation (due to mode-independence violation) in a pure state is the same as violation of product form since its optimal decomposition is itself up to global phase, and violation of product form is the definition of entanglement in a pure state.",
"All of this prompts the question: Can a strictly classical state violate mode independence with its decomposition states alone?",
"In other words, are there (diagonal) strictly classical states with $\\hat{\\mathcal {P}}= 0$ but $\\hat{\\mathcal {S}}>0$ ?",
"The answer is yes, but it depends on which basis states correspond to nonzero eigenvalues, and is limited by $r$ .",
"For a strictly classical two-qubit state $\\rho $ , the case of $r=1$ always involves a single pure computational basis state, which always has product form, so $\\hat{\\mathcal {S}}=0$ and $\\hat{\\mathcal {P}}= 0$ .",
"For $r=4$ , the complete set of computational basis states will always form a mode-independent (MI) set for some of the permutation unitaries $U$ , so $\\hat{\\mathcal {S}}=0$ while $\\hat{\\mathcal {P}}\\geqslant 0$ .",
"For $r=2$ and $r=3$ , the strictly-classical limitation of $D=r$ means that the probabilities will always appear MI, since for example a general set of $\\lbrace p_1, p_2 ,p_3\\rbrace $ can be interpreted as $\\lbrace p_1^{(1)}p_1^{(2)} ,p_1^{(1)}p_2^{(2)} , p_1^{(1)}p_3^{(2)}\\rbrace $ , where $p_1^{(1)}=1$ and $\\sum _{j_2 =1}^{3}p_{j_2}^{(2)}=1$ where $\\mathbf {D}=(1,3)$ since $D_1 D_2 =D$ .",
"Therefore for $r=2$ and $r=3$ we get $\\hat{\\mathcal {P}}= 0$ always.",
"But for statance, in $r=3$ no group of three computational basis states can form an MI set and all 3-level permutation unitaries yield the same result.",
"For example, suppose the decomposition states involved are $\\begin{array}{*{20}l}{\\rho _1 } &\\!\\!",
"{ \\equiv \\rho _{(1,1)} } &\\!\\!",
"{ = |1\\rangle \\langle 1| \\otimes |1\\rangle \\langle 1|} \\\\{\\rho _2 } &\\!\\!",
"{ \\equiv \\rho _{(1,2)} } &\\!\\!",
"{ = |1\\rangle \\langle 1| \\otimes |2\\rangle \\langle 2|} \\\\{\\rho _3 } &\\!\\!",
"{ \\equiv \\rho _{(1,3)} } &\\!\\!",
"{ = |2\\rangle \\langle 2| \\otimes |1\\rangle \\langle 1|.}",
"\\\\\\end{array}$ Then, putting (eq.30) into (eq.24) gives (for the 3-level $U=I$ ), $\\begin{array}{*{20}l}{\\mu _1 } &\\!\\!",
"{ = \\frac{2}{3}|1\\rangle \\langle 1| \\otimes |1\\rangle \\langle 1| +\\!",
"\\frac{1}{3}|2\\rangle \\langle 2| \\otimes |1\\rangle \\langle 1|} \\\\{\\mu _2 } &\\!\\!",
"{ = \\frac{2}{3}|1\\rangle \\langle 1| \\otimes |2\\rangle \\langle 2| +\\!",
"\\frac{1}{3}|2\\rangle \\langle 2| \\otimes |2\\rangle \\langle 2|} \\\\{\\mu _3 } &\\!\\!",
"{ = \\frac{2}{3}|1\\rangle \\langle 1| \\otimes |1\\rangle \\langle 1| +\\!",
"\\frac{1}{3}|2\\rangle \\langle 2| \\otimes |1\\rangle \\langle 1|,} \\\\\\end{array}$ which, when put into (eq.23) and (eq.22) yields $\\hat{\\mathcal {S}}(\\rho ) \\equiv \\frac{1}{{\\mathcal {N}_{\\mathcal {S}} }}\\mathop {\\min }\\limits _{\\lbrace U\\rbrace } [\\mathcal {S}(\\rho )] = \\frac{4}{3{\\mathcal {N}_{\\mathcal {S}} }} > 0,$ since $\\mathcal {S}(\\rho )=\\frac{4}{3}$ here for all 3-level permutation unitaries, and thus we get another powerful result: Theorem4: Strictly classical states can have nonzero statance even though all of the decomposition states are computational basis states.",
"For $r\\!=\\!2$ , the factorizability of the set of basis states belonging to nonzero eigenvalues determines the statance.",
"For example if $\\rho \\!=\\!p_{1}|1\\rangle \\langle 1| \\otimes |1\\rangle \\langle 1| \\!+\\!p_{2}|1\\rangle \\langle 1| \\otimes |2\\rangle \\langle 2|$ , then $\\hat{\\mathcal {S}}\\!=\\!0$ since the mode-1 state is a common factor, but if $\\rho \\!=\\!p_{1}|1\\rangle \\langle 1| \\otimes |1\\rangle \\langle 1| \\!+\\!p_{2}|2\\rangle \\langle 2| \\otimes |2\\rangle \\langle 2|$ , then $\\hat{\\mathcal {S}}\\!>\\!0$ .",
"However, it is important to acknowledge that mixed strictly classical states are by necessity epistemic, so that the probabilities strictly represent observer ignorance of the actual ontic state.",
"Since the ontic state of a strictly classical system is always pure, it can have no probability correlation, and since it is also a computational basis state, then it has no entanglement, and by Theorem3 it also has no decomposition-state correlation.",
"Therefore we can summarize all of this as: Theorem5: All ontic strictly classical states have no nonlocal correlation of any kind.",
"(Where again, ontic means the actual state, which is instantaneously pure as defined in [sec:III.C]Sec.sec:III.C.)",
"Proof: See the above text.",
"Theorem6: Mixed strictly classical states (which are epistemic by necessity) can have nonlocal correlation as any combination of decomposition-state correlation and probability correlation, but can never have entanglement correlation, and all of this nonlocal correlation is strictly observer-induced.",
"(Again, epistemic means that this is a state of observer knowledge, so any mixture of a strictly classical state is induced by the observer's ignorance of the actual state.)",
"Proof: See the text before Theorem5.",
"Note that since strict classicality forbids superposition, reduction cannot induce mixture from pure strictly classical states; it can only remix the pre-existing observer-induced mixture of a mixed strictly classical parent state.",
"Theorem5 is exactly what we should expect of the ontic state of a strictly classical system; nonlocality is impossible.",
"Theorem6 is significant because it shows that nonlocality can arise in mixed strictly classical states.",
"However, while these theorems are true for strictly classical states (which are all diagonal) they do not generally also apply to diagonal quantum states (for which general decomposition unitaries are allowed in statance and probablance calculations, not merely permutation unitaries).",
"So far, we have developed one general measure of nonlocal correlation (the correlance $\\mathcal {X}$ ), and two measures of specific kinds of nonlocal correlation (statance $\\hat{\\mathcal {S}}$ for decomposition-state correlation, and probablance $\\hat{\\mathcal {P}}$ for probability correlation).",
"For entanglement correlation, we can use any valid entanglement measure, such as the ent $\\Upsilon $ [18] or its generalization for distinguishing distinctly different types of multipartite entanglement as the ent-concurrence $\\Upsilon _C$ [22].",
"Thus, we have measures for each of the main mechanisms of nonlocal correlation from [sec:I.A]Sec.sec:I.A, although among these measures, only the correlance $\\mathcal {X}$ is computable for mixed states at this time.",
"Now we return to the essential question that prompted the idea of quantum discord $\\mathcal {Q}$ ; is it possible to construct a measure of nonlocal correlation that can distinguish between quantum and classical nonlocal correlation?"
],
[
"STRONG DISCORDANCE, DISCORDANCE, AND DIAGONAL DISCORDANCE",
"Before we can answer the question at the end of [sec:IV.C]Sec.sec:IV.C, we need to define what we mean by classical correlations.",
"Generalizing from Werner [32], any fully $N$ -partite separable state in the form of (eq.4) is “classically correlated,” however we cannot accept this as accurate terminology here because separable quantum states in general fail the requirements for being strictly classical states in [sec:III.C]Sec.sec:III.C, even when such states are diagonal, as explained in [sec:III]Sec.sec:III.",
"Instead, we must recognize that strictly classical states are statistical mixtures built up from estimators of probabilities generated from using measurement time windows that are too long to show the true nature of the classical system as a step function of pure computational basis states, and contain no quantum superposition in any reference frame.",
"In other words, the most general truly classical state is represented by a strictly classical epistemic mixed diagonal density matrix.",
"However, since ontic quantum states can have the same form as epistemic strictly classical mixed states (with the key difference that no choice of measurement time window could reveal a quantum state to be a step function of pure computational basis states unless that were the reality of the system) then we get another theorem: Theorem7: Decomposition-state correlation and probability correlation are not exclusively strictly-classical phenomena.",
"Proof: There exist ontic quantum states of the same form as epistemic strictly classical mixed states but with inherently different meaning that nevertheless have nonentanglement-decomposition-state correlation and/or probability correlation as their only source of nonlocal correlation, such as the diagonal two-qubit reductions of some fully $N$ -partite entangled $N$ -qubit state [similar to (eq.21) but where the reduction is a two-qubit system].",
"In other words; reductions of ontic quantum states can lead to ontic quantum mixed states of the same form as epistemic strictly classical mixed states."
],
[
"Strong Discordance: A Measure of Nonlocal Correlation Beyond that Achievable by a Strictly Classical State",
"Despite the fact that Theorem4, Theorem6, and Theorem7 seem to indicate that quantum states cannot be so easily distinguished from strictly classical states, we can still use these theorems as motivation to create a measure that reveals whether a quantum state has correlations beyond those of a strictly classical strength, which we will refer to as distinctly non-strictly-classical nonlocal correlation, or more compactly, distinctly nonclassical nonlocal correlation, or just distinctly quantum nonlocal correlation.",
"(Essentially, this means that we acknowledge Theorem7 but will still consider any nonlocal correlation in quantum states that is not greater than the correlation achievable by a strictly classical state to have classical strength regardless of the inherently quantum physical origins that cause it.)",
"Therefore we define the strong discordance of $\\rho $ as a measure of nonlocal correlation beyond that achievable by a strictly classical state, given by $\\mathcal {D}_{S}(\\rho ) \\equiv \\frac{1}{{\\mathcal {N}_{\\mathcal {D}_{S}} }}\\max \\lbrace 0,\\mathcal {X}(\\rho ) - \\mathcal {X}(\\rho _{D_{\\max } } )\\rbrace ,$ where $\\mathcal {X}(\\rho )$ is the correlance from (eq.9), with normalization $\\mathcal {N}_{\\mathcal {D}_{S}} \\equiv \\mathop {\\max }\\limits _{\\lbrace \\rho ^{\\prime }\\rbrace } \\lbrace \\mathcal {X}(\\rho ^{\\prime }) - \\mathcal {X}(\\rho _{D_{\\max } } )\\rbrace = 1 - \\frac{2^{N-1}-1}{2^{N}\\mathcal {N}_{\\mathcal {X}}},$ where $\\rho _{D_{\\max } }$ are any diagonal states that maximize $\\mathcal {X}_{D}(\\rho )$ such as (eq.17), and $\\mathcal {X}(\\rho _{D_{\\max } } ) = \\widetilde{\\mathcal {X}}(\\rho _{D_{\\max } } )/\\mathcal {N}_{\\mathcal {X}}$ , where $\\widetilde{\\mathcal {X}}(\\rho _{D_{\\max } } )=\\mathcal {N}_{\\mathcal {X}_D } = \\frac{1}{2} - \\frac{1}{{2^N }}$ from (eq.18), and $\\mathcal {N}_{\\mathcal {X}}$ is from (eq.12), (eq.13), or (eq.K.57), so then (eq.33) becomes $\\mathcal {D}_{S}(\\rho ) = \\frac{2^{N}\\mathcal {N}_{\\mathcal {X}}}{{2^{N}\\mathcal {N}_{\\mathcal {X}}-2^{N - 1} + 1}}\\max \\!",
"\\left\\lbrace {0,\\mathcal {X}(\\rho ) - \\frac{2^{N-1}-1}{2^{N}\\mathcal {N}_{\\mathcal {X}}}} \\right\\rbrace \\!.$ Thus, (eq.35) gives strong discordance in exactly computable form for all pure and mixed $N$ -partite states.",
"The definition of $\\mathcal {D}_{S}(\\rho )$ is such that the only states that can attain $\\mathcal {D}_{S}(\\rho )>0$ are those with correlance $\\mathcal {X}(\\rho )$ that exceeds that of the correlance-maximizing diagonal states $\\rho _{D_{\\max } }$ (since those have the form of the most correlated strictly classical states).",
"Thus, strong discordance $\\mathcal {D}_{S}(\\rho )$ is a measure of how much nonlocal correlation exists in $\\rho $ beyond that achievable by a mixed strictly classical state.",
"$\\mathcal {D}_{S}(\\rho )=0$ iff there exists a mixed strictly classical state that can achieve the same amount of nonlocal correlation as $\\rho $ .",
"Similarly, if $\\mathcal {D}_{S}(\\rho )>0$ , then there is definitely some nonlocal correlation that is distinctly quantum and not achievable by a strictly classical state, and $\\mathcal {D}_{S}(\\rho )=1$ iff $\\rho $ is maximally entangled.",
"However, having $\\mathcal {D}_{S}(\\rho )=0$ does not guarantee that there is no nonlocal correlation, and it does not necessarily mean that the nonlocal correlation has nonquantum origins; as stated earlier, states with $\\mathcal {D}_{S}(\\rho )=0$ can still have nonlocal correlation arising from distinctly quantum mechanisms such as entanglement, but the key point is that for these states, the correlation is weak enough that the same amount could be generated by a mixed strictly classical state, and that is why we consider it to be “not distinctly quantum.” Also as mentioned earlier, states with $\\mathcal {D}_{S}(\\rho )>0$ do not necessarily have exclusively quantum origins of nonlocal correlation; there may be multiple mechanisms, some of which can be nonquantum in nature (such as probability correlation arising from an epistemic mixture of nondiagonal quantum states).",
"[fig:8]Figure fig:8 tests strong discordance $\\mathcal {D}_{S}(\\rho )$ for the same families as [fig:2]Fig.fig:2.",
"Figure: (color online) Strong discordance of arbitrary two-qubit mixed states for each of the six families of nonlocal correlation of [tab:1]Table tab:1, 10 5 10^5 states each (colors not related to [fig:1]Fig.fig:1).",
"This suggests that Families 3–6 (all separable) have 𝒟 S =0\\mathcal {D}_{S}=0, and only if we allow entanglement, as in Families 1 and 2, can states achieve the full range of strong discordance up to 𝒟 S =1\\mathcal {D}_{S}=1.",
"Thus, Werner's observation is valid, but should be restated as “nonlocal correlation in all separable states is not stronger than that of strictly classical mixed states.”In summary, a few caveats for strong discordance are: [leftmargin=*,labelindent=4pt] 1.",
"The definition of $\\mathcal {D}_{S}(\\rho )$ in (eq.33–eq.35) implies that $\\mathcal {D}_{S}(\\rho )>0$ guarantees some distinctly nonclassical nonlocal correlation, but it does not necessarily mean there are no sources of strictly classical nonlocal correlation contributing to the total correlation.",
"2.",
"A state for which $\\mathcal {D}_{S}(\\rho )=0$ is not necessarily a strictly classical state or even a diagonal state, but rather it is a state with no nonlocal correlations stronger than those of a strictly classical state, meaning that it may have any combination of probability correlation and decomposition-state correlation (including entanglement), but that there exists a strictly classical state that could achieve the same amount of nonlocal correlation using a combination of only probability correlation and nonentanglement-decomposition-state correlation.",
"3.",
"A value of $\\mathcal {D}_{S}(\\rho )>0$ guarantees some nondiagonality of $\\rho $ , but some nondiagonal states can have $\\mathcal {D}_{S}(\\rho )=0$ (so $\\mathcal {D}_{S}(\\rho )=0$ does not imply diagonality).",
"Proof: The maximum correlance of all diagonal states is the threshold for $\\mathcal {D}_{S}(\\rho )=0$ , so any states with $\\mathcal {D}_{S}(\\rho )>0$ must have more correlance than the most correlated diagonal states, and therefore must not be diagonal.",
"For the second claim, a proof by example is that $\\mathcal {D}_{S}(\\rho )=0$ for a product state of nondiagonal mode states such as $\\rho =\\frac{1}{2}({\\hspace{-2.0pt}}~_{1}^{1}{\\hspace{1.0pt}}~_{1}^{1})\\otimes \\frac{1}{2}({\\hspace{-2.0pt}}~_{-1}^{\\phantom{-}1}{\\hspace{1.0pt}}~_{\\phantom{-}1}^{-1})$ .",
"4.",
"The evidence that $\\mathcal {D}_{S}(\\rho )=0$ for all separable states in [fig:8]Fig.fig:8 strongly supports the requirement that strictly classical states must be diagonal, because their value of $\\mathcal {X}(\\rho _{D_{\\max }})$ defines the threshold of distinctly quantum correlation strength in (eq.33).",
"Therefore, strong discordance is a more selective measure than quantum discord, but because it rejects some weakly entangled states, it is too strong to be a workable alternative to quantum discord.",
"Therefore, next we develop a more inclusive measure that is closer in spirit to quantum discord, but more conceptually consistent."
],
[
"Discordance: A Measure of Nonlocal Correlation in a Distinguishably Quantum State",
"Here, we define a more inclusive measure of nonlocal correlation that can never report a zero for distinctly quantum states such as entangled states, even if strictly classical states exist that have the same correlance.",
"To achieve this, we need a way to distinguish strictly classical states from nonlocally correlated quantum states with novel quantum features.",
"A practical feature for this purpose is coherence (here, coherence means that $\\rho $ has some nonzero off-diagonal elements in the computational basis).",
"Even though superposition and coherence do not imply nonlocal correlation, they do imply the presence of novel quantum effects, and are necessary for some nonlocal correlations such as entanglement.",
"One caveat here is that, as we showed in (eq.21), diagonal quantum states can have “superposition without coherence” (such as reductions of maximally entangled states), so if we use coherence as a criterion for being a distinguishably quantum state, we would still be rejecting diagonal quantum states with nonzero correlance.",
"However, as discussed in [sec:III.C]Sec.sec:III.C, it is not generally possible to determine whether a diagonal quantum state is quantum or strictly classical without doing a potentially impossible set of experiments or having more information beyond the state, such as its parent state before a reduction.",
"Therefore, for the purpose of measuring nonlocal correlation in a state that is definitely quantum using the state alone, the best we can do is use coherence as a criterion to reject strictly classical states at the expense of also rejecting diagonal quantum states that may have some nonlocal correlation.",
"As a preliminary step, we define raw nondiagonality as $\\widetilde{\\nu }(\\rho ) \\equiv \\text{tr}[(\\rho - \\Delta )^2 ],$ where $\\Delta $ is the maximally dephased input state, $\\begin{array}{*{20}l}{\\Delta } &\\!\\!",
"{\\equiv \\Delta (\\rho ) \\equiv \\sum \\limits _{a = 1}^n {\\rho _{a,a} |a\\rangle \\langle a|} ,} \\\\\\end{array}$ so that $\\widetilde{\\nu }(\\rho )=0$ iff $\\rho $ is diagonal, and $\\widetilde{\\nu }(\\rho )>0$ iff $\\rho $ is not diagonal.",
"We could normalize $\\widetilde{\\nu }(\\rho )$ either over all states [by dividing $\\widetilde{\\nu }(\\rho )$ by $1-\\frac{1}{n}$ ] or to only the subspace for which the diagonal elements are nonzero, but for our purposes, we are only interested in the sign of $\\widetilde{\\nu }(\\rho )$ , so we do not need to normalize it.",
"Then, we can define a measure of nonlocal correlation in a distinguishably quantum state as the discordance, $\\mathcal {D}(\\rho ) \\equiv \\text{sgn}[\\widetilde{\\nu }(\\rho )]\\mathcal {X}(\\rho ),$ where $\\widetilde{\\nu }(\\rho )$ is from (eq.36) and $\\mathcal {X}(\\rho )$ is the correlance from (eq.9).",
"Thus, only states with some nondiagonality can have nonzero discordance, which ensures that all entangled states are recognized has having discordance, and only diagonal quantum states with nonzero correlance have their nonlocal correlation unrecognized (with the justifiability for that being that such states have density matrices with identical form to a strictly classical state, meaning that the state alone is not sufficient to distinguish them from strictly classical systems).",
"$\\mathcal {D}(\\rho )=0$ if $\\rho $ has no nonlocal correlation or if the state is diagonal (whether it has nonlocal correlation or not), and $\\mathcal {D}(\\rho )=1$ iff $\\rho $ is maximally entangled.",
"[fig:9]Figure fig:9 plots the discordance for the families of [tab:1]Table tab:1.",
"Figure: (color online) Discordance 𝒟\\mathcal {D} of arbitrary two-qubit mixed states for each of the six families of nonlocal correlation of [tab:1]Table tab:1, 10 5 10^5 states each (colors not related to [fig:1]Fig.fig:1).",
"This shows that only Family 6 (product states, pure and mixed) has 𝒟=0\\mathcal {D}=0 always, while Families 3–5 (all separable) can have 𝒟>0\\mathcal {D}>0 (indicating some instances of quantum nonlocality without entanglement , , ).",
"Essentially, discordance is correlance recognized for nondiagonal states only.",
"Again, only if entanglement is allowed, as in Families 1 and 2, can states achieve the full range of discordance up to 𝒟=1\\mathcal {D}=1.Thus, we now have three main computable measures on all mixed and pure states to measure different degrees of general nonlocal correlation; [leftmargin=*,labelindent=4pt] 1.",
"Strong discordance $\\mathcal {D}_{S}(\\rho )$ (nonzero for states with more nonlocal correlation [as measured by correlance] than what is achievable by any strictly classical state).",
"2.",
"Discordance $\\mathcal {D}(\\rho )$ (nonzero for quantum states with nonlocal correlation [as measured by correlance] that have the distinguishably quantum feature of coherence and are thus nondiagonal).",
"3.",
"Correlance $\\mathcal {X}(\\rho )$ (nonzero for any nonlocally correlated state, whether quantum or strictly classical).",
"Ironically, as far as we know, all evidence suggests that there is really no such thing as a strictly classical state in the real world, and that all states are truly quantum.",
"Therefore, correlance is really all we ever need to measure the presence of any nonlocal correlation in a state.",
"However, for the purpose of distinguishing quantum from strictly classical scenarios, the above measures give us powerful tools with different degrees of specificity.",
"In fact, we will show that they obey a similar relationship to (eq.2), but first we will do some simple examples to compare these measures to quantum discord."
],
[
"Comparisons of Discordance Measures with Quantum Discord",
"Here, we briefly compare correlance, discordance, and strong discordance with quantum discord, and also concurrence for reference.",
"Keep in mind that none of these new measures is meant to calculate quantum discord, but rather they are intended as different measures of the same intended qualitative features, with various restrictions that might be useful in particular applications.",
"First, we consider two-qubit Werner states [32], $\\rho = a|\\Psi ^ - \\rangle \\langle \\Psi ^ - | + \\frac{{1 - a}}{4}I,$ where $a \\in [0,1]$ , $|\\Psi ^ - \\rangle \\equiv \\frac{1}{{\\sqrt{2} }}(|1,2\\rangle - |2,1\\rangle )$ is the (maximally entangled) singlet state (where the generic basis for each qubit is $\\lbrace |1\\rangle ,|2\\rangle \\rbrace $ ), and $\\frac{1}{4}I$ is the maximally mixed state.",
"Using the quantum discord from [13], [fig:10]Fig.fig:10 compares the various measures, showing that discordance and correlance behave similarly to quantum discord.",
"Figure: (color online) Comparison of strong discordance 𝒟 S \\mathcal {D}_{S}, discordance 𝒟\\mathcal {D}, and correlance 𝒳\\mathcal {X} with quantum discord 𝒬\\mathcal {Q}, and concurrence CC shown for reference for the Werner states of (eq.39).",
"Note that 𝒟 S \\mathcal {D}_{S} is the most exclusive, rejecting entangled states weak enough that some strictly classical states have the same correlance (𝒟 S =0\\mathcal {D}_{S}=0 when C>0C>0, proving Caveat 2 by example), while 𝒟\\mathcal {D} and 𝒳\\mathcal {X} both monotonically grow similarly to 𝒬\\mathcal {Q} (their exact values relative to 𝒬\\mathcal {Q} do not matter since they are not meant to calculate it; only their similar nonzero behavior is relevant).",
"Here, 𝒟=𝒳\\mathcal {D}=\\mathcal {X} since the state is nondiagonal except at a=0a=0 where both are 0 anyway.Next, consider the mixture (from [13]), $\\rho = \\frac{1}{3}[(1 - a)|1,1\\rangle \\langle 1,1| + 2|\\Psi ^ + \\rangle \\langle \\Psi ^ + | + a|2,2\\rangle \\langle 2,2|],$ where $a \\in [0,1]$ and $|\\Psi ^ + \\rangle \\equiv \\frac{1}{{\\sqrt{2} }}(|1,2\\rangle + |2,1\\rangle )$ .",
"Using its $\\mathcal {Q}$ from [13], [fig:11]Fig.fig:11 compares it to our various measures, again showing that $\\mathcal {D}$ and $\\mathcal {X}$ have similar behavior to $\\mathcal {Q}$ .",
"Figure: (color online) Comparison of strong discordance 𝒟 S \\mathcal {D}_{S}, discordance 𝒟\\mathcal {D}, and correlance 𝒳\\mathcal {X} with quantum discord 𝒬\\mathcal {Q}, and concurrence CC (for reference) for the states of (eq.40).",
"Notice that 𝒟\\mathcal {D} and 𝒳\\mathcal {X} both monotonically grow symmetrically about a=0.5a=0.5 similarly to 𝒬\\mathcal {Q} (again, their exact values relative to 𝒬\\mathcal {Q} are not important; only their similar nonzero behavior matters).",
"Here, 𝒟=𝒳\\mathcal {D}=\\mathcal {X} since the state is always nondiagonal.Thus, these (limited) tests in [fig:10]Fig.fig:10 and [fig:11]Fig.fig:11 agree with our earlier reasoning that for the purpose of finding a substitute for quantum discord $\\mathcal {Q}$ , strong discordance $\\mathcal {D}_{S}$ is too exclusive while correlance $\\mathcal {X}$ is too inclusive (since it recognizes nonlocal correlation in diagonal states, which is not encountered in these examples).",
"However, discordance $\\mathcal {D}$ seems to be the most inclusive it is possible to be without having more information beyond the state itself, making it the most appropriate substitute for quantum discord.",
"In fact, all of these measures are related in an elegant way that formally parallels the theory of quantum discord, as we will show next."
],
[
"Analogous Quantities to Quantum Mutual Information, Classical Correlation, and Quantum Discord",
"Since discordance $\\mathcal {D}$ only recognizes the nonlocal correlation of nondiagonal states, here we define a measure for only diagonal states as the diagonal discordance, $\\mathcal {D}_{D}(\\rho ) \\equiv (1 - \\text{sgn}[\\widetilde{\\nu }(\\rho )])\\mathcal {X}(\\rho ),$ which is just the correlance $\\mathcal {X}(\\rho )$ of (eq.9) with a sifting factor to map all nondiagonal states to 0, where $\\widetilde{\\nu }(\\rho )$ is the raw nondiagonality from (eq.36).",
"[We could define (eq.41) in terms of $\\mathcal {X}_{D}$ from (eq.15), but as we show next, it is more useful to leave it in this “unnormalized” form.",
"Also, $\\mathcal {X}_{D}$ does not have the sifting factor $(1 - \\text{sgn}[\\widetilde{\\nu }(\\rho )])$ because $\\mathcal {X}_{D}$ is a stand-alone measure on diagonal states.",
"The sifting factors in $\\mathcal {D}_{D}$ and $\\mathcal {D}$ are justified because they are meant to be used together, as shown next.]",
"Thus, diagonal discordance $\\mathcal {D}_{D}$ measures the nonlocal correlation in diagonal states, be they strictly classical or diagonal quantum, and is loosely the analog of “classical correlation” $\\mathcal {C}$ from the quantum discord theory.",
"Interestingly, comparing (eq.38) and (eq.41), we see that $\\mathcal {X}(\\rho ) = \\mathcal {D}_{D}(\\rho ) + \\mathcal {D}(\\rho ),$ in formal analogy to (eq.2) which was $\\mathcal {I}(\\rho )=\\mathcal {C}(\\rho )+\\mathcal {Q}(\\rho )$ where $\\mathcal {I}$ is the quantum mutual information, $\\mathcal {C}$ is the classical correlation, and $\\mathcal {Q}$ is the quantum discord.",
"However, the similarity of (eq.42) to (eq.2) is a deceptive parallel for several reasons; [leftmargin=*,labelindent=4pt] 1.",
"Classical correlation $\\mathcal {C}$ inherently contains a conceptual flaw: the rank-1 projection operators (of the von Neumann measurements involved in its definition) are generally allowed to have superposition, which sabotages the goal of measuring distinctly classical correlation by allowing it to depend on novel quantum properties.",
"In contrast, diagonal discordance $\\mathcal {D}_{D}$ only registers as nonzero for diagonal states, which have no coherence, making them indistinguishable from strictly classical states, so $\\mathcal {D}_{D}$ is a much more appropriate measure of classical correlation (with the caveat that diagonal quantum states can have “superposition without coherence” as discussed in [sec:III]Sec.sec:III).",
"2.",
"Since quantum discord $\\mathcal {Q}$ is defined as $\\mathcal {Q}\\equiv \\mathcal {I}-\\mathcal {C}$ , it inherits the conceptual flaw of $\\mathcal {C}$ from Reason 1.",
"In contrast, $\\mathcal {D}$ of (eq.38) is designed to be the most general and inclusive measure of nonlocal correlation in distinguishably quantum states through its rejection of diagonal states, since, barring further information beyond the state, nondiagonality is the defining feature separating quantum from classical.",
"Again, we acknowledge the existence of diagonal quantum states, but they are indistinguishable from strictly classical states unless further information is known.",
"3.",
"Classical correlation $\\mathcal {C}$ and quantum discord $\\mathcal {Q}$ are not mutually exclusive since they can be simultaneously nonzero.",
"While this is not necessarily a problem since one can imagine simultaneous influences of classical and quantum mechanisms of correlation, the conceptual flaw mentioned in Reason 1 raises the question of whether there could be some overlap of what these two measures detect.",
"Meanwhile, $\\mathcal {D}_{D}$ and $\\mathcal {D}$ are mutually exclusive, since they depend on whether or not the state has coherence (with the value also depending on $\\mathcal {X}$ ).",
"However, as mentioned earlier, since all states really are fundamentally quantum, it makes more sense to divide based on the novel quantum feature of coherence (nondiagonality) than it does to define classicality based on measurements allowing projectors with superposition as in the definition of $\\mathcal {C}$ and thus $\\mathcal {Q}$ .",
"4.",
"Quantum mutual information $\\mathcal {I}$ is only defined for bipartite systems, whereas correlance $\\mathcal {X}$ can handle $N$ -partite systems (and is extended to further multipartite generalizations in [sec:VII]Sec.sec:VII).",
"Therefore, while (eq.42) does not exactly parallel the quantum-discord relation of (eq.2), it does have a solid grounding in the well-defined notion of nonlocal correlation in terms of actual quantum states and their distance from achieving mode-independence, which is the basis of correlance $\\mathcal {X}$ .",
"It makes sense to use diagonality as the criterion for classicality since that is the form of strictly classical states of [sec:III]Sec.sec:III which are carefully defined to avoid superposition both internally and through transformation.",
"Furthermore, since we already showed in Theorem4, Theorem6, and Theorem7 that statance $\\hat{\\mathcal {S}}$ and probablance $\\hat{\\mathcal {P}}$ can arise in both diagonal and nondiagonal states, it makes sense that the general measure of correlance $\\mathcal {X}$ is used to calculate the actual value of correlation for both diagonal states in $\\mathcal {D}_{D}$ and nondiagonal states in $\\mathcal {D}$ , since $\\mathcal {X}$ measures all forms of nonlocal correlation whether or not the state is quantum or classical.",
"Thus, our definitions divide correlations in a much more conceptually consistent way than quantum discord simply by focusing on the state itself and carefully acknowledging the distinctly quantum feature of coherence."
],
[
"Hidden Variables Are Not Another Kind of Nonlocal Correlation",
"Throughout this paper, we use the term nonlocal correlation to mean a state's inability to be factored into a mode-independent product form as defined in [sec:I.A]Sec.sec:I.A.",
"However, there is a more subtle way to achieve nonlocal correlation that we must consider.",
"It is often said that “correlation does not imply causation” (meaning that correlation does not imply variable-dependence), but that does not mean that causation cannot give rise to correlation (meaning that variable-dependence can lead to correlation, just not always).",
"This is essentially the idea of hidden-variable (HV) theories [33], [34], [35], [30], where all states of quantum theory can be recast for each observable as diagonal states with variable-dependent probabilities that yield the same mean values of that observable as quantum theory, implying all possible correlations, including those perceived as entanglement.",
"Furthermore, HV models produce deterministic theories, in which the future is exactly predictable if one knows the initial conditions of some set of “hidden variables,” whose trajectories, when sampled, yield the variable-dependent probabilities.",
"Hidden-variable theories are generally regarded with skepticism for many reasons [36], [37], [38], and a full treatment of this subject is beyond the scope of this paper.",
"However, since correlance measures nonlocal correlation as a state's distance from mode-independent product form, then the variable-dependence of an implied quantum state $\\rho $ constructed from a complete HV theory only affects nonlocal correlation insofar as it affects the state's closeness to mode-independent product form.",
"In other words, HV theories do not constitute a new mechanism of nonlocal correlation, so all measures in this paper also apply to states of HV theories."
],
[
"Multipartite Generalizations Beyond $N$ -Mode Correlations",
"So far, correlance, diagonal correlance, statance, probablance, strong discordance, discordance, and diagonal discordance only measure nonlocal correlations over all $N$ modes of an $N$ -mode coincidence system (see [18], App.A).",
"For instance, the correlance $\\mathcal {X}(\\rho )$ of (eq.9) measures how far $\\rho $ is from being a product state of $N$ modes as its distance from its own reduction product (since only product states are their own reduction products).",
"But in general, nonlocal correlation can arise in multipartite systems in many different ways, and they are not all equivalent.",
"For example, from [22], a 4-qubit GHZ state [39], [40], [41] and a Bell-product state can have the same full $N$ -partite entanglement by some necessary and sufficient measure of full $N$ -partite entanglement, but the separability of the two Bell states in the Bell-product state shows that it has a distinctly different kind of multipartite entanglement than the GHZ state for which no partitions of the $N$ modes are separable.",
"Furthermore, we may only be interested in determining a correlation property for a subset of the total modes, or we may want to view groups of modes as if they were single modes and determine that property between those groups, or between several groups and a single mode.",
"In [22], a preliminary study was done on how best to adapt the ent from an $N$ -mode entanglement measure to a more general measure called the ent-concurrence to measure distinctly multipartite entanglement.",
"The ent-concurrence (equal to the concurrence [42], [43] for the case of two qubits) was shown to have the ability to distinguish between distinctly different types of multipartite entanglement for states that may have the same amount of full $N$ -partite entanglement via the ent, such as a 4-qubit GHZ state and a Bell-product state.",
"This section explains how to generalize the correlance to measure the more general distinctly multipartite features described above, based on the formalism of [22].",
"Generalizations of diagonal correlance, statance, probablance, strong discordance, discordance, and diagonal discordance follow that of correlance by direct analogy, and are all computable on all states, except for generalizations of statance and probablance, for which computable expressions are not yet known."
],
[
"Multicorrelance",
"The multicorrelance for any pure or mixed $\\rho $ is $\\mathcal {M}_\\mathcal {X}(\\rho ) \\equiv \\frac{1}{\\mathcal {N}_{\\mathcal {M}_\\mathcal {X}} }\\sum \\limits _{k = 2}^N {\\mathcal {M}_{\\mathcal {X}} ^{(\\mathbf {N}^{(\\mathbf {k})} )} (\\rho )} ,$ where $\\mathcal {N}_{\\mathcal {M}_\\mathcal {X}}$ is a normalization factor from maximizing the unnormalized $\\mathcal {M}_\\mathcal {X}(\\rho )$ over all possible input states, and we define the $N$ -mode $k$ -partitional multicorrelance, $\\mathcal {M}_{\\mathcal {X}} ^{(\\mathbf {N}^{(\\mathbf {k})} )} (\\rho ) \\equiv \\frac{1}{{\\mathcal {N}_{\\mathcal {M}_{\\mathcal {X}_k} } }}\\sum \\nolimits _{h = 1}^{\\lbrace _k^N \\rbrace } {\\sqrt{{\\mathcal {X}} ^{(\\mathbf {N}_h^{(\\mathbf {k})} )} (\\rho )} } ,$ where $\\mathcal {N}_{\\mathcal {M}_{\\mathcal {X}_k} }$ is a normalization factor again over all possible input states, $\\lbrace _k^N \\rbrace \\equiv \\frac{1}{{k!",
"}}\\sum \\nolimits _{j = 0}^k {( - 1)^{k - j} (_j^k )j^N } $ are Stirling numbers of the second kind where $(_j^k ) \\equiv \\frac{{k!}}{{j!",
"(k - j)!",
"}}$ , and $\\lbrace \\mathcal {X}^{(\\mathbf {N}_h^{(\\mathbf {k})} )} (\\rho )\\rbrace $ is the set of all $N$ -mode $k$ -partitional correlances (defined below), where $h$ is an index arbitrarily assigning a label to a given unique partition of modes.",
"To define the $N$ -mode $k$ -partitional correlances, first we need a more explicit definition of correlance as $\\mathcal {X}(\\rho ) \\equiv \\mathcal {X}(\\rho ,\\mathbf {n}) \\equiv \\frac{1}{{\\mathcal {N}_{\\mathcal {X}} (\\mathbf {n})}}\\text{tr}\\left( {[\\rho - \\varsigma (\\rho ,\\mathbf {n})]^2 } \\right),$ where $\\mathcal {N}_{\\mathcal {X}} (\\mathbf {n}) \\equiv \\max _{\\forall \\rho ^{\\prime } \\in \\mathcal {H}} \\lbrace \\text{tr}([\\rho ^{\\prime } - \\varsigma (\\rho ^{\\prime },\\mathbf {n})]^2 )\\rbrace $ , where the maximizing states $\\rho ^{\\prime }_{\\max } \\in \\lbrace \\lbrace \\rho _{\\text{ME}} \\rbrace ,\\lbrace \\rho _{\\text{ME}_{\\text{TGX}} } \\rbrace \\rbrace $ are maximally full-$N$ -partite-entangled states of an $N$ -mode $n$ -level system where $\\mathbf {n} \\equiv (n_1 , \\ldots ,n_N )$ , mode $m$ has $n_m$ levels, $n \\equiv n_1 \\cdots n_N $ , and the reduction product is $\\varsigma (\\rho ,\\mathbf {n}) \\equiv \\mathop \\otimes \\limits _{m = 1}^N {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)} .$ Then, recalling that ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}$ is the $n_m$ -level single-mode reduction of $\\rho $ for mode $m$ , and that ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})}$ is the multimode reduction where $\\mathbf {m} \\equiv (m_1 , \\ldots ,m_S )$ for $S\\in 1,\\ldots ,N$ , we see that the indices of argument $\\mathbf {n}$ are the mode labels $\\mathbf {m}$ (with $S=N$ ) that label the reductions in $\\varsigma (\\rho ,\\mathbf {n})$ , so that $\\mathbf {n}$ implicitly governs the tensor product in (eq.46).",
"Given the above, let the partitional correlance be $\\begin{array}{*{20}l}{\\mathcal {X}^{(\\mathbf {m}^{(\\mathbf {T})} )} (\\rho )} &\\!\\!",
"{\\hspace{-1.0pt}} { \\equiv \\mathcal {X}({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})} ,\\mathbf {n}^{(\\mathbf {m}^{(\\mathbf {T})} )} )} \\\\{} &\\!\\!",
"{\\hspace{-1.0pt}} { \\equiv \\frac{1}{{\\mathcal {N}_{\\mathcal {X}} (\\mathbf {n}^{(\\mathbf {m}^{(\\mathbf {T})} )} )}}\\text{tr}{\\hspace{-1.5pt}}\\left( {[{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})}{\\hspace{-1.7pt}} -{\\hspace{-1.0pt}} \\varsigma ({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})} ,\\mathbf {n}^{(\\mathbf {m}^{(\\mathbf {T})} )} )]^2 } \\right){\\hspace{-2.0pt}},} \\\\\\end{array}$ which lets us specify as input any reduction to mode group $\\mathbf {m}$ of the input state with any repartitioning of its mode structure into new mode groups as $\\mathbf {m}^{(\\mathbf {T})} {\\hspace{-1.2pt}}\\equiv {\\hspace{-1.2pt}} (\\mathbf {m}^{(1)} | \\ldots |\\mathbf {m}^{(T)} )$ , with levels $\\mathbf {n}^{(\\mathbf {m}^{(\\mathbf {T})} )} {\\hspace{-1.2pt}}\\equiv $ $ (n_{\\mathbf {m}^{(1)} } , \\ldots ,n_{\\mathbf {m}^{(T)} } )$ where $T {\\hspace{-1.2pt}}\\in {\\hspace{-1.2pt}} 1, \\ldots ,S$ , and $\\mathcal {N}_{\\mathcal {X}} (\\mathbf {n}^{(\\mathbf {m}^{(\\mathbf {T})} )} ) $ $\\equiv {\\hspace{-1.2pt}} \\max _{\\forall \\rho ^{(\\mathbf {m})\\prime } \\in \\mathcal {H}^{(\\mathbf {m})} } \\lbrace \\text{tr}([\\rho ^{(\\mathbf {m})\\prime } {\\hspace{-1.5pt}}-{\\hspace{-1.5pt}} \\varsigma (\\rho ^{(\\mathbf {m})\\prime } ,\\mathbf {n}^{(\\mathbf {m}^{(\\mathbf {T})} )} )]^2 )\\rbrace $ where the maximizing states $\\rho _{\\max }^{(\\mathbf {m})\\prime } \\in \\lbrace \\lbrace \\rho _{\\text{ME}}^{(\\mathbf {m})} \\rbrace ,\\lbrace \\rho _{\\text{ME}_{\\text{TGX}} }^{(\\mathbf {m})} \\rbrace \\rbrace $ are maximally $T$ -partite entangled states of a $T$ -mode $n_{\\mathbf {m}^{(\\mathbf {T})} } \\equiv n_{\\mathbf {m}^{(1)} } \\cdots n_{\\mathbf {m}^{(T)} } $ -level system of mode-structure $\\mathbf {n}^{(\\mathbf {m}^{(\\mathbf {T})} )} \\equiv (n_{\\mathbf {m}^{(1)} } , \\ldots ,n_{\\mathbf {m}^{(T)} } )$, where the input states can be specified in terms of the general reduction structure and underlying original modes as ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})}$ where $\\mathbf {m} \\equiv (m_1 , \\ldots ,m_S )$ since it is true by definition that $n_{\\mathbf {m}}=n^{(\\mathbf {m}^{(\\mathbf {T})} )}$ where $\\mathbf {n}^{(\\mathbf {m})} \\equiv (n_{m_1 } , \\ldots ,n_{m_S } )$ and $n_{\\mathbf {m}} \\equiv n_{m_1 } \\cdots n_{m_S } $ .",
"See ([22], App.C) for more details about the notation.",
"We do not use the reduction symbol over density matrices in the normalization because this maximization is over all states in the Hilbert space $\\mathcal {H}^{(\\mathbf {m})}$ of the reduced system, not merely reductions from the parent state.",
"The reduction product in (eq.47) is then $\\varsigma ({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})} ,\\mathbf {n}^{(\\mathbf {m}^{(\\mathbf {T})} )} ) \\equiv \\mathop \\otimes \\limits _{q = 1}^T {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m}^{(q)})} ,$ where each multimode reduction ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m}^{(q)})}$ has internal mode structure $\\mathbf {m}^{(q)} \\!",
"\\equiv \\!",
"(m_1^{(q)} , \\ldots ,m_{G^{(q)} }^{(q)} )$ where $G^{(q)} \\in 1, \\ldots ,S$, in terms of the original indivisible modes $m_j$ such that all of them appear exactly once among all new mode groups $\\mathbf {m}^{(q)}$ for $q \\in 1, \\ldots ,T$ .",
"Thus, in the case of the nonreduction ($S=N$ ), the input state to the partitional correlance is the full state ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {N})}={\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1,\\ldots ,N)}=\\rho $ , and the new mode-group vector is (in this particular case) any element of the set $\\mathbf {m}^{(\\mathbf {T})} \\in \\lbrace \\mathbf {N}_h^{(\\mathbf {k})} \\rbrace \\equiv $ $ \\lbrace (\\mathbf {N}_h^{(1)} | \\ldots |\\mathbf {N}_h^{(k)} )\\rbrace |_{h = 1}^{\\lbrace _k^N \\rbrace } $, meaning it is the set of all $\\lbrace _k^N \\rbrace $ unique partitions of $N$ objects partitioned into $k$ groups.",
"As an example of how the notation works, for a tripartite system, $N=3$ and $\\mathbf {N}\\equiv (1,\\ldots ,N)=(1,2,3)$ , so the possible $N$ -mode partitions into $k=2$ groups are $\\begin{array}{*{20}l}{\\mathbf {m}^{(\\mathbf {T})} } &\\!\\!",
"{ \\in \\lbrace (1|2,3),(2|1,3),(3|1,2)\\rbrace } \\\\{} &\\!\\!",
"{ \\equiv \\lbrace (\\mathbf {N}_1^{(1)} |\\mathbf {N}_1^{(2)} ),(\\mathbf {N}_2^{(1)} |\\mathbf {N}_2^{(2)} ),(\\mathbf {N}_3^{(1)} |\\mathbf {N}_3^{(2)} )\\rbrace } \\\\{} &\\!\\!",
"{ \\equiv \\lbrace \\mathbf {N}_h^{(\\mathbf {2})} \\rbrace |_{h = 1}^{\\lbrace _2^3 \\rbrace } \\equiv \\lbrace \\mathbf {N}_h^{(\\mathbf {k})} \\rbrace , } \\\\\\end{array}$ where notice that reordering is not considered unique here, so for instance we do not list $(3|2,1)$ as a unique option since we have already listed $(3|1,2)$ .",
"Therefore, specifying $\\mathbf {m}^{(\\mathbf {T})} \\in \\lbrace \\mathbf {N}_h^{(\\mathbf {k})} \\rbrace $ in the partitional correlance gives the $N$ -mode $k$ -partitional correlances as $\\begin{array}{*{20}l}{\\mathcal {X}^{(\\mathbf {N}_h^{(\\mathbf {k})} )} (\\rho )} &\\!\\!",
"{ \\equiv \\mathcal {X}({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {N})} ,\\mathbf {n}^{(\\mathbf {N}_h^{(\\mathbf {k})} )} ) = \\mathcal {X}(\\rho ,\\mathbf {n}^{(\\mathbf {N}_h^{(\\mathbf {k})} )} )} \\\\{} &\\!\\!",
"{ \\equiv \\frac{1}{{\\mathcal {N}_{\\mathcal {X}} (\\mathbf {n}^{(\\mathbf {N}_h^{(\\mathbf {k})} )} )}}\\text{tr}\\left( {[\\rho - \\varsigma (\\rho ,\\mathbf {n}^{(\\mathbf {N}_h^{(\\mathbf {k})} )} )]^2 } \\right),} \\\\\\end{array}$ with $N$ -mode $k$ -partitional reduction products, $\\varsigma (\\rho ,\\mathbf {n}^{(\\mathbf {N}_h^{(\\mathbf {k})} )} ) \\equiv \\mathop \\otimes \\limits _{q = 1}^k {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {N}_h^{(q)})} .$ Therefore, (eq.50) appears in the terms of (eq.44).",
"The multicorrelance $\\mathcal {M}_{\\mathcal {X}}(\\rho )$ measures the simultaneous amount of all $N$ -mode $k$ -partitional multicorrelances $\\mathcal {M}_{\\mathcal {X}}^{(\\mathbf {N}^{(\\mathbf {k})})}(\\rho )$, while each individual $\\mathcal {M}_{\\mathcal {X}}^{(\\mathbf {N}^{(\\mathbf {k})})}(\\rho )$ measures not only the combination of all possible $N$ -mode $k$ -partitional correlances, but it also measures how equally distributed they are by use of the square root.",
"Alternatively, we may also define the $h$ th $N$ -mode $k$ -partitional root-correlance as $\\mathcal {M}_{\\mathcal {X}} ^{(\\mathbf {N}_h^{(\\mathbf {k})} )} (\\rho ) \\equiv \\sqrt{\\mathcal {X}^{(\\mathbf {N}_h^{(\\mathbf {k})} )} (\\rho )} ,$ which simplifies the notation for later concepts.",
"Note that in this nomenclature, the prefix multi implies a sum over all varieties of something, whereas the absence of multi means no sum over varieties.",
"For example, the $N$ -mode $k$ -partitional multicorrelance implies a sum over all $N$ -mode $k$ -partitional root-correlances which are each single-term quantities for a specific $N$ -mode $k$ -partition.",
"The multicorrelance $\\mathcal {M}_\\mathcal {X}(\\rho )$ is built to reveal whether there is any nonlocal correlation at all in a given state at its $N$ -mode scale, even including different perspectives created by grouping those modes together in a way that the total size of the input state remains the same.",
"As an example, for a 4-partite system, the (unnormalized) multicorrelance is, from (eq.43), $\\widetilde{\\mathcal {M}}_{\\mathcal {X}} (\\rho ) = {\\mathcal {M}}_{\\mathcal {X}} ^{(\\mathbf {N}^{(\\mathbf {2})} )} (\\rho ) + {\\mathcal {M}}_{\\mathcal {X}} ^{(\\mathbf {N}^{(\\mathbf {3})} )} (\\rho ) + {\\mathcal {M}}_{\\mathcal {X}} ^{(\\mathbf {N}^{(\\mathbf {4})} )} (\\rho ),$ where $\\mathcal {M}_{\\mathcal {X}} ^{(\\mathbf {N}_h^{(\\mathbf {k})} )} (\\rho )\\equiv \\widetilde{\\mathcal {M}}_{\\mathcal {X}} ^{(\\mathbf {N}^{(\\mathbf {k})} )} (\\rho )/\\mathcal {N}_{\\mathcal {M}_{\\mathcal {X}_k} }$ are the $N$ -mode $k$ -partitional multicorrelances from (eq.44), given in unnormalized form (with correlance inputs suppressed) by $\\begin{array}{*{20}l}{\\widetilde{\\mathcal {M}}_{\\mathcal {X}} ^{(\\mathbf {N}^{(\\mathbf {2})} )} (\\rho ) = } &\\!\\!",
"{\\sqrt{\\mathcal {X}^{(1|2,3,4)} } + \\sqrt{\\mathcal {X}^{(2|1,3,4)} } + \\sqrt{\\mathcal {X}^{(3|1,2,4)} } } \\\\{} &\\!\\!",
"{+ \\sqrt{\\mathcal {X}^{(4|1,2,3)} } + \\sqrt{\\mathcal {X}^{(1,2|3,4)} } + \\sqrt{\\mathcal {X}^{(1,3|2,4)} } } \\\\{} &\\!\\!",
"{ + \\sqrt{\\mathcal {X}^{(1,4|2,3)} } } \\\\{\\widetilde{\\mathcal {M}}_{\\mathcal {X}} ^{(\\mathbf {N}^{(\\mathbf {3})} )} (\\rho ) = } &\\!\\!",
"{\\sqrt{\\mathcal {X}^{(1|2|3,4)} } + \\sqrt{\\mathcal {X}^{(1|3|2,4)} } + \\sqrt{\\mathcal {X}^{(1|4|2,3)} } } \\\\{} &\\!\\!",
"{ + \\sqrt{\\mathcal {X}^{(2|3|1,4)} } + \\sqrt{\\mathcal {X}^{(2|4|1,3)} } + \\sqrt{\\mathcal {X}^{(3|4|1,2)} } } \\\\{\\widetilde{\\mathcal {M}}_{\\mathcal {X}} ^{(\\mathbf {N}^{(\\mathbf {4})} )} (\\rho ) = } &\\!\\!",
"{\\sqrt{\\mathcal {X}^{(1|2|3|4)} } = \\sqrt{\\mathcal {X}(\\rho )} ,} \\\\\\end{array}$ where the radicands above are obtained from (eq.50).",
"The specification of the structure vectors for each term then determines how each one is specifically calculated by controlling which effective modes are recognized for each term.",
"The result is that if any nonlocal correlation exists between any partitioning of the $N$ modes, the multicorrelance will report the presence of that correlation as a value relative to some maximum over all states.",
"The square root of each partitional correlance is used here in analogy with the definition of ent-concurrence from [22], where it was shown to be able to distinguish different types of distinctly multipartite entanglement, whereas without the square root, states like the 4-qubit GHZ state and Bell-product states did not appear to have different multipartite entanglement.",
"In this context, tests of the 4-qubit tier-1, tier-2, and tier-3 maximally entangled states of [22] have shown that the square root is necessary to distinguish the $N$ -mode 3-partitional multicorrelances of the tier-2 and tier-3 states (GHZ state and Bell-product states) which are identical if the square root is omitted.",
"See [22] for mathematical details about why the square root is appropriate for the ent-concurrence; the same argument applies here as well.",
"The purpose of the detailed notation is just to keep track of partitions for the purpose of telling the correlance function over which groups of modes we want to measure nonlocal correlations.",
"As we will soon see, this degree of specificity will allow us to extract information about all possible nonlocal correlations (with the limitation that the correlance cannot distinguish between particular types of nonlocal correlation such as free entanglement and bound entanglement [44], [45], [46]).",
"For a more in-depth exposition of partitions in this same notation, see ([22], App.C), and for further details on multipartite reductions see ([18], App.B)."
],
[
"$N$ -mode Partitional Multicorrelance Vector",
"For a finer-grained picture of nonlocal correlation that tells us between which mode groups correlation exists, we define the $N$ -mode partitional multicorrelance vector, $\\Xi _{\\mathcal {M}_{\\mathcal {X}} }^{(\\mathbf {N})} (\\rho ) \\equiv \\left( {\\begin{array}{*{20}c}{\\lbrace \\mathcal {M}_{\\mathcal {X}} ^{(\\mathbf {N}_h^{(\\mathbf {2})} )} (\\rho )\\rbrace } \\\\\\vdots \\\\{\\lbrace \\mathcal {M}_{\\mathcal {X}} ^{(\\mathbf {N}_h^{(\\mathbf {N})} )} (\\rho )\\rbrace } \\\\\\end{array}} \\right){\\hspace{-1.5pt}},$ where $\\lbrace \\mathcal {M}_{\\mathcal {X}} ^{(\\mathbf {N}_h^{(\\mathbf {k})} )} (\\rho )\\rbrace $ is the set of all $N$ -mode $k$ -partitional root-correlances, and is valid for both pure and mixed states $\\rho $ .",
"For example, in a 4-mode system (suppressing input arguments), $\\scalebox {0.89}{\\begin{array}{l}\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(\\mathbf {4})} \\equiv \\Xi _{\\mathcal {M}_\\mathcal {X}}^{(1,2,3,4)}= \\\\\\rule {0pt}{28.5pt}\\!\\!\\left(\\!\\!",
"{\\hspace{-0.6pt}} {\\begin{array}{*{20}c}{\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(1|2,\\!3,\\!4)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(2|1,\\!3,\\!4)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(3|1,\\!2,\\!4)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(4|1,\\!2,\\!3)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(1,\\!2|3,\\!4)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(1,\\!3|2,\\!4)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(1,\\!4|2,\\!3)} } \\\\{\\rule {0pt}{12.5pt}\\!\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(1|2|3,\\!4)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(1|3|2,\\!4)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(1|4|2,\\!3)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(2|3|1,\\!4)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(2|4|1,\\!3)} {\\hspace{1.0pt}}\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(3|4|1,\\!2)} } \\\\{\\rule {0pt}{12.5pt}\\!\\textnormal {\\scalebox {0.70}{\\mathcal {M}}}_\\mathcal {X}^{(1|2|3|4)} } \\\\\\end{array}}\\!\\!",
"\\right)\\!",
"{\\hspace{-1.0pt}}, \\\\\\end{array}}$ where each particular $\\mathcal {M}_{\\mathcal {X}} ^{(\\mathbf {N}_h^{(\\mathbf {k})} )} (\\rho )$ is given by (eq.52).",
"The top row gives each $N$ -mode 2-partitional root-correlance, and each row farther down treats increasing partitions until the bottom row gives the $N$ -mode $N$ -partitional root-correlance which is just the square root of the actual correlance from (eq.9).",
"Thus, $\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(\\mathbf {N})}$ gives a fine-grained, location-specific view of nonlocal correlation.",
"For a more intermediate picture of nonlocal correlation, the $N$ -mode $k$ -partitional multicorrelances from (eq.44), expressible as [with the help of (eq.52)], $\\mathcal {M}_\\mathcal {X}^{(\\mathbf {N}^{(\\mathbf {k})} )} (\\rho ) \\equiv \\frac{1}{{\\mathcal {N}_{\\mathcal {M}_{\\mathcal {X}_k} } }}\\sum \\nolimits _{h = 1}^{\\lbrace _k^N \\rbrace } {\\mathcal {M}_\\mathcal {X}^{(\\mathbf {N}_h^{(\\mathbf {k})} )} (\\rho )} ,$ give a measure of the total possible nonlocal correlation over all partitions of a certain $k$ -value for an $N$ -mode system, such as all possible bipartitions.",
"Note that the more aggregated measures such as multicorrelance and $N$ -mode $k$ -partitional multicorrelance do not necessarily imply that all of these nonlocal correlation resources are available for use simultaneously; rather they indicate that such correlations are present, and typically only some of them may be used simultaneously."
],
[
"Absolute Multicorrelance",
"Again following [22], while the multicorrelance $\\mathcal {M}_\\mathcal {X}(\\rho )$ and its more specific $N$ -mode partitional multicorrelance vector $\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(\\mathbf {N})}(\\rho )$ and the even more specific $N$ -mode $k$ -partitional root-correlances $\\mathcal {M}_{\\mathcal {X}} ^{(\\mathbf {N}_h^{(\\mathbf {k})} )} (\\rho )$ give us a sense of the nonlocal correlation present in the full input state within its full Hilbert space, we can get an even more in-depth picture of the resources available in a state by evaluating the nonlocal correlations within reductions of the input state.",
"To this end, rather than look at all $N$ modes, we focus on a subset of modes denoted by mode group $\\mathbf {m}\\equiv (m_{1},\\ldots ,m_{S})$ for $S\\in 2,\\ldots ,N$ (we exclude $S = 1$ here since nonlocal correlation is correlation between at least two modes), and thus we now want to look for multipartite nonlocal correlations of some $S$ -mode reduction of $N$ -mode state $\\rho $ .",
"Thus, we define the $S$ -mode partitional multicorrelance vector as $\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(\\mathbf {m})} (\\rho ) \\equiv \\left( {\\begin{array}{*{20}c}{\\lbrace \\mathcal {M}_\\mathcal {X}^{(\\mathbf {m}_h^{(\\mathbf {2})} )} (\\rho )\\rbrace } \\\\\\vdots \\\\{\\lbrace \\mathcal {M}_\\mathcal {X}^{(\\mathbf {m}_h^{(\\mathbf {S})} )} (\\rho )\\rbrace } \\\\\\end{array}} \\right),$ where $\\lbrace \\mathcal {M}_\\mathcal {X}^{(\\mathbf {m}_h^{(\\mathbf {k})} )} (\\rho )\\rbrace $ is the set of all $S$ -mode $T$ -partitional root-correlances of a given reduction ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})}$ , where $T\\in 2,\\ldots ,S$ here and each particular partitioning is labeled by $h$ , and each $S$ -mode $T$ -partitional root-correlance is $\\mathcal {M}_\\mathcal {X}^{(\\mathbf {m}_h^{(\\mathbf {T})} )} (\\rho ) \\equiv \\sqrt{\\mathcal {X}^{(\\mathbf {m}_h^{(\\mathbf {T})} )} (\\rho )},$ where the $h$ th $S$ -mode $T$ -partitional correlance is $\\begin{array}{*{20}l}{\\mathcal {X}^{(\\mathbf {m}_h^{(\\mathbf {T})} )} (\\rho )}{\\hspace{-2.0pt}} &\\!\\!",
"{ \\equiv {\\hspace{-1.5pt}} \\mathcal {X}({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})} ,\\mathbf {n}^{(\\mathbf {m}_h^{(\\mathbf {T})} )} )} \\\\{}{\\hspace{-2.0pt}} &\\!\\!",
"{ \\equiv {\\hspace{-1.5pt}} \\frac{1}{{\\mathcal {N}_\\mathcal {X}(\\mathbf {n}^{(\\mathbf {m}_h^{(\\mathbf {T})} )} )}}\\text{tr}{\\hspace{-1.5pt}}\\left( {[{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})} {\\hspace{-1.5pt}}-{\\hspace{-1.5pt}} \\varsigma ({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})} ,\\mathbf {n}^{(\\mathbf {m}_h^{(\\mathbf {T})} )} )]^2 } \\right){\\hspace{-1.5pt}},} \\\\\\end{array}$ with $S$ -mode $T$ -partitional reduction products, $\\varsigma (\\rho ^{(\\mathbf {m})} ,\\mathbf {n}^{(\\mathbf {m}_h^{(\\mathbf {T})} )} ) \\equiv \\mathop \\otimes \\limits _{q = 1}^T {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m}_h^{(q)})} ,$ where (eq.61) is written in terms of a general input state $\\rho ^{(\\mathbf {m})}$ in the Hilbert space of the reduced system to help show that the reductions of this state must be taken to calculate this quantity.",
"Therefore, in (eq.60) we are taking reductions of reductions to calculate $\\varsigma ({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})} ,\\mathbf {n}^{(\\mathbf {m}_h^{(\\mathbf {T})} )} )$ .",
"Again, since correlance can handle mixed states, all of these definitions work for general states both pure and mixed.",
"Thus, in (eq.58), row $T-1$ is a list of all $S$ -mode $T$ -partitional root-correlances of ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})}$ which is the mode-$\\mathbf {m}$ reduction of $\\rho $ .",
"Then, to account for the fact that we can make $\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(\\mathbf {m})} (\\rho )$ for many different reduction mode-groups $\\mathbf {m}$ for a given parent state $\\rho $ , we can collect them all in a larger object called the multicorrelance array $\\widetilde{\\nabla } _{\\mathcal {M}_\\mathcal {X}} $ (not a gradient), the elements of which are all possible $S$ -mode partitional multicorrelance vectors, $(\\widetilde{\\nabla }_{\\mathcal {M}_\\mathcal {X}} )_{k,l} (\\rho ) \\equiv \\Xi _{\\mathcal {M}_\\mathcal {X}}^{([\\text{nCk}(\\mathbf {N},k)]_{l, \\cdots } )} (\\rho ),$ where again $\\mathbf {N}\\equiv (1,\\ldots ,N)$ , and $k\\in 2,\\ldots ,N$ , and $l \\in 1, \\ldots ,(_k^N )$ , where $(_k^N ) \\equiv \\frac{{N!}}{{k!",
"(N - k)!",
"}}$ , and $\\text{nCk}(\\mathbf {N},k)$ is the vectorized “$n$ -choose-$k$ ” function yielding the matrix whose rows are each unique combinations of the elements of $\\mathbf {N}$ chosen $k$ at a time, and we use the notation $A_{l,\\cdots }$ to mean the $l$ th row of matrix $A$ .",
"Note that the tilde in (eq.62), in keeping with earlier notation, implies that this quantity has not been normalized over all states.",
"As we will soon see, this is actually preferable in this case, since each smallest element of the resulting object is already normalized, and therefore it is most useful to just see the actual root-correlance values as they are.",
"Continuing our example from earlier, a 4-partite multicorrelance array has the form (suppressing inputs) $\\widetilde{\\nabla } _{\\mathcal {M}_\\mathcal {X}} (\\rho ){\\hspace{-1.0pt}} ={\\hspace{-4.0pt}} \\left({\\hspace{-5.0pt}} {\\begin{array}{*{20}c}{\\begin{array}{*{20}c}{\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(1,2)} } & {\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(1,3)} } & {\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(1,4)} } & {\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(2,3)} } \\hfill & {\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(2,4)} } & {\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(3,4)} } \\\\\\end{array}} \\\\{\\begin{array}{*{20}c}{\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(1,2,3)} } & {\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(1,2,4)} } & {\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(1,3,4)} } & {\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(2,3,4)} } \\\\\\end{array}} \\\\{\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(1,2,3,4)} } \\\\\\end{array}}{\\hspace{-5.0pt}} \\right){\\hspace{-3.0pt}},$ where the 4-mode partitional multicorrelance vector $\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(1,2,3,4)}$ was given in (eq.56), while the 3-mode partitional multicorrelance vectors $\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(a,b,c)}$ have the form $\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(a,b,c)} = \\left( {\\begin{array}{*{20}c}{\\begin{array}{*{20}c}{\\mathcal {M}_\\mathcal {X}^{(a|b,c)} } & {\\mathcal {M}_\\mathcal {X}^{(b|a,c)} } & {\\mathcal {M}_\\mathcal {X}^{(c|a,b)} } \\\\\\end{array}} \\\\{\\mathcal {M}_\\mathcal {X}^{(a|b|c)} } \\\\\\end{array}} \\right){\\hspace{-1.5pt}},$ and the 2-mode partitional multicorrelance vectors all have just one element of the form $\\Xi _{\\mathcal {M}_\\mathcal {X}}^{(a,b)} = \\mathcal {M}_\\mathcal {X}^{(a|b)} = \\mathcal {M}_\\mathcal {X}^{(a,b)} .$ The multicorrelance array $\\widetilde{\\nabla } _{\\mathcal {M}_\\mathcal {X}} (\\rho )$ gives the most detailed picture of the nonlocal correlations available in $\\rho $ , showing in which reductions such correlations exist, as well as where (between which mode groups) in those reductions they exist.",
"Again, it is unlikely that all correlations within a state are available to be exploited simultaneously, but this treatment may give us a powerful way to categorize states for their potential resources of nonlocal correlation.",
"Lastly, for the most aggregated measure of the total potential nonlocal correlation resources within a state, we can define the absolute multicorrelance as $\\mathcal {M}_{\\mathcal {X}_{\\text{abs}} } (\\rho ) \\equiv \\frac{{||\\widetilde{\\nabla } _{\\mathcal {M}_\\mathcal {X}} (\\rho )||_1 }}{{\\max _{\\forall \\rho ^{\\prime } \\in \\mathcal {H}} \\lbrace ||\\widetilde{\\nabla } _{\\mathcal {M}_\\mathcal {X}} (\\rho ^{\\prime })||_1 \\rbrace }},$ which is the normalized 1-norm of the multicorrelance array, where the 1-norm of vector $\\mathbf {v}$ is $||\\mathbf {v}||_1 \\equiv \\sum \\nolimits _{k = 1}^{\\dim (\\mathbf {v})} {|v_k |} $ , and the 1-norm is taken over the smallest scalar elements of $\\widetilde{\\nabla } _{\\mathcal {M}_\\mathcal {X}} (\\rho )$ , meaning for example that even though $\\widetilde{\\nabla } _{\\mathcal {M}_\\mathcal {X}} (\\rho )$ in (eq.63) has 11 “elements,” the fact that each of them is an object containing several scalar elements means that the total number of scalar-element terms in the 1-norm in $\\mathcal {M}_{\\mathcal {X}_{\\text{abs}} } (\\rho )$ for a 4-partite system is 36.",
"Note that all of this multipartite generalization follows the formalism of [22] and [18] closely, with the exception that here there is no difficulty in handling mixed states and our measure considers all nonlocal correlation, not just entanglement.",
"Keep in mind that all state measures rate states based on different criteria, and in general, the decision for which measure to use depends on the application.",
"For example, if we only intend on using a particular state $\\rho $ in its full $N$ -mode form for the correlations that exist between partitions of its $N$ modes, then none of the measures in [sec:VII.C]Sec.sec:VII.C are directly relevant for that application, and the measures of [sec:VII.A]Sec.sec:VII.A and [sec:VII.B]Sec.sec:VII.B are more appropriate.",
"However, even then, if we do not care about the total available correlations among the $N$ modes and instead want it only between a specific partition of mode groups, then we would only need a particular $N$ -mode $k$ -partitional root-correlance as our measure of choice to compare candidate states for our application.",
"Therefore, care should be taken to precisely define the needs of a given application before jumping in and applying one of these measures; otherwise the value of “goodness” they give on a scale of 0 to 1 may mean something unsuited to a given application.",
"Finally, all of the definitions in [sec:VII]Sec.sec:VII can be applied to diagonal correlance, strong discordance, discordance, and diagonal discordance as well (and also statance and probablance although they are not yet generally computable); to do so in these equations simply replace all occurrences of $\\mathcal {X}$ with $\\mathcal {L}$ where $\\mathcal {L}$ is the label for that particular measure such as $\\mathcal {D}$ for discordance or $\\mathcal {D}_{D}$ for diagonal discordance etc.",
"For the nomenclature, just replace all occurrences of “correlance” (including within compound words) with the name of new measure, such as in “multidiscordance” or “multi-diagonal-discordance,” etc."
],
[
"Conclusions",
"In this paper, we have introduced several measures of nonlocal correlation for discrete $N$ -partite systems, with explicit generalizations of them for multipartite correlations beyond merely $N$ -partite correlations in an $N$ -mode system.",
"Most of these measures are computable for all states both mixed and pure, giving them distinct advantages over other measures such as quantum discord.",
"To facilitate this goal, in [sec:I.A]Sec.sec:I.A we first defined several mechanisms of nonlocal correlation, not all of which are mutually exclusive, but all of which are important to acknowledge.",
"The two main kinds identified are called decomposition-state correlation and probability correlation.",
"We also identified entanglement correlation (which we later showed to be a special kind of decomposition-state correlation), and we defined general mixed product states as the absence of nonlocal correlation.",
"From the above general mechanisms of nonlocal correlation, we were able to identify the Six Families of Nonlocal Correlation in [tab:1]Table tab:1 as a set of forms that density matrices can take that yield nonlocal correlation.",
"While these families are not all mutually exclusive, identifying them was an extremely helpful conceptual aid to investigating nonlocal correlations in multipartite systems.",
"From these initial observations, we immediately constructed the main measure of interest as the correlance $\\mathcal {X}$ in [sec:I.B]Sec.sec:I.B, which measures all possible $N$ -mode nonlocal correlation by gauging a state's distance from having mode-independent product form, meaning its distance from its own reduction product (since only product states are their own reduction products), the conditions for which are established in [sec:I.A]Sec.sec:I.A.",
"The only drawback of $\\mathcal {X}$ is that it cannot distinguish between different kinds of correlation.",
"Its advantages are that it is computable for all states, and it is capable of detecting all nonlocal correlation (including, for instance, the notoriously difficult-to-detect bound entanglement [44], [45], [46]).",
"In [sec:II]Sec.sec:II, we proved the validity of and derived the normalization factor of $\\mathcal {X}$ (contained in various appendices), and did extensive numerical tests in a wide variety of multipartite systems as a set of necessary checks against the proofs.",
"Furthermore, we constructed a related measure as the diagonal correlance $\\mathcal {X}_{D}$ intended for only diagonal states and we sketched a similar proof of its normalization from which we derived its normalization factor, and numerically tested its normalization as well.",
"We also compared $\\mathcal {X}_{D}$ to the well-known Pearson correlation and gave extensive instructions for how to handle classical data for use with $\\mathcal {X}_{D}$ in [sec:App.M]App.sec:App.M.",
"The diagonal correlance $\\mathcal {X}_{D}$ is intended for classical probability distributions only, since those must always be diagonal since superposition and thus coherence (nonzero off-diagonal elements of a density matrix) are not possible in classical physics.",
"(This also provided a nice segue into a discussion of what we really mean by classical states; a crucial topic for any paper on nonlocal correlations including quantum discord and notions of classicality.)",
"[sec:III]Section sec:III delves deeply into the distinction between classical and quantum, and ultimately proposes the new term strictly classical to provide a well-defined and conceptually consistent meaning for true classicality beyond the often carelessly and ambiguously used term “classical.” Ultimately, we conclude that the main defining feature for strict classicality must be the absence of quanutm superposition, both intrinsically and through transformation.",
"This notion is a departure from the popular belief that coherent states are the most appropriate standards of classicality despite having quantum superposition.",
"This topic is discussed in depth in [sec:App.O]App.sec:App.O which describes the consistency of strict classicality with established physics and also explains why using coherent states as standards of classicality is not conceptually consistent.",
"[sec:III]Section sec:III also makes the important observation that there exist diagonal states that are quantum, and identifies the property of a state having “superposition without coherence,” meaning that in cases such as when a diagonal state is a reduction of a larger maximally entangled parent state, the probabilities of the reduction are directly inherited from the superposition coefficients of the parent state, which are themselves overlaps of the parent state with the basis states.",
"Thus we can have diagonal states that still have superposition in an inherited sense.",
"All of these ideas relating classicality to the edge of distinctly quantum properties form powerful conceptual tools for the remaining discussions.",
"In [sec:IV]Sec.sec:IV, we present the hard-to-compute measures statance $\\hat{\\mathcal {S}}(\\rho )$ and probablance $\\hat{\\mathcal {P}}(\\rho )$ , which measure the decomposition-state correlation and probability correlation respectively.",
"From these definitions, we learn several very useful new facts, one of the most interesting of which is that strictly classical states (which are diagonal) can have nonzero $\\hat{\\mathcal {S}}(\\rho )$ and $\\hat{\\mathcal {P}}(\\rho )$ , just as general quantum states can.",
"Furthermore, we showed that these types of correlation can exist independently of each other in strictly classical states as well.",
"This taught us that correlations of a classical kind cannot be purely ascribed to one mechanism such as probability correlation or decomposition-state correlation, because in the cases of diagonal quantum states where we have “superposition without coherence,” both of these types of nonlocal correlation are entirely quantum in origin, having come from the superposition of the parent state before the reduction.",
"Therefore, since there is no fundamental way to distinguish strictly classical states from diagonal quantum states without having more information beyond the state, this established that diagonal quantum states are not distinguishable from strictly classical states.",
"[sec:V]Section sec:V then began by defining the strong discordance $\\mathcal {D}_{S}$ as a measure of nonlocal correlation beyond that achievable by a strictly classical state, as a reasonable first step towards constructing a measure sensitive to uniquely quantum correlations.",
"However, despite being exactly computable, we soon found that $\\mathcal {D}_{S}$ is too exclusive, since it rejects weak quantum correlation such as entanglement that produces a value of correlance $\\mathcal {X}$ that is also achievable by some strictly classical states.",
"Therefore, since we desired a measure that can detect all novel quantum correlation such as entanglement, and since we established that diagonality is the fundamental defining feature of strictly classical states but also the point at which diagonal quantum states cannot be distinguished from strictly classical states, then this prompted the definition of discordance $\\mathcal {D}$ in [sec:V.B]Sec.sec:V.B as a measure of nonlocal correlation in a distinguishably quantum state.",
"The fully computable measure of discordance $\\mathcal {D}$ uses nondiagonality as the threshold for recognizing nonlocal correlation, so that it is guaranteed to report “zero discordance” for strictly classical states (which are all diagonal), while it is also guaranteed to correctly report the net correlation for all entangled states as well as for all quantum states that are distinguishably quantum by being nondiagonal.",
"Its only weakness is that $\\mathcal {D}$ rejects the nonlocal correlation of diagonal quantum states as being “not quantum enough” to qualify as discordance.",
"However, that prompted the definition in [sec:V.D]Sec.sec:V.D of another computable measure, the diagonal discordance $\\mathcal {D}_{D}$ which only recognizes the nonlocal correlation in diagonal states, and rejects all distinguishably quantum states (nondiagonal states) as being “too quantum.” Together, the three computable measures of correlance $\\mathcal {X}$ , diagonal discordance $\\mathcal {D}_{D}$ , and discordance $\\mathcal {D}$ were shown to obey the relation $\\mathcal {X}=\\mathcal {D}_{D}+\\mathcal {D}$ in (eq.42), which formally parallels the quantum-discord relation from (eq.2), $\\mathcal {I}=\\mathcal {C}+\\mathcal {Q}$ , where $\\mathcal {I}$ is the quantum mutual information, $\\mathcal {C}$ is the classical correlation, and $\\mathcal {Q}$ is the quantum discord.",
"We then showed that this similarity is only one of form, but not meaning, since we identified the main conceptual flaw of quantum discord $\\mathcal {Q}$ is that it is defined in terms of classical correlation $\\mathcal {C}$ which is based on von Neumann projectors that are allowed to have quantum superposition, something that is completely at odds with classical physics.",
"Therefore, it may be that $\\mathcal {C}$ and $\\mathcal {Q}$ have some conceptually inconsistent overlaps in what they measure, making them conceptually less preferable, beyond the fact that they are difficult calculate.",
"Alternatively, we could treat the raw strong discordance $\\widetilde{\\mathcal {D}}_{S}(\\rho )\\equiv \\max \\lbrace 0,\\mathcal {X}(\\rho )-\\mathcal {X}(\\rho _{D_{\\max }})\\rbrace $ as an analog of $\\mathcal {Q}$ , and then define an analog to $\\mathcal {C}$ as the raw diagonal strong discordance $\\widetilde{\\mathcal {D}}_{DS}(\\rho )\\equiv \\mathcal {X}(\\rho )-\\widetilde{\\mathcal {D}}_{S}(\\rho )=$ $\\min \\lbrace \\mathcal {X}(\\rho ),\\mathcal {X}(\\rho _{D_{\\max }})\\rbrace $ .",
"Then $\\mathcal {X}=\\widetilde{\\mathcal {D}}_{DS}+\\widetilde{\\mathcal {D}}_{S}$ would parallel the original idea of $\\mathcal {I}=\\mathcal {C}+\\mathcal {Q}$ more closely since strong-discordance states (those with $\\mathcal {D}_{S}>0$ ) would simultaneously have nonzero values of both $\\widetilde{\\mathcal {D}}_{DS}$ and $\\widetilde{\\mathcal {D}}_{S}$ (all $\\mathcal {D}_{S}>0$ states would have maximal $\\widetilde{\\mathcal {D}}_{DS}$ ).",
"However, the $\\widetilde{\\mathcal {D}}_{S}$ part would still behave discontinuously, yielding zero for weakly entangled states until the maximally-correlated diagonal-state threshold, as for $\\mathcal {D}_{S}$ in [fig:10]Fig.fig:10.",
"Thus, discordance $\\mathcal {D}$ is a better replacement for quantum discord since $\\mathcal {D}$ never ignores entanglement, no matter how weak.",
"Nevertheless, this alternative trio of measures $\\lbrace \\mathcal {X},\\widetilde{\\mathcal {D}}_{DS},\\widetilde{\\mathcal {D}}_{S}\\rbrace $ may be useful in some applications.",
"Some possible extensions and improvements of these ideas would be their generalization to systems of infinite levels, and then to continuous systems.",
"Beyond that, hybrid systems such as those where some modes are discrete and some modes are continuous would be another interesting area to explore.",
"It would also be nice to have a more explicit proof of the normalization of diagonal correlance $\\mathcal {X}_{D}$ rather than the sketch given in [sec:App.L]App.sec:App.L, even though the numerical tests seem to vindicate it to excellent precision.",
"However, no such difficulty arose for correlance $\\mathcal {X}$ , the normalization of which was explicitly proved in [sec:App.J]App.sec:App.J, and that is the more important measure since it covers all states regardless of diagonality.",
"Note that while the explicit normalization factors given are exact, as system size gets larger (either in number of levels $n$ or number of modes $N$ or both), the accumulated numerical errors in typical computers may lead to increasingly large errors in normalization.",
"Therefore, in those cases it may be beneficial to find a maximally correlated state of the desired type to act as a normalization standard, which may help compensate for the errors since then the standard state must undergo the same pre-normalized computation sequence as the input state.",
"In closing, this paper provides several useful measures of nonlocal correlation that are carefully motivated by the possible mechanisms of nonlocal correlation and also the distinctly quantum feature of superposition and its related feature of coherence.",
"The main measures of correlance $\\mathcal {X}$ , diagonal discordance $\\mathcal {D}_{D}$ , and discordance $\\mathcal {D}$ have the advantages of being conceptually consistent ways to distinguish quantum correlations from classical correlations through their acknowledgment that quantum superposition cannot be allowed in any reasonable definition of classicality, which is the bedrock of the notion of strict classicality.",
"Since these measures are also exactly computable for both pure and mixed states and work in all multipartite discrete systems, they provide us with extremely powerful tools for studying nonlocal correlation."
],
[
"Brief Review of Reduced States",
"Let the multimode reduction from $N$ -mode parent state $\\rho $ to a composite subsystem of $S{\\hspace{-1.0pt}}\\in {\\hspace{-1.0pt}} 1,\\ldots ,N$ possibly noncontiguous and reordered modes $\\mathbf {m} \\equiv (m_1 , \\ldots ,m_S)$ be ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})} \\equiv \\text{tr}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} (\\rho ),$ where the $\\raisebox {-4pt}{\\scalebox {1.2}{̌}}$ symbol in ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})}$ indicates that it is a reduction of $\\rho $ (and not merely an isolated system of the same size as mode group $\\mathbf {m}$ ), and the bar in $\\mathbf {{\\mathop {m}\\limits ^{{\\hspace{-3.5pt}}~_{\\overline{{\\hspace{7.0pt}}}}}}}$ means “not $\\mathbf {m}$ ,” meaning that we trace over all modes whose labels are not in $\\mathbf {m}$ .",
"See App.B of [18] for details."
],
[
"Quantum Discord Definition",
"A suggested definition for quantum discord [11], [13] is $\\mathcal {Q}(\\rho )\\equiv \\mathcal {I}(\\rho )-\\mathcal {C}(\\rho ),$ with $\\mathcal {I}(\\rho )$ from (eq.1), where the classical correlation is $\\mathcal {C}(\\rho )\\equiv \\mathop {\\max }\\limits _{\\lbrace P_k^{(2)} \\rbrace } \\left[ {\\mathcal {I}(\\rho |\\lbrace P_k^{(2)} \\rbrace )} \\right],$ where we define $\\mathcal {I}(\\rho |\\lbrace P_k^{(2)} \\rbrace )\\equiv S({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}) - S(\\rho |\\lbrace P_k^{(2)} \\rbrace )$ as the quantum mutual information of the von Neumann measurement with projection operators $\\lbrace P_k^{(2)} \\rbrace $ for mode 2, where $S(\\rho |\\lbrace P_k^{(2)} \\rbrace ) \\equiv \\sum \\nolimits _k {p_k } S(\\rho _k )$ is the conditional von Neumann entropy given this measurement, where $p_k {\\hspace{-1.0pt}}\\equiv {\\hspace{-1.0pt}} \\text{tr}[(I^{(1)} {\\hspace{-1.0pt}} \\otimes {\\hspace{-1.0pt}} P_k^{(2)} )\\rho (I^{(1)} {\\hspace{-1.0pt}} \\otimes {\\hspace{-1.0pt}} P_k^{(2)} )^ \\dag ]$ are the probabilities of conditional measurement-outcome states $\\rho _k \\equiv \\frac{1}{{p_k }}(I^{(1)} \\otimes P_k^{(2)} )\\rho (I^{(1)} \\otimes P_k^{(2)} )^ \\dag $ ."
],
[
"Purity",
"Recall that the purity of $\\rho $ is the trace of its square, $P \\equiv P(\\rho ) \\equiv \\text{tr}(\\rho ^2 ),$ with range $P \\in [\\frac{1}{n},1]$ for an isolated system, and $\\rho $ is pure iff $P=1$ .",
"Otherwise, if $P<1$ then $\\rho $ is mixed.",
"Thus, for mode $m$ , $P(\\rho ^{(m)} ) \\in [\\frac{1}{{n_m }},1]$ for an isolated mode $m$ .",
"Beware: if the parent state $\\rho $ is not a product state, then its reductions ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}$ can have constraints on the limits of their purity, as proved in [18].",
"The reason (eq.3) has no nonlocal correlation is that its tensor product ensures that each reduction (marginal state) is ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}=\\rho ^{(m)} $ , and therefore has no dependence on any other modes."
],
[
"Some Basic Details about Entanglement",
"In separable states, as in (eq.4), notice that the probabilities $p_j$ do not need any special structure for (eq.4) to be satisfied, and that the decomposition states $\\rho _j$ merely need product form, but not mode independence.",
"Therefore, product form of the full state is not necessary for separability in general.",
"However, for pure states, product form is both necessary and sufficient for separability.",
"For example, in a two-qubit system where each qubit has a generic basis with labels starting on 1 (our convention in this paper) as $\\lbrace |1\\rangle ,|2\\rangle \\rbrace $ , the pure state $|\\psi \\rangle = ac|1\\rangle \\otimes |1\\rangle + ad|1\\rangle \\otimes |2\\rangle + bc|2\\rangle \\otimes |1\\rangle + bd|2\\rangle \\otimes |2\\rangle $ is separable because it can be factored as $|\\psi \\rangle = (a|1\\rangle + b|2\\rangle ) \\otimes (c|1\\rangle + d|2\\rangle )$ , so that $\\rho = |\\psi \\rangle \\langle \\psi | = \\rho ^{(1)} \\otimes \\rho ^{(2)}$ where $\\rho ^{(m)} = |\\psi ^{(m)} \\rangle \\langle \\psi ^{(m)} |$ , and $|\\psi ^{(1)} \\rangle = a|1\\rangle + b|2\\rangle $ and $|\\psi ^{(2)} \\rangle = c|1\\rangle + d|2\\rangle $ .",
"In contrast, the pure state $|\\Phi ^ + \\rangle = \\frac{1}{{\\sqrt{2} }}(|1\\rangle \\otimes |1\\rangle + |2\\rangle \\otimes |2\\rangle )$ is not separable so it is entangled (and happens to be maximally entangled) because it cannot be factored into a product form (since this example is for pure states)."
],
[
"Example of Product Form",
"To get product form for a two-qubit state $\\rho = \\sum \\nolimits _j {p_j \\rho _j }$ (and to demonstrate mode independence), Theorem1 requires (eq.5–eq.8) by setting $\\rho _j \\!\\equiv \\!",
"\\rho _j^{(1)} \\!\\otimes \\!",
"\\rho _j^{(2)}$ and $\\!\\!\\scalebox {0.95}{\\begin{array}{*{20}l}{p_1 } &\\!\\!\\!",
"{ =\\!",
"p_{(1,1)} } &\\!\\!\\!",
"{ \\equiv \\!",
"p_1^{(1)} p_1^{(2)} ,\\;\\;} &\\!\\!\\!",
"{\\rho _1^{(1)} } &\\!\\!\\!",
"{ =\\!",
"\\rho _{(1,1)}^{(1)} } &\\!\\!\\!",
"{ \\equiv \\!",
"\\rho _{(1)}^{(1)} ,\\;\\;} &\\!\\!\\!",
"{\\rho _1^{(2)} } &\\!\\!\\!",
"{ =\\!",
"\\rho _{(1,1)}^{(2)} } &\\!\\!\\!",
"{ \\equiv \\!",
"\\rho _{(1)}^{(2)} ,} \\\\{p_2 } &\\!\\!\\!",
"{ =\\!",
"p_{(1,2)} } &\\!\\!\\!",
"{ \\equiv \\!",
"p_1^{(1)} p_2^{(2)} ,\\;\\;} &\\!\\!\\!",
"{\\rho _2^{(1)} } &\\!\\!\\!",
"{ =\\!",
"\\rho _{(1,2)}^{(1)} } &\\!\\!\\!",
"{ \\equiv \\!",
"\\rho _{(1)}^{(1)} ,\\;\\;} &\\!\\!\\!",
"{\\rho _2^{(2)} } &\\!\\!\\!",
"{ =\\!",
"\\rho _{(1,2)}^{(2)} } &\\!\\!\\!",
"{ \\equiv \\!",
"\\rho _{(2)}^{(2)} ,} \\\\{p_3 } &\\!\\!\\!",
"{ =\\!",
"p_{(2,1)} } &\\!\\!\\!",
"{ \\equiv \\!",
"p_2^{(1)} p_1^{(2)} ,\\;\\;} &\\!\\!\\!",
"{\\rho _3^{(1)} } &\\!\\!\\!",
"{ =\\!",
"\\rho _{(2,1)}^{(1)} } &\\!\\!\\!",
"{ \\equiv \\!",
"\\rho _{(2)}^{(1)} ,\\;\\;} &\\!\\!\\!",
"{\\rho _3^{(2)} } &\\!\\!\\!",
"{ =\\!",
"\\rho _{(2,1)}^{(2)} } &\\!\\!\\!",
"{ \\equiv \\!",
"\\rho _{(1)}^{(2)} ,} \\\\{p_4 } &\\!\\!\\!",
"{ =\\!",
"p_{(2,2)} } &\\!\\!\\!",
"{ \\equiv \\!",
"p_2^{(1)} p_2^{(2)} ,\\;\\;} &\\!\\!\\!",
"{\\rho _4^{(1)} } &\\!\\!\\!",
"{ =\\!",
"\\rho _{(2,2)}^{(1)} } &\\!\\!\\!",
"{ \\equiv \\!",
"\\rho _{(2)}^{(1)} ,\\;\\;} &\\!\\!\\!",
"{\\rho _4^{(2)} } &\\!\\!\\!",
"{ =\\!",
"\\rho _{(2,2)}^{(2)} } &\\!\\!\\!",
"{ \\equiv \\!",
"\\rho _{(2)}^{(2)} ,} \\\\\\end{array}}$ where all states in (eq.E.1) are pure.",
"Then $\\rho $ becomes $\\begin{array}{*{20}l}\\rho &\\!\\!",
"= &\\!\\!",
"{p_1^{(1)} p_1^{(2)} \\rho _{(1)}^{(1)} \\otimes \\rho _{(1)}^{(2)} + p_1^{(1)} p_2^{(2)} \\rho _{(1)}^{(1)} \\otimes \\rho _{(2)}^{(2)} } \\\\{} &\\!\\!",
"{} &\\!\\!",
"{\\!+ p_2^{(1)} p_1^{(2)} \\rho _{(2)}^{(1)} \\otimes \\rho _{(1)}^{(2)} + p_2^{(1)} p_2^{(2)} \\rho _{(2)}^{(1)} \\otimes \\rho _{(2)}^{(2)} } \\\\{} &\\!\\!",
"= &\\!\\!",
"{p_1^{(1)} \\rho _{(1)}^{(1)} \\otimes (p_1^{(2)} \\rho _{(1)}^{(2)} + p_2^{(2)} \\rho _{(2)}^{(2)} )} \\\\{} &\\!\\!",
"{} &\\!\\!",
"{\\!+ p_2^{(1)} \\rho _{(2)}^{(1)} \\otimes (p_1^{(2)} \\rho _{(1)}^{(2)} + p_2^{(2)} \\rho _{(2)}^{(2)} )} \\\\{} &\\!\\!",
"= &\\!\\!",
"{(p_1^{(1)} \\rho _{(1)}^{(1)} + p_2^{(1)} \\rho _{(2)}^{(1)} ) \\otimes (p_1^{(2)} \\rho _{(1)}^{(2)} + p_2^{(2)} \\rho _{(2)}^{(2)} )} \\\\{} &\\!\\!",
"= &\\!\\!",
"{\\rho ^{(1)} \\otimes \\rho ^{(2)}, } \\\\\\end{array}$ with $\\rho ^{(1)} {\\hspace{-1.5pt}} \\equiv {\\hspace{-1.0pt}} p_1^{(1)} \\rho _{(1)}^{(1)} {\\hspace{-1.5pt}}+{\\hspace{-1.0pt}} p_2^{(1)} \\rho _{(2)}^{(1)} $ and $\\rho ^{(2)} {\\hspace{-1.5pt}}\\equiv {\\hspace{-1.0pt}} p_1^{(2)} \\rho _{(1)}^{(2)} {\\hspace{-1.5pt}}+{\\hspace{-1.0pt}} p_2^{(2)} \\rho _{(2)}^{(2)} $ .",
"Notice the redundant probabilities and decomposition states in (eq.E.1); this is a consequence of applying Theorem1.",
"Also, if either or both mode reductions have lower rank, some of the probabilities in (eq.E.1) will be zero.",
"Thus, the smallest maximum number of decomposition states required for any mixed product state is $r_{1}\\cdots r_{N}=r,$ (which is also their minimum number of decomposition states) where the ranks of each single-mode reduction of the product state are $r_{m}\\in 1,\\ldots ,n_{m}$ , and $r\\equiv \\text{rank}(\\rho )$ , which is just an application the well-known property for tensor products that $\\text{rank}(A\\otimes B)=\\text{rank}(A)\\text{rank}(B)$ .",
"In fact, by making mode-independent (MI) tensor products of the eigenstates of the reductions, and forming the corresponding probabilities from MI products of the eigenvalues of the reductions, we can always construct an $r$ -member MI decomposition of any product state."
],
[
"Proof of ",
"Starting with a general mixed state, $\\rho = \\sum \\nolimits _j {p_j \\rho _j },$ applying (eq.5) gives a separable state, $\\rho = \\sum \\nolimits _j {p_j \\rho _j^{(1)} \\otimes \\cdots } \\otimes \\rho _j^{(N)},$ which is not enough to definitely get product form.",
"If we then also use (eq.7), we get $\\begin{array}{*{20}l}{\\rho } &\\!\\!",
"{=\\sum \\nolimits _j {(p_j^{(1)} \\cdots p_j^{(N)} )\\rho _j^{(1)} \\otimes \\cdots \\otimes \\rho _j^{(N)} }} \\\\{} &\\!\\!",
"{= \\sum \\nolimits _j {p_j^{(1)} \\rho _j^{(1)} \\otimes \\cdots \\otimes p_j^{(N)} \\rho _j^{(N)} },} \\\\\\end{array}$ which is still not enough for product form because the indices are “locked” across the modes, indicating nonlocal correlation.",
"Therefore, vectorizing the indices as $\\rho ={\\hspace{-4.5pt}} \\sum \\limits _{j_1 , \\ldots ,j_N } {}{\\hspace{-4.5pt}} p_{(j_1 , \\ldots ,j_N )}^{(1)} \\rho _{(j_1 , \\ldots ,j_N )}^{(1)} {\\hspace{-1.0pt}} \\otimes \\cdots \\otimes p_{(j_1 , \\ldots ,j_N ) }^{(N)} \\rho _{(j_1 , \\ldots ,j_N )}^{(N)},$ and then applying both (eq.8) and (eq.6), we get $\\begin{array}{*{20}l}{\\rho } &\\!\\!",
"{= \\sum \\nolimits _{j_1 } { \\cdots \\sum \\nolimits _{j_N } {p_{j_1 }^{(1)} \\rho _{j_1 }^{(1)} \\otimes \\cdots \\otimes p_{j_N }^{(N)} \\rho _{j_N }^{(N)} } }} \\\\{} &\\!\\!",
"{= (\\sum \\nolimits _{j_1 } {p_{j_1 }^{(1)} \\rho _{j_1 }^{(1)} } ) \\otimes \\cdots \\otimes (\\sum \\nolimits _{j_N } {p_{j_N }^{(N)} \\rho _{j_N }^{(N)} } )} \\\\{} &\\!\\!",
"{= \\rho ^{(1)} \\otimes \\cdots \\otimes \\rho ^{(N)}} \\\\\\end{array}$ where $\\rho ^{(m)} \\equiv \\sum \\nolimits _{j_m } {p_{j_m }^{(m)} \\rho _{j_m }^{(m)} } $ , which proves the sufficiency of (eq.5–eq.8) for product form, and thus the absence of all nonlocal correlation.",
"To prove the necessity of (eq.5–eq.8) for product form, we simply start from the product-form definition in (eq.3), $\\rho = \\rho ^{(1)} \\otimes \\cdots \\otimes \\rho ^{(N)} .$ Then, since each mode's state can be generally mixed as $\\rho ^{(m)} = \\sum \\nolimits _{j_m } {p_{j_m }^{(m)} \\rho _{j_m }^{(m)} } ,$ we can put (eq.F.7) into (eq.F.6) to get $\\rho = (\\sum \\nolimits _{j_1 } {p_{j_1 }^{(1)} \\rho _{j_1 }^{(1)} } ) \\otimes \\cdots \\otimes (\\sum \\nolimits _{j_N } {p_{j_N }^{(N)} \\rho _{j_N }^{(N)} } ).$ Rearranging shows that product states must have form $\\begin{array}{*{20}l}{\\rho } &\\!\\!",
"{= \\sum \\nolimits _{j_1 } { \\cdots \\sum \\nolimits _{j_N } {p_{j_1 }^{(1)} \\rho _{j_1 }^{(1)} \\otimes \\cdots \\otimes p_{j_N }^{(N)} \\rho _{j_N }^{(N)} } }} \\\\{} &\\!\\!",
"{= \\sum \\limits _{j_1 , \\ldots ,j_N } {(p_{j_1 }^{(1)} \\cdots p_{j_N }^{(N)} )\\rho _{j_1 }^{(1)} \\otimes \\cdots \\otimes \\rho _{j_N }^{(N)} }.}",
"\\\\\\end{array}$ Thus, line 2 of (eq.F.9) verifies that this form is equivalent to product-form.",
"Then, to show how (eq.F.9) relates to the notation of a general mixed state, we can unify these all with a vector index only if each mode has redundantly defined quantities of the forms $p_{j_m }^{(m)} = p_{(j_1 , \\ldots ,j_N )}^{(m)} $ and $\\rho _{j_m }^{(m)} = \\rho _{(j_1 , \\ldots ,j_N )}^{(m)} $ , regardless of the values of the indices of labels other than $m$ , so then (eq.F.9) becomes $\\begin{array}{*{20}l}{\\rho } &\\!\\!",
"{= {\\hspace{-20.0pt}}\\sum \\limits _{{\\hspace{17.0pt}}j_1 , \\ldots ,j_N \\rule {0pt}{6.5pt}} {{\\hspace{-18.0pt}}(p_{(j_1 , \\ldots ,j_N )}^{(1)} {\\hspace{-3.0pt}}\\cdots {\\hspace{-2.0pt}} p_{(j_1 , \\ldots ,j_N )}^{(N)} )\\rho _{(j_1 , \\ldots ,j_N )}^{(1)} {\\hspace{-2.0pt}}\\otimes {\\hspace{-3.0pt}} \\cdots {\\hspace{-3.0pt}} \\otimes {\\hspace{-2.0pt}} \\rho _{(j_1 , \\ldots ,j_N )}^{(N)} }} \\\\{} &\\!\\!",
"{= \\sum \\nolimits _j {(p_j^{(1)} \\cdots p_j^{(N)} )} \\rho _j^{(1)} \\otimes \\cdots \\otimes \\rho _j^{(N)},} \\\\\\end{array}$ and then if we abbreviate $p_j=p_j^{(1)} \\cdots p_j^{(N)}$ and $\\rho _j =$ $ \\rho _j^{(1)} \\otimes \\cdots \\otimes \\rho _j^{(N)}$ , then (eq.F.10) takes the general form $\\rho = \\sum \\nolimits _j {p_j \\rho _j } ,$ which shows that since we started from product states in (eq.F.6) and used (eq.5–eq.8) to recover the form of a general state in (eq.F.11), then (eq.5–eq.8) are necessary for product form.",
"Thus, we have now proven that (eq.5–eq.8) are necessary and sufficient conditions for a state to have product form."
],
[
"Proofs and Caveats for the Six Families of Nonlocal Correlation",
"Here we develop some important facts about the six families of nonlocal correlation from [tab:1]Table tab:1."
],
[
"Proof that All Decomposition Probabilities are Expressible in Product Form Without Mode Normalization",
"Given a set of normalized decomposition probabilities $\\lbrace p_j \\rbrace $ , we can always express them in product form as $p_j = \\prod \\nolimits _{m = 1}^N {p_j ^{(m)} } = p_j ^{(1)} \\cdots p_j ^{(N)}\\;\\; \\forall j,$ by using the parameterization $p_j ^{(m)} = p_j ^{x_m^2 ({\\theta } ^{\\lbrace j\\rbrace } )}\\;\\; \\forall j;\\;\\;m \\in 1, \\ldots ,N,$ where for each $j$ , the $x_m ({\\theta } ^{\\lbrace j\\rbrace } )$ for $m \\in 1, \\ldots ,N$ are $N$ -dimensional unit-hyperspherical coordinates [31] with $N-1$ angles ${\\theta } ^{\\lbrace j\\rbrace }\\equiv (\\theta _1^{\\lbrace j\\rbrace } , \\ldots ,\\theta _{N - 1}^{\\lbrace j\\rbrace } )$ which can be restricted to $\\theta _k^{\\lbrace j\\rbrace } \\in [0,\\frac{\\pi }{2}]$ for this application, with a different set for each $j$ .",
"These coordinates have the property that $\\sum \\nolimits _{m = 1}^N {x_m^2 ({\\theta } ^{\\lbrace j\\rbrace } )} = 1\\;\\;\\forall j$ .",
"Notice that in (eq.G.2), we are setting each mode-specific factor $p_j ^{(m)}$ equal to the full probability $p_j$ raised to the power $x_m^2 ({\\theta } ^{\\lbrace j\\rbrace } )$ .",
"Thus, $\\begin{array}{*{20}l}{p_j } &\\!\\!",
"{= p_j ^1 = p_j ^{x_1^2 ({\\theta } ^{\\lbrace j\\rbrace } ) + \\cdots + x_N^2 ({\\theta } ^{\\lbrace j\\rbrace } )} = p_j ^{x_1^2 ({\\theta } ^{\\lbrace j\\rbrace } )} \\cdots p_j ^{x_N^2 ({\\theta } ^{\\lbrace j\\rbrace } )}} \\\\{} &\\!\\!",
"{= p_j ^{(1)} \\cdots p_j ^{(N)}\\;\\;\\forall j,} \\\\\\end{array}$ which also works right to left, given (eq.G.2).",
"Therefore, we have proven that all decomposition probabilities are always expressible in product form as in (eq.G.1), where we do not require mode normalization over $j$ as $\\sum \\nolimits _j {p_j ^{(m)} } {\\hspace{-2.0pt}}={\\hspace{-2.0pt}} 1\\;\\forall m$ , which is actually impossible as we prove in [sec:App.G.2]App.sec:App.G.2.",
"The consequence of (eq.G.3) is that we do not need to define separate sets of families based on whether they have decompositions with product-form probabilities as $p_j = p_j ^{(1)} \\cdots p_j ^{(N)}$ , since such families would be redundant to other families with general probabilities $p_j$ and the same decomposition-state forms.",
"For example, defining a family as $\\rho = \\sum \\nolimits _j (p_j ^{(1)} \\cdots p_j ^{(N)})\\rho _j$ would be exactly the same as Family 1, which is $\\rho = \\sum \\nolimits _j p_{j}\\rho _j$ of [tab:1]Table tab:1."
],
[
"Proof that Product-Form Probabilities with Mode Normalization on the Full Decomposition Index Are Impossible Except Trivially",
"Suppose we require that in addition to having product form (PF), the decomposition probabilities $p_j$ must also have mode normalization over the full decomposition index $j$ , so that $p_j = \\prod \\nolimits _{m = 1}^N {p_j ^{(m)} } = p_j ^{(1)} \\cdots p_j ^{(N)} \\;\\;\\forall j;\\;\\;\\sum \\nolimits _j {p_j ^{(m)} } = 1\\;\\;\\forall m,$ which is not the same thing as mode independence (MI) from (eq.8) due to the sum over the full $j$ rather than mode-specific index $j_m$ where $j \\equiv (j_1 , \\ldots ,j_N )$ .",
"First, for the PF part, we can use the parameterization of [sec:App.G.1]App.sec:App.G.1, which, together with the mode normalization over $j$ , requires that $\\sum \\nolimits _j {p_j ^{(m)} } = \\sum \\nolimits _j {p_j ^{x_m^2 ({\\theta } ^{\\lbrace j\\rbrace } )} } = 1\\;\\;\\forall m.$ For decompositions with $D$ states, this requires that $p_1 ^{x_m^2 ({\\theta } ^{\\lbrace 1\\rbrace } )} + \\cdots + p_D ^{x_m^2 ({\\theta } ^{\\lbrace D\\rbrace } )} = 1\\;\\;\\forall m.$ Now, given that any collection of $D$ real numbers on $(0,1]$ such as $\\lbrace p_{j}\\rbrace $ can be viewed as eigenvalues of some physical $D$ -level state, then they are always exactly determined by a set of $D$ equations for their power sums as $p_1^k + \\cdots + p_D^k = \\sum \\nolimits _{j=1}^{D} {p_j^k } \\equiv P_k ,$ for integers $k \\in 1, \\ldots ,D$ .",
"Thus, (eq.G.6) constitutes a set of generally additional constraints beyond the main set in (eq.G.7), so together (eq.G.6) and (eq.G.7) form a generally overdetermined set of nonlinear equations.",
"It turns out that this overdetermined set does have solutions, but only for cases that are irrelevant for the purpose of defining nontrivial families of decompositions for probability correlation, as we now briefly explain.",
"One way to get a solution to an overdetermined set is to find conditions for which the additional constraints simplify to the main constraints (eq.G.7).",
"Since the powers of $p_j$ in (eq.G.6) are squared unit-hyperspherical coordinates and therefore are each no greater than 1, then for a given $m$ in (eq.G.6), we can never achieve the condition of having both all powers being equal and all being integers, without preventing that for all other $m$ .",
"Specifically, for a particular $m$ , the only way to involve integer powers is to set $x_m^2 ({\\theta } ^{\\lbrace j\\rbrace } ) = 1\\;\\forall j$ and thus satisfy (eq.G.6) for that $m$ , but then for all other $m$, we would have $x_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}^2 ({\\theta } ^{\\lbrace j\\rbrace } ) = 0\\;\\forall j$ , so that we would have $N-1$ unsatisfied equations of the form of (eq.G.6).",
"There are only two ways that could work.",
"It could work for a unipartite system (one with $N=1$ mode, meaning no physical coincidence behavior, which is a system that can never have nonlocal correlation of any kind).",
"Alternatively, it could work with multiple modes if $D=1$ , but that means the state is pure, in which case the question of achieving a special new kind of decomposition probability correlation through (eq.G.4) is irrelevant since there is only one unique decomposition state with probability 1.",
"Therefore, we have outlined the proof that expressing decomposition probabilities $p_j$ as products of $N$ factors where each is separately normalized over full decomposition index $j$ is impossible in all cases except pure states or unipartite states, neither of which can have probability correlation.",
"(An alternative proof would be to use $p$ -norms to achieve $DN$ separate inequalities, which lead to the same trivial exceptions.)"
],
[
"Quasi-Families",
"Since state decompositions are generally not unique, it may happen that for some states, they belong to multiple partially-intersecting families from [tab:1]Table tab:1, but only through different decompositions, so that they do not belong to the intersection of those families since they do not have a single decomposition in that intersection.",
"For example, since Family 2 is $\\rho = \\sum \\nolimits _j {(p_{j_1 }^{(1)} \\cdots p_{j_N }^{(N)} )\\rho _j } $ , while Family 3 is $\\rho {\\hspace{-1.0pt}} ={\\hspace{-3.0pt}} \\sum \\nolimits _j {p_j \\rho _j^{(1)} {\\hspace{-2.5pt}}\\otimes {\\hspace{-1.5pt}}\\cdots {\\hspace{-1.5pt}} \\otimes {\\hspace{-1.0pt}} \\rho _j^{(N)} } $, we see that the probabilities of Family 2 are a strict subset of Family 3, but the decomposition states of Family 2 are a strict superset of Family 3, meaning that neither Family 2 nor Family 3 can be strict subsets of each other, although they can have an intersection, which is Family 4 $\\rho = \\sum \\nolimits _j {(p_{j_1 }^{(1)} \\cdots p_{j_N }^{(N)} )\\rho _j^{(1)} \\otimes \\cdots \\otimes \\rho _j^{(N)} } $ .",
"While any state with a decomposition in Family 4 is in both Family 2 and Family 3, there may exist some states that have different decompositions in Family 2 and Family 3 separately, but no decomposition in Family 4.",
"Thus we define a quasi-family as being a set of states with the property of belonging to multiple partially intersecting families without belonging to their intersection.",
"To identify all quasi-families, we have to check every possible set of every possible number of families, keeping in mind that if any family is a proper subset of another family, then that pair does not constitute a quasi-family, since then membership in the subset guarantees membership in the superset for the same decomposition.",
"Checking all family combinations in [tab:1]Table tab:1 shows that there are only three quasi-families in this group, $\\begin{array}{*{20}l}{\\text{Quasi-Family}} & {2|3,} \\\\{\\text{Quasi-Family}} & {2|5,} \\\\{\\text{Quasi-Family}} & {4|5,} \\\\\\end{array}$ where Quasi-Family $x_{1}|\\cdots |x_{M}$ is a quasi-family involving $M$ families; Family $x_{1}$ through Family $x_{M}$ .",
"Note that due the various subset memberships in this small set of families, there are no quasi-families between more than two families in [tab:1]Table tab:1.",
"The quasi-families may not be as important as the main families.",
"In particular, being able to satisfy multiple family definitions with a single decomposition is what leads to such extreme behavior as achieving product form, whereas a state that only achieved mode-independent (MI) probabilities and MI decomposition states with separate decompositions alone would not achieve product form.",
"Therefore, the fact that a state can have membership to multiple families but not their intersection does not diminish the significance of the families regarding their roles in producing nonlocal correlation.",
"Ultimately, since a state needs to have at least a single decomposition that satisfies a given family definition for it to have the nonlocal correlation achievable by that family, then quasi-families are not as important as families regarding nonlocal correlation.",
"Furthermore, due to the difficulty of constructing states that have different decompositions of particular forms, we do not have examples of quasi-family states at this time (keeping in mind that states within the intersection of the two families involved in a quasi-family are not part of the quasi-family, so we cannot use such intersections to generate examples of quasi-family members).",
"It may be that no states exist in any quasi-families, in which case they can be ignored as physically irrelevant.",
"Nevertheless, we mention quasi-families here in case they turn out to be physically meaningful in some way, and thus this is a possible area for further research."
],
[
"True-Generalized X (TGX) States",
"True-generalized X (TGX) states are defined as a special family of states that are conjectured to be related to all general states (both pure and mixed) by an entanglement-preserving unitary (EPU) transformation, so that the TGX state and the general state connected by such an EPU have the same entanglement, a property called EPU equivalence.",
"The name TGX means “the true generalization of X states with respect to entanglement for all systems as big as or larger than two qubits,” meaning that, just as the X states are EPU equivalent to general states for $2\\times 2$ systems (which is now proven by two independent methods as detailed below), TGX states (if they exist) are generally the larger-system analog of that two-qubit family, having the defining property of EPU equivalence with the set of general states.",
"The leading candidates for TGX states in all systems are called simple states, defined as those states for which their $N$ single-mode reductions are all diagonal in the computational basis, such that all of the off-diagonal parent-state matrix elements appearing in the formal off-diagonals of those reductions are identically zero (meaning that those parent elements do not merely add to zero, but are each themselves zero).",
"As a simple nontrivial example of TGX states, in $2\\times 3$ , since the single-mode reductions are $\\begin{array}{*{20}l}{{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}\\! }",
"&\\!\\!",
"{ =\\!",
"\\left(\\!",
"{\\begin{array}{*{20}c}{\\rho _{1,1} + \\rho _{2,2} + \\rho _{3,3} } & {\\rho _{1,4} + \\rho _{2,5} + \\rho _{3,6} } \\\\{\\rho _{4,1} + \\rho _{5,2} + \\rho _{6,3} } & {\\rho _{4,4} + \\rho _{5,5} + \\rho _{6,6} } \\\\\\end{array}}\\!",
"\\right)} \\\\{{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(2)}\\! }",
"&\\!\\!",
"{ =\\!",
"\\left(\\!",
"{\\begin{array}{*{20}c}{\\rho _{1,1} + \\rho _{4,4} } & {\\rho _{1,2} + \\rho _{4,5} } & {\\rho _{1,3} + \\rho _{4,6} } \\\\{\\rho _{2,1} + \\rho _{5,4} } & {\\rho _{2,2} + \\rho _{5,5} } & {\\rho _{2,3} + \\rho _{5,6} } \\\\{\\rho _{3,1} + \\rho _{6,4} } & {\\rho _{3,2} + \\rho _{6,5} } & {\\rho _{3,3} + \\rho _{6,6} } \\\\\\end{array}}\\!",
"\\right)\\!,} \\\\\\end{array}$ then the parent elements contributing to the off-diagonals of these reductions are $\\rho _{4,1}$ , $\\rho _{5,2}$ , $\\rho _{6,3}$ , $\\rho _{2,1}$ , $\\rho _{5,4}$ , $\\rho _{3,1}$ , $\\rho _{6,4}$ , $\\rho _{3,2}$ , $\\rho _{6,5}$ , and their index-swapped counterparts, so setting all of these to zero not only makes the reductions diagonal, but defines a simple parent state as $\\rho =\\!",
"\\left( {\\begin{array}{*{20}c}{\\rho _{1,1} } & \\cdot & \\cdot & \\cdot & {\\rho _{1,5} } & {\\rho _{1,6} } \\\\\\cdot & {\\rho _{2,2} } & \\cdot & {\\rho _{2,4} } & \\cdot & {\\rho _{2,6} } \\\\\\cdot & \\cdot & {\\rho _{3,3} } & {\\rho _{3,4} } & {\\rho _{3,5} } & \\cdot \\\\\\cdot & {\\rho _{4,2} } & {\\rho _{4,3} } & {\\rho _{4,4} } & \\cdot & \\cdot \\\\{\\rho _{5,1} } & \\cdot & {\\rho _{5,3} } & \\cdot & {\\rho _{5,5} } & \\cdot \\\\{\\rho _{6,1} } & {\\rho _{6,2} } & \\cdot & \\cdot & \\cdot & {\\rho _{6,6} } \\\\\\end{array}} \\right)\\!,$ which we take as a working hypothesis to be the family of $2\\times 3$ TGX states (where dots represent zeros to help show its form).",
"Note that in all of the work on TGX states so far, all evidence strongly supports the hypothesis that simple states are TGX states, so the two terms are often used interchangeably.",
"However, if simple states are ever proved not to have EPU equivalence, the idea of TGX states can then be reserved for EPU equivalent states if they exist.",
"See [17] for many other examples.",
"A Brief History of TGX States: [leftmargin=*,labelindent=4pt] 1.",
"(2013) [17] gave the first definition of TGX states, and the general form was conjectured to be that of simple states.",
"The idea of EPU equivalence was also introduced, and strong numerical evidence was shown that simple TGX states are EPU equivalent to all states for $2\\times 2$ and $2\\times 3$ systems, and this property was conjectured to hold for TGX states in all quantum systems.",
"Numerical evidence was also given showing that literal X states cannot in general be EPU equivalent to general states, with respect to negativity.",
"The maximally entangled-basis (MEB) theorem was conjectured and shown to be fulfilled by TGX states for several example systems.",
"2.",
"(2014) [47] presented the Bloch-vector form of simple candidates for TGX states.",
"3.",
"(2014) [48] proved the conjecture of [17] for the $2\\times 2$ case by showing the implicit existence of an EPU connecting all general states to X states (which are TGX states in $2\\times 2$ systems).",
"4.",
"(2016) [18] presented the multipartite entanglement measure the ent.",
"TGX states (simple candidates) were used to prove the MEB theorem for all discrete quantum systems.",
"It was proved that ME TGX states have the special property of having balanced superposition.",
"Furthermore, ME TGX states were shown to yield indexing patterns that can function as a multipartite Schmidt decomposition state for full $N$ -partite entanglement.",
"This also presented the 13-step algorithm as a method for deterministically constructing all possible ME TGX states in all discrete quantum systems.",
"5.",
"(2017) [49] proved that in $2\\times 3$ systems, literal X states definitely cannot achieve EPU equivalence to general states with respect to negativity, which also proves that literal X states cannot have EPU equivalence in general systems if negativity is a valid measure of entanglement in $2\\times 2$ and $2\\times 3$ .",
"This study also added further numerical evidence agreeing with that of [17] suggesting that the TGX states may indeed achieve EPU equivalence in $2\\times 3$ systems.",
"6.",
"(2018) [50] presented an explicit family of $2\\times 2$ X states parameterized by concurrence and spectrum and proved it to be EPU-equivalent to the set of all states, providing an explicit proof of the original conjecture of [17], and proving the existence of an explicit formula for the EPU of the transformation, as well as yielding an explicit ready-to-use EPU-equivalent state family."
],
[
"Proof that Correlance is a Necessary and Sufficient Measure of All Nonlocal Correlation",
"First, from the definition of correlance $\\mathcal {X}$ in (eq.9–eq.12), $\\mathcal {X}(\\rho ) = 0\\;\\;\\text{iff}\\;\\;\\rho = \\varsigma (\\rho ),$ since $\\mathcal {X}(\\rho )=\\text{tr}[(\\rho - \\varsigma )^2 ]/\\mathcal {N}_{\\mathcal {X}}$ is proportional to the square magnitude of the difference of the Bloch vectors of $\\rho $ and $\\varsigma (\\rho )$ , which is zero iff $\\rho =\\varsigma (\\rho )$ .",
"Next, by Theorem1, proven in [sec:App.F]App.sec:App.F, $\\rho = \\varsigma (\\rho )\\;\\;\\text{iff}\\;\\;\\rho \\;\\;\\text{has product form}.$ Furthermore, from (eq.3), $\\mbox{$\\rho $ has no nonlocal correlation \\,iff\\, $\\rho $ has product form,}$ which is true since discarding any modes by partial tracing leaves the states of the remaining modes unchanged.",
"Then, putting (eq.I.3) into (eq.I.2) and the result into (eq.I.1) gives $\\mbox{$\\mathcal {X}(\\rho )=0$ \\,iff\\, $\\rho $ has no nonlocal correlation,}$ which is what we set out to prove.",
"This means we can use $\\mathcal {X}$ to detect any and all kinds of nonlocal correlation.",
"The only drawback is that it cannot tell us which kind(s) of correlation is(are) present.",
"Nevertheless, if $\\mathcal {X}(\\rho )>0$ , we are guaranteed that $\\rho $ has some nonlocal correlation."
],
[
"Proof that Correlance is Properly Normalized",
"Here we prove that it is valid to normalize correlance $\\mathcal {X}$ with maximally entangled (ME) states (see [sec:App.K]App.sec:App.K for a derivation of the normalization factor).",
"For ease of display, the proof's steps are given as numbered facts.",
"[leftmargin=*,labelindent=4pt] 1.",
"$\\mathcal {X}$ measures nonlocal correlation, as proved in [sec:App.I]App.sec:App.I.",
"(“Nonlocal correlation” abbreviates “full $N$ -partite nonlocal correlation” here; see [sec:VII]Sec.sec:VII for generalizations to distinctly multipartite nonlocal correlation).",
"2.",
"From (eq.10), the raw (unnormalized) correlance $\\widetilde{\\mathcal {X}}(\\rho )\\equiv $ $\\text{tr}[(\\rho -\\varsigma )^{2}]=\\text{tr}(\\rho ^{2})-2\\text{tr}(\\rho \\varsigma )+\\text{tr}(\\varsigma ^{2})$ is a function of input state $\\rho $ and its reduction product $\\varsigma \\equiv \\varsigma (\\rho )\\equiv {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}\\otimes \\cdots \\otimes {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(N)}$ .",
"As we will see later, $\\widetilde{\\mathcal {X}}(\\rho )$ is simply a function of the squared Euclidean distance between the Bloch vectors of $\\rho $ and $\\varsigma (\\rho )$ .",
"Thus correlance $\\mathcal {X}(\\rho )\\equiv \\widetilde{\\mathcal {X}}(\\rho )/{\\mathcal {N}_\\mathcal {X}}$ is a measure of distance between $\\rho $ and its reduction product $\\varsigma $ .",
"3.",
"Pure states can be used to maximize $\\mathcal {X}(\\rho )$ .",
"Proof: $\\mathcal {X}(\\rho )$ is proportional to a squared distance between Bloch vectors (BVs) as $\\mathcal {X}(\\rho ) = \\frac{1}{\\mathcal {N}_{\\mathcal {X}}}\\frac{n - 1}{n}|{\\Gamma } - {\\Gamma }_\\varsigma |^2 $ (see [sec:App.K]App.sec:App.K for details about our BV notation) where ${\\Gamma }$ is the BV of $\\rho $ and ${\\Gamma }_\\varsigma $ is the BV of $\\varsigma \\equiv \\varsigma (\\rho )$ , both of which have real components in a Hermitian operator basis.",
"$\\mathcal {X}(\\rho )$ can be adapted for BV input as $\\mathcal {X}^{\\prime }({\\Gamma })\\equiv \\mathcal {X}(\\rho )$ , and since both terms in the reduction difference vector ${\\Delta }\\equiv {\\Delta }({\\Gamma })\\equiv {\\Gamma }-{\\Gamma }_\\varsigma $ depend on the input state, we can rewrite it as $\\mathcal {X}^{\\prime \\prime }({\\Delta })\\equiv \\mathcal {X}^{\\prime }({\\Gamma })\\equiv \\mathcal {X}(\\rho )=\\frac{1}{\\mathcal {N}_{\\mathcal {X}}}\\frac{n - 1}{n}|{\\Delta }|^2$ .",
"Then, recall that $f(x)$ is strongly convex iff for all $x$ and $y$ in its domain and $p\\geqslant 0$ , there exists some scalar $m\\geqslant 0$ such that $f(px + (1 - p)y) \\leqslant pf(x) + (1 - p)f(y) - \\frac{1}{2}mp(1 - p)||x - y||_2^2 $ , and that any strongly convex function is also convex.",
"Thus, given reduction difference vectors $\\mathbf {X}$ and $\\mathbf {Y}$ , since the quantities $\\mathcal {X}^{\\prime \\prime }(p\\mathbf {X}+(1-p)\\mathbf {Y})$ and $p\\mathcal {X}^{\\prime \\prime }(\\mathbf {X}) + (1 - p)\\mathcal {X}^{\\prime \\prime }(\\mathbf {Y}) - \\frac{1}{2}mp(1 - p)|\\mathbf {X} - \\mathbf {Y}|^2 $ are equal if $m=\\frac{2}{\\mathcal {N}_{\\mathcal {X}}}\\frac{n - 1}{n}$ , then $\\mathcal {X}^{\\prime \\prime }({\\Delta })$ and thus $\\mathcal {X}(\\rho )$ are both strongly convex and convex.",
"Then, recalling Jensen's inequality for convex $f(\\mathbf {x})$ , that $f(\\sum \\nolimits _{j = 1}^{{\\hspace{1.0pt}}D} { p_j \\mathbf {x}_j } ) \\leqslant \\sum \\nolimits _{j = 1}^D { p_j f(\\mathbf {x}_j )}$ , to which a corollary is $f(\\sum \\nolimits _{j = 1}^{{\\hspace{1.0pt}}D} { p_j \\mathbf {x}_j } ) \\leqslant \\max \\lbrace f(\\mathbf {x}_1 ), \\ldots ,f(\\mathbf {x}_D )\\rbrace $, then for any mixed input $\\rho $ , $\\mathcal {X}(\\rho ) \\leqslant \\max \\lbrace \\mathcal {X}(\\rho _1) , \\ldots ,\\mathcal {X}(\\rho _D ) \\rbrace $ where $\\rho _j$ are pure decomposition states of $\\rho $ .",
"Thus maximizers of $\\mathcal {X}(\\rho )$ over all $\\rho $ are pure.",
"4.",
"Since pure states have the trivial decomposition probability of $p_{1}=1$ , with only one pure decomposition state up to global phase, then the only mechanism for nonlocal correlation in pure states is the nonfactorizability of the state itself, meaning its entanglement.",
"5.",
"Since entanglement is the only nonlocal correlation possible for pure states by Fact 4, the pure states of highest $\\mathcal {X}$ are those of highest entanglement.",
"(Since [sec:App.I]App.sec:App.I proved that nonlocal correlation is what $\\mathcal {X}$ measures, this also means that for pure states, the states of highest nonlocal correlation are the states of highest entanglement.)",
"6.",
"The pure states with the highest entanglement are any $\\rho $ for which all of its single-mode reductions ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}$ have the lowest simultaneous purities possible for them to have, given their pure parent state $\\rho $ .",
"We define these states as maximally full-$N$ -partite-entangled states, or just “maximally entangled” (ME) states here, and represent them as $\\rho _{\\text{ME}}$ .",
"(This definition led to the derivation of the automatically normalized entanglement measure the ent $\\Upsilon (\\rho )$ in [18].)",
"7.",
"Therefore, by Fact 5 and Fact 6, the highest $\\mathcal {X}$ for pure states is achieved by pure ME states $\\rho _{\\text{ME}}$ , so then by Fact 3, the pure ME states maximize $\\mathcal {X}$ over all states (mixed and pure), which proves the part of (eq.12) that says $\\mathcal {N}_{\\mathcal {X}}=\\widetilde{\\mathcal {X}}(\\rho _{\\text{ME}})$ .",
"8.",
"In [18] it was shown that the simplest maximally full-$N$ -partite-entangled states are ME TGX states $\\rho _{\\text{ME}_{\\text{TGX}}}$ , since they achieve the minimum simultaneous single-mode purities while also having equal superposition coefficients, with not all levels being nonzero.",
"Therefore, since ME TGX states have the same entanglement as general ME states, then by Fact 7, this proves the part of (eq.12) that says $\\mathcal {N}_{\\mathcal {X}}=\\widetilde{\\mathcal {X}}(\\rho _{\\text{ME}_{\\text{TGX}}})$ .",
"Thus, we have proven that any pure ME state can be used to normalize $\\mathcal {X}$ over all states, both mixed and pure.",
"The reason for using ME TGX states is that they are generally simpler than general ME states, since it was proved in [18] that ME TGX states always have balanced superposition and not all levels are nonzero.",
"Furthermore, the ME TGX states can be methodically generated, using the 13-step algorithm of [18].",
"Regarding the distance interpretation of $\\mathcal {X}$ from Fact 2, see [51] for the intimate connection between distance measures of entanglement and convex-roof extensions of entanglement monotones."
],
[
"Proof and Calculation of Explicit Correlance Normalization Factors",
"This proof makes extensive use of multipartite Bloch vectors [52], [47], and therefore this appendix has two parts; [sec:App.K.1]App.sec:App.K.1 reviews multipartite-Bloch-vector formalism, and [sec:App.K.2]App.sec:App.K.2 derives the normalization factors."
],
[
"Review of Bloch-Vector Quantities",
"The concept of a Bloch vector is simply to use a set of operators that is somehow complete in that it allows us to expand any operator as a linear combination of that set of operators.",
"The Bloch vector is then the list of scalar coefficients of that expansion.",
"The idea of Bloch vectors actually originated as Stokes parameters in 1852 [53], which were used as an operational method of describing classical light.",
"However, the quantum-mechanical density matrix was not invented until 1927, by von Neumann [54], and the modern idea of Bloch vectors for quantum states came from Felix Bloch's 1946 treatment of mixed-state qubits [52], which were soon-after connected with the density matrix.",
"The 1961 development of the Gell-Mann (GM) matrices by Ne'eman and Gell-Mann [55], [56] paved the way for describing states larger than a qubit, but it was not until 1981 that a unipartite $n$ -level Bloch vector was devised, by Hioe and Eberly [57].",
"Soon after, many multipartite descriptions were attempted and many works treat simple cases of these.",
"Therefore, here we present a brief general treatment of multipartite Bloch vectors, from the more complete work in [47] from 2014.",
"Consider a multipartite system of $N$ subsystems (modes), with Hilbert space $\\mathcal {H}^{(1)}\\otimes \\mathcal {H}^{(2)}\\otimes \\cdots \\otimes \\mathcal {H}^{(N)}$ , where $\\mathcal {H}^{(m)}$ is the Hilbert space of mode $m$ , where $\\mathcal {H}$ has $n$ total levels such that $n = n_1 \\cdots n_N $ , where $n_m$ is the number of levels of mode $m$ .",
"Now let $\\lbrace \\nu _k ^{(m)} \\rbrace $ be a complete basis of $n_m^2$ operators for mode $m$ , such that all operators in $\\mathcal {H}^{(m)}$ can be expanded as linear combinations of $\\lbrace \\nu _k ^{(m)} \\rbrace $ , and where $\\nu _0 ^{(m)} \\equiv I^{(m)} $ is the identity for $\\mathcal {H}^{(m)}$ .",
"Furthermore, under the Hilbert-Schmidt (HS) inner product $A\\cdot B\\equiv \\text{tr}(A^{\\dag }B)$ , suppose that $\\lbrace \\nu _k ^{(m)} \\rbrace $ has uniform orthogonality, $\\text{tr}(\\nu _{j_{m}}^{(m)\\dag } \\nu _{k_{m}}^{(m)} ) = n_{m} \\delta _{{j_{m}},{k_{m}}},$ for all ${j_{m}},{k_{m}}\\in 0,\\ldots ,n_{m}^{2}-1$ , where “uniform” means that only one case is needed to cover all indices, which allows the simplest transition to a multipartite basis.",
"Thus, we can define a multipartite basis as $\\nu _{\\mathbf {k}}\\equiv \\nu _{k_1 , \\ldots ,k_N }^{(1,\\ldots ,N)} \\equiv \\nu _{k_1 }^{(1)} \\otimes \\cdots \\otimes \\nu _{k_N }^{(N)}$ for $k_{m}\\in 0,\\ldots ,n_{m}^{2}-1\\;\\,\\forall m\\in \\lbrace 1,\\ldots ,N\\rbrace $ , where the vector-index subscript indicates the multipartite nature of the basis, where the number of elements in the vector is the number of modes $N$ .",
"The set $\\lbrace \\nu _{\\mathbf {k}}\\rbrace $ inherits the uniform orthogonality of its modes as $\\text{tr}(\\nu _{\\mathbf {j}}^{\\dag } \\nu _{\\mathbf {k}} ) = n\\delta _{\\mathbf {j},\\mathbf {k}},$ valid for all $k_{m}\\in 0,\\ldots ,n_{m}^{2}-1\\;\\,\\forall m\\in \\lbrace 1,\\ldots ,N\\rbrace $ , and where $\\delta _{\\mathbf {j},\\mathbf {k}}\\equiv \\delta _{j_1 ,k_1 } \\cdots \\delta _{j_N ,k_N }$ .",
"The HS completeness of $\\lbrace \\nu _{\\mathbf {k}}\\rbrace $ lets us express all density operators as $\\rho =\\sum \\nolimits _{\\mathbf {k}}{c_{\\mathbf {k}}\\nu _{\\mathbf {k}} } = \\frac{1}{n}\\!",
"( {\\nu _{\\mathbf {0}} + A \\sum \\nolimits _{\\mathbf {k}\\ne \\mathbf {0}} {\\Gamma _{\\mathbf {k}} \\nu _{\\mathbf {k}} } } )$ , where $c_{\\mathbf {k}}$ are generally complex scalars, and $\\Gamma _{\\mathbf {k}}\\equiv \\Gamma _{\\mathbf {k}\\ne \\mathbf {0}}\\equiv \\Gamma _{k_{1},\\ldots ,k_{N}}\\equiv \\frac{1}{A}\\frac{c_{k_{1},\\ldots ,k_{N}}}{c_{0,\\ldots ,0}}$ are the $n^2 -1$ scalars known as Bloch components that constitute a Bloch vector ${\\Gamma }$ , and we choose $\\Gamma _{0,\\ldots ,0}\\equiv \\frac{c_{0,\\ldots ,0}}{c_{0,\\ldots ,0}}=1$ by convention.",
"Then, computing the purity $P\\equiv \\text{tr}(\\rho ^2)$ and applying the unitization condition that $|{\\Gamma }|=1$ for pure states $\\rho $ , then $A=\\sqrt{n-1}$ , and we obtain the multipartite Bloch-vector expansion of $\\rho $ as $\\begin{array}{*{20}c}{\\rho } &\\!\\!",
"{=\\frac{1}{n}(\\nu _{\\mathbf {0}} + \\sqrt{n - 1} {\\Gamma }\\cdot {\\nu } ),} \\\\\\end{array}$ where ${\\Gamma }$ is the list of scalars $\\lbrace \\Gamma _{\\mathbf {k} \\ne \\mathbf {0}}\\rbrace $ , and ${\\nu }$ is the list of operators $\\lbrace \\nu _{\\mathbf {k}\\ne \\mathbf {0}}\\rbrace $ in the same order as $\\lbrace \\Gamma _{\\mathbf {k} \\ne \\mathbf {0}}\\rbrace $ , and the dot product in the context of (eq.K.4) is just an abbreviation for $\\sum \\nolimits _{\\mathbf {k}\\ne \\mathbf {0}} {\\Gamma _{\\mathbf {k}} \\nu _{\\mathbf {k}} }$ .",
"Then, applying (eq.K.3) to (eq.K.4) gives the multipartite Bloch components as $\\begin{array}{*{20}c}{\\Gamma _{\\mathbf {k} \\ne \\mathbf {0}}} &\\!\\!",
"{= \\frac{1}{\\sqrt{n - 1}}\\text{tr}(\\rho \\nu _{\\mathbf {k}}^{\\dag } ).}",
"\\\\\\end{array}$ Note that if $\\lbrace \\nu _{\\mathbf {k}}\\rbrace $ consists entirely of Hermitian operators, then the $\\Gamma _{\\mathbf {k} \\ne \\mathbf {0}}$ will all be real.",
"The Bloch vector ${\\Gamma }$ contains all of the same information as $\\rho $ and can be used as an alternative method of representing any physical state.",
"The purity of $\\rho $ is $P \\equiv \\text{tr}(\\rho ^2 ) = \\frac{1}{n}(1 + (n - 1)|{\\Gamma } |^2 )$ , so we can define the Bloch purity as $\\begin{array}{*{20}c}{P_{\\text{B}}} &\\!\\!",
"{\\equiv |{\\Gamma } |^2 = \\sum \\limits _{\\mathbf {k} \\ne \\mathbf {0}} {|\\Gamma _{\\mathbf {k}} |^2 } = \\frac{n\\text{tr}(\\rho ^2 ) - 1}{n - 1},} \\\\\\end{array}$ which obeys $0\\leqslant P_{\\text{B}}\\leqslant 1$ , such that $P_{\\text{B}}=1$ for pure states, $0\\leqslant P_{\\text{B}}<1$ for general strictly mixed states, and $P_{\\text{B}}=0$ for the maximally mixed state.",
"So far, we have merely specified properties of $\\lbrace \\nu _{\\mathbf {k}}\\rbrace $ without explaining how to make it.",
"One simple way to construct it is, for each $m\\in \\lbrace 1,\\ldots ,N\\rbrace $ , let $\\begin{array}{*{20}c}{\\nu _0^{(m)} \\equiv \\lambda _0^{(m)} \\equiv I^{(m)}} & {\\text{and}} & {\\nu _{k_{m}}^{(m)} \\equiv \\sqrt{\\frac{n_{m}}{2}} \\lambda _{k_{m}}^{(m)},} \\\\\\end{array}$ where ${k_{m}}\\in 1,\\ldots ,n_{m}^{2}-1$ , and the $\\lambda _{k_{m}}^{(m)}$ are generalized Gell-Mann (GM) matrices in mode $m$ of $n_m$ levels, given by the implicit equations $\\begin{array}{*{20}l}{\\lambda _{\\scriptstyle \\alpha ^2 - 2(\\alpha -\\beta ) - 1 \\hfill \\atop \\scriptstyle \\alpha ^2 - 2(\\alpha -\\beta ) \\hfill }^{(m)} \\equiv \\begin{array}{*{20}c}{\\; E_{(\\alpha ,\\beta )}^{[n_m]}+E_{(\\beta ,\\alpha )}^{[n_m]} } \\\\{\\!",
"i\\!\\left({E_{(\\alpha ,\\beta )}^{[n_m]}-E_{(\\beta ,\\alpha )}^{[n_m]} }\\right) } \\\\\\end{array}\\!\\!",
"; \\;\\alpha > \\beta ,} \\\\{\\lambda _{\\alpha ^2 - 1}^{(m)} \\equiv \\sqrt{\\frac{2}{\\alpha (\\alpha - 1)}} \\left(\\!",
"{ - (\\alpha - 1)E_{(\\alpha ,\\alpha )}^{[n_m]} +\\!",
"\\sum \\limits _{q = 1}^{\\alpha - 1} {E_{(q,q)}^{[n_m]} } }\\!",
"\\right)\\!",
",} \\\\\\end{array}$ where $2 \\leqslant \\alpha \\leqslant n_m$ and $1 \\leqslant \\beta \\leqslant \\alpha -1$ , and $E_{(\\alpha ,\\beta )}^{[n_m]}\\equiv |\\alpha \\rangle \\langle \\beta |$ is the $n_m\\times n_m$ matrix with a 1 in the row-$\\alpha $ , column-$\\beta $ entry and 0 elsewhere, given that the top row is row 1, and the left column is column 1, and $\\lambda _{0}^{(m)}\\equiv I^{[n_m]}$ is the $n_m\\times n_m$ identity matrix for mode $m$ .",
"A given pair of integers $\\alpha ,\\beta $ determines the conventional label for each GM matrix.",
"Note that the GM matrices are not preferable as a basis for general multipartite Bloch vectors (though they are often used for that), because their orthogonality relations require two cases to include the identity, resulting in $2^N$ cases for the orthogonality of a general $N$ -partite system.",
"Thus, putting (eq.K.8) into (eq.K.7) and using that in (eq.K.2), we obtain a realization for the multipartite basis $\\lbrace \\nu _{\\mathbf {k}}\\rbrace $ , which has only one orthogonality case, which is given by (eq.K.3).",
"We now develop some formalism that will be useful in establishing the results we will use to prove the normalization factors of the correlance.",
"For multipartite systems, the implicit-basis representation of ${\\Gamma }$ is not intuitive, since any vector or matrix representation uses relative positions on a page to encode the basis, so listing out components with vector indices is not helpful.",
"Therefore, we will simply keep basis-explicit notation using a vector-indexed basis, and to that end we introduce a uniform standard basis (USB) as $\\mathbf {b}_{\\mathbf {k}} \\equiv \\mathbf {b}_{k_1 , \\ldots ,k_N }^{(1, \\ldots ,N)} \\equiv \\frac{1}{{\\sqrt{n} }}\\nu _{\\mathbf {k}} = \\frac{1}{{\\sqrt{n} }}\\nu _{k_1 }^{(1)} \\otimes \\cdots \\otimes \\nu _{k_N }^{(N)} .$ If we then define the Hilbert-Schmidt inner product between Bloch vectors $\\mathbf {A}$ and $\\mathbf {B}$ in the USB as $\\mathbf {A} \\cdot \\mathbf {B} \\equiv \\text{tr}(\\mathbf {A}^{\\dag } \\mathbf {B}),$ then any USB as defined above has orthonormality $\\mathbf {b}_\\mathbf {j} \\cdot \\mathbf {b}_\\mathbf {k} = \\delta _{\\mathbf {j},\\mathbf {k}} .$ Then, in the USB, Bloch vectors have explicit-basis form, ${\\Gamma } = \\sum \\limits _{\\mathbf {k} \\ne \\mathbf {0}}^\\mathbf {K} {\\Gamma _\\mathbf {k} \\mathbf {b}_\\mathbf {k} },$ where $K_m \\equiv n_{m}^{2}-1$ and $k_m \\in 0,\\ldots ,K_m$ , so only the case of all $k_m =0$ is excluded from the sum.",
"Thus, in the USB, ${\\Gamma }$ is a matrix, and $\\Gamma _\\mathbf {k}$ have the same values as in (eq.K.5) but are now given by $\\Gamma _{\\mathbf {k} \\ne \\mathbf {0}} = {\\Gamma } \\cdot \\mathbf {b}_\\mathbf {k} = \\text{tr}({\\Gamma } \\mathbf {b}_\\mathbf {k} ^{\\dag } ) = \\sqrt{\\frac{n}{{n - 1}}} \\text{tr}(\\rho \\mathbf {b}_\\mathbf {k} ^{\\dag } ),$ and the density matrix can be expanded more simply as $\\rho = \\frac{1}{n}(I + \\sqrt{n(n - 1)} {\\Gamma } ),$ where note that we use boldness to distinguish Bloch-vector objects from density matrices, despite both being represented as matrices here.",
"The (informal) overlap of any two states $\\rho _A$ and $\\rho _B$ is then (using their Hermiticity in the HS inner product), $\\text{tr}(\\rho _A \\rho _B ) = \\frac{1}{n}\\left( {1 + (n - 1){\\Gamma } _A \\cdot {\\Gamma } _B } \\right),$ where, since all ${\\Gamma }$ are also Hermitian, using (eq.K.12) and supposing that the $\\mathbf {b}_\\mathbf {k}$ are Hermitian as well so that all Bloch components are real, then ${\\Gamma } _A \\cdot {\\Gamma } _B = \\text{tr}({\\Gamma } _A {\\Gamma } _B ) = \\sum \\limits _{\\mathbf {k} \\ne \\mathbf {0}}^\\mathbf {K} {(\\Gamma _A )_\\mathbf {k} (\\Gamma _B )_\\mathbf {k} } ,$ and so the Bloch purity of a Bloch vector in the USB is $P_\\text{B} \\equiv |{\\Gamma } |^2 = {\\Gamma } \\cdot {\\Gamma } = \\text{tr}({\\Gamma }^2 ) = \\sum \\limits _{\\mathbf {k} \\ne \\mathbf {0}}^\\mathbf {K} {|\\Gamma _\\mathbf {k} |^2 } .$ For reference, a useful way to obtain ${\\Gamma }$ in the USB is ${\\Gamma } = \\frac{1}{{\\sqrt{n(n - 1)} }}(n\\rho - I).$ The multipartite reduction to $S\\in 1,\\ldots , N$ modes notated by mode-label vector $\\mathbf {m}\\equiv (m_{1},\\ldots ,m_{S})$ is given in density-matrix form expanded by its USB reduced Bloch vector ${{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}$ as ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(\\mathbf {m})} = \\frac{1}{{n_{\\mathbf {m}} }}(I^{(\\mathbf {m})} + \\sqrt{n_{\\mathbf {m}} (n_{\\mathbf {m}} - 1)} {\\hspace{1.0pt}}{{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})} ),$ where $n_{\\mathbf {m}}\\equiv n_{m_1 } \\cdots n_{m_S }$ , and ${{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}$ has components ${\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}_{\\mathbf {k}_{\\mathbf {m}} \\ne \\mathbf {0}_\\mathbf {m}}={{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}\\cdot \\mathbf {b}_{\\mathbf {k}_\\mathbf {m} }^{(\\mathbf {m})} ,$ where $\\mathbf {k}_{\\mathbf {m}}\\equiv (k_{m_1 } , \\ldots ,k_{m_S } )$ , $\\mathbf {0}_{\\mathbf {m}}\\equiv (0_{m_1 } , \\ldots ,0_{m_S } )$ , and the USB of the reduced system is $\\mathbf {b}_{\\mathbf {k}_\\mathbf {m} }^{(\\mathbf {m})}\\equiv \\frac{1}{{\\sqrt{n_\\mathbf {m} } }}\\nu _{k_{m_1 } }^{(m_1 )} \\otimes \\cdots \\otimes \\nu _{k_{m_S } }^{(m_S )} ,$ where the mode-specific basis operators $\\nu _{k_{m } }^{(m )}$ are defined in (eq.K.7).",
"Note that modes in $\\mathbf {m}$ are not necessarily contiguous or ordered, but if order is changed, permutation unitaries are needed; see ([18], App.B).",
"Here is where our formalism will start to show benefits (with more to follow below).",
"First, note that these definitions cause ${{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}$ to be automatically unitized, meaning that $|{{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}|=1$ iff ${{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}$ is a pure state and $|{{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}|=0$ iff ${{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}$ is ideally maximally mixed for a general system of mode structure $\\mathbf {m}$ .",
"In particular, we get a simple relationship that connects the Bloch components of the reduction to the full Bloch-vector components by a common factor as $\\begin{array}{*{20}l}{{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}_{\\mathbf {k}_\\mathbf {m}\\ne \\mathbf {0}_\\mathbf {m}} } &\\!\\!",
"{=\\sqrt{\\frac{{n - 1}}{{n_\\mathbf {m} - 1}}} \\Gamma _{\\mathbf {k}\\delta _{\\mathbf {k}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} ,\\mathbf {0}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} } }} \\\\{} &\\!\\!",
"{= \\sqrt{\\frac{{n - 1}}{{n_\\mathbf {m} - 1}}} \\Gamma _{0, \\ldots ,0,k_{m_1 } ,0, \\ldots ,0,k_{m_S } ,0, \\ldots ,0} } \\\\\\end{array},$ where $\\mathbf {{\\mathop {m}\\limits ^{{\\hspace{-3.5pt}}~_{\\overline{{\\hspace{7.0pt}}}}}}}$ is the ordered set of all full-system mode labels not in $\\mathbf {m}$ , and where $\\mathbf {k}\\equiv (k_1 , \\ldots ,k_N )$ , $\\mathbf {k}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}}\\equiv (k_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}_1 } , \\ldots ,k_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}_{N - S} } )$ , and $\\delta _{\\mathbf {k}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} ,\\mathbf {0}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} }\\equiv \\delta _{k_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}_1 } ,0} \\cdots \\delta _{k_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}_{N - S} } ,0}$ , where ${\\mathop {m}\\limits ^{{\\hspace{-3.5pt}}~_{\\overline{{\\hspace{7.0pt}}}}}}_j$ is the $j$ th mode label in $\\mathbf {{\\mathop {m}\\limits ^{{\\hspace{-3.5pt}}~_{\\overline{{\\hspace{7.0pt}}}}}}}$ .",
"Thus, the explicit-basis form of a multipartite reduced Bloch vector is ${{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})} {\\hspace{-1.0pt}}={\\hspace{-4.0pt}} \\sum \\limits _{\\mathbf {k}_\\mathbf {m} \\ne \\mathbf {0}_\\mathbf {m} }^{\\mathbf {K}_\\mathbf {m} } {{\\hspace{-7.0pt}}{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}_{\\mathbf {k}_{\\mathbf {m}}} \\mathbf {b}_{\\mathbf {k}_\\mathbf {m} }^{(\\mathbf {m})} } {\\hspace{-1.0pt}} ={\\hspace{-1.0pt}} \\sqrt{{\\hspace{-2.0pt}}\\frac{{n - 1}}{{n_\\mathbf {m} - 1}}}{\\hspace{-2.0pt}} \\sum \\limits _{\\mathbf {k}_\\mathbf {m} \\ne \\mathbf {0}_\\mathbf {m} }^{\\mathbf {K}_\\mathbf {m} } {{\\hspace{-7.0pt}}\\Gamma _{\\mathbf {k}\\delta _{\\mathbf {k}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} ,\\mathbf {0}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} } } \\mathbf {b}_{\\mathbf {k}_\\mathbf {m} }^{(\\mathbf {m})} } ,$ where $\\mathbf {K}_{\\mathbf {m}}\\equiv (K_{m_{1}},\\ldots ,K_{m_{S}})$ , where $K_{m_{j}}\\equiv n_{m_{j}}^2 -1$ .",
"The Bloch purity of a reduction in a Hermitian USB is ${P}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}_\\text{B} {\\hspace{-2.0pt}}\\equiv {\\hspace{-1.0pt}} |{{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})} |^2 {\\hspace{-2.0pt}}={\\hspace{-2.0pt}} {{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})} \\cdot {{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})} {\\hspace{-2.0pt}}={\\hspace{-1.0pt}} \\text{tr}({{{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}}^{2}) {\\hspace{-1.0pt}}={\\hspace{-5.0pt}}\\sum \\limits _{\\mathbf {k}_\\mathbf {m} \\ne \\mathbf {0}_\\mathbf {m}}^{\\mathbf {K}_\\mathbf {m}} {{\\hspace{-7.0pt}}|{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})} _{\\mathbf {k}_\\mathbf {m}} |^2 } {\\hspace{-1.0pt}}.$ It is often tidier to simply organize the full Bloch-vector components corresponding to certain reductions into groups by defining correlation vectors (in the USB of the full Bloch vector) as $\\mathbf {X}^{(\\mathbf {m})}\\equiv \\sum \\limits _{\\mathbf {k}_\\mathbf {m} = \\mathbf {1}_\\mathbf {m} }^{\\mathbf {K}_\\mathbf {m} } {{\\hspace{-7.0pt}}\\Gamma _{\\mathbf {k}\\delta _{\\mathbf {k}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} ,\\mathbf {0}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} } }\\mathbf {b}_{\\mathbf {k}\\delta _{\\mathbf {k}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} ,\\mathbf {0}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} } }},$ where $\\mathbf {1}_{\\mathbf {m}}\\equiv (1_{m_{1}},\\ldots ,1_{m_{S}})$ , with the property that ${\\Gamma } = \\sum \\limits _{k = 1}^N {\\sum \\limits _{l = 1}^{\\binom{N}{k}} {\\mathbf {X}^{((\\text{nCk}[\\mathbf {c},k])_{l, \\cdots } )} } } ,$ where $\\mathbf {c}\\equiv (1,\\ldots ,N)$ , and $\\text{nCk}[\\mathbf {v},k]$ is the vectorized $n$ -choose-$k$ function that gives a matrix whose rows are the unique combinations of the elements of $\\mathbf {v}$ chosen $k$ at a time, and $A_{l,\\cdots }$ is the $l$ th row of a matrix $A$ .",
"So, for example, for a tripartite system (meaning $N=3$ ), $\\begin{array}{*{20}l}{{\\Gamma } = } &\\!\\!",
"{\\mathbf {X}^{(1)} + \\mathbf {X}^{(2)} + \\mathbf {X}^{(3)} } \\\\{} &\\!\\!",
"{ + \\mathbf {X}^{(1,2)} + \\mathbf {X}^{(2,3)} + \\mathbf {X}^{(1,3)} } \\\\{} &\\!\\!",
"{ + \\mathbf {X}^{(1,2,3)}, } \\\\\\end{array}$ where, just expanding a few terms as examples, ${\\hspace{-4.0pt}}\\begin{array}{*{20}c}{\\begin{array}{*{20}c}{\\mathbf {X}^{(2)} {\\hspace{-1.0pt}}={\\hspace{-5.0pt}} \\sum \\limits _{k_2 = 1}^{K_2 } {{\\hspace{-3.0pt}}\\Gamma _{0,k_2 ,0} \\mathbf {b}_{0,k_2 ,0} },} & {\\mathbf {X}^{(1,3)} {\\hspace{-1.0pt}} ={\\hspace{-5.0pt}} \\sum \\limits _{k_1 ,k_3 = 1,1}^{K_1 ,K_3 } {{\\hspace{-7.0pt}}\\Gamma _{k_1 ,0,k_3 } \\mathbf {b}_{k_1 ,0,k_3 } },} \\\\\\end{array}} \\\\{\\mathbf {X}^{(1,2,3)} {\\hspace{-1.0pt}} ={\\hspace{-5.0pt}} \\sum \\limits _{k_1 ,k_2 ,k_3 = 1,1,1}^{K_1 ,K_2 ,K_3 } {{\\hspace{-9.0pt}}\\Gamma _{k_1 ,k_2 ,k_3 } \\mathbf {b}_{k_1 ,k_2 ,k_3 } }.}",
"\\\\\\end{array}{\\hspace{-4.0pt}}$ We can even define unitized correlation vectors as $\\widetilde{\\mathbf {X}}^{(\\mathbf {m})}\\equiv \\sqrt{\\frac{n-1}{n_{\\mathbf {m}}-1}}\\mathbf {X}^{(\\mathbf {m})}=\\sum \\limits _{\\mathbf {k}_\\mathbf {m} = \\mathbf {1}_\\mathbf {m} }^{\\mathbf {K}_\\mathbf {m} } {{\\hspace{-7.0pt}}{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}_{\\mathbf {k}_\\mathbf {m}}\\mathbf {b}_{\\mathbf {k}\\delta _{\\mathbf {k}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} ,\\mathbf {0}_{\\mathbf {{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} } }},$ with the property that the Bloch purity of $\\widetilde{\\mathbf {X}}^{(\\mathbf {m})}$ matches that of ${{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}$ so that $|\\widetilde{\\mathbf {X}}^{(\\mathbf {m})}|^2 = |{{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}|^2$ , and the only difference between them is that the matrix basis of $\\widetilde{\\mathbf {X}}^{(\\mathbf {m})}$ lives in the full space of parent state ${\\Gamma }$ , whereas the matrix basis of ${{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(\\mathbf {m})}$ lives in the space of mode group $\\mathbf {m}$ .",
"As proved in [47], we only need the single-mode reductions to quantify full $N$ -partite entanglement, and as such, it is useful to define the liaison vector, ${\\Lambda } \\equiv {\\Gamma } - \\sum \\limits _{m = 1}^N {\\mathbf {X}^{(m)} } = {\\Gamma } - \\sum \\limits _{m = 1}^N {\\sqrt{\\frac{{n_m - 1}}{{n - 1}}} \\widetilde{\\mathbf {X}}^{(m)} } ,$ which is the sum of all strictly multipartite reductions, so for example, in a tripartite system, ${\\Lambda } = \\mathbf {X}^{(1,2)} + \\mathbf {X}^{(2,3)} + \\mathbf {X}^{(1,3)} + \\mathbf {X}^{(1,2,3)}.$ Since ${\\Lambda }$ is the group of all strictly multimode Bloch components, the vector indices $\\mathbf {k}$ of its components $\\Lambda _{\\mathbf {k}}$ always have at least two nonzero indices $k_m$ .",
"For example, $\\mathbf {X}^{(1,3)}$ and $\\mathbf {X}^{(1,2,3)}$ of (eq.K.28) both have two or more nonzero indices, as do all correlation vectors that make up ${\\Lambda }$ .",
"Note that the Bloch purity is expressible as $|{\\Gamma }|^2 = |{\\Lambda }|^2 + \\sum \\limits _{m = 1}^N {|\\mathbf {X}^{(m)} |^2 } ,$ which we use in our correlance-normalization proof.",
"Another useful fact is that all separable states obey $\\Gamma _{\\mathbf {k} \\ne \\mathbf {0}} = \\frac{{A_\\mathbf {k} }}{{\\sqrt{n - 1} }}\\sum \\nolimits _j {p_j {\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(1)}_{j|k_1 } \\cdots {\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(N)}_{j|k_N } } ,$ where ${\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{j|k_m }\\equiv ({{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{j})_{k_m}$ , and $A_{\\mathbf {k}}\\equiv A_{k_{1}}\\cdots A_{k_{N}};{\\hspace{8.0pt}}A_{k_{m}}\\equiv (\\sqrt{n_{m}-1}{\\hspace{1.0pt}})^{\\text{sgn}(k_{m})}.$ For product-form states, using (eq.5–eq.8) in (eq.K.33) leads to $\\Gamma _{\\mathbf {k} \\ne \\mathbf {0}} = \\frac{{A_\\mathbf {k} }}{{\\sqrt{n - 1} }}{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(1)}_{k_1 } \\cdots {\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(N)}_{k_N } ,$ where we used the fact that ${\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{k_m }\\equiv \\sum \\nolimits _{j_m}{p_{j_m}{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{{j_m}|k_m }}$ .",
"There are many useful applications of this formalism (and much more to say about it), which was introduced in the present form in [47], but this will suffice as a good working reference both here and for future research."
],
[
"Proof of Explicit Correlance Normalization Factors",
"Expanding (eq.12) in its simplest form in ME TGX states (since they have nice properties such as balanced superposition, multiple levels of zero probability, and diagonal reductions), we get $\\mathcal {N}_{\\mathcal {X}} = 1 - 2\\text{tr}(\\rho _{\\text{ME}_{\\text{TGX}} } \\varsigma _{\\text{ME}_{\\text{TGX}} } ) + \\text{tr}(\\varsigma _{\\text{ME}_{\\text{TGX}} } ^2 ),$ where $\\varsigma _{\\text{ME}_{\\text{TGX}} }\\equiv \\varsigma (\\rho _{\\text{ME}_{\\text{TGX}} })$ where $\\varsigma (\\rho )$ is the reduction product from (eq.11).",
"Since the reduction product has product form, then its purity also has product form as $\\begin{array}{*{20}l}{\\text{tr}(\\varsigma _{\\text{ME}_{\\text{TGX}} } ^2 )} &\\!\\!",
"{ = P({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{\\text{ME}_{\\text{TGX}} }) \\cdots P({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(N)}_{\\text{ME}_{\\text{TGX}} })} \\\\{} &\\!\\!",
"{ = \\prod \\nolimits _{m = 1}^N {P_{\\text{MP}}^{(m)} (L_* )},\\rule {0pt}{11pt} } \\\\\\end{array}$ where $P(\\rho )\\equiv \\text{tr}(\\rho ^{2})$ , and $P_{\\text{MP}}^{(m)} (L_* )$ is the minimum physical reduction purity of mode $m$ given a pure maximally full-$N$ -partite-entangled parent state with $L_*$ levels of equal nonzero probabilities.",
"To calculate each $P_{\\text{MP}}^{(m)} (L_* )$ , first make the more general function, $\\begin{array}{*{20}l}{P_{\\text{MP}}^{(m)} (L)\\equiv } &\\!\\!",
"{ \\bmod (L,n_m )\\left( {\\frac{{1 + \\text{floor} (L/n_m )}}{L}} \\right)^2 } \\\\{} &\\!\\!",
"{ + (n_m - \\bmod (L,n_m ))\\left( {\\frac{{\\text{floor}(L/n_m )}}{L}} \\right)^2 ,} \\\\\\end{array}$ where $\\bmod (a,b) \\equiv a - \\text{floor}(a/b)b$ , and use that to define $M(L) \\equiv 1 - \\frac{1}{N}\\sum \\limits _{m = 1}^N {\\frac{{n_m P_{\\text{MP}}^{(m)} (L) - 1}}{{n_m - 1}}} .$ Then find the set $\\mathbf {L}_{*}\\equiv \\lbrace L_{*}\\rbrace $ which are the values of $L$ that satisfy $\\mathop {\\min }\\limits _{L \\in 2, \\ldots ,n_{\\,\\overline{{\\hspace{-1.8pt}}\\max ^{~^{~^{~}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"}}\\,} [1 - M(L)],$ where $n_{\\,\\overline{{\\hspace{-1.8pt}}\\max ^{~^{~^{~}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"}}\\,\\!\\equiv \\!",
"n/n_{\\max }$ and $n_{\\max }\\!\\equiv \\!\\max (\\mathbf {n})\\!=\\!\\lbrace n_1,\\ldots ,n_N\\rbrace $ .",
"Then by convention let $L_{*}\\equiv \\min (\\mathbf {L}_{*})$ and use that to compute each $P_{\\text{MP}}^{(m)} (L_* )$ .",
"Note that we could simplify things slightly, but these quantities have physical significance in the context of the entanglement measure the ent [18], so we use these forms for conceptual consistency.",
"Thus, putting (eq.K.37) into (eq.K.36) gives $\\mathcal {N}_{\\mathcal {X}} = 1 - 2\\text{tr}(\\rho _{\\text{ME}_{\\text{TGX}} } \\varsigma _{\\text{ME}_{\\text{TGX}} } ) + \\prod \\limits _{m = 1}^N {P_{\\text{MP}}^{(m)} (L_* )}.$ To get the overlap term in (eq.K.41), it is helpful to use a Bloch-vector formalism, such as that given in [sec:App.K.1]App.sec:App.K.1.",
"First, we note that Bloch components of $\\varsigma $ have the product form of (eq.K.35) as $\\Gamma _{\\mathbf {k} \\ne \\mathbf {0}}(\\varsigma ) = \\frac{{A_\\mathbf {k} }}{{\\sqrt{n - 1} }}{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(1)}_{k_1 }(\\varsigma ) \\cdots {\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(N)}_{k_N }(\\varsigma ) ,$ where $\\mathbf {k}\\equiv (k_{1},\\ldots ,k_{N})$ is a vector index for multipartite Bloch vectors where $k_{m}\\in 0,\\ldots ,n_{m}^2 -1$ for $m\\in 1,\\ldots ,N$ , (so $\\mathbf {k}\\ne \\mathbf {0}$ means only the case of all $k_m =0$ is excluded), $A_\\mathbf {k}$ is given in (eq.K.34), and ${\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{k_m }(\\varsigma )$ are Bloch components of the mode-$m$ reduction of ${\\Gamma }(\\varsigma )$ which is the Bloch vector of $\\varsigma $ (see [sec:App.K.1]App.sec:App.K.1 for more details).",
"Now, consider the following facts: [leftmargin=*,labelindent=4pt] 1.",
"If the parent state is an ME TGX state and multiple modes have size $n_{\\max }$ , then, as proved in ([18], App.D.4.c), all mode-$m$ reductions are ideally maximally mixed, having purities $P({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)})=\\frac{1}{n_{m}}\\;\\forall m\\in 1,\\ldots ,N$ .",
"2.",
"If the parent state is an ME TGX state and exactly one mode (call it mode $N$ ) has size $n_{\\max }$ , then, as proved in ([18], App.D.4.c), all nonlargest mode-$m$ reductions are ideally maximally mixed, with purities $P({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)})=\\frac{1}{n_{m}}\\;\\forall m\\in 1,\\ldots ,N-1$ , while ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(N)}$ has purity $P_{\\text{MP}}^{(m)}(L_{*})$ as given in (eq.K.38–eq.K.40), which is larger than that of the ideal maximally mixed state for an isolated system of size $n_m$ levels, which is due to the purity and maximal entanglement of its parent state (see [18], App.D for full explanations).",
"Note that $P_{\\text{MP}}^{(m)}(L_{*})$ simplifies to the correct value for all modes in this case, not just the largest mode.",
"3.",
"From Step 1 and Step 2, all maximally full-$N$ -partite entangled states have at least one single-mode reduction that is ideally maximally mixed.",
"4.",
"As given in [sec:App.K.1]App.sec:App.K.1, any mode-$m$ ideally maximally-mixed Bloch vector has magnitude $|{{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}|=0$ , and thus all its components are ${\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{k_{m}>0}=0$ .",
"5.",
"From Steps 1, 2, 3, and 4, for ME states, there always exist at least $N-1$ reduced Bloch vectors with $|{{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}(\\rho _{\\text{ME}})|=0$ and thus ${\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{k_{m}>0}(\\rho _{\\text{ME}})=0$ for $m\\in 1,\\ldots ,N-1$ .",
"(Note, here and throughout we use the convention that the system is organized in increasing mode size, so that mode $N$ is always the largest even if there are multiple largest modes.)",
"6.",
"The liaison vector ${\\Lambda }$ from (eq.K.30) is composed entirely of components of the full Bloch vector ${\\Gamma }$ that have two or more nonzero Bloch indices $k_m$ .",
"7.",
"From (eq.K.42) and Step 6, the components of ${\\Lambda }(\\varsigma _{\\text{ME}})$ take the form $\\Lambda _{\\mathbf {k} \\ne \\mathbf {0}}(\\varsigma _{\\text{ME}}) = \\frac{{A_\\mathbf {k} }}{{\\sqrt{n - 1} }}{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(1)}_{k_1 }(\\varsigma _{\\text{ME}}) \\cdots {\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(N)}_{k_N }(\\varsigma _{\\text{ME}}) =0 ,$ where $\\varsigma _{\\text{ME}}\\equiv \\varsigma (\\rho _{\\text{ME}})$ , because even though the largest mode can have nonzero components ${\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(N)}_{k_N >0 }(\\varsigma _{\\text{ME}})$ (if it is the only largest mode), all other modes have ${\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m<N)}_{k_m >0 }(\\varsigma _{\\text{ME}})=0$ , so since $\\mathbf {k}$ of $\\Lambda _\\mathbf {k}$ must always have at least two nonzero indices, at least one of the factors in $\\Lambda _\\mathbf {k}(\\varsigma _{\\text{ME}})$ will always be zero.",
"In the case of multiple largest modes, at least two factors in $\\Lambda _\\mathbf {k}(\\varsigma _{\\text{ME}})$ will always be zero, so (eq.K.43) holds true for all systems.",
"Therefore (eq.K.43) means that ${\\Lambda }(\\varsigma _{\\text{ME}})=\\mathbf {0}.$ Now, putting (eq.K.44) into (eq.K.30), we see that the full Bloch vector of $\\varsigma _{\\text{ME}}$ is a function of only single-mode reduced Bloch components as ${\\Gamma }(\\varsigma _\\text{ME} ) = \\mathbf {X}^{(1)} (\\varsigma _\\text{ME} ) + \\cdots + \\mathbf {X}^{(N)} (\\varsigma _\\text{ME} ),$ where from putting (eq.K.22) into (eq.K.25), we see that, in general, $\\mathbf {X}^{(m)}$ are Bloch vectors whose only nonzero components are proportional to single-mode reduction Bloch components, but whose basis matrices are in the space of the full Bloch vector, given by $\\mathbf {X}^{(m)}\\equiv \\sqrt{\\frac{n_{m}-1}{n-1}}\\sum \\limits _{k_m = 1 }^{K_m } {{\\hspace{-2.0pt}}{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{k_m}\\mathbf {b}_{\\mathbf {k}\\delta _{\\mathbf {k}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} ,\\mathbf {0}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} } }},$ where $\\mathbf {b}_{\\mathbf {k}\\delta _{\\mathbf {k}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} ,\\mathbf {0}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} } }\\equiv \\mathbf {b}_{0_{1},\\ldots ,0_{m-1},k_{m},0_{m+1},\\ldots ,0_{N}}$ is part of a matrix basis for the full Bloch vector, as defined in [sec:App.K.1]App.sec:App.K.1.",
"Thus, (eq.K.45) becomes ${\\Gamma }(\\varsigma _\\text{ME} ) {\\hspace{-1.0pt}}={\\hspace{-2.0pt}}\\sum \\limits _{m=1}^{N}{{\\hspace{-3.0pt}}{\\textstyle \\sqrt{\\frac{n_{m}-1}{n-1}}}{\\hspace{-2.0pt}}\\sum \\limits _{k_m = 1 }^{K_m } {{\\hspace{-3.5pt}}{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{k_m}(\\rho _\\text{ME} )\\mathbf {b}_{\\mathbf {k}\\delta _{\\mathbf {k}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} ,\\mathbf {0}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} } }}}{\\hspace{-2.0pt}}.$ where we used the fact that ${\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{k_m}(\\varsigma _\\text{ME} )={\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{k_m}(\\rho _\\text{ME} )$ since the reduction product $\\varsigma _{\\text{ME}}$ is constructed from a tensor product of the reductions of $\\rho _{\\text{ME}}$ and therefore it has exactly the same reduced Bloch components as $\\rho _{\\text{ME}}$ , with the main difference between the two states being that ${\\Lambda }(\\rho _{\\text{ME}})\\ne \\mathbf {0}$ .",
"Thus, we also have, using (eq.K.30), ${\\Gamma }(\\rho _\\text{ME} ) {\\hspace{-1.0pt}}={\\hspace{-2.0pt}}{\\Lambda }(\\rho _{\\text{ME}})+{\\hspace{-3.0pt}}\\sum \\limits _{m=1}^{N}{{\\hspace{-3.0pt}}{\\textstyle \\sqrt{\\frac{n_{m}-1}{n-1}}}{\\hspace{-2.0pt}}\\sum \\limits _{k_m = 1 }^{K_m } {{\\hspace{-3.5pt}}{\\Gamma }{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)}_{k_m}(\\rho _\\text{ME} )\\mathbf {b}_{\\mathbf {k}\\delta _{\\mathbf {k}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} ,\\mathbf {0}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} } }}}{\\hspace{-2.0pt}}.$ In the computation of the overlap of these two states, we will need to take the dot product of their Bloch vectors.",
"However, due to (eq.K.44), the liaison vector of $\\rho _{\\text{ME}}$ will have no contribution to the overlap, and furthermore, since all of the remaining terms in (eq.K.47) and (eq.K.48) are identical, then the Bloch overlap of these states is simply ${\\Gamma }(\\rho _{\\text{ME}} ) \\cdot {\\Gamma } (\\varsigma _{\\text{ME}} ) = \\sum \\limits _{m = 1}^N {\\frac{{n_m - 1}}{{n - 1}}|{{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)} (\\rho _{\\text{ME}} )|^2 }.$ However, this is the same as the Bloch purity of $\\varsigma _{\\text{ME}}$ , $|{\\Gamma }(\\varsigma _{\\text{ME}} )|^2 ={\\Gamma }(\\varsigma _{\\text{ME}} ) \\cdot {\\Gamma } (\\varsigma _{\\text{ME}} ) = \\sum \\limits _{m = 1}^N {\\frac{{n_m - 1}}{{n - 1}}|{{\\Gamma }}{\\hspace{-6.9pt}}{{\\textnormal {\\raisebox {1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-3.1pt}}{~}^{(m)} (\\rho _{\\text{ME}} )|^2 },$ so we have ${\\Gamma }(\\rho _{\\text{ME}} ) \\cdot {\\Gamma } (\\varsigma _{\\text{ME}} ) ={\\Gamma }(\\varsigma _{\\text{ME}} ) \\cdot {\\Gamma } (\\varsigma _{\\text{ME}} ).$ Then, putting (eq.K.51) into the fact from (eq.K.15) that $\\text{tr}(\\rho _A \\rho _B ) = \\frac{1}{n}\\left( {1 + (n - 1){\\Gamma } _A \\cdot {\\Gamma } _B } \\right)$ , we obtain the important result that $\\text{tr}(\\rho _{\\text{ME}} \\varsigma (\\rho _{\\text{ME}} )) = P(\\varsigma (\\rho _{\\text{ME}} )).$ Then, since this holds true for pure ME TGX states as well, then using (eq.K.37) in (eq.K.52) gives $\\text{tr}(\\rho _{\\text{ME}_{\\text{TGX}}} \\varsigma (\\rho _{\\text{ME}_{\\text{TGX}}})) =\\prod \\limits _{m = 1}^N {P_{\\text{MP}}^{(m)} (L_* )},$ which, when put into (eq.K.41) yields $\\mathcal {N}_{\\mathcal {X}} = 1 - \\prod \\limits _{m = 1}^N {P_{\\text{MP}}^{(m)} (L_* )},$ which proves the result from (eq.13), valid for all systems, where $P_{\\text{MP}}^{(m)} (L_* )$ is given by (eq.K.38) with $L=L_{*}$ where $L_{*}$ is determined from (eq.K.40).",
"For the special-case systems where more than one mode has size $n_{\\max }$ , we can use the fact, proved in ([18], App.D.4.c), that for maximally entangled states, all reductions are ideally maximally mixed, so then (eq.K.53) is $\\begin{array}{*{20}l}{\\prod \\nolimits _{m = 1}^N {P_{\\text{MP}}^{(m)} (L_* )}} &\\!\\!",
"{ = \\text{tr}[(\\frac{1}{{n_1 }}I^{(1)} \\otimes \\cdots \\otimes \\frac{1}{{n_N }}I^{(N)} )^2 ]} \\\\{} &\\!\\!",
"{ = \\frac{1}{{(n_1 \\cdots n_N )^2 }}\\text{tr}[I^{(1)} \\otimes \\cdots \\otimes I^{(N)} ]\\rule {0pt}{11pt}} \\\\{} &\\!\\!",
"{ = \\frac{1}{{n^2 }}n\\rule {0pt}{11pt}} \\\\{} &\\!\\!",
"{ = \\frac{1}{n},\\rule {0pt}{11pt}} \\\\\\end{array}$ which, when put into (eq.K.54) gives $\\mathcal {N}_{\\mathcal {X}} = 1 - \\frac{1}{n},$ in agreement with (eq.14), but again, is only valid for systems with multiple modes of size $n_{\\max }$ .",
"For systems with exactly one largest mode, we must use (eq.K.54).",
"Finally, since all cases of mode structures have the property that all modes except for the nominally largest mode (which we will call $m_{\\max }$ here for generality, meaning the single mode with size $n_{\\max }$ designated as the nominally largest mode regardless of whether other modes also have size $n_{\\max }$ ) have minimal reductions that are ideally maximally mixed when the parent is maximally full-$N$ -partite entangled, then (eq.K.54) can be simplified to $\\mathcal {N}_{\\mathcal {X}} = 1 - \\frac{P_{\\text{MP}}^{(m_{\\max })} (L_* )}{n_{\\,\\overline{{\\hspace{-1.8pt}}\\max ^{~^{~^{~}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"}}\\,},$ which may be used as an alternative to (eq.13) or (eq.K.54), where $n_{\\,\\overline{{\\hspace{-1.8pt}}\\max ^{~^{~^{~}}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"}}\\,\\equiv n/n_{\\max }$ ."
],
[
"Proof-Sketch and Derivation of the Normalization of Diagonal Correlance",
"Here we sketch a proof of why (eq.17) is valid for finding the normalization of diagonal correlance $\\mathcal {X}_D$ , and derive the explicit result in (eq.18).",
"[leftmargin=*,labelindent=4pt] 1.",
"Since all diagonal pure states are pure product states with no nonlocal correlation, that disqualifies pure states as maximizers of $\\mathcal {X}_D$ .",
"2.",
"Therefore, by Step 1, maximizers of $\\mathcal {X}_D$ (called $\\rho _{D_{\\max }}$ ) must have rank 2 or higher.",
"3.",
"By definition, $\\rho _{D_{\\max }}$ maximize $\\widetilde{\\mathcal {X}}_D$ as in (eq.16).",
"Therefore, while $\\rho _{D_{\\max }}$ must have rank $r\\geqslant 2$ by Step 2, they must also not have product form by definition, and their Bloch vectors must also have the largest distance from their own reduction products $\\varsigma _{D_{\\max }}\\equiv \\varsigma (\\rho _{D_{\\max }})$ under the Hilbert-Schmidt inner product, out of all states compared to their own reduction products.",
"(Note: $\\mathcal {X}_D$ and $\\mathcal {X}$ are not measures of distance to the set of all product states from the given input $\\rho $ ; instead they are measures of the distance between $\\rho $ and its own reduction product.",
"Therefore, all $\\rho _{D_{\\max }}$ have the biggest distance between themselves and their own $\\varsigma _{D_{\\max }}$ out of all diagonal states $\\rho _D$ , but their $\\varsigma _{D_{\\max }}$ are generally neither the farthest nor the closest product states from themselves.)",
"4.",
"Since all $\\rho _D$ are diagonal, then by the definition of partial trace, their reductions ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}_{\\!D}$ are also diagonal.",
"5.",
"From (eq.16), the function to be maximized by $\\rho _{D_{\\max }}$ is $\\widetilde{\\mathcal {X}}(\\rho _{D} )\\!=\\!P(\\rho _{D})\\!+\\!P({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{\\!D})\\cdots P({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(N)}_{\\!D})\\!-2\\text{tr}[\\rho _{D}\\varsigma _{D}]$ over all $\\rho _{D}$ , where $P(\\sigma )\\equiv \\text{tr}(\\sigma ^2)$ is the purity of $\\sigma $ , and the overlap is $\\text{tr}[\\rho _{D}\\varsigma _{D}]=\\text{tr}[\\rho _{D}({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{\\!D}\\otimes \\cdots \\otimes {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(N)}_{\\!D})]$ .",
"6.",
"By Step 5, maximizers of $\\widetilde{\\mathcal {X}}(\\rho _{D} )$ need to simultaneously fulfill the conditions of having highest purity, highest single-mode reduction purities, and lowest overlap of parent state and reduction product.",
"7.",
"By Step 6, we need to minimize $\\text{tr}[\\rho _{D}\\varsigma _{D}]$ (while also keeping the purity terms as high as possible).",
"So first, expanding it as $\\text{tr}[\\rho _{D}\\varsigma _{D}]=\\sum \\nolimits _{a = 1}^n {} (\\rho _D )_{a,a} (\\varsigma _D )_{a,a}=\\sum \\nolimits _{a_{1},\\ldots ,a_{N}=1,\\ldots ,1}^{n_{1},\\ldots ,n_{N}} {(\\rho _D )_{\\mathbf {a},\\mathbf {a}}[({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{\\!D})_{a_{1},a_{1}}\\cdots ({\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(N)}_{\\!D})_{a_{N},a_{N}}]}$ where $\\mathbf {a}\\equiv (a_{1},\\ldots ,a_{N})$ , shows that each term is a product of the probabilities of $\\rho _D$ with the probabilities of each of the $N$ single-mode reductions of $\\rho _D$ (which are also functions of probabilities of $\\rho _D$ ), by Step 4.",
"This suggests that the more diagonal elements of $\\rho _D$ are zero, the lower this overlap function will be, which would also increase the purity of $\\rho _D$ , but not necessarily the reduction purity product.",
"Therefore this suggests that we reduce rank as much as possible while also looking at the particular ways to choose nonzero elements to maximize $\\widetilde{\\mathcal {X}}(\\rho _{D} )$ .",
"For example, in $2\\times 3$ , abbreviating with $\\rho \\equiv \\rho _D$ and $\\varsigma _{D}\\equiv \\varsigma (\\rho _{D})$ , $\\begin{array}{*{20}l}{(\\varsigma _D )_{1,1} ={\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{1,1}{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(2)}_{1,1}} &\\!\\!",
"{ = (\\rho _{1,1} + \\rho _{2,2} + \\rho _{3,3} )(\\rho _{1,1} + \\rho _{4,4} )} \\\\{(\\varsigma _D )_{2,2} ={\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{1,1}{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(2)}_{2,2}} &\\!\\!",
"{ = (\\rho _{1,1} + \\rho _{2,2} + \\rho _{3,3} )(\\rho _{2,2} + \\rho _{5,5} )} \\\\{(\\varsigma _D )_{3,3} ={\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{1,1}{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(2)}_{3,3}} &\\!\\!",
"{ = (\\rho _{1,1} + \\rho _{2,2} + \\rho _{3,3} )(\\rho _{3,3} + \\rho _{6,6} )} \\\\{(\\varsigma _D )_{4,4} ={\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{2,2}{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(2)}_{1,1}} &\\!\\!",
"{ = (\\rho _{4,4} + \\rho _{5,5} + \\rho _{6,6} )(\\rho _{1,1} + \\rho _{4,4} )} \\\\{(\\varsigma _D )_{5,5} ={\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{2,2}{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(2)}_{2,2}} &\\!\\!",
"{ = (\\rho _{4,4} + \\rho _{5,5} + \\rho _{6,6} )(\\rho _{2,2} + \\rho _{5,5} )} \\\\{(\\varsigma _D )_{6,6} ={\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(1)}_{2,2}{\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(2)}_{3,3}} &\\!\\!",
"{ = (\\rho _{4,4} + \\rho _{5,5} + \\rho _{6,6} )(\\rho _{3,3} + \\rho _{6,6} ),} \\\\\\end{array}$ which shows that some choices of nonzero parent elements lead to more nonzero elements in $\\varsigma _D$ than others.",
"Since the rank of $\\rho $ determines the number of terms in $\\text{tr}[\\rho _{D}\\varsigma _{D}]$ anyway, then to minimize it for a given rank, we need to choose the nonzero parent elements in a way that maximizes the number of nonzero terms in $\\varsigma _D$ so that their values are lower due to normalization, which allows a lower value of $\\text{tr}[\\rho _{D}\\varsigma _{D}]$ .",
"8.",
"Considering rank and choice of nonzero elements, notice that for all states, $\\rho _{1,1}$ always appears in ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}_{1,1}$ and $\\rho _{n,n}$ always appears in ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}_{{n_m},{n_m}}\\;\\forall m\\in 1,\\ldots ,N$ .",
"Thus for any number of $N$ modes, if a rank-2 parent state only has $\\rho _{1,1}\\ne 0$ and $\\rho _{n,n}\\ne 0$ , then $\\varsigma _{1,1}$ and $\\varsigma _{n,n}$ will always be nonzero and several other elements of $\\varsigma $ may still be nonzero as well, as in (eq.L.1) which has four nonzero elements in this case.",
"This is important because since the parent state has all other $n-2$ diagonal elements as 0, then $n-2$ terms of the negative overlap term in Step 5 vanish as seen by Step 7, reducing its ability to lessen the objective function, while the factors from $\\varsigma $ are lower than they would be for other choices of nonzero parent elements, since there are more than two nonzero elements of $\\varsigma $ .",
"Also, the parent purity has its largest minimum since $P(\\rho )\\in [\\frac{1}{2},1)$ , while $P(\\varsigma )\\in [\\frac{1}{n},1)$ with possibly a larger minimum due to the parent state's correlation (as shown for reduction products of maximally entangled parent states in [18]), where we used $\\rho \\equiv \\rho _D$ and $\\varsigma \\equiv \\varsigma _D$ .",
"9.",
"By Step 7, since every term of the objective function $\\widetilde{\\mathcal {X}}_D$ (for any rank) is a function of parent elements, their normalization and nonnegativity makes $\\widetilde{\\mathcal {X}}_D$ a linear combination of products of squares of $r$ unit-hyperspherical coordinates $\\lbrace x_k\\rbrace $ .",
"It is well-known that the sum of even powers of $\\lbrace x_k\\rbrace $ is minimized when $x_k=\\frac{1}{\\sqrt{r}}\\;\\forall k\\in 1,\\ldots ,r$ (see [18], App.D.2).",
"Here, this same solution maximizes the rank-2 case of $\\widetilde{\\mathcal {X}}_D$ , and for ranks $r\\geqslant 3$ , neither balanced probabilities nor any other combination of values of hyperspherical coordinates can cause $\\widetilde{\\mathcal {X}}_D$ to get as large as the rank-2 case, because they generally cause a smaller parent-state purity and a smaller reduction-purity product than the rank-2 case.",
"The overlap term behaves less consistently, however its combination with the other terms in $\\widetilde{\\mathcal {X}}_D$ is such that the rank-2 value of $\\widetilde{\\mathcal {X}}_D$ is always the largest when maximized over all combinations of nonzero-levels for the equal-probabilities case.",
"This was confirmed by brute-force combinatorial comparison of all ranks and nonzero-element combinations for several multipartite systems in equal-probability states, while [fig:4]Fig.fig:4 provides strong numerical evidence that it is true over all $\\rho _D$ .",
"However, a rigorous proof of this is still lacking.",
"10.",
"Therefore, Step 9 gives evidence that $\\rho _{D_{\\max } }$ must have rank 2 and equal probabilities (note that if either probability is larger, the state would be closer to being a pure product state, so it would have nonmaximal correlation).",
"Furthermore, Step 8 suggests one way to chose particular nonzero elements that will minimize the overlap the most out of all possible pairs, since it causes the most terms in the overlap to be zero, while causing the most terms of $\\varsigma $ to be nonzero which lowers the values of the surviving overlap terms.",
"Therefore, our candidate maximizer states are $\\rho _{D_{\\max } } = \\text{diag}\\lbrace \\frac{1}{2},0, \\ldots ,0,\\frac{1}{2}\\rbrace =\\frac{1}{2}(|1\\rangle \\langle 1| + |n\\rangle \\langle n|)$ .",
"11.",
"To see that the candidate $\\rho _{D_{\\max } }$ from Step 10 also satisfies the requirement from Step 3 that it not have product form, note that in any diagonal state of equal nonzero probabilities, the factorizability of basis elements of the nonzero terms completely determines whether it has product form.",
"A rank-2 diagonal state of equal nonzero elements that does not have product form in all multipartite systems is $\\rho _{D_{\\max }}=\\frac{1}{2}(|1\\rangle \\langle 1| + |n\\rangle \\langle n|)$ of (eq.17), since its basis states of nonzero probabilities yield the coincidence form $\\rho _{D_{\\max }}=\\frac{1}{2}(|1^{(1)}\\rangle \\langle 1^{(1)}|\\otimes \\cdots \\otimes |1^{(N)}\\rangle \\langle 1^{(N)}|+|n_{1}^{(1)}\\rangle \\langle n_{1}^{(1)}|\\otimes \\cdots \\otimes |n_{N}^{(N)}\\rangle \\langle n_{N}^{(N)}|)$ , which is unfactorizable because the projectors in each mode are different in each term, preventing any single-mode projectors from being factored out.",
"Using the first and last basis element guarantees that the projectors will always have different labels in each mode between the two terms, which is the simplest way to guarantee this property for all systems.",
"Thus, since this is exactly the candidate state of Step 10, we have justified why it is a prototypical $\\rho _{D_{\\max } }$ .",
"Now that we have a motivation for why $\\rho _{D_{\\max }}$ from Step 11 is a nonfactorizable diagonal state furthest from its own reduction product, we need to calculate its raw diagonal correlance to get the normalization factor $\\mathcal {N}_{\\mathcal {X}_D}$ of (eq.18).",
"Therefore, we start with the raw correlance of $\\rho _{D_{\\max }}=\\frac{1}{2}(|1\\rangle \\langle 1| + |n\\rangle \\langle n|)$ from Step 11 above, as $\\begin{array}{*{20}l}{\\mathcal {N}_{\\mathcal {X}_D } } &\\!\\!",
"{ \\equiv \\widetilde{\\mathcal {X}}(\\rho _{D_{\\max } } ) = \\text{tr}[(\\rho _{D_{\\max } } - \\varsigma _{D_{\\max } } )^2 ]} \\\\{} &\\!\\!",
"{ = P(\\rho _{D_{\\max } } ) - 2\\text{tr}[\\rho _{D_{\\max } } \\varsigma _{D_{\\max } } ] + P(\\varsigma _{D_{\\max } } ),} \\\\\\end{array}$ where $\\varsigma _{D_{\\max } } \\equiv \\varsigma (\\rho _{D_{\\max } } )$ , and $\\rho _{D_{\\max } } = \\text{diag}\\lbrace \\frac{1}{2}_1 ,0_2 , \\ldots ,0_{n - 1} ,\\frac{1}{2}_n \\rbrace ,$ (where we use subscripts to help indicate which basis element corresponds to each matrix element), and thus $P(\\rho _{D_{\\max } } ) = \\frac{1}{2}.$ Then, using the fact from Step 8 that writing any single-mode reduction in terms of the matrix elements of the parent state always features the first and last diagonal parent matrix elements in separate diagonal matrix elements of the reduced states (specifically the first and last element of each, where the multipartite basis is ordered by the standard register-counting convention), then each mode-$m$ reduction mirrors the parent state as ${\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}_{D_{\\max } } = \\text{diag}\\lbrace \\frac{1}{2}_1 ,0_2 , \\ldots ,0_{n_m - 1} ,\\frac{1}{2}_{n_m } \\rbrace ,$ which means that the reduction product is $\\varsigma _{D_{\\max } } \\equiv \\mathop \\otimes \\limits _{m = 1}^N {\\rho }{\\hspace{-5.3pt}}{{\\textnormal {\\raisebox {-1.2pt}{\\scalebox {1.2}{̌}}}}}{\\hspace{-4.2pt}}{~}^{(m)}_{D_{\\max } } = \\mathop \\otimes \\limits _{m = 1}^N \\text{diag}\\lbrace \\frac{1}{2}_1 ,0_2 , \\ldots ,0_{n_m - 1} ,\\frac{1}{2}_{n_m } \\rbrace ,$ and therefore has purity $P(\\varsigma _{D_{\\max } } ) = \\prod \\limits _{m = 1}^N {\\frac{1}{2}} = \\frac{1}{{2^N }}.$ For the overlap in (eq.L.2), since all but the first and last elements of $\\rho _{D_{\\max } }$ are zero, only the first and last elements of $\\varsigma _{D_{\\max } }$ affect the overlap, and since those elements are both $\\frac{1}{{2^N }}$ by expanding (eq.L.6), then $\\text{tr}(\\rho _{D_{\\max } } \\varsigma _{D_{\\max } } ) = \\text{tr}(\\text{diag}\\lbrace \\frac{1}{2} \\cdot \\frac{1}{{2^N }},0 , \\ldots ,0 ,\\frac{1}{2} \\cdot \\frac{1}{{2^N }}\\rbrace ) = \\frac{1}{{2^N }}.$ Thus, putting (eq.L.4), (eq.L.7), and (eq.L.8) into (eq.L.2) gives $\\mathcal {N}_{\\mathcal {X}_D }= \\frac{1}{2} - 2\\frac{1}{{2^N }} + \\frac{1}{{2^N }} = \\frac{1}{2} - \\frac{1}{{2^N }},$ which is the result in (eq.18).",
"Note that any given diagonal state $\\rho _D$ can be converted to another diagonal state of equal $\\mathcal {X}_D$ by a local-permutation unitary (LPU) (which can generally be complex), meaning a tensor-product of permutation unitaries of each mode, since these operators cause neither superpositions between modes nor superpositions within the modes, preserving both locality and diagonality.",
"Local unitaries (LU) also preserve $\\mathcal {X}_D$ , but not diagonality, so they cannot be used to reach all states of a given $\\mathcal {X}_D$ from a single $\\rho _D$ .",
"Thus, since $\\rho _{D_{\\max }}$ only has rank 2 and balanced probabilities, the set of all $\\rho _{D_{\\max }}$ is found by applying any LPU to the prototypical $\\rho _{D_{\\max }}$ of (eq.17)."
],
[
"Algorithm $\\mathcal {A}_{\\rho }$ to Form a Multipartite Density Matrix from Strictly Classical Data",
"The purpose of this algorithm is to generate a density-matrix estimator $\\rho $ from the data set of measurements of a strictly classical multivariable system, so that we can then measure the general nonlocal correlation in the data using the diagonal correlance $\\mathcal {X}_D (\\rho )$ of (eq.15).",
"Continuing from the description leading up to (eq.19), the main idea of this algorithm is that the data for each of the $N$ random variables (RVs) $\\lbrace x^{(m)}\\rbrace $ is first quantized separately, and then combined as a set of $n_{S}$ quantized $N$ -tuple data points $\\lbrace \\mathbf {x}_j ^{\\prime } \\rbrace $ .",
"During the quantization, we make a list of the center values of every bin of each mode as representative bin values $\\widetilde{\\mathbf {x}}^{(m)}\\equiv (\\widetilde{x}^{(m)}_{1},\\ldots ,\\widetilde{x}^{(m)}_{n_{m}})^T$ .",
"Then, we create the density matrix $\\rho $ for the quantized data by cycling through $N$ -tuples of representative bin values in “register format” (for example, $\\lbrace (\\widetilde{x}^{(1)}_{1},\\widetilde{x}^{(2)}_{1}),(\\widetilde{x}^{(1)}_{1},\\widetilde{x}^{(2)}_{2}),(\\widetilde{x}^{(1)}_{2},\\widetilde{x}^{(2)}_{1}),(\\widetilde{x}^{(1)}_{2},\\widetilde{x}^{(2)}_{2})\\rbrace $ in the simplest nontrivial case), and counting the number of quantized data points that match each of them as an entire $N$ -tuple, forming a multipartite histogram, which is then normalized and placed in the main-diagonal elements of an $n\\times n$ matrix, giving us $\\rho $ .",
"The following steps constitute algorithm $\\mathcal {A}_{\\rho }$ : [leftmargin=*,labelindent=4pt] 1.",
"Identify the $N$ RVs of a given data set as $\\mathbf {x} \\equiv (x^{(1)} , \\ldots ,x^{(N)} ).$ 2.",
"Organize the data as the $n_S \\times N$ matrix, $X{\\hspace{-1.5pt}} \\equiv {\\hspace{-4.0pt}} \\left({\\hspace{-3.0pt}} {\\begin{array}{*{20}c}{\\mathbf {x}_1 } \\\\\\vdots \\\\{\\mathbf {x}_{n_S } } \\\\\\end{array}}{\\hspace{-3.0pt}} \\right) {\\hspace{-4.0pt}}={\\hspace{-4.0pt}} \\left({\\hspace{-2.0pt}} {\\begin{array}{*{20}c}{(x_1^{(1)} , \\ldots ,x_1^{(N)} )} \\\\\\vdots \\\\{(x_{n_S }^{(1)} , \\ldots ,x_{n_S }^{(N)} )} \\\\\\end{array}}{\\hspace{-2.0pt}} \\right) {\\hspace{-4.0pt}}\\equiv {\\hspace{-4.0pt}} \\left({\\hspace{-3.0pt}} {\\begin{array}{*{20}c}{X_{1,1} } &{\\hspace{-7.0pt}} \\cdots &{\\hspace{-7.0pt}} {X_{1,N} } \\\\{} &{\\hspace{-7.0pt}} \\vdots &{\\hspace{-7.0pt}} {} \\\\{X_{n_S ,1} } &{\\hspace{-7.0pt}} \\cdots &{\\hspace{-7.0pt}} {X_{n_S ,N} } \\\\\\end{array}}{\\hspace{-4.0pt}} \\right){\\hspace{-2.5pt}},$ where the $j$ th $N$ -dimensional data point $\\mathbf {x}_j$ is an $N$ -tuple of measured values, for a total of $n_S$ measurements of $N$ values each.",
"Thus, elements of $X$ are $X_{j,m} \\equiv x_j^{(m)} $ for $j \\in 1, \\ldots ,n_S $ and $m \\in 1, \\ldots ,N$ .",
"3.",
"Choose quantization (bin) numbers $n_m$ for each RV as $\\mathbf {n} \\equiv (n_1 , \\ldots ,n_N ).$ For discrete RVs, $n_m$ is the number of possible outcomes for RV $x^{(m)}$ .",
"For continuous RVs, $n_m$ is the finite number of recognized possible values for RV $x^{(m)}$ , and there are many techniques for choosing $n_m$ appropriately based on the data.",
"Choice of $n_m$ sets the bin number in a histogram of the data in column $m$ of $X$ .",
"Note: in most cases, the precision of the measuring devices used to obtain the data already imposes some quantization on the RVs.",
"4.",
"Define the domain of each RV as a bounded a pair of extreme values $[x_{\\min }^{(m)},x_{\\max }^{(m)}]$ .",
"In some cases it might be preferable to choose bounds that hug the extremes of the data itself, such as by using $\\begin{array}{*{20}c}{\\begin{array}{*{20}c}{x_{\\min }^{(m)} \\equiv \\min \\lbrace X_{:,m} \\rbrace ,} \\\\{x_{\\max }^{(m)} \\equiv \\max \\lbrace X_{:,m} \\rbrace ,} \\\\\\end{array}} & {X_{:,m} \\equiv \\left({\\hspace{-2.0pt}} {\\begin{array}{*{20}c}{X_{1,m} } \\\\\\vdots \\\\{X_{n_S ,m} } \\\\\\end{array}}{\\hspace{-2.0pt}} \\right)\\!.}",
"\\\\\\end{array}$ However, note that defining $[x_{\\min }^{(m)},x_{\\max }^{(m)}]$ as the extreme values of what could happen for each RV (regardless of whether data reaches those values) is technically the more correct method here, so these should generally be specified inputs instead of using (eq.M.4).",
"5.",
"Get bin-edge lists $\\mathbf {e}^{(m)} $ for each RV as $\\mathbf {e}^{(m)} \\equiv (e_1^{(m)} , \\ldots ,e_{n_m + 1}^{(m)} )^T ,$ where the $n_m +1$ bin edges for RV $x^{(m)}$ are $e_{k_m }^{(m)} \\equiv x_{\\min }^{(m)} + \\frac{{x_{\\max }^{(m)} - x_{\\min }^{(m)} }}{{n_m }}(k_m - 1),$ for $k_m \\in 1, \\ldots ,n_m + 1$ .",
"6.",
"Get representative bin-value lists for each RV as $\\widetilde{\\mathbf {x}}^{(m)} \\equiv (\\widetilde{x}_1^{(m)} , \\ldots ,\\widetilde{x}_{n_m }^{(m)} )^T ,$ with elements $\\widetilde{x}_q^{(m)} \\equiv \\frac{{e_q^{(m)} + e_{q + 1}^{(m)} }}{2};\\quad q \\in 1, \\ldots ,n_m .$ 7.",
"Quantize the data as an $n_S \\times N$ matrix $X^{\\prime }$ using $\\begin{array}{l}X^{\\prime } = 0^{[n_S \\times N]} \\\\{\\hspace{-2.0pt}}\\begin{array}{*{20}l}{\\text{for}\\;m \\in 1, \\ldots ,N} \\hfill \\\\{{\\hspace{8.0pt}}\\text{for}\\;j \\in 1, \\ldots ,n_S } \\hfill \\\\{{\\hspace{16.0pt}}\\text{for}\\;q \\in 1, \\ldots ,n_m } \\hfill \\\\{{\\hspace{24.0pt}}\\text{if}\\;(q {\\hspace{-1.0pt}}=={\\hspace{-1.0pt}} 1)\\;\\&\\; (X_{j,m} {\\hspace{-1.0pt}}<{\\hspace{-1.0pt}} e_2^{(m)} )} \\hfill \\\\{{\\hspace{32.0pt}}X^{\\prime }_{j,m} = \\widetilde{x}_1^{(m)} } \\hfill \\\\{{\\hspace{24.0pt}}\\text{elseif}\\;[(q {\\hspace{-1.0pt}}>{\\hspace{-1.0pt}} 1)\\;\\&\\; (q {\\hspace{-1.0pt}}<{\\hspace{-1.0pt}} n_m )} \\hfill \\\\{{\\hspace{52.0pt}}\\&\\; (X_{j,m} {\\hspace{-1.0pt}}\\geqslant {\\hspace{-1.0pt}} e_q^{(m)} )\\;\\&\\; (X_{j,m} {\\hspace{-1.0pt}}<{\\hspace{-1.0pt}} e_{q + 1}^{(m)} )]} \\hfill \\\\{{\\hspace{32.0pt}}X^{\\prime }_{j,m} = \\widetilde{x}_q^{(m)} } \\hfill \\\\{{\\hspace{24.0pt}}\\text{elseif}\\;(q {\\hspace{-1.0pt}}=={\\hspace{-1.0pt}} n_m )\\;\\& \\;(X_{j,m} {\\hspace{-1.0pt}}\\geqslant {\\hspace{-1.0pt}} e_{n_m }^{(m)} )} \\hfill \\\\{{\\hspace{32.0pt}}X^{\\prime }_{j,m} = \\widetilde{x}_{n_m }^{(m)} } \\hfill \\\\\\end{array} \\\\\\end{array}$ where the notation $M^{[a\\times b]}$ means an $a\\times b$ matrix $M$ .",
"Notice that for bin 1 and bin $n_m$ , the conditions assign data outside the RV extremes to those end bins; that is to allow variations in measured values to go under or over the extremes, but again, this may not be applicable in all situations, so modify as appropriate.",
"8.",
"Generate the multipartite histogrammic $n\\times n$ density matrix $\\rho $ (where $n=n_{1}\\cdots n_{N}$ ) using $\\begin{array}{l}\\rho = 0^{[n\\times n]} \\\\\\text{for}\\;a \\in 1, \\ldots ,n \\\\{\\hspace{8.0pt}}\\mathbf {a} = \\mathbf {a}_a^{\\lbrace N,\\mathbf {n}\\rbrace } \\\\{\\hspace{8.0pt}}\\text{for}\\;j \\in 1, \\ldots ,n_{S} \\\\{\\hspace{16.0pt}}\\widetilde{\\mathbf {x}}_j = X^{\\prime }_{j, \\cdots } \\\\{\\hspace{16.0pt}}\\mathbf {v} = 0^{[1 \\times N]} \\\\{\\hspace{16.0pt}}\\text{for}\\;m \\in 1, \\ldots ,N \\\\{\\hspace{24.0pt}}v_{m} = \\widetilde{x}_{a_m }^{(m)} \\\\{\\hspace{16.0pt}}\\text{if}\\;\\text{sum}(\\widetilde{\\mathbf {x}}_j == \\mathbf {v}) == N \\\\{\\hspace{24.0pt}}\\rho _{a,a} = \\rho _{a,a} + 1 \\\\\\rho = \\rho /\\text{tr}(\\rho ) \\\\\\end{array}$ where $\\mathbf {a}_a^{\\lbrace N,\\mathbf {n}\\rbrace }$ is the inverse register function given in [sec:App.U]App.sec:App.U that maps scalar index $a$ to vector index $\\mathbf {a}\\equiv (a_{1},\\ldots ,a_{N})$ , and $\\mathbf {v}\\equiv (v_1,\\ldots ,v_N)$ is a temporary vector to hold the multipartite representative bin value corresponding to scalar index $a$ .",
"This step counts all the occurrences of each multipartite representative bin value that arises in the quantized data $X^{\\prime }$ and tabulates it as a relative frequency in the main diagonal of a density matrix indexed with the corresponding scalar row-column indices for that value.",
"Note that, being diagonal by design, this $\\rho $ can be stored as a vector of length $n$ , since it is just a classical discrete probability density function, but we keep it in matrix form here for conceptual continuity with the full quantum problem.",
"The result of this algorithm is $\\rho $ , an estimator of the strictly classical density matrix, which can then be used to calculate the diagonal correlance $\\mathcal {X}_D (\\rho )$ in (eq.15).",
"The above steps may not necessarily be the most efficient implementation of $\\mathcal {A}_{\\rho }$ ; however in this form, the steps are at least conceptually clear."
],
[
"Pearson Correlation Coefficient",
"Consider a system of $N$ classical random variables (RVs) $\\mathbf {x} \\equiv (x^{(1)} , \\ldots ,x^{(N)} )$ , for a dataset $X$ defined as a matrix of $N$ columns representing the RVs, and $n_S$ rows each consisting of an $N$ -tuple sample of particular measured (observed) values $\\mathbf {x}_j \\equiv (x_j^{(1)} , \\ldots ,x_j^{(N)} )$ for the $N$ RVs, for a total sample number of $n_S$ measured $N$ -tuples [see (eq.M.2)].",
"Thus, $X_{j,m}$ is the $j$ th observed value for the $m$ th RV $x^{(m)}$ .",
"The sample Pearson correlation coefficient [19], [20], [21], which is defined only for the case of two variables, is given for any two RVs $x^{(a)}$ and $x^{(b)}$ by $r_{\\text{P}} \\equiv r_{\\text{P}} ^{(a,b)} \\equiv \\frac{{V_S ^{(a,b)} }}{{s^{(a)} s^{(b)} }},$ where $V_S ^{(a,b)}\\equiv \\text{cov}_{S}(x^{(a)},x^{(b)})$ is the (unbiased) sample covariance between $x^{(a)}$ and $x^{(b)}$ , $\\begin{array}{*{20}l}{V_S ^{(a,b)} } &\\!\\!",
"{\\equiv \\frac{1}{{n_S - 1}}\\sum \\limits _{j = 1}^{n_S } {(x_j ^{(a)} - \\mu _S^{(a)} )} (x_j ^{(b)} - \\mu _S^{(b)} )} \\\\{} &\\!\\!",
"{ = \\frac{1}{{n_S - 1}}\\sum \\limits _{j = 1}^{n_S } {(X_{j,a} - \\mu _S^{(a)} )} (X_{j,b} - \\mu _S^{(b)} ),} \\\\\\end{array}$ where the sample mean of RV $x^{(m)}$ is $\\mu _S^{(m)} \\equiv \\overline{x}^{(m)} \\equiv \\frac{1}{{n_S }}\\sum \\limits _{j = 1}^{n_S } {x_j^{(m)} } = \\frac{1}{{n_S }}\\sum \\limits _{j = 1}^{n_S } {X_{j,m} } ,$ and the sample standard deviation is $\\begin{array}{*{20}l}{s^{(m)} \\equiv \\sqrt{V_S ^{(m,m)} } } &\\!\\!",
"{ \\equiv \\sqrt{\\frac{1}{{n_S - 1}}\\sum \\limits _{j = 1}^{n_S } {(x_j ^{(m)} - \\mu _S^{(m)} )^2 } } } \\\\{} &\\!\\!",
"{ = \\sqrt{\\frac{1}{{n_S - 1}}\\sum \\limits _{j = 1}^{n_S } {(X_{j,m} - \\mu _S^{(m)} )^2 } } .}",
"\\\\\\end{array}$ We can also form an $N$ -by-$N$ sample covariance matrix $V_S$ with elements $(V_S )_{a,b} \\equiv V_S ^{(a,b)} $ and an $N$ -by-$N$ sample Pearson correlation matrix $r_\\text{P}$ with elements $(r_\\text{P} )_{a,b} \\equiv r_\\text{P}^{(a,b)} $ , each containing all pairwise covariances and Pearson correlations between the $N$ RVs.",
"However, it is common practice to simply write $r_{\\text{P}} \\equiv r_{\\text{P}} ^{(a,b)}$ , as in (eq.N.1), to denote the scalar value of the sample Pearson correlation coefficient between two particular variables when those are the only two variables in the problem and no Pearson correlation matrix is used elsewhere in the analysis, which is the convention we use.",
"Also, it is not standard to use the “$\\text{P}$ ” subscript, but we use it to distinguish it from our use of $r$ to mean rank.",
"The Pearson correlation has range $[-1,1]$ , and is merely a measure of linear correlation between two RVs only.",
"Thus, two variables can have strong nonlinear correlations and yet produce $r_{\\text{P}}=0$ , so it is not a measure of general correlation.",
"Furthermore, the old adage that “correlation does not imply causation” means that having a nonzero $r_\\text{P}$ does not mean that the RVs share some functional dependence on a common parameter, but rather it means that the data of the two RVs have a linear correspondence whether they are functions of the same underlying variables or not."
],
[
"Consistency of Strict Classicality with Established Physics",
"Here, we list a few additional reasons why the strictly classical states defined in [sec:III.C]Sec.sec:III.C are compatible with existing ideas in physics, and are a much more appropriate standard for classicality than coherent states.",
"[leftmargin=*,labelindent=4pt] 1.",
"For multiple modes of the same size, the set of all distinct pure strictly classical states has the same form as the set of possible wave functions that yield Maxwell-Boltzmann statistics [58], [59], which arise from a collection of identical but distinguishable particles, and is a regime of statistics that would apply if the universe were not quantum-mechanical, i.e.",
"in a strictly classical universe.",
"For example, for two identical, distinguishable two-level particles, the possible wave functions are $\\lbrace \\psi _1 (x_1 ) \\otimes \\psi _1 (x_2 ),$ $\\psi _1 (x_1 ) \\otimes \\psi _2 (x_2 ),$ $\\psi _2 (x_1 ) \\otimes \\psi _1 (x_2 ),$ $\\psi _2 (x_1 ) \\otimes \\psi _2 (x_2 )\\rbrace $ , where $\\psi _a (x_m ) \\equiv $ $ \\langle x^{(m)} |\\psi _a^{(m)} \\rangle $ , $|\\psi _a^{(m)} \\rangle $ is the $a$ th possible state of particle $m$ , and $|x^{(m)} \\rangle $ is the position eigenstate of particle $m$ .",
"These states form a set of pure computational basis states, which qualify (in form) as strictly classical states as defined in [sec:III.C]Sec.sec:III.C.",
"2.",
"In regards to probability representing the lack of knowledge of the observer, there are really two general kinds of states with different meanings [28], [29], [30]; [leftmargin=*,labelindent=4pt] a. Ontic states: the actual state of a system, part of fundamental reality.",
"b. Epistemic states: the state in the context of an observer, given limitations and lack of full knowledge; is usually a statistical mixture of ontic states each assigned probabilities based on observer ignorance.",
"Pure strictly classical states are ontic, since the state is definite.",
"Mixed strictly classical states are epistemic, since by the definition of strict classicality, any probability distributions arise from lack of specific knowledge (i.e., taking sample measurements with time windows too long to notice the fact that the system is instantaneously pure).",
"This dichotomy fits nicely with the classical notion of probability; the only kind of probability in a strictly classical system is the ignorance-induced kind, and there is no fundamental quantum probability such as that which arises in pure quantum states with superposition (we exclude the possibility of hidden variable theories until [sec:VI]Sec.sec:VI).",
"Furthermore, since pure strictly classical states are pure product states of computational basis states in all reference frames, nonlocal correlation is impossible in an ontic strictly classical state.",
"In [sec:III]Sec.sec:III–[sec:V]Sec.sec:V we discuss the more interesting case of mixed strictly classical nonlocal correlation in detail.",
"3.",
"Quantum coherence functions arise from finding mean values of quantities composed of electric field operators in analogy to classical coherence functions.",
"What this leads to is a certification of whether or not the state of the electric field is a coherent state (which is only a necessary and sufficient certification of being in a coherent state if the field has $m$ th-order quantum coherence values of $|g ^{(m)} (x_1 , \\ldots ,x_m )| = 1$ $\\forall m \\in \\lbrace 1, \\ldots ,\\infty \\rbrace $ , meaning that it is “infinite-order coherent” [27], where here $x_m\\equiv ({\\eta }_m,t_m)$ where ${\\eta }_m \\equiv \\mathbf {r}-\\mathbf {r}_m$ is a separation vector from source point $\\mathbf {r}_m$ to field point $\\mathbf {r}$ and $t_m \\equiv t-\\frac{|{{\\eta }_m}|}{c}$ is the retarded time for light to reach $\\mathbf {r}$ from $\\mathbf {r}_m$ ).",
"However, measuring how close a state is to infinite-order coherence does not prove anything about classicality; rather it is a test of how close the state is to being a coherent state (with lower-order failures such as $|g ^{(2)} (x_1 ,x_2 )| \\ne 1$ being sufficient to conclude that a state is not a coherent state, thereby making such a test practical to use).",
"The fallacy of assuming that a state's classicality is determined by its similarity to a coherent state as quantified by quantum coherence functions is evident for several reasons.",
"The main red flag indicating this fallacy is that a coherent state requires quantum superposition in the Fock basis, an impossibility in truly classical physics.",
"Another red flag is that quantum-coherence-function results are often interpreted in conjunction with quantum-phase space probability distributions where nonclassicality is said to be evident by the appearance of negative probabilities, which are not truly allowed in quantum mechanics; a density matrix with any negative diagonal elements is nonphysical.",
"In contrast, the definition of strict classicality forbids quantum superposition, a restriction that is a necessary requirement of classicality not satisfied by coherent states, and strict classicality requires no negative probabilities to certify it.",
"Regarding Reason 3 above, the classical coherence functions arise by defining abbreviations for quantities not naturally expressible in terms of electric field intensities at single spacetime points alone.",
"The quantum coherence functions simply promote the classical field quantities to operators and promote the time average to a quantum ensemble average; thus its main claim to classicality is through a generalization of Ehrenfest's theorem [60], [61], which is merely an approximate condition for correspondence of quantum operators to classical quantities, but not sufficient (nor exactly necessary) to determine classicality since it is not a fundamental law of quantum mechanics.",
"In other words; even though construction of quantum coherence functions is permissible, the ability of certain quantum states (such as coherent states) to produce mean values that we expect in a classical theory is not sufficient to conclude that such states are truly classical; at best we could say that they exhibit classical behavior, and as such, coherent states might be considered the most classical of all quantum states that possess superposition in a computational basis."
],
[
"Limits of Decomposition Indices",
"In [sec:IV.A]Sec.sec:IV.A, the decomposition unitary $U$ is limited to dimension $D$ of at least $r\\equiv \\text{rank}(\\rho )$ (guaranteed by the existence of the spectral decomposition), and a minimax of at most $r^2$ (proved by P. Horodecki using the Caratheodory theorem justified by a Bloch-vector expansion [44], [62], which sets the maximum number of decomposition states needed to describe a separable state with mode-independent (MI) decomposition states [Family 5 from [tab:1]Table tab:1]), so that the number of pure decomposition states can always be limited to $D \\in r, \\ldots ,r^2 $ , even though we are free to use $D \\in r, \\ldots ,\\infty $ .",
"Note that in [44], the term “dimension” is used to mean “rank” which can be seen in a later work by the same author [63] in which the upper decomposition limit is clearly identified as $r^2$ .",
"The fact that this limit should be $r^2$ rather than $n^2$ is clear when we consider the case of pure states, which have $r=1$ and only need one decomposition state (the pure state itself up to global phase), regardless of dimension.",
"However, if for some reason this interpretation is incorrect and the use of $\\max (D)=r^2$ turns out to be too restrictive, then we would simply use $\\max (D)=n^2$ instead.",
"Since none of the main results in this paper rely on actually computing statance or probablance, this issue has no bearing on the main results.",
"Then, since $j \\equiv (j_1 , \\ldots ,j_N )$ is our index for decomposition states of an $N$ -mode state, we need to determine the bounds of each $j_m$ such that we achieve a given total of $D$ decomposition states.",
"Therefore, if we let the range of each mode label be $j_m \\in 1, \\ldots ,D_m $ , we can define a vector $\\mathbf {D}\\equiv (D_{1},\\ldots ,D_{N})$ such that $D = \\prod \\nolimits _{m = 1}^N {D_m } = D_1 \\cdots D_N $ [due to the nested sums in quantities like (eq.24)] so that as $j \\equiv (j_1 , \\ldots ,j_N )$ counts over all allowed values, it has exactly $D$ of them.",
"Now we just need to determine the value of a given $D_m$ .",
"By the Caratheodory theorem [44], [62], the upper necessary limit $D_m$ for each mode label for a Family-5 state is $n_{m}^2$ , where $n_{m}$ is the number of levels of mode $m$ .",
"Thus, by this line of reasoning, we need to use $D_m \\in 1, \\ldots ,n_m ^2 \\;\\;\\,\\text{s.t.",
"}\\;\\prod \\limits _{m = 1}^N {D_m } = D\\;\\;\\text{for}\\;\\;D \\in r, \\ldots ,r^2 .$ However, note that in some cases, the Caratheodory theorem's application by P. Horodecki may be too restrictive for the following reason.",
"Recall that in Wootters's full concurrence paper [43], he included an explicit method showing that it is always possible to decompose any separable two-qubit state with exactly $r$ separable decomposition states (he used $n$ to mean rank there) of the form of Family 3 from [tab:1]Table tab:1.",
"Yet in P. Horodecki's paper [44], he placed an upper bound on the number of necessary separable decomposition states of $r^2$ , but he used states of the form of Family 5 from [tab:1]Table tab:1, which are only a small subset of all separable states, the full set being Family 3.",
"The interesting thing about Wootters's decomposition is that it also applies to the Family-5 states, meaning that even states for which statance is zero (since they have MI decomposition states) have a separable decomposition of the Family-5 form with only $r$ members!",
"But there is no contradiction here; the reason the Wootters decomposition works for all separable states is that it is based merely on minimizing average entanglement, a special kind of decomposition-state correlation, which means that at least for two qubits, all states with an MI set of decomposition states (Families 5 and 6 from [tab:1]Table tab:1) are guaranteed to have a separable decomposition of only $r$ members.",
"But because statance measures all decomposition-state correlation, of which entanglement is only a part, the decomposition that minimizes statance is generally more restrictive and therefore requires more members, so that is why we must use the Caratheodory theorem to set the limits in the statance calculation.",
"Basically, P. Horodecki used a set of states that is only a subset of separable states; his proof applies only to Family 5 and its subset, Family 6.",
"However, his result is still correct; it is just too loose of an upper bound for more general separable states such as those of Families 3 and 4 that are not also in Families 5 and 6.",
"To see how (eq.P.1) fits into the decomposition of zero-statance states (since they are the standard by which zero statance is achieved), [tab:3]Table tab:3 shows all the possible decomposition vectors $\\mathbf {D}$ allowable for two qubits for the purpose of keeping $D \\in r,\\ldots ,r^2$ .",
"Table: Allowable minimally necessary numbers of two-qubit decomposition states DD given r≡rank(ρ)r{\\hspace{-1.0pt}}\\equiv {\\hspace{-1.0pt}}\\text{rank}(\\rho ), and vectors of possible mode-independent single-mode index limits 𝐃≡(D 1 ,D 2 )\\mathbf {D}{\\hspace{-1.0pt}}\\equiv {\\hspace{-1.0pt}}(D_{1},D_{2}) such that ∏ m=1 N D m =D\\prod \\nolimits _{m=1}^{N}D_{m}{\\hspace{-1.0pt}}={\\hspace{-1.0pt}}D for D∈r,...,r 2 D{\\hspace{-1.0pt}}\\in {\\hspace{-1.0pt}} r,\\ldots ,r^2 as in (eq.P.1).Notice that $D$ is never a prime number for $D> n$ , and for the same reason, any numbers whose only factors include values that exceed any $n_{m}^2$ are also excluded, such as $D=10$ whose only factors are $1\\cdot 10$ and $2\\cdot 5$ , since both 10 and 5 exceed $n_{2}^2=4$ .",
"We can use decompositions such as that, but if P. Horodecki's proof is correct, then we are guaranteed to always be able to find a decomposition of member numbers from the set in [tab:3]Table tab:3 instead.",
"For example, if a decomposition for a rank-3 state is known to have 10 MI states, we should always be able to find a different MI decomposition with only $D\\in \\lbrace 3,4,6,8,9\\rbrace $ MI states.",
"However, in a rank-4 state, starting with a known decomposition of 10 MI states may require that we increase to $D=12$ or $D=16$ states, depending on whether a different MI decomposition with fewer MI states than 10 exists for the input state.",
"But if we allow things like $D=10$ here, then our single-mode limits would need to exceed the single-mode Caratheodory bounds.",
"Therefore, the single-mode Caratheodory bounds of $n_{m}^2$ may actually lead to larger decompositions (or they may instead always allow a smaller decomposition), but we do not have a proof for the convertibility of such decompositions at this time.",
"To see how $\\mathbf {D}$ relates to a state with zero statance, consider a two-qubit state such as $\\begin{array}{*{20}r}{\\rho = } &\\!\\!\\!",
"{p_1 \\rho _1^{(1)} \\otimes \\rho _1^{(2)} +} &\\!\\!\\!",
"{ p_6 \\rho _2^{(1)} \\otimes \\rho _1^{(2)} } \\\\{} &\\!\\!\\!",
"{ + p_2 \\rho _1^{(1)} \\otimes \\rho _2^{(2)} +} &\\!\\!\\!",
"{ p_7 \\rho _2^{(1)} \\otimes \\rho _2^{(2)} } \\\\{} &\\!\\!\\!",
"{ + p_3 \\rho _1^{(1)} \\otimes \\rho _3^{(2)} +} &\\!\\!\\!",
"{ p_8 \\rho _2^{(1)} \\otimes \\rho _3^{(2)} } \\\\{} &\\!\\!\\!",
"{ + p_4 \\rho _1^{(1)} \\otimes \\rho _4^{(2)} +} &\\!\\!\\!",
"{ p_9 \\rho _2^{(1)} \\otimes \\rho _4^{(2)} } \\\\{} &\\!\\!\\!",
"{ + p_5 \\rho _1^{(1)} \\otimes \\rho _5^{(2)} +} &\\!\\!\\!",
"{ p_{10} \\rho _2^{(1)} \\otimes \\rho _5^{(2)} } \\\\\\end{array}{\\hspace{-2.0pt}}.$ Since statance ignores the decomposition probabilities, then regardless of what they are, we can merely look at the decomposition states themselves, and here we notice that we can generate the same set from $\\lbrace \\rho _1^{(1)} ,\\rho _2^{(1)} \\rbrace \\otimes \\lbrace \\rho _1^{(2)} ,\\rho _2^{(2)} ,\\rho _3^{(2)} ,\\rho _4^{(2)} ,\\rho _5^{(2)} \\rbrace \\equiv \\lbrace \\rho _{(j_1 ,j_2 )} \\rbrace ,$ where the tensor product distributes over commas (if you prefer, the commas can be replaced by plus signs, and then the terms of the resulting sum comprise $\\lbrace \\rho _{(j_1 ,j_2 )} \\rbrace $ .",
"We can simplify (eq.P.3) further using the notation $\\begin{array}{*{20}l}{\\lbrace 1,2\\rbrace \\otimes \\lbrace 1,2,3,4,5\\rbrace } &\\!\\!\\!",
"{ \\equiv \\!",
"\\lbrace (j_1 ,j_2 )\\rbrace } \\\\{} &\\!\\!\\!",
"{ =\\!",
"\\lbrace (1,1),(1,2),(1,3),(1,4),(1,5),} \\\\{} &\\!\\!\\!",
"{\\phantom{=\\!\\lbrace }(2,1),(2,2),(2,3),(2,4),(2,5)\\rbrace .}",
"\\\\\\end{array}\\!\\!$ The decomposition vector $\\mathbf {D}\\equiv (D_{1},\\ldots ,D_{N})$ is a list of the maximal index labels in each mode that are needed to decompose a zero-statance state whose optimal decomposition has $D=D_{1}\\cdots D_{N}$ MI decomposition states.",
"Thus, in this example, the decomposition in (eq.P.2) has $\\mathbf {D}\\equiv (2,5),$ [which are also the sizes of the mode sets in (eq.P.4)].",
"According to P. Horodecki's decomposition limit, which applies to states such as this, $D_2 =5$ is larger than is needed for the mode-2 limit, so we should be able to find a different decomposition with mode-2 limit $D_2 \\leqslant 4$ .",
"If $r=3$ , then [tab:3]Table tab:3 shows that we should be guaranteed to find such an MI decomposition of fewer members where $D\\in \\lbrace 3,4,6,8,9\\rbrace $ , but if $r=4$ , then we may or may not be able to find a different MI decomposition with fewer members, as mentioned earlier, though we are guaranteed to find one that has at most $D=r^2 =16$ MI members.",
"Regarding the discrepancy between Wootters's decompositions and P. Horodecki's limits, we may hypothesize that Wootters's result is true for all systems (not just two qubits); that all full $N$ -partite separable states have a decomposition of exactly $r$ separable pure decomposition states in the form of Family 3, but that for states with zero statance, they require at least between $r$ and $r^2$ pure decomposition states to achieve a set of MI decomposition states (and furthermore, states belonging to Families 5 and 6 all have both types of decompositions; an $r$ -member separable decomposition, and an $(r,\\ldots ,r^2)$ -member zero-statance decomposition).",
"Note that in cases where the spectral decomposition is also a statance-minimizing decomposition, it is not necessarily $U=I^{[r]}$ that achieves this (where square bracketed superscripts indicate matrix dimension), but rather it is the $r$ -level permutation unitary that minimizes statance.",
"This is important when constructing examples using orthogonal decomposition states, since the descending-order convention (DOC) of eigenvalues used to define their labels does not always produce the statance-minimizing decomposition when $U=I^{[r]}$ ; therefore even when constructing such states, we still need to search the order of their labels to truly minimize the statance for that decomposition.",
"Furthermore, since the eigenvalues in the probability formula $p_j = \\sum \\nolimits _{k = 1}^r {\\lambda _k |U_{j,k} |^2 } $ in the DOC prevent any columns of $U$ above column $r$ from affecting the probability, the factors $|U_{j,k} |^2$ must not all be zero for a given $j$ so that we consider only sets where $p_j > 0\\;\\;\\forall j$ .",
"A necessary and sufficient way to ensure this is to specify that $\\sum \\nolimits _{k = 1}^r {|U_{j,k} |^2 } \\ne 0\\;\\;\\forall j$ .",
"Then, since there is always at least one nonzero eigenvalue, all probabilities are nonzero."
],
[
"Proof that Statance is a Necessary and Sufficient Measure of Decomposition-State Correlation",
"Here we prove that achieving zero statance $\\hat{\\mathcal {S}}(\\rho )=0$ is necessary and sufficient for any mixed or pure state $\\rho $ to have no decomposition-state correlation as defined in [sec:I.A]Sec.sec:I.A, and that having $\\hat{\\mathcal {S}}(\\rho )>0$ is necessary and sufficient for $\\rho $ to have some decomposition-state correlation, proving that statance is a necessary and sufficient measure of decomposition-state correlation.",
"First, the unoptimized statance $\\mathcal {S}(\\rho )$ of (eq.23) has form $\\mathcal {S}(\\rho ) \\equiv ||\\mathbf {w}||_{1} = \\sum \\nolimits _{j = 1}^D {|w_j |} ,$ which is just a 1-norm of the vector $\\mathbf {w}$ with components $w_j \\equiv \\text{tr}[(\\rho _j - \\mu _j )^2 ],$ where the absolute value $|w_j |$ in (eq.Q.1) is because the operator $\\Delta _j\\equiv \\rho _j - \\mu _j$ is Hermitian and therefore has only real eigenvalues, so the trace of its square as $w_j \\equiv \\text{tr}[\\Delta _j^2]$ in (eq.Q.2) is just the sum of the squares of its real eigenvalues, which is always nonnegative.",
"Thus, it is always true that $w_j =|w_j |$ .",
"Therefore, since the necessary and sufficient condition for any 1-norm of a vector $\\mathbf {w}$ to be zero is that $\\mathbf {w}=\\mathbf {0}$ which means that all of its components are zero $w_j =0\\;\\forall j$ , then the necessary and sufficient condition for $\\mathcal {S}(\\rho )$ to be zero is that $w_j =0\\;\\forall j$ , or equivalently, that $\\rho _j =\\mu _j\\;\\forall j$ (since that is the unique condition that causes $\\rho _j - \\mu _j=0$ which means that $w_j = \\text{tr}[(\\rho _j - \\mu _j )^2] = 0\\;\\forall j$ ).",
"Then, due to the definition in (eq.22) of statance $\\hat{\\mathcal {S}}(\\rho )$ as being the minimum value of $\\mathcal {S}(\\rho )$ over all decompositions of $\\rho $ , then the necessary and sufficient condition for $\\hat{\\mathcal {S}}(\\rho )=0$ can be stated as $\\hat{\\mathcal {S}}(\\rho ) = 0\\;\\;\\text{iff}\\;\\;\\exists \\lbrace p_j ,\\rho _j \\rbrace \\;\\;\\text{s.t.",
"}\\;\\;\\rho _j = \\mu _j \\;\\forall j,$ where $\\lbrace p_j ,\\rho _j \\rbrace $ is some particular decomposition set of $\\rho $ .",
"To show that statance $\\hat{\\mathcal {S}}(\\rho )$ is a necessary and sufficient measure of decomposition-state correlation, we will first show that a sufficient condition for $\\hat{\\mathcal {S}}(\\rho )=0$ is that $\\rho $ has a decomposition with mode-independent decomposition states ($\\text{MI}\\lbrace \\rho _j\\rbrace $ ), which means that $\\hat{\\mathcal {S}}(\\rho )=0$ is necessary for $\\text{MI}\\lbrace \\rho _j\\rbrace $ .",
"Then, we will show that $\\hat{\\mathcal {S}}(\\rho )=0$ is also sufficient for $\\text{MI}\\lbrace \\rho _j\\rbrace $ , which means that $\\text{MI}\\lbrace \\rho _j\\rbrace $ is necessary for $\\hat{\\mathcal {S}}(\\rho )=0$ .",
"To prove that having $\\text{MI}\\lbrace \\rho _j\\rbrace $ is sufficient for $\\hat{\\mathcal {S}}(\\rho )=0$ , we need to show that a $\\rho $ with $\\text{MI}\\lbrace \\rho _j\\rbrace $ causes $\\rho _j = \\mu _j \\;\\forall j$ .",
"Therefore, suppose that $\\rho $ has an optimal decomposition with $\\text{MI}\\lbrace \\rho _j\\rbrace $ .",
"Then, by definition of MI, combining (eq.5) and (eq.6) (using a different dummy index to keep things distinct), the decomposition states take the special form $\\rho _j \\equiv \\rho _{(j_1 , \\ldots ,j_N )} = \\mathop \\otimes \\limits _{q = 1}^N \\rho _{j_q }^{(q)} \\;\\;\\forall j.$ For $\\mu _j \\equiv \\mu _{(j_1 , \\ldots ,j_N )} $ in (eq.24), we need the quantity $\\rho _{(j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } )} = \\mathop \\otimes \\limits _{q = 1}^N \\rho _{j_q ^{\\lbrace m\\rbrace } }^{(q)} ,$ so then, the quantity $R_{j_m }^{(m)} \\equiv {\\hspace{-27.0pt}}\\sum \\limits _{{\\hspace{27.0pt}}j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } = 1, \\ldots ,1}^{D_1 , \\ldots ,D_N } {{\\hspace{-27.0pt}}\\delta _{j_m ^{\\lbrace m\\rbrace } ,j_m } \\rho _{(j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } )} }$ becomes $\\begin{array}{*{20}l}{R_{j_m }^{(m)} }\\!",
"&\\!\\!\\!",
"{ = {\\hspace{-27.0pt}}\\sum \\limits _{{\\hspace{27.0pt}}\\rule {0pt}{9pt}j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } = 1, \\ldots ,1}^{D_1 , \\ldots ,D_N } {{\\hspace{-29.0pt}}\\delta _{j_m ^{\\lbrace m\\rbrace } ,j_m } \\mathop \\otimes \\limits _{q = 1}^N \\rho _{j_q ^{\\lbrace m\\rbrace } }^{(q)} } } \\\\{}\\!",
"&\\!\\!\\!",
"{ =\\rule {0pt}{22pt} {\\hspace{-134.0pt}}\\sum \\limits _{{\\hspace{130.0pt}}\\rule {0pt}{13pt}j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_{m - 1} ^{\\lbrace m\\rbrace } ,j_{m + 1} ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } = 1, \\ldots ,1,1, \\ldots ,1}^{{\\hspace{78.0pt}}{D_1 , \\ldots ,D_{m - 1} ,D_{m + 1} , \\ldots ,D_N }_{\\vphantom{\\rule {0pt}{7pt}}}} {{\\hspace{-134.0pt}}\\textnormal {\\scalebox {0.94}{\\rho _{j_1 ^{\\lbrace m\\rbrace } }^{(1)} {\\hspace{-2.0pt}} \\otimes {\\hspace{-2.0pt}} \\cdots {\\hspace{-2.0pt}} \\otimes {\\hspace{-1.0pt}} \\rho _{j_{m - 1} ^{\\lbrace m\\rbrace } }^{(m - 1)} {\\hspace{-2.0pt}} \\otimes {\\hspace{-1.0pt}} \\rho _{j_m }^{(m)} {\\hspace{-2.0pt}} \\otimes {\\hspace{-1.0pt}} \\rho _{j_{m + 1} ^{\\lbrace m\\rbrace } }^{(m + 1)} {\\hspace{-2.0pt}} \\otimes {\\hspace{-2.0pt}} \\cdots {\\hspace{-2.0pt}} \\otimes {\\hspace{-1.0pt}} \\rho _{j_N ^{\\lbrace m\\rbrace } }^{(N)} }}} } \\\\{}\\!",
"&\\!\\!\\!",
"{ =\\rule {0pt}{24pt}{\\hspace{-5.0pt}}\\left[ {{\\hspace{-3.0pt}}\\left( {\\sum \\limits _{j_1 ^{\\lbrace m\\rbrace } = 1}^{D_1 } {{\\hspace{-3.0pt}}\\rho _{j_1 ^{\\lbrace m\\rbrace } }^{(1)} } } \\right){\\hspace{-4.0pt}} \\otimes {\\hspace{-2.0pt}} \\cdots {\\hspace{-2.0pt}} \\otimes {\\hspace{-4.0pt}} \\left( {\\sum \\limits _{j_{m - 1} ^{\\lbrace m\\rbrace } = 1}^{D_{m - 1} } {{\\hspace{-3.0pt}}\\rho _{j_{m - 1} ^{\\lbrace m\\rbrace } }^{(m - 1)} } } \\right){\\hspace{-4.0pt}} \\otimes {\\hspace{-1.0pt}} \\rho _{j_m }^{(m)} } \\right.}",
"\\\\{}\\!",
"&\\!\\!\\!",
"{\\left.",
"{ {\\hspace{10.0pt}}\\otimes {\\hspace{-4.0pt}} \\left( {\\sum \\limits _{j_{m + 1} ^{\\lbrace m\\rbrace } = 1}^{D_{m + 1} } {{\\hspace{-3.0pt}}\\rho _{j_{m + 1} ^{\\lbrace m\\rbrace } }^{(m + 1)} } } \\right) {\\hspace{-4.0pt}}\\otimes \\cdots \\otimes {\\hspace{-4.0pt}}\\left( {\\sum \\limits _{j_N ^{\\lbrace m\\rbrace } = 1}^{D_N } {{\\hspace{-3.0pt}}\\rho _{j_N ^{\\lbrace m\\rbrace } }^{(N)} } } \\right){\\hspace{-3.0pt}}} \\right]{\\hspace{-3.0pt}},} \\\\\\end{array}$ and since (eq.24) has the form $\\mu _j \\equiv \\mathop \\otimes \\limits _{m = 1}^N \\text{tr}_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} \\left( {\\frac{1}{{D_1 \\cdots D_{m - 1} D_{m + 1} \\cdots D_N }}R_{j_m }^{(m)} } \\right),$ then putting (eq.Q.7) into (eq.Q.8) gives $\\begin{array}{*{20}l}{\\mu _j } &\\!\\!",
"{ =\\!\\!",
"\\mathop \\otimes \\limits _{m = 1}^N {\\hspace{-4.0pt}}\\left[ {\\text{tr}{\\hspace{-3.0pt}}\\left( {{\\hspace{-3.0pt}}\\frac{1}{{D_1 }}{\\hspace{-6.0pt}}\\sum \\limits _{\\rule {0pt}{9pt}j_1 ^{\\lbrace m\\rbrace } = 1}^{D_1 } {{\\hspace{-6.0pt}}\\rho _{j_1 ^{\\lbrace m\\rbrace } }^{(1)} } {\\hspace{-3.0pt}}} \\right) {\\hspace{-4.0pt}}\\cdots {\\hspace{-1.0pt}} \\text{tr}{\\hspace{-3.0pt}}\\left( {{\\hspace{-3.0pt}}\\frac{1}{{D_{m - 1} }}{\\hspace{-6.0pt}}\\sum \\limits _{\\rule {0pt}{9pt}j_{m - 1} ^{\\lbrace m\\rbrace } = 1}^{D_{m - 1} } {{\\hspace{-6.0pt}}\\rho _{j_{m - 1} ^{\\lbrace m\\rbrace } }^{(m - 1)} } {\\hspace{-3.0pt}}} \\right){\\hspace{-2.0pt}}\\rho _{j_m }^{(m)} } \\right.}",
"\\\\{} &\\!\\!\\!",
"{\\left.",
"{ {\\hspace{28.0pt}}\\times {\\hspace{-1.0pt}}\\text{tr}{\\hspace{-3.0pt}}\\left( {{\\hspace{-3.0pt}}\\frac{1}{{D_{m + 1} }}{\\hspace{-6.0pt}}\\sum \\limits _{\\rule {0pt}{9pt}j_{m + 1} ^{\\lbrace m\\rbrace } = 1}^{D_{m + 1} } {{\\hspace{-6.0pt}}\\rho _{j_{m + 1} ^{\\lbrace m\\rbrace } }^{(m + 1)} } {\\hspace{-3.0pt}}} \\right) {\\hspace{-4.0pt}}\\cdots {\\hspace{-1.0pt}} \\text{tr}{\\hspace{-3.0pt}}\\left( {{\\hspace{-3.0pt}}\\frac{1}{{D_N }}{\\hspace{-6.0pt}}\\sum \\limits _{\\rule {0pt}{9pt}j_N ^{\\lbrace m\\rbrace } = 1}^{D_N } {{\\hspace{-6.0pt}}\\rho _{j_N ^{\\lbrace m\\rbrace } }^{(N)} } {\\hspace{-3.0pt}}} \\right){\\hspace{-4.0pt}}} \\right]} \\\\{} &\\!\\!",
"{ =\\!\\!",
"\\mathop \\otimes \\limits _{m = 1}^N \\rho _{j_m }^{(m)} } \\\\\\end{array}$ and since (eq.Q.4) holds for each $j$ , then (eq.Q.9) does as well, so then putting (eq.Q.9) into (eq.Q.4) we see that $\\rho _j = \\mu _j \\;\\forall j,$ which, by (eq.Q.3) proves that having $\\text{MI}\\lbrace \\rho _j\\rbrace $ is sufficient to cause $\\hat{\\mathcal {S}}(\\rho )=0$ .",
"To show that having $\\text{MI}\\lbrace \\rho _j\\rbrace $ is also necessary to cause $\\hat{\\mathcal {S}}(\\rho )=0$ , we will simply show that $\\hat{\\mathcal {S}}(\\rho )=0$ is sufficient for $\\rho $ to have $\\text{MI}\\lbrace \\rho _j\\rbrace $ .",
"Therefore, supposing that $\\hat{\\mathcal {S}}(\\rho )=0$ , then by (eq.Q.3), $\\rho _j = \\mu _j \\;\\forall j$ , which by (eq.24) is $\\rho _{(j_1 , \\ldots ,j_N )} = \\mu _{(j_1 , \\ldots ,j_N )} = \\mathop \\otimes \\limits _{m = 1}^N \\beta _{j_m }^{(m)} ,$ where each $\\beta _{j_m }^{(m)}$ is a physical state in mode $m$ given by $\\beta _{j_m }^{(m)} \\equiv \\text{tr}_{{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}}} {\\hspace{-3.0pt}}\\left( {{\\hspace{-2.0pt}}\\frac{1}{{D_{{{\\hspace{0.0pt}}\\mathop {m}\\limits ^{{\\hspace{-0.3pt}}{\\overline{{\\hspace{5.5pt}}}}}{\\hspace{0.0pt}}}} }}{\\hspace{-130.0pt}}\\sum \\limits _{{\\hspace{130.0pt}}\\rule {0pt}{11pt}j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_{m - 1} ^{\\lbrace m\\rbrace } ,j_{m + 1} ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } = 1, \\ldots ,1,1, \\ldots ,1}^{{\\hspace{78.0pt}}{D_1 , \\ldots ,D_{m - 1} ,D_{m + 1} , \\ldots ,D_N }_{\\vphantom{\\rule {0pt}{2pt}}}} {{\\hspace{-128.0pt}}\\rho _{(j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_{m - 1} ^{\\lbrace m\\rbrace } ,j_m ,j_{m + 1} ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } )} } {\\hspace{5.0pt}}} \\right){\\hspace{-2.0pt}},$ where note that in (eq.Q.12) we did not expand the decomposition states further because we are making no assumption about their form in this part of the proof.",
"Thus, (eq.Q.11) shows that the condition of $\\hat{\\mathcal {S}}(\\rho )=0$ automatically leads to a situation where the $\\rho _j \\equiv \\rho _{(j_1 , \\ldots ,j_N )} $ have a (tensor) product form of $N$ factors guaranteed for all $j$ , and each factor depends on a different mode-specific index $j_m$ , which is the definition of the form of mode independence for decomposition states.",
"Furthermore, since each $\\beta _{j_m }^{(m)} $ is a convex sum of physical states (which are just the pure decomposition states with special mode-specific index arguments) with equal probabilities such that $\\text{tr}(\\beta _{j_m }^{(m)} ) = 1$ , then each $\\beta _{j_m }^{(m)} $ qualifies as a physical state, and we can simply rename them as states $\\beta _{j_m }^{(m)} \\equiv \\rho _{j_m }^{(m)} ,$ which, put into (eq.Q.11), is the definition of $\\text{MI}\\lbrace \\rho _j\\rbrace $ , and so we have proven that $\\hat{\\mathcal {S}}(\\rho )=0$ is a sufficient condition for $\\text{MI}\\lbrace \\rho _j\\rbrace $ , and thus $\\text{MI}\\lbrace \\rho _j\\rbrace $ is necessary for $\\hat{\\mathcal {S}}(\\rho )=0$ .",
"We have now shown that the condition of a state having a decomposition with mode-independent decomposition states ($\\text{MI}\\lbrace \\rho _j\\rbrace $ ) is both necessary and sufficient to cause $\\hat{\\mathcal {S}}(\\rho )=0$ .",
"Then, since a violation of $\\text{MI}\\lbrace \\rho _j\\rbrace $ implies $\\hat{\\mathcal {S}}(\\rho )\\ne 0$ , and by its definition this means that $\\hat{\\mathcal {S}}(\\rho )>0$ , we have proven that statance $\\hat{\\mathcal {S}}(\\rho )$ is a valid measure of decomposition-state correlation, since the violation of $\\text{MI}\\lbrace \\rho _j\\rbrace $ is the definition of decomposition-state correlation, and the achievement of $\\text{MI}\\lbrace \\rho _j\\rbrace $ is the necessary and sufficient condition for $\\hat{\\mathcal {S}}(\\rho )=0$ ."
],
[
"Statance Example",
"Suppose we have a two-qubit state of the type in Family 5 of [tab:1]Table tab:1, with the statance-minimizing decomposition $\\rho = \\sum \\nolimits _j {p_j \\rho _{j_1 }^{(1)} \\otimes \\rho _{j_2 }^{(2)} } $ , where the probabilities $p_j \\equiv p_{(j_1 , j_2 )} $ in this example happen to not have mode independence, while by definition the decomposition states do have mode independence.",
"Therefore, in this particular example, the decomposition states are $\\begin{array}{*{20}l}{\\rho _1 } &\\!\\!",
"{ \\equiv \\rho _{(1,1)} } &\\!\\!",
"{ = \\rho _1^{(1)} \\otimes \\rho _1^{(2)} } \\\\{\\rho _2 } &\\!\\!",
"{ \\equiv \\rho _{(1,2)} } &\\!\\!",
"{ = \\rho _1^{(1)} \\otimes \\rho _2^{(2)} } \\\\{\\rho _3 } &\\!\\!",
"{ \\equiv \\rho _{(2,1)} } &\\!\\!",
"{ = \\rho _2^{(1)} \\otimes \\rho _1^{(2)} } \\\\{\\rho _4 } &\\!\\!",
"{ \\equiv \\rho _{(2,2)} } &\\!\\!",
"{ = \\rho _2^{(1)} \\otimes \\rho _2^{(2)}. }",
"\\\\\\end{array}$ To find the statance of $\\rho $ , first write the general form of $\\mu _j \\equiv \\mu _{(j_1 ,j_2 )} $ from (eq.24) with $\\mathbf {D}=(2,2)$ as $\\begin{array}{*{20}l}{\\mu _j} &\\!\\!",
"{\\equiv \\text{tr}_{\\overline{1}} \\left( {\\frac{1}{2}[\\rho _{(j_1 ,1)} {\\hspace{-1.0pt}}+{\\hspace{-1.0pt}} \\rho _{(j_1 ,2)} ]} \\right){\\hspace{-2.0pt}} \\otimes \\text{tr}_{\\overline{2}} \\left( {\\frac{1}{2}[\\rho _{(1,j_2 )} {\\hspace{-1.0pt}}+{\\hspace{-1.0pt}} \\rho _{(2,j_2 )} ]} \\right){\\hspace{-1.5pt}}.}",
"\\\\\\end{array}$ Writing these out for each index and then plugging-in the particular states in (eq.R.1) gives $\\begin{array}{*{20}l}{\\mu _1 } &\\!\\!",
"{ = \\text{tr}_{\\overline{1}} (\\rho _1^{(1)} \\otimes \\frac{1}{2}[\\rho _1^{(2)} {\\hspace{-1.5pt}}+{\\hspace{-1.0pt}} \\rho _2^{(2)} ]) \\otimes \\text{tr}_{\\overline{2}} (\\frac{1}{2}[\\rho _1^{(1)} {\\hspace{-1.5pt}}+{\\hspace{-1.0pt}} \\rho _2^{(1)} ] \\otimes \\rho _1^{(2)} )} \\\\{} &\\!\\!",
"{ = \\rho _1^{(1)} \\otimes \\rho _1^{(2)} } \\\\{\\mu _2 } &\\!\\!",
"{ = \\text{tr}_{\\overline{1}} (\\rho _1^{(1)} \\otimes \\frac{1}{2}[\\rho _1^{(2)} {\\hspace{-1.5pt}}+{\\hspace{-1.0pt}} \\rho _2^{(2)} ]) \\otimes \\text{tr}_{\\overline{2}} (\\frac{1}{2}[\\rho _1^{(1)} {\\hspace{-1.5pt}}+{\\hspace{-1.0pt}} \\rho _2^{(1)} ] \\otimes \\rho _2^{(2)} )} \\\\{} &\\!\\!",
"{ = \\rho _1^{(1)} \\otimes \\rho _2^{(2)} } \\\\{\\mu _3 } &\\!\\!",
"{ = \\text{tr}_{\\overline{1}} (\\rho _2^{(1)} \\otimes \\frac{1}{2}[\\rho _1^{(2)} {\\hspace{-1.5pt}}+{\\hspace{-1.0pt}} \\rho _2^{(2)} ]) \\otimes \\text{tr}_{\\overline{2}} (\\frac{1}{2}[\\rho _1^{(1)} {\\hspace{-1.5pt}}+{\\hspace{-1.0pt}} \\rho _2^{(1)} ] \\otimes \\rho _1^{(2)} )} \\\\{} &\\!\\!",
"{ = \\rho _2^{(1)} \\otimes \\rho _1^{(2)} } \\\\{\\mu _4 } &\\!\\!",
"{ = \\text{tr}_{\\overline{1}} (\\rho _2^{(1)} \\otimes \\frac{1}{2}[\\rho _1^{(2)} {\\hspace{-1.5pt}}+{\\hspace{-1.0pt}} \\rho _2^{(2)} ]) \\otimes \\text{tr}_{\\overline{2}} (\\frac{1}{2}[\\rho _1^{(1)} {\\hspace{-1.5pt}}+{\\hspace{-1.0pt}} \\rho _2^{(1)} ] \\otimes \\rho _2^{(2)} )} \\\\{} &\\!\\!",
"{ = \\rho _2^{(1)} \\otimes \\rho _2^{(2)} .}",
"\\\\\\end{array}$ Then, putting (eq.R.1) and (eq.R.3) into (eq.23), we get $\\begin{array}{*{20}l}{\\mathcal {S}(\\rho )} &\\!\\!",
"{ \\equiv \\sum \\limits _{j_1 ,j_2 =1,1}^{D_{1},D_{2}} {{\\hspace{-4.0pt}}\\text{tr}[(\\rho _{(j_1 ,j_2 )} - \\mu _{(j_1 ,j_2 )} )^2 ]} } \\\\{} &\\!\\!",
"{ = \\sum \\limits _{j_1 ,j_2 =1,1}^{D_{1},D_{2}} {{\\hspace{-4.0pt}}\\text{tr}[(\\rho _{j_1 }^{(1)} \\otimes \\rho _{j_2 }^{(2)} - \\rho _{j_1 }^{(1)} \\otimes \\rho _{j_2 }^{(2)} )^2 ]} = 0,} \\\\\\end{array}$ since for this example, each $\\mu _j$ is exactly equal to $\\rho _j$ regardless of what the probabilities were.",
"Then, since (eq.R.4) was calculated with an optimal decomposition [because it minimizes the unoptimized statance $\\mathcal {S}(\\rho )$ ], $\\mathcal {S}(\\rho )$ yields the statance from (eq.22) as $\\hat{\\mathcal {S}}(\\rho ) \\equiv \\frac{1}{{\\mathcal {N}_{\\mathcal {S}} }}\\mathop {\\min }\\limits _{\\lbrace U\\rbrace } [\\mathcal {S}(\\rho )] = 0.$ Thus, this example shows that any states belonging to Families 5 or 6 from [tab:1]Table tab:1 have zero statance regardless of their decomposition probabilities, because those states all have mode-independent sets of decomposition states for their statance-minimizing decompositions."
],
[
"Proof that Probablance is a Necessary and Sufficient Measure of Probability Correlation",
"Here we prove that achieving zero probablance $\\hat{\\mathcal {P}}(\\rho )=0$ is necessary and sufficient for any mixed or pure state $\\rho $ to have no probability correlation as defined in [sec:I.A]Sec.sec:I.A, and that having $\\hat{\\mathcal {P}}(\\rho )>0$ is necessary and sufficient for the state to have some probability correlation, proving that probablance is a necessary and sufficient measure of probability correlation.",
"First, note that the unoptimized probablance $\\mathcal {P}(\\rho )$ from (eq.27) can be written as $\\mathcal {P}(\\rho ) \\equiv |\\mathbf {v}|^2 = \\sum \\nolimits _j {|v_j |^2 } ,$ which is just a nonzero scalar times a 2-norm of the vector $\\mathbf {v}\\equiv \\mathbf {p}^{\\prime }-\\mathbf {q}^{\\prime },$ where $\\mathbf {p}^{\\prime }$ has components $p_j^{\\prime } \\equiv p_{(j_1 , \\ldots ,j_N )}^{\\prime } $ which is the set of decomposition probabilities of an arbitrary statance-minimizing (SM) decomposition of $\\rho $ as given by (eq.29), and $\\mathbf {q}^{\\prime }$ has components $q_j^{\\prime } \\equiv q_{(j_1 , \\ldots ,j_N )}^{\\prime } $ defined in (eq.28), so $\\mathbf {v}$ has components $v_j \\equiv p_j^{\\prime } - q_j^{\\prime } $ .",
"Therefore, since the necessary and sufficient condition for any 2-norm of a vector $\\mathbf {v}$ to be zero is that $\\mathbf {v}=\\mathbf {0}$ , which means that all of its components are zero $v_j \\equiv 0\\;\\forall j$ , then the necessary and sufficient condition for $\\mathcal {P}(\\rho )$ to be zero is that $v_j \\equiv 0\\;\\forall j$ , or equivalently, that $p_j^{\\prime } = q_j^{\\prime } \\;\\forall j$ .",
"Then, due to the definition in (eq.26) of $\\hat{\\mathcal {P}}(\\rho )$ as being the minimum value of $\\mathcal {P}(\\rho )$ over all SM decompositions of $\\rho $ , the necessary and sufficient condition for $\\hat{\\mathcal {P}}(\\rho )=0$ can be stated as $\\hat{\\mathcal {P}}(\\rho ) \\!=\\!",
"0{\\hspace{5.0pt}} \\text{iff} {\\hspace{5.0pt}} \\exists \\lbrace p_j^{\\prime } ,\\rho _j^{\\prime } \\rbrace {\\hspace{5.0pt}}\\text{s.t.",
"}{\\hspace{5.0pt}}p_j^{\\prime } \\!",
"=\\!",
"q_j^{\\prime } \\;\\forall j{\\hspace{5.0pt}}\\text{and}{\\hspace{5.0pt}}\\hat{\\mathcal {S}}(\\rho )\\!=\\!\\hat{\\mathcal {S}}_{\\min },$ where $\\lbrace p_j^{\\prime } ,\\rho _j^{\\prime } \\rbrace $ is some particular SM decomposition of $\\rho $ , $\\hat{\\mathcal {S}}(\\rho )$ is the statance from [sec:IV.A]Sec.sec:IV.A, and $\\hat{\\mathcal {S}}_{\\min }$ is the minimal $\\hat{\\mathcal {S}}(\\rho )$ over all decompositions of $\\rho $ .",
"(The requirement that $\\lbrace p_j^{\\prime } ,\\rho _j^{\\prime } \\rbrace $ causes $\\hat{\\mathcal {S}}_{\\min }$ is actually already part of the definition of $\\lbrace p_j^{\\prime } ,\\rho _j^{\\prime } \\rbrace $ , so it is redundant in (eq.S.3), but stated anyway to emphasize its importance).",
"To show that $\\hat{\\mathcal {P}}(\\rho )$ is a necessary and sufficient measure of probability correlation, we will first show that a sufficient condition for $\\hat{\\mathcal {P}}(\\rho )=0$ is that $\\rho $ has an SM decomposition with mode-independent probabilities ($\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}\\equiv \\text{MI}\\lbrace p_{j}^{\\prime }\\rbrace $ ), meaning that $\\hat{\\mathcal {P}}(\\rho )=0$ is necessary for $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ .",
"Then, we will show that $\\hat{\\mathcal {P}}(\\rho )=0$ is also sufficient for $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ , meaning that $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ is also necessary for $\\hat{\\mathcal {P}}(\\rho )=0$ .",
"To prove that having $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ is sufficient for $\\hat{\\mathcal {P}}(\\rho )=0$ , we need to show that a state $\\rho $ with $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ causes $p_j^{\\prime } = q_j^{\\prime } \\;\\forall j$ .",
"Therefore, suppose that $\\rho $ has an SM decomposition with $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ .",
"Then, by definition, combining (eq.7) and (eq.8) (using a different dummy index to keep things distinct), $p_j^{\\prime } \\equiv p_{(j_1 , \\ldots ,j_N )}^{\\prime } = \\prod \\nolimits _{q = 1}^N p_{j_q }^{(q)} \\;\\;\\forall j,$ where $\\sum \\nolimits _{j_q } {p_{j_q }^{(q)} } = 1;\\;\\;\\forall q \\in 1, \\ldots ,N$ .",
"For $q_j^{\\prime } \\equiv q_{(j_1 , \\ldots ,j_N )}^{\\prime }$ in (eq.28), we need the quantity $p_{(j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } )}^{\\prime } = \\prod \\nolimits _{q = 1}^N p_{j_q ^{\\lbrace m\\rbrace } }^{(q)} ,$ so then, the quantity $Q_{j_m }^{(m)} \\equiv {\\hspace{-27.0pt}}\\sum \\limits _{{\\hspace{27.0pt}}j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace }= 1, \\ldots ,1 }^{D_1 , \\ldots ,D_N } {{\\hspace{-27.0pt}}\\delta _{j_m ^{\\lbrace m\\rbrace } ,j_m } p_{(j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } )}^{\\prime } }$ takes on the value $\\begin{array}{*{20}l}{Q_{j_m }^{(m)} } &\\!\\!",
"{ = {\\hspace{-27.0pt}}\\sum \\limits _{\\rule {0pt}{10pt}{\\hspace{27.0pt}}j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace }= 1, \\ldots ,1 }^{D_1 , \\ldots ,D_N } {{\\hspace{-27.0pt}}\\delta _{j_m ^{\\lbrace m\\rbrace } ,j_m } \\prod \\nolimits _{q = 1}^N {p_{j_q ^{\\lbrace m\\rbrace } }^{(q)} } } } \\\\{} &\\!\\!",
"{ = \\rule {0pt}{22pt} {\\hspace{-134.0pt}}\\sum \\limits _{{\\hspace{130.0pt}}\\rule {0pt}{14pt}j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_{m - 1} ^{\\lbrace m\\rbrace } ,j_{m + 1} ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } = 1, \\ldots ,1,1, \\ldots ,1}^{{\\hspace{78.0pt}}{D_1 , \\ldots ,D_{m - 1} ,D_{m + 1} , \\ldots ,D_N }_{\\vphantom{\\rule {0pt}{8pt}}}} {{\\hspace{-131.0pt}}p_{j_1 ^{\\lbrace m\\rbrace } }^{(1)} \\cdots p_{j_{m - 1} ^{\\lbrace m\\rbrace } }^{(m - 1)} p_{j_m }^{(m)} p_{j_{m + 1} ^{\\lbrace m\\rbrace } }^{(m + 1)} \\cdots p_{j_N ^{\\lbrace m\\rbrace } }^{(N)} } } \\\\{} &\\!\\!",
"{= \\left[ {{\\hspace{-3.0pt}}\\left({\\sum \\limits _{j_1 ^{\\lbrace m\\rbrace } =1}^{D_{1}} {{\\hspace{-3.0pt}}p_{j_1 ^{\\lbrace m\\rbrace } }^{(1)} } }\\right) \\cdots \\left({\\sum \\limits _{j_{m - 1} ^{\\lbrace m\\rbrace } =1}^{D_{m-1}} {{\\hspace{-3.0pt}}p_{j_{m - 1} ^{\\lbrace m\\rbrace } }^{(m - 1)} } }\\right) p_{j_m }^{(m)} } \\right.}",
"\\\\{} &\\!\\!",
"{\\left.",
"{ {\\hspace{13.0pt}}\\times {\\hspace{-5.0pt}}\\left({\\sum \\limits _{j_{m + 1} ^{\\lbrace m\\rbrace } =1}^{D_{m+1}} {{\\hspace{-3.0pt}}p_{j_{m + 1} ^{\\lbrace m\\rbrace } }^{(m + 1)} } }\\right) \\cdots \\left({\\sum \\limits _{j_N ^{\\lbrace m\\rbrace } =1}^{D_{N}} {{\\hspace{-3.0pt}}p_{j_N ^{\\lbrace m\\rbrace } }^{(N)} } }\\right){\\hspace{-3.0pt}}} \\right]} \\\\{} &\\!\\!",
"{ = p_{j_m }^{(m)} ,} \\\\\\end{array}$ which, when put into (eq.28), gives $q_j^{\\prime } \\equiv q_{(j_1 , \\ldots ,j_N )}^{\\prime } = \\prod \\nolimits _{m = 1}^N p_{j_m }^{(m)} .$ Since (eq.S.4) holds for each $j$ , then (eq.S.8) does as well, so then putting (eq.S.8) into (eq.S.4), we see that $p_j^{\\prime } = q_j^{\\prime } \\;\\forall j,$ which, by (eq.S.3) proves that having $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ is sufficient to cause $\\hat{\\mathcal {P}}(\\rho )=0$ .",
"To show that having $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ is also necessary to cause $\\hat{\\mathcal {P}}(\\rho )=0$ , we will simply show that $\\hat{\\mathcal {P}}(\\rho )=0$ is sufficient for $\\rho $ to have $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ .",
"Therefore, supposing that $\\hat{\\mathcal {P}}(\\rho )=0$ , then by (eq.S.1) and (eq.S.2), $p_j^{\\prime } = q_j^{\\prime } \\;\\forall j$ , which expands using (eq.28) as $p_{(j_1 , \\ldots ,j_N )}^{\\prime } = \\prod \\nolimits _{m = 1}^N {b_{j_m }^{(m)} } ,$ where we define the quantity $b_{j_m }^{(m)} \\equiv {\\hspace{-132.0pt}}\\sum \\limits _{{\\hspace{130.0pt}}\\rule {0pt}{11pt}j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_{m - 1} ^{\\lbrace m\\rbrace } ,j_{m + 1} ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } = 1, \\ldots ,1,1, \\ldots ,1}^{{\\hspace{78.0pt}}{D_1 , \\ldots ,D_{m - 1} ,D_{m + 1} , \\ldots ,D_N }_{\\vphantom{\\rule {0pt}{3pt}}}} {{\\hspace{-128.0pt}}p_{(j_1 ^{\\lbrace m\\rbrace } , \\ldots ,j_{m - 1} ^{\\lbrace m\\rbrace } ,j_m ,j_{m + 1} ^{\\lbrace m\\rbrace } , \\ldots ,j_N ^{\\lbrace m\\rbrace } )}^{\\prime } } ,$ where we cannot simplify the probabilities further because we are making no assumptions about their form here.",
"Also, since the sum of all the $p_j^{\\prime } \\equiv p_{(j_1 , \\ldots ,j_N )}^{\\prime } $ is 1, and the sum in (eq.S.11) is missing one of the mode sums that would complete that sum to 1, and since the terms in (eq.S.11) are all probabilities existing on $[0,1]$ , then these quantities $b_{j_m }^{(m)}$ generally satisfy $0 \\leqslant b_{j_m }^{(m)} \\leqslant 1.$ Thus, (eq.S.10) shows that the condition of $\\hat{\\mathcal {P}}(\\rho )=0$ automatically leads to a situation where each $p_j^{\\prime } \\equiv p_{(j_1 , \\ldots ,j_N )}^{\\prime } $ has a product form of $N$ factors guaranteed for all $j$ , and each factor depends on a different mode-specific index $j_m$ , which is the definition of the form of mode independence.",
"Furthermore, since $b_{j_m }^{(m)} \\in [0,1]$ by (eq.S.12), then by the normalization property of the $p_j^{\\prime } \\equiv p_{(j_1 , \\ldots ,j_N )}^{\\prime } $ , the mode-independence leads to all the sums producing positive factors, seen by summing (eq.S.10) as $\\begin{array}{*{20}l}{{\\hspace{-27.0pt}}\\sum \\limits _{\\rule {0pt}{8pt}{\\hspace{27.0pt}}j_1 , \\ldots ,j_N = 1, \\ldots ,1}^{D_1 , \\ldots ,D_N } {{\\hspace{-31.0pt}}p_{(j_1 , \\ldots ,j_N )} } } &\\!\\!",
"{ = {\\hspace{-4.0pt}}\\left({\\sum \\limits _{j_1 =1}^{D_{1}} {b_{j_1 }^{(1)} } }\\right) \\cdots \\left({\\sum \\limits _{j_N =1}^{D_{N}} {b_{j_N }^{(N)} } }\\right)} \\\\{{\\hspace{40.0pt}}1} &\\!\\!",
"{ = B^{(1)} \\cdots B^{(N)} ,} \\\\\\end{array}$ where $B^{(m)} \\equiv \\sum \\nolimits _{j_m } {b_{j_m }^{(m)} } $ such that $B^{(m)} \\in [0,1]$ .",
"Then, the only way that the right-side product, with each factor on $[0,1]$ , could equal the left is if all of the factors are 1, meaning $B^{(m)} =1\\;\\forall m$ , which means that each $b_{j_m }^{(m)}$ then satisfies all of the conditions for being a probability since $b_{j_m }^{(m)} \\in [0,1]$ and $\\sum \\nolimits _{j_m } {b_{j_m }^{(m)} }=1$ .",
"[Also, $\\sum \\nolimits _{j_m } {b_{j_m }^{(m)} }=1$ due to the definition in (eq.S.11) because it supplies the missing mode sum for the full probabilities to sum to 1.]",
"Therefore, we can rename them as mode-specific probabilities $b_{j_m }^{(m)} \\equiv p_{j_m }^{(m)} ,$ obeying $\\sum \\nolimits _{j_m } {p_{j_m }^{(m)} }=1$ so that (eq.S.10) becomes $p_{(j_1 , \\ldots ,j_N )}^{\\prime } = \\prod \\nolimits _{m = 1}^N {p_{j_m }^{(m)} };\\;\\;\\sum \\nolimits _{j_m } {p_{j_m }^{(m)} }=1\\;\\;\\forall m \\;\\forall j ,$ which is the definition of $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ , and therefore we have proven that $\\hat{\\mathcal {P}}(\\rho )=0$ is a sufficient condition for $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ , and thus $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ is necessary for $\\hat{\\mathcal {P}}(\\rho )=0$ .",
"We have now shown that the condition of a state having an SM decomposition with mode-independent decomposition probabilities ($\\text{MI}\\lbrace p_{j}\\rbrace $ ) is both necessary and sufficient to cause $\\hat{\\mathcal {P}}(\\rho )=0$ .",
"Then, since a violation of $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ implies $\\hat{\\mathcal {P}}(\\rho )\\ne 0$ , and by its definition this means that $\\hat{\\mathcal {P}}(\\rho )> 0$ , we have proven that probablance $\\hat{\\mathcal {P}}(\\rho )$ is a valid measure of probability correlation, since violation of $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ is the definition of probability correlation, and achievement of $\\text{MI}\\lbrace p_{j}\\rbrace _{\\text{SM}}$ is the necessary and sufficient condition for $\\hat{\\mathcal {P}}(\\rho )=0$ .",
"See [sec:IV.B]Sec.sec:IV.B for the explanation of why we must use a statance-minimizing decomposition to compute probablance."
],
[
"Probablance Example",
"To see how probablance works for a state with mode-independent (MI) probabilities of a statance-minimizing (SM) decomposition, consider the special case of some product-form two-qubit input mixed state $\\rho $ , which, due to its product form, has an optimal decomposition [one that minimizes the unoptimized probablance of (eq.27)] with probabilities of the form $\\begin{array}{*{20}l}{p_{(1,1)}^{\\prime } } &\\!\\!",
"{ = p_1^{(1)} p_1^{(2)} } \\\\{p_{(1,2)}^{\\prime } } &\\!\\!",
"{ = p_1^{(1)} p_2^{(2)} } \\\\{p_{(2,1)}^{\\prime } } &\\!\\!",
"{ = p_2^{(1)} p_1^{(2)} } \\\\{p_{(2,2)}^{\\prime } } &\\!\\!",
"{ = p_2^{(1)} p_2^{(2)}, } \\\\\\end{array}$ where $p_1^{(m)} + p_2^{(m)} = 1$ , and these $p_{(j_1 , \\ldots ,j_N )}^{\\prime } $ are found from solving the optimization problem over all the unitaries $U^{\\prime }$ of an SM decomposition that produce $p_{(j_1 , \\ldots ,j_N )}^{\\prime } $ from (eq.29) to minimize $\\mathcal {P}(\\rho )$ in (eq.27) to give the probablance $\\hat{\\mathcal {P}}(\\rho )$ in (eq.26).",
"(In practice, we would just get a list of numbers that have this structure, but we would not know it at this point in the calculation; it would only be clear when we finish computing the probablance.)",
"In [sec:App.E]App.sec:App.E we proved that any mixed product-form state has an optimal decomposition of $r=\\text{rank}(\\rho )$ decomposition states, where the rank of each single-mode reduction is $r_m$ and $r_1 \\cdots r_N = r$ .",
"Using this fact, then $\\mathbf {D}=(2,2)$ , and the overall form of $q_{j}^{\\prime }$ from (eq.28) is $q_{(j_1 ,j_2 )}^{\\prime }=(p_{(j_1 ,1)}^{\\prime } + p_{(j_1 ,2)}^{\\prime } )(p_{(1,j_2 )}^{\\prime } + p_{(2,j_2 )}^{\\prime } ).$ Then, using the particular values of (eq.T.1) in (eq.T.2) gives $\\begin{array}{*{20}l}{q_{(1,1)}^{\\prime } } &\\!\\!",
"{ = (p_1^{(1)} p_1^{(2)} + p_1^{(1)} p_2^{(2)} )(p_1^{(1)} p_1^{(2)} + p_2^{(1)} p_1^{(2)} )} \\\\{} &\\!\\!",
"{ = p_1^{(1)} (p_1^{(2)} + p_2^{(2)} )(p_1^{(1)} + p_2^{(1)} )p_1^{(2)} } \\\\{} &\\!\\!",
"{ = p_1^{(1)} p_1^{(2)}, } \\\\\\end{array}$ and similarly, $\\begin{array}{*{20}l}{q_{(1,2)}^{\\prime } } &\\!\\!",
"{ = (p_1^{(1)} p_1^{(2)} \\!\\!",
"+\\!",
"p_1^{(1)} p_2^{(2)} )(p_1^{(1)} p_2^{(2)} \\!\\!",
"+\\!",
"p_2^{(1)} p_2^{(2)} )} &\\!\\!",
"{ = p_1^{(1)} p_2^{(2)} } \\\\{q_{(2,1)}^{\\prime } } &\\!\\!",
"{ = (p_2^{(1)} p_1^{(2)} \\!\\!",
"+\\!",
"p_2^{(1)} p_2^{(2)} )(p_1^{(1)} p_1^{(2)} \\!\\!",
"+\\!",
"p_2^{(1)} p_1^{(2)} )} &\\!\\!",
"{ = p_2^{(1)} p_1^{(2)} } \\\\{q_{(2,2)}^{\\prime } } &\\!\\!",
"{ = (p_2^{(1)} p_1^{(2)} \\!\\!",
"+\\!",
"p_2^{(1)} p_2^{(2)} )(p_1^{(1)} p_2^{(2)} \\!\\!",
"+\\!",
"p_2^{(1)} p_2^{(2)} )} &\\!\\!",
"{ = p_2^{(1)} p_2^{(2)} ,} \\\\\\end{array}$ which are exactly the same as the optimal decomposition probabilities themselves [which only happens when the optimal decomposition probabilities have mode independence as in (eq.T.1)].",
"Thus, putting (eq.T.3–eq.T.4) into (eq.27), $\\mathcal {P}(\\rho ) = \\sum \\limits _{j_1 ,j_2 =1,1}^{D_{1},D_{2}} {|p_{(j_1 ,j_2 )}^{\\prime } - q_{(j_1 ,j_2 )}^{\\prime } |^2 } = 0,$ which also yields the fully optimized probablance as $\\hat{\\mathcal {P}}(\\rho )=0$ here, since we specified that the $U^{\\prime }$ we used to get the $p_{(j_1 ,j_2 )}^{\\prime }$ produced an SM decomposition, and since 0 is the lowest value that $\\mathcal {P}(\\rho )$ can have, then this decomposition minimizes the unoptimized probablance.",
"Thus, this example shows the mechanism by which states with SM decompositions of mode-independent product-form probabilities produce a probablance of 0."
],
[
"Register and Inverse Register Functions",
"Register counting is any ordered set of sets of characters for which the rightmost character increases in its ordered set until it reaches the end, and then the character to its immediate left can increment by one member in its ordered set, but the character to its right simultaneously resets and must go through another cycle before the character to its left can increment again.",
"For example, in a two-character word where each character has an alphabet of two letters (a system with structure $2\\times 2$ such as two qubits), register counting goes as $\\lbrace (1,1),(1,2),(2,1),(2,2)\\rbrace $ .",
"We can relabel this vector-index list with the scalar-index labels $\\lbrace 1,2,3,4\\rbrace $ .",
"For an $N$ -mode discrete quantum system with structure $\\mathbf {n}=(n_1 ,\\ldots ,n_{N})$ , where mode $m$ has $n_m$ outcomes labeled in increasing order from 1 to $n_m$ , the map from a particular vector-index outcome labeled as $\\mathbf {a}=(a_1 ,\\ldots ,a_{N})$ , where $a_m \\in 1, \\ldots ,n_m \\;\\forall m\\in 1,\\ldots ,N$ to the scalar-index outcome in the set $1,\\ldots ,n$ , where $n\\equiv n_{1}\\cdots n_N$ , is given by the indical register function [47], [18], $\\begin{array}{*{20}l}{R_\\mathbf {a}^{\\lbrace N,\\mathbf {n}\\rbrace } } &\\!\\!",
"{ \\equiv 1 +\\!",
"\\sum \\limits _{m = 1}^N {\\left( {(a_m - 1)\\!\\!\\!\\!\\prod \\limits _{j = m + 1}^N \\!\\!\\!\\!",
"{n_j } } \\right)}} \\\\{} &\\!\\!",
"{ = (a_1 - 1)(n_2 \\cdots n_N )\\!",
"+\\!",
"(a_2 - 1)(n_3 \\cdots n_N ) \\!+ \\cdots } \\\\{} &\\!\\!",
"{\\;\\;\\;\\, + (a_{N - 2} - 1)(n_{N - 1} n_N )\\!",
"+\\!",
"(a_{N - 1} - 1)n_N \\!",
"+\\!",
"a_N ,} \\\\\\end{array}$ which maps the vector index $\\mathbf {a}$ to scalar index $a$ as $a=R_\\mathbf {a}^{\\lbrace N,\\mathbf {n}\\rbrace }$ .",
"For example, in a two-qubit system, for which $\\mathbf {n}=(2,2)$ , (eq.U.1) maps the vector index $(2,1)$ to the scalar index 3.",
"In other situations, we know a scalar-index value $a$ and want to know to which vector-index $\\mathbf {a}$ it corresponds.",
"For that, we use the inverse indical register function [18], $\\mathbf {a}_a^{\\lbrace N,\\mathbf {n}\\rbrace } = (a_1 , \\ldots a_N );\\;\\;\\left\\lbrace {\\begin{array}{*{20}l}{a_m } &\\!\\!",
"{ = \\text{floor}(\\frac{v_{m - 1} - 1}{d_{m}})+1} \\\\{v_m } &\\!\\!",
"{ = v_{m - 1} - (a_m - 1)d_m ,} \\\\\\end{array}} \\right.$ for $m\\in 1,\\ldots ,N$ , where $v_0 \\equiv a$ , and $d_m \\equiv \\Pi _{q=m+1}^{N}n_{q}$ .",
"Thus (eq.U.2) maps scalar index $a$ to vector index $\\mathbf {a}$ as $\\mathbf {a}=\\mathbf {a}_a^{\\lbrace N,\\mathbf {n}\\rbrace }$ .",
"For example, in a two-qubit system, (eq.U.2) maps scalar index 2 to vector index $(1,2)$ .",
"Note that throughout this paper, although $\\mathbf {n}$ is the mode-size vector to use for register counting of the basis functions with (eq.U.1) and (eq.U.2), in the case of register counting the decomposition indices $j\\equiv (j_{1},\\ldots ,j_{N})$ , we need to use the decomposition-limit vector $\\mathbf {D}\\equiv (D_{1},\\ldots ,D_{N})$ [see [sec:App.P]App.sec:App.P] instead of $\\mathbf {n}$ in (eq.U.1) and (eq.U.2) to convert between scalar decomposition indices $j$ and vector decomposition indices $(j_{1},\\ldots ,j_{N})$ [where again all indices start on 1].",
"Thus we have $j=R_\\mathbf {j}^{\\lbrace N,\\mathbf {D}\\rbrace }$ and $\\mathbf {j}=\\mathbf {j}_j^{\\lbrace N,\\mathbf {D}\\rbrace }$ , where $n_m \\rightarrow D_m$ , $a_m \\rightarrow j_m$ , $v_0 \\equiv j$ , and we use $\\mathbf {j}$ to specifically mean the vector index as a collection of many indices $\\mathbf {j}\\equiv (j_{1},\\ldots ,j_{N})$ rather than as a single equivalent number $j$ ."
]
] | 2001.03453 |
[
[
"Path independence of the additive functionals for stochastic\n differential equations driven by G-L\\'evy processes"
],
[
"Abstract In the paper, we consider a type of stochastic differential equations driven by G-L\\'evy processes.",
"We prove that a kind of their additive functionals has path independence and extend some known results."
],
[
"Introduction",
"Recently, development of mathematical finance forces the appearance of a type of processes-G-Brownian motions([5]).",
"And then the related theory, such as stochastic calculus and stochastic differential equations (SDEs in short) driven by G-Brownian motions, are widely studied([1], [5], [6], [8]).",
"However, in some financial models, volatility uncertainty makes G-Brownian motions insufficient for simulating these models.",
"One important reason lies in the continuity of their paths with respect to the time variable.",
"So, Hu-Peng [2] solved the problem by introducing G-Lévy processes.",
"And the type of processes has discontinuous (right continuous with left limits) paths.",
"Later, Paczka [3] defined the Itô-Lévy stochastic integrals, deduced the Itô formula, established SDEs driven by G-Lévy processes and stated the existence and uniqueness of solutions for these equations under lipschitz conditions.",
"Most recently, under non-lipschitz conditions, Wang-Gao [14] proved well-definedness of SDEs driven by G-Lévy processes and investigated exponential stability of their solutions.",
"Here, we follows up the line in [14], define the additive functionals of SDEs driven by G-Lévy processes and study their path independence.",
"Concretely speaking, we consider the following SDEs on ${\\mathbb {R}}^d$ : $\\mathrm {d}Y_t=b(t,Y_t)\\mathrm {d}t +h_{ij}(t,Y_t)\\mathrm {d}{\\langle }B^i, B^j{\\rangle }_t+\\sigma (t,Y_t)\\mathrm {d}B_t+\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }f(t,Y_t,u)L(\\mathrm {d}t,\\mathrm {d}u),$ where $B$ is a G-Brownian motion, ${\\langle }B^i, B^j{\\rangle }_t$ is the mutual variation process of $B^i$ and $B^j$ for $i,j=1,2,\\cdots ,d$ and $L(\\mathrm {d}t,\\mathrm {d}u)$ is a G-random measure (See Subsection REF ).",
"The coefficients $b: [0,T]\\times {\\mathbb {R}}^d\\mapsto {\\mathbb {R}}^d$ , $h_{ij}=h_{ji}: [0,T]\\times {\\mathbb {R}}^d\\mapsto {\\mathbb {R}}^d$ , $\\sigma : [0,T]\\times {\\mathbb {R}}^d\\mapsto {\\mathbb {R}}^{d\\times d}$ and $f: [0,T]\\times {\\mathbb {R}}^d\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )\\mapsto {\\mathbb {R}}^d$ are Borel measurable.",
"Here and hereafter we use the convention that the repeated indices stand for the summation.",
"Thus, under (${\\bf H}^1_{b,h,\\sigma ,f}$ )-(${\\bf H}^2_{b,h,\\sigma ,f}$ ) in Subsection REF , by [14], we know that Eq.",
"(REF ) has a unique solution $Y_t$ .",
"And then we introduce the additive functionals of $Y_t$ and define path independence of these functionals.",
"Finally, we prove that these functionals have path independence under some assumption.",
"Next, we say our motivations.",
"First, we mention that Ren-Yang [12] proved path independence of additive functionals for SDEs driven by G-Brownian motions.",
"Since these equations can not satisfy the actual demand very well, to extend them becomes one of our motivations.",
"Second, by analyzing some special cases, we surprisingly find that, we can express explicitly these additive functionals.",
"However, in our known results ([9], [10], [11]), it is difficult to express explicitly additive functionals.",
"Therefore, this is the other of our motivations.",
"This paper is arranged as follows.",
"In Section , we introduce G-Lévy processes, the Itô-Lévy stochastic integrals, SDEs driven by G-Lévy processes, additive functionals, path independence and some related results.",
"The main results and their proofs are placed in Section .",
"Moreover, we analysis some special cases and compare our result with some known results ([9], [10], [11], [12]) in Subsection REF ."
],
[
"Preliminary",
"In the section, we introduce some concepts and results used in the sequel."
],
[
"Notation",
"In the subsection, we introduce notations used in the sequel.",
"For convenience, we shall use $\\mid \\cdot \\mid $ and $\\parallel \\cdot \\parallel $ for norms of vectors and matrices, respectively.",
"Furthermore, let $\\langle \\cdot $ , $\\cdot \\rangle $ denote the scalar product in ${\\mathbb {R}}^d$ .",
"Let $Q^*$ denote the transpose of the matrix $Q$ .",
"Let $lip({\\mathbb {R}}^n)$ be the set of all Lipschitz continuous functions on ${\\mathbb {R}}^n$ and $C_{b, lip}({\\mathbb {R}}^d)$ be the collection of all bounded and Lipschitz continuous functions on ${\\mathbb {R}}^d$ .",
"Let $C_b^3({\\mathbb {R}}^d)$ be the space of bounded and three times continuously differentiable functions with bounded derivatives of all orders less than or equal to 3."
],
[
"G-Lévy processes",
"In the subsection, we introduce G-Lévy processes.(c.f.",
"[2]) Let $\\Omega $ be a given set and ${\\mathcal {H}}$ be a linear space of real functions defined on $\\Omega $ such that if $X_1,\\dots ,X_n\\in {\\mathcal {H}}$ , then $\\phi (X_1,\\dots ,X_n)\\in {\\mathcal {H}}$ for each $\\phi \\in lip({\\mathbb {R}}^n)$ .",
"If $X\\in {\\mathcal {H}}$ , we call $X$ as a random variable.",
"Definition 2.1 If a functional $\\bar{{\\mathbb {E}}}: {\\mathcal {H}}\\mapsto {\\mathbb {R}}$ satisfies: for $X, Y\\in {\\mathcal {H}}$ , (i) $X\\geqslant Y, \\bar{{\\mathbb {E}}}[X]\\geqslant \\bar{{\\mathbb {E}}}[Y]$ , (ii) $\\bar{{\\mathbb {E}}}[X+Y]\\leqslant \\bar{{\\mathbb {E}}}[X]+\\bar{{\\mathbb {E}}}[Y]$ , (iii)$ \\mbox{ for all}~\\lambda \\geqslant 0, \\bar{{\\mathbb {E}}}[\\lambda X]=\\lambda \\bar{{\\mathbb {E}}}[X]$ , (iv)$ \\mbox{ for all}~c\\in {\\mathbb {R}}, \\bar{{\\mathbb {E}}}[X+c]=\\bar{{\\mathbb {E}}}[X]+c$ , we call $\\bar{{\\mathbb {E}}}$ as a sublinear expectation on ${\\mathcal {H}}$ and $(\\Omega , {\\mathcal {H}}, \\bar{{\\mathbb {E}}})$ as a sublinear expectation space.",
"Next, we define the distribution of a random vector on $(\\Omega , {\\mathcal {H}}, \\bar{{\\mathbb {E}}})$ .",
"For a $n$ -dimensional random vector $X=(X_1, X_2, \\cdots , X_n)$ for $X_i\\in {\\mathcal {H}}$ , $i=1,2, \\cdots , n$ , set $F_X(\\phi ):=\\bar{{\\mathbb {E}}}(\\phi (X)), \\qquad \\phi \\in lip({\\mathbb {R}}^n),$ and then we call $F_X$ as the distribution of $X$ .",
"Definition 2.2 Assume that $X_1, X_2$ are two $n$ -dimensional random vectors defined on different sublinear expectation spaces.",
"If for all $\\phi \\in lip({\\mathbb {R}}^n)$ , $F_{X_1}(\\phi )=F_{X_2}(\\phi ),$ we say that the distributions of $X_1, X_2$ are the same.",
"Definition 2.3 For two random vectors $Y=(Y_1, Y_2, \\cdots , Y_m)$ for $Y_j\\in {\\mathcal {H}}$ and $X=(X_1, X_2, \\cdots , X_n)$ for $X_i\\in {\\mathcal {H}}$ , if for all $\\phi \\in lip({\\mathbb {R}}^n\\times {\\mathbb {R}}^m)$ , $\\bar{{\\mathbb {E}}}[\\phi (X,Y)]=\\bar{{\\mathbb {E}}}[\\bar{{\\mathbb {E}}}[\\phi (x,Y)]_{x=X}],$ we say that $Y$ is independent from $X$ .",
"Here, we use two above concepts to define Lévy processes on $(\\Omega , {\\mathcal {H}}, \\bar{{\\mathbb {E}}})$ .",
"Definition 2.4 Let $X=(X_t)_{t\\geqslant 0}$ be a $d$ -dimensional càdlàg process on $(\\Omega , {\\mathcal {H}}, \\bar{{\\mathbb {E}}})$ .",
"If $X$ satisfies (i) $X_0=0$ ; (ii) for $t, s\\geqslant 0$ , the increment $X_{s+t}-X_t$ is independent from $(X_{t_1}, X_{t_2}, \\cdots , X_{t_n})$ , for any $n$ and $0\\leqslant t_1<t_2\\cdots <t_n\\leqslant t$ ; (iii) the distribution of $X_{s+t}-X_t$ does not depend on $t$ ; we call $X$ as a Lévy process.",
"Definition 2.5 Assume that $X$ is a $d$ -dimensional Lévy process.",
"If there exists a decomposition $X_t=X_t^c+X_t^d$ for $t\\geqslant 0$ , where $(X_t^c, X_t^d)$ is a $2d$ -dimensional Lévy process satisfying $\\lim \\limits _{t\\downarrow 0}\\frac{\\bar{{\\mathbb {E}}}|X_t^c|^3}{t}=0, \\quad \\bar{{\\mathbb {E}}}|X_t^d|\\leqslant Ct, \\quad t\\geqslant 0, \\quad C\\geqslant 0,$ we call $X$ as a G-Lévy process.",
"In the following, we characterize G-Lévy processes by partial differential equations.",
"Theorem 2.6 Assume that $X$ is a $d$ -dimensional G-Lévy process.",
"Then for $g\\in C_b^3({\\mathbb {R}}^d)$ with $g(0)=0$ , set $G_X[g(\\cdot )]:=\\lim \\limits _{t\\downarrow 0}\\frac{\\bar{{\\mathbb {E}}}[g(X_t)]}{t},$ and then $G_X$ has the following Lévy-Khintchine representation $G_X[g(\\cdot )]=\\sup \\limits _{(\\nu ,\\zeta ,Q)\\in {\\mathcal {U}}}\\left\\lbrace \\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }g(u)\\nu (\\mathrm {d}u)+{\\langle }\\partial _x g(0), \\zeta {\\rangle }+\\frac{1}{2}tr[\\partial _x^2g(0)QQ^*]\\right\\rbrace ,$ where ${\\mathcal {U}}$ is a subset of ${\\mathcal {M}}({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )\\times {\\mathbb {R}}^d\\times {\\mathbb {R}}^{d\\times d}$ , ${\\mathcal {M}}({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )$ is the collection of all measures on $({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace , {B}({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ))$ , ${\\mathbb {R}}^{d\\times d}$ is the set of all $d\\times d$ matrices and ${\\mathcal {U}}$ satisfies $\\sup \\limits _{(\\nu ,\\zeta ,Q)\\in {\\mathcal {U}}}\\left\\lbrace \\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }|u|\\nu (\\mathrm {d}u)+|\\zeta |+\\frac{1}{2}tr[QQ^*]\\right\\rbrace <\\infty .$ Theorem 2.7 Suppose that $X$ is a $d$ -dimensional G-Lévy process.",
"Then for $\\phi \\in C_{b, lip}({\\mathbb {R}}^d)$ , $v(t,x):=\\bar{{\\mathbb {E}}}[\\phi (x+X_t)]$ is the unique viscosity solution of the following partial integro-differential equation: $0&=&\\partial _t v(t,x)-G_X[v(t,x+\\cdot )-v(t,x)]\\\\&=&\\partial _t v(t,x)-\\sup \\limits _{(\\nu ,\\zeta ,Q)\\in {\\mathcal {U}}}\\bigg \\lbrace \\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }[v(t,x+u)-v(t,x)]\\nu (\\mathrm {d}u)+{\\langle }\\partial _x v(t,x), \\zeta {\\rangle }\\\\&&\\qquad \\qquad \\qquad \\qquad +\\frac{1}{2}tr[\\partial _x^2v(t,x)QQ^*]\\bigg \\rbrace $ with the initial condition $v(0,x)=\\phi (x)$ .",
"Conversely, if we have a set ${\\mathcal {U}}$ satisfying (REF ), is there a $d$ -dimensional G-Lévy process having the Lévy-Khintchine representation (REF ) with the same set ${\\mathcal {U}}$ ?",
"The answer is affirmed.",
"We take $\\Omega :=D_0({\\mathbb {R}}^+, {\\mathbb {R}}^d)$ , where $D_0({\\mathbb {R}}^+, {\\mathbb {R}}^d)$ is the space of all càdlàg functions ${\\mathbb {R}}_+\\ni t\\mapsto \\omega _t\\in {\\mathbb {R}}^d$ with $\\omega _0=0$ , equipped with the Skorokhod topology.",
"Theorem 2.8 Suppose that ${\\mathcal {U}}$ satisfies (REF ).",
"Then there exists a sublinear expectation $\\bar{{\\mathbb {E}}}$ on $\\Omega $ such that the canonical process $X$ is a $d$ -dimensional G-Lévy process having the Lévy-Khintchine representation (REF ) with the same set ${\\mathcal {U}}$ ."
],
[
"A capacity",
"In the subsection, we introduce a capacity and related definitions.",
"First of all, fix a set ${\\mathcal {U}}$ satisfying (REF ) and $T>0$ and take $\\Omega _T:=D_0([0,T], {\\mathbb {R}}^d)$ and the sublinear expectation $\\bar{{\\mathbb {E}}}$ in Theorem REF .",
"Thus, we know that $(\\Omega _T, {\\mathcal {H}}, \\bar{{\\mathbb {E}}})$ is a sublinear expectation space.",
"Here, we work on the space.",
"Let $L_G^p(\\Omega _T)$ be the completion of $lip(\\Omega _T)$ under the norm $\\Vert \\cdot \\Vert _p:=(\\bar{{\\mathbb {E}}}|\\cdot |^p)^{1/p}, p\\geqslant 1$ .",
"Let ${\\mathcal {V}}:=\\lbrace \\nu \\in {\\mathcal {M}}({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ): \\exists (\\zeta , Q)\\in {\\mathbb {R}}^d\\times {\\mathbb {R}}^{d\\times d} ~\\mbox{such ~ that}~ (\\nu , \\zeta , Q)\\in {\\mathcal {U}}\\rbrace $ and let ${\\mathcal {G}}$ be the set of all the Borel measurable functions $g: {\\mathbb {R}}^d\\mapsto {\\mathbb {R}}^d$ with $g(0)=0$ .",
"Assumption: There exists a measure $\\mu \\in {\\mathcal {M}}({\\mathbb {R}}^d)$ such that $\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }|z|\\mu (\\mathrm {d}z)<\\infty , \\quad \\mu (\\lbrace 0\\rbrace )=0，$ and for all $\\nu \\in {\\mathcal {V}}$ there exists a function $g_{\\nu }\\in {\\mathcal {G}}$ satisfying $\\nu (A)=\\mu (g_{\\nu }^{-1}(A)), \\quad \\forall A\\in {B}({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ).$ There exists $0<q<1$ such that $\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\int _{0<|z|<1}|z|^q\\nu (\\mathrm {d}z)<\\infty .$ $\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\nu ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )<\\infty .$ Let $(\\tilde{\\Omega }, {F}, {\\mathbb {P}})$ be a probability space supporting a Brownian motion $W$ and a Poisson random measure $N(\\mathrm {d}t, \\mathrm {d}z)$ with the intensity measure $\\mu (\\mathrm {d}z)\\mathrm {d}t$ .",
"Let ${F}_t:=\\sigma \\left\\lbrace W_s, N((0,s],A): 0\\leqslant s\\leqslant t, A\\in {B}({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )\\right\\rbrace \\vee {\\mathcal {N}}, \\quad {\\mathcal {N}}:=\\lbrace U\\in {F}, {\\mathbb {P}}(U)=0\\rbrace .$ We introduce the following set.",
"Definition 2.9 ${\\mathcal {A}}_{0,T}^{{\\mathcal {U}}}$ is a set of all the processes $\\theta _t=(\\theta ^d_t, \\theta _t^{1,c}, \\theta _t^{2,c})$ for $t\\in [0,T]$ satisfying (i) $(\\theta _t^{1,c}, \\theta _t^{2,c})$ is a ${F}_t$ -adapted process and $\\theta ^d$ is a ${F}_t$ -predictable random field on $[0,T]\\times {\\mathbb {R}}^d$ , (ii) For ${\\mathbb {P}}$ -a.s. $\\omega $ and a.e.",
"$t\\in [0,T]$ , $(\\theta ^d(t,\\cdot )(\\omega ), \\theta ^{1,c}_t(\\omega ), \\theta ^{2,c}_t(\\omega ))\\in \\left\\lbrace (g_{\\nu },\\zeta ,Q)\\in {\\mathcal {G}}\\times {\\mathbb {R}}^d\\times {\\mathbb {R}}^{d\\times d}: (\\nu ,\\zeta ,Q)\\in {\\mathcal {U}}\\right\\rbrace ,$ (iii) ${\\mathbb {E}}^{{\\mathbb {P}}}\\left[\\int _0^T{\\Big (}|\\theta ^{1,c}_t|+\\Vert \\theta ^{2,c}_t\\Vert ^2+\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }|\\theta ^d(t,z)|\\mu (\\mathrm {d}z){\\Big )}\\mathrm {d}t\\right]<\\infty .$ For $\\theta \\in {\\mathcal {A}}_{0,T}^{{\\mathcal {U}}}$ , set $B_t^{0,\\theta }:=\\int _0^t\\theta ^{1,c}_s\\mathrm {d}s+\\int _0^t\\theta ^{2,c}_s\\mathrm {d}W_s+\\int _0^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }\\theta ^d(s,z)N(\\mathrm {d}s,\\mathrm {d}z), \\quad t\\in [0,T],$ and by Corollary 14 in [3], it holds that for $\\xi \\in L_G^1(\\Omega _T)$ $\\bar{{\\mathbb {E}}}[\\xi ]=\\sup \\limits _{\\theta \\in {\\mathcal {A}}_{0,T}^{{\\mathcal {U}}}}{\\mathbb {E}}^{{\\mathbb {P}}^{\\theta }}[\\xi ], \\quad {\\mathbb {P}}^{\\theta }={\\mathbb {P}}\\circ (B_{\\cdot }^{0,\\theta })^{-1}.$ And then define $\\bar{C}(D):=\\sup \\limits _{\\theta \\in {\\mathcal {A}}_{0,T}^{{\\mathcal {U}}}}{\\mathbb {P}}^{\\theta }(D), \\quad D\\in {B}(\\Omega _T),$ and $\\bar{C}$ is a capacity.",
"For $D\\in {B}(\\Omega _T)$ , if $\\bar{C}(D)=0$ , we call $D$ as a polar set.",
"So, if a property holds outside a polar set, we say that the property holds quasi-surely (q.s.",
"in short)."
],
[
"The Itô integrals with respect to G-Lévy processes",
"In the subsection, we introduce the Itô integrals with respect to G-Lévy processes under the framework of the above subsection.",
"Let $X$ denote the canonical process on the space, i.e.",
"$X_t(\\omega )=\\omega _t, t\\in [0,T]$ .",
"So, $X$ is a $d$ -dimensional G-Lévy process.",
"Although the Itô integrals with respect to G-Brownian motions have been introduced in [8], we need to introduce two related spaces used in the sequel.",
"Take $0=t_0<t_1<\\cdots <t_N=T$ .",
"Let $p\\geqslant 1$ be fixed.",
"Define ${\\mathcal {M}}^{p,0}_G(0,T):=\\Big \\lbrace \\eta _t(\\omega )=\\sum \\limits _{j=1}^N\\xi _{j-1}(\\omega )1_{[t_{j-1},t_j)}(t);\\xi _{j-1}(\\omega )\\in L^p_G(\\Omega _{t_{j-1}})\\Big \\rbrace .$ Let ${\\mathcal {M}}^p_G(0,T)$ and ${\\mathcal {H}}_G^p(0,T)$ denote the completion of ${\\mathcal {M}}^{p,0}_G(0,T)$ under the norm $\\Vert \\eta \\Vert _{{\\mathcal {M}}^p_G(0,T)}=\\left(\\int _0^T\\bar{{\\mathbb {E}}}|\\eta _t|^p\\mathrm {d}t\\right)^{\\frac{1}{p}} ~\\mbox{and}~\\Vert \\eta \\Vert _{{\\mathcal {H}}^p_G(0,T)}=\\left(\\bar{{\\mathbb {E}}}\\left(\\int _0^T|\\eta _t|^2\\mathrm {d}t\\right)^{\\frac{p}{2}}\\right)^{\\frac{1}{p}},$ respectively.",
"Let ${\\mathcal {M}}^p_G([0,T], {\\mathbb {R}}^d)$ and ${\\mathcal {H}}^p_G([0,T], {\\mathbb {R}}^d)$ be the collection of all the processes $\\eta _t=(\\eta ^1_t, \\eta ^2_t, \\cdots , \\eta ^d_t), \\quad t\\in [0,T], \\quad \\eta ^i\\in {\\mathcal {M}}^{p}_G(0,T) ~\\mbox{and}~ {\\mathcal {H}}_G^p(0,T),$ respectively.",
"Next, we introduce the Itô integrals with respect to random measures.",
"First, define a random measure: for any $0\\leqslant t\\leqslant T$ and $A\\in {B}({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )$ , $\\kappa _t:=X_t-X_{t-}, \\quad L((0,t], A):=\\sum \\limits _{0<s\\leqslant t}I_A(\\kappa _s), \\quad q.s..$ And then we define the Itô integral with respect to the random measure $L(\\mathrm {d}t, \\mathrm {d}u)$ .",
"Let ${\\mathcal {H}}^S_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ))$ be the collection of all the processes defined on $[0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )\\times \\Omega $ with the form $f(s,u)(\\omega )=\\sum \\limits _{k=1}^{n-1}\\sum \\limits _{l=1}^m\\phi _{k,l}(X_{t_1}, X_{t_2}-X_{t_1}, \\cdots , X_{t_k}-X_{t_{k-1}})I_{[t_k, t_{k+1})}(s)\\psi _l(u), n,m\\in {\\mathbb {N}},$ where $0\\leqslant t_1<\\cdots <t_n\\leqslant T$ is a partition of $[0,T]$ , $\\phi _{k,l}\\in C_{b, lip}({\\mathbb {R}}^{d\\times k})$ and $\\lbrace \\psi _l\\rbrace _{l=1}^m\\subset C_{b, lip}({\\mathbb {R}}^d)$ are functions with disjoint supports and $\\psi _l(0)=0$ .",
"Definition 2.10 For any $f\\in {\\mathcal {H}}^S_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ))$ , set $\\int _0^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }f(s,u)L(\\mathrm {d}s, \\mathrm {d}u):=\\sum \\limits _{0<s\\leqslant t}f(s,\\kappa _s)I_{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }(\\kappa _s), \\quad q.s..$ By Theorem 28 in [3], we have that $\\int _0^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }f(s,u)L(\\mathrm {d}s, \\mathrm {d}u)\\in L_G^2(\\Omega _T)$ .",
"Let ${\\mathcal {H}}^2_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ))$ be the completion of ${\\mathcal {H}}^S_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ))$ with respect to the norm $\\Vert \\cdot \\Vert _{{\\mathcal {H}}^2_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ))}$ , where $\\Vert f\\Vert _{{\\mathcal {H}}^2_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ))}:=\\bar{{\\mathbb {E}}}\\left[\\int _0^T\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }|f(s,u)|^2\\nu (\\mathrm {d}u)\\mathrm {d}s\\right]^{1/2}, \\quad f\\in {\\mathcal {H}}^S_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )).$ Thus, Corollary 29 in [3] admits us to get that for $f\\in {\\mathcal {H}}^2_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ))$ , $\\int _0^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }f(s,u)L(\\mathrm {d}s, \\mathrm {d}u)=\\sum \\limits _{0<s\\leqslant t}f(s,\\kappa _s)I_{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }(\\kappa _s), \\quad q.s..$ Let ${\\mathcal {H}}^2_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ), {\\mathbb {R}}^d)$ be the space of all the processes $f(t,u)={\\Big (}f^1(t,u), f^2(t,u), \\cdots , f^d(t,u){\\Big )}, \\quad f^i\\in {\\mathcal {H}}^2_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )).$"
],
[
"Stochastic differential equations driven by G-Lévy processes",
"In the subsection, we introduce stochastic differential equations driven by G-Lévy processes and related additive functionals.",
"First, we introduce some notations.",
"Let ${\\mathbb {S}}^d$ be the space of all $d\\times d$ symmetric matrices.",
"For $A\\in {\\mathbb {S}}^d$ , set $G(A):=\\frac{1}{2}\\sup \\limits _{Q\\in {\\mathcal {Q}}}{\\rm tr}[QQ^*A],$ where ${\\mathcal {Q}}$ is a nonempty, bounded, closed and convex subset of ${\\mathbb {R}}^{d\\times d}$ .",
"And then $G: {\\mathbb {S}}^d\\mapsto {\\mathbb {R}}$ is a monotonic, sublinear and positive homogeneous functional([6]).",
"We choose ${\\mathcal {U}}\\subset {\\mathcal {M}}({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )\\times \\lbrace 0\\rbrace \\times {\\mathcal {Q}}$ satisfying (REF ) and still work under the framework of Subsection REF .",
"So, the canonical process $X_t$ can be represented as $X_t=B_t+X_t^d$ , where $B_t$ is a G-Brownian motion associated with ${\\mathcal {Q}}$ and $X_t^d$ is a pure jump G-Lévy process associated with ${\\mathcal {M}}({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )$ .",
"Next, we consider Eq.",
"(REF ) and assume: There exists a constant $C_1>0$ such that for any $t\\in [0,T]$ and $x,y\\in {\\mathbb {R}}^d$ , $&&|b(t,x)-b(t,y)|^2+|h_{ij}(t,x)-h_{ij}(t,y)|^2+\\Vert \\sigma (t,x)-\\sigma (t,y)\\Vert ^2\\\\&+&\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }|f(t,x,u)-f(t,y,u)|^2\\nu (\\mathrm {d}u)\\leqslant C_1\\rho (|x-y|^2),$ where $\\rho :(0,+\\infty )\\mapsto (0,+\\infty )$ is a continuous, increasing and concave function so that $\\rho (0+)=0, \\quad \\int _0^1\\frac{dr}{\\rho (r)}=+\\infty .$ There exists a constant $C_2>0$ such that for any $t\\in [0,T]$ $|b(t,0)|^2+|h_{ij}(t,0)|^2+\\Vert \\sigma (t,0)\\Vert ^2+\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }|f(t,0,u)|^2\\nu (\\mathrm {d}u)\\leqslant C_2.$ By [14], we know that under (${\\bf H}^1_{b,h,\\sigma ,f}$ )-(${\\bf H}^2_{b,h,\\sigma ,f}$ ), Eq.",
"(REF ) has a unique solution $Y_t$ with $\\bar{{\\mathbb {E}}}\\left[\\sup \\limits _{t\\in [0,T]}|Y_t|^2\\right]<\\infty .$ And then we introduce the following additive functional $F_{s,t}&:=&\\alpha \\int _s^t G(g_1)(r,Y_r)\\mathrm {d}r+\\beta \\int _s^t g^{ij}_1(r,Y_r)\\mathrm {d}{\\langle }B^i,B^j{\\rangle }_r+\\int _s^t {\\langle }g_2(r,Y_r),\\mathrm {d}B_r{\\rangle }\\nonumber \\\\&&+\\int _s^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }g_3(r,Y_r, u)L(\\mathrm {d}r, \\mathrm {d}u)+\\int _s^t\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }\\gamma g_3(r,Y_r, u)\\nu (\\mathrm {d}u)\\mathrm {d}r, \\nonumber \\\\&&\\quad 0\\leqslant s<t\\leqslant T,$ where $\\alpha , \\beta , \\gamma \\in {\\mathbb {R}}$ are three constants and $&&g_1: [0,T]\\times {\\mathbb {R}}^d\\mapsto {\\mathbb {R}}^{d\\times d}, \\quad g_1^{ij}=g_1^{ji}, \\\\&&g_2:[0,T]\\times {\\mathbb {R}}^d\\mapsto {\\mathbb {R}}^d, \\\\&&g_3:[0,T]\\times {\\mathbb {R}}^d\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )\\mapsto {\\mathbb {R}},$ are Borel measurable so that $F_{s,t}$ is well-defined.",
"Definition 2.11 The additive functional $F_{s,t}$ is called path independent, if there exists a function $V: [0,T]\\times {\\mathbb {R}}^d\\mapsto {\\mathbb {R}},$ such that for any $s\\in [0, T]$ and $Y_s\\in L^2_G(\\Omega _T)$ , the solution $(Y_t)_{t\\in [s,T]}$ of Eq.",
"(REF ) satisfies $F_{s,t}=V(t,Y_t)-V(s,Y_s).$"
],
[
"Main results and their proofs",
"In the section, we state and prove the main results under the framework of Subsection REF .",
"And then we analysis some special cases and compare our result with some known results."
],
[
"Main results",
"In the subsection, we state and prove the main results.",
"Let us begin with a key lemma.",
"Lemma 3.1 Assume that ${\\mathcal {Q}}$ is bounded away from 0 and $Z_t$ is a 1-dimensional G-Itô-Lévy process, i.e.",
"$Z_t=\\int _0^t\\Gamma _s\\mathrm {d}s+\\int _0^t\\Phi _{ij}(s)\\mathrm {d}{\\langle }B^i, B^j{\\rangle }_s+\\int _0^t{\\langle }\\Psi _s,\\mathrm {d}B_s{\\rangle }+\\int _0^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }K(s,u)L(\\mathrm {d}s, \\mathrm {d}u),$ where $\\Gamma \\in {\\mathcal {M}}^1_G(0,T), \\Phi _{ij}\\in {\\mathcal {M}}^1_G(0,T), \\Phi _{ij}=\\Phi _{ji}, i,j=1,2,\\cdots ,d, \\Psi \\in {\\mathcal {H}}^1_G([0,T], {\\mathbb {R}}^d), K\\in {\\mathcal {H}}^2_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace ))$ .",
"Then $Z_t=0$ for all $t\\in [0,T]$ q.s.",
"if and only if $\\Gamma _t=0, \\Phi _{ij}(t)=0, \\Psi _t=0$ a.e.$\\times $ q.s.",
"on $[0,T]\\times \\Omega _T$ and $K(t,u)=0$ a.e.$\\times $ a.e.$\\times $ q.s.",
"on $[0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )\\times \\Omega _T$ .",
"Sufficiency is direct if one inserts $\\Gamma _t=0, \\Phi _{ij}(t)=0, \\Psi _t=0, K(t,u)=0$ into (REF ).",
"Let us prove necessity.",
"If $Z_t=0$ for any $t\\in [0,T]$ , we get that $0=\\int _0^t\\Gamma _s\\mathrm {d}s+\\int _0^t\\Phi _{ij}(s)\\mathrm {d}{\\langle }B^i, B^j{\\rangle }_s+\\int _0^t\\Psi _s^*\\mathrm {d}B_s+\\int _0^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }K(s,u)L(\\mathrm {d}s, \\mathrm {d}u).$ By taking the quadratic process with $\\int _0^t{\\langle }\\Psi _s, \\mathrm {d}B_s{\\rangle }$ on two sides of (REF ), it holds that $0&=&{\\langle }\\int _0^{\\cdot }\\Psi ^*_s\\mathrm {d}B_s, \\int _0^{\\cdot }\\Psi ^*_s\\mathrm {d}B_s{\\rangle }_t=\\int _0^t\\Psi ^{i}_s\\Psi ^{j}_s\\mathrm {d}{\\langle }B^i, B^j{\\rangle }_s=\\int _0^t{\\rm tr}(\\Psi _s\\Psi ^*_s\\mathrm {d}{\\langle }B{\\rangle }_s)\\\\&=&\\int _0^t{\\rm tr}(\\mathrm {d}{\\langle }B{\\rangle }_s\\Psi _s\\Psi ^*_s)=\\int _0^t{\\langle }\\mathrm {d}{\\langle }B{\\rangle }_s\\Psi _s, \\Psi _s{\\rangle },$ where ${\\langle }B{\\rangle }:=\\left(\\begin{array}{c}{\\langle }B^1,B^1{\\rangle }\\quad {\\langle }B^1,B^2{\\rangle }\\cdots {\\langle }B^1,B^d{\\rangle }\\\\\\vdots \\qquad \\qquad \\vdots \\qquad \\qquad \\vdots \\\\{\\langle }B^d,B^1{\\rangle }\\quad {\\langle }B^d,B^2{\\rangle }\\cdots {\\langle }B^d,B^d{\\rangle }\\end{array}\\right).$ Note that ${\\mathcal {Q}}$ is bounded away from 0.",
"Thus, there exists a constant $\\iota >0$ such that ${\\langle }B{\\rangle }_s\\geqslant \\iota sI_d$ and then $0=\\int _0^t{\\langle }\\mathrm {d}{\\langle }B{\\rangle }_s\\Psi _s, \\Psi _s{\\rangle }\\geqslant \\iota \\int _0^t{\\langle }\\Psi _s, \\Psi _s{\\rangle }\\mathrm {d}s.$ From this, we know that $\\Psi _t=0$ a.e.$\\times $ q.s..",
"So, (REF ) becomes $0=\\int _0^t\\Gamma _s\\mathrm {d}s+\\int _0^t\\Phi _{ij}(s)\\mathrm {d}{\\langle }B^i, B^j{\\rangle }_s+\\int _0^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }K(s,u)L(\\mathrm {d}s, \\mathrm {d}u).$ Next, set $\\tau _0=0, \\quad \\tau _n:=\\inf \\lbrace t>\\tau _{n-1}: \\kappa _t\\ne 0 \\rbrace , \\quad n=1,2,\\cdots ,$ and $\\lbrace \\tau _n\\rbrace $ is a stopping time sequence with respect to $({B}_t)_{t\\geqslant 0}$ and $\\tau _n\\uparrow \\infty $ as $n\\rightarrow \\infty $ q.s.(c.f.",
"[3]).",
"So, by (REF ) it holds that for $t\\in [0, \\tau _1\\wedge T)$ , $0=\\int _0^{\\tau _1\\wedge T}\\Gamma _s\\mathrm {d}s+\\int _0^{\\tau _1\\wedge T}\\Phi _{ij}(s)\\mathrm {d}{\\langle }B^i, B^j{\\rangle }_s,$ i.e.",
"$-\\int _0^{\\tau _1\\wedge T}\\Gamma _s\\mathrm {d}s=\\int _0^{\\tau _1\\wedge T}\\Phi _{ij}(s)\\mathrm {d}{\\langle }B^i, B^j{\\rangle }_s.$ Thus, by the similar deduction to that in [12] one can have that $\\bar{{\\mathbb {E}}}\\int _0^{\\tau _1\\wedge T}\\left({\\rm tr}[\\Phi _s\\Phi _s]\\right)^{1/2}\\mathrm {d}s=\\bar{{\\mathbb {E}}}\\int _0^{\\tau _1\\wedge T}|\\Gamma _s|\\mathrm {d}s=0.$ Based on this, we know that $\\Phi _t=0, \\Gamma _t=0$ for $t\\in [0, \\tau _1\\wedge T)$ .",
"If $\\tau _1\\geqslant T$ , the proof is over; if $\\tau _1<T$ , we continue.",
"For $t=\\tau _1$ , (REF ) goes to $\\Phi _t=0, \\quad \\Gamma _t=0, \\quad K(t,\\kappa _t)=0.$ For $t\\in [\\tau _1, \\tau _2\\wedge T)$ , by the same means to the above for $t\\in [0, \\tau _1\\wedge T)$ , we get that $\\Phi _t=0, \\Gamma _t=0$ for $t\\in [\\tau _1, \\tau _2\\wedge T)$ .",
"If $\\tau _2\\geqslant T$ , the proof is over; if $\\tau _2<T$ , we continue till $T\\leqslant \\tau _n$ .",
"Thus, we obtain that $\\Gamma _t=0, \\Phi _{ij}(t)=0, \\Psi _t=0$ a.e.$\\times $ q.s.",
"on $[0,T]\\times \\Omega _T$ and $K(t,u)=0$ a.e.$\\times $ a.e.$\\times $ q.s.",
"on $[0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )\\times \\Omega _T$ .",
"The proof is complete.",
"The main result in the section is the following theorem.",
"Theorem 3.2 Assume that ${\\mathcal {Q}}$ is bounded away from 0 and $b, h, \\sigma , f$ satisfy (${\\bf H}^1_{b,h,\\sigma ,f}$ )-(${\\bf H}^2_{b,h,\\sigma ,f}$ ).",
"Then for $V\\in C^{1,2}_b([0,T]\\times {\\mathbb {R}}^d)$ , $F_{s,t}$ is path independent in the sense of (REF ) if and only if $(V, g_1, g_2, g_3)$ satisfies the partial integral-differential equation $\\left\\lbrace \\begin{array}{ll}\\partial _tV(t,x)+{\\langle }\\partial _xV(t,x), b(t,x){\\rangle }=\\alpha G(g_1)(t,x)+\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }\\gamma g_3(t,x, u)\\nu (\\mathrm {d}u),\\\\{\\langle }\\partial _xV(t,x), h_{ij}(t,x){\\rangle }+\\frac{1}{2}{\\langle }\\partial ^2_xV(t,x)\\sigma ^i(t,x), \\sigma ^j(t,x){\\rangle }=\\beta g^{ij}_1(t, x),\\\\(\\sigma ^T\\partial _x V)(t, x)=g_2(t, x),\\\\V{\\Big (}t, x+f(t,x,u){\\Big )}-V(t, x)=g_3(t, x,u), \\\\t\\in [0,T], x\\in {\\mathbb {R}}^d, u\\in {\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace .\\end{array}\\right.$ First, we prove necessity.",
"On one hand, since $F_{s,t}$ is path independent in the sense of (REF ), by Definition REF it holds that $V(t,Y_t)-V(s,Y_s)&=&\\alpha \\int _s^t G(g_1)(r,Y_r)\\mathrm {d}r+\\beta \\int _s^t g^{ij}_1(r,Y_r)\\mathrm {d}{\\langle }B^i,B^j{\\rangle }_r+\\int _s^t {\\langle }g_2(r,Y_r),\\mathrm {d}B_r{\\rangle }\\nonumber \\\\&&+\\int _s^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }g_3(r,Y_r, u)L(\\mathrm {d}r, \\mathrm {d}u)+\\int _s^t\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }\\gamma g_3(r,Y_r, u)\\nu (\\mathrm {d}u)\\mathrm {d}r.\\nonumber \\\\$ On the other hand, applying the Itô formula for G-Itô-Lévy processes ([3]) to $V(t,Y_t)$ , one can obtain that $V(t,Y_t)-V(s,Y_s)&=&\\int _s^t\\partial _rV(r,Y_r)\\mathrm {d}r+\\int _s^t\\partial _kV(r,Y_r)b^k(r,Y_r)\\mathrm {d}r\\nonumber \\\\&&+\\int _s^t\\partial _kV(r,Y_r)h^k_{ij}(r,Y_r)\\mathrm {d}{\\langle }B^i, B^j{\\rangle }_r+\\int _s^t{\\langle }(\\sigma ^T\\partial _xV)(r,Y_r),\\mathrm {d}B_r{\\rangle }\\nonumber \\\\&&+\\int _s^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }{\\Big (}V(r,Y_r+f(r,Y_r,u))-V(r,Y_r){\\Big )}L(\\mathrm {d}r,\\mathrm {d}u)\\nonumber \\\\&&+\\frac{1}{2}\\int _s^t\\partial _{kl}V(r,Y_r)\\sigma ^{ki}(r,Y_r)\\sigma ^{lj}(r,Y_r)\\mathrm {d}{\\langle }B^i, B^j{\\rangle }_r.$ By (REF ) (${\\bf H}^1_{b,h,\\sigma ,f}$ )-(${\\bf H}^2_{b,h,\\sigma ,f}$ ), one can verify that $&&\\partial _rV(r,Y_r)+\\partial _kV(r,Y_r)b^k(r,Y_r)\\in {\\mathcal {M}}^1_G(0,T),\\\\&&\\partial _kV(r,Y_r)h^k_{ij}(r,Y_r)+\\frac{1}{2}\\partial _{kl}V(r,Y_r)\\sigma ^{ki}(r,Y_r)\\sigma ^{lj}(r,Y_r)\\in {\\mathcal {M}}^1_G(0,T),\\\\&&(\\sigma ^T\\partial _xV)(r,Y_r)\\in {\\mathcal {H}}^1_G([0,T], {\\mathbb {R}}^d),\\\\&&V(r,Y_r+f(r,Y_r,u))-V(r,Y_r)\\in {\\mathcal {H}}^2_G([0,T]\\times ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )).$ Thus, by (REF ) (REF ) and Lemma REF we know that $\\left\\lbrace \\begin{array}{ll}\\partial _rV(r,Y_r)+{\\langle }\\partial _xV(r,Y_r), b(r,Y_r){\\rangle }=\\alpha G(g_1)(r,Y_r)+\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }\\gamma g_3(r,Y_r, u)\\nu (\\mathrm {d}u),\\\\{\\langle }\\partial _xV(r,Y_r), h_{ij}(r,Y_r){\\rangle }+\\frac{1}{2}{\\langle }\\partial ^2_xV(r,Y_r)\\sigma ^i(r,Y_r), \\sigma ^j(r,Y_r){\\rangle }=\\beta g^{ij}_1(r,Y_r),\\\\(\\sigma ^T\\partial _x V)(r,Y_r)=g_2(r,Y_r),\\\\V{\\Big (}r,Y_r+f(r,Y_r,u){\\Big )}-V(r,Y_r)=g_3(r,Y_r,u), \\quad a.e.\\times q.s..\\end{array}\\right.$ Now, we insert $r=s, Y_s=x\\in {\\mathbb {R}}^d$ into the above equalities and get that $\\left\\lbrace \\begin{array}{ll}\\partial _sV(s,x)+{\\langle }\\partial _xV(s,x), b(s,x){\\rangle }=\\alpha G(g_1)(s,x)+\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }\\gamma g_3(s,x, u)\\nu (\\mathrm {d}u),\\\\{\\langle }\\partial _xV(s,x), h_{ij}(s,x){\\rangle }+\\frac{1}{2}{\\langle }\\partial ^2_xV(s,x)\\sigma ^i(s,x), \\sigma ^j(s,x){\\rangle }=\\beta g^{ij}_1(s, x),\\\\(\\sigma ^T\\partial _x V)(s, x)=g_2(s, x),\\\\V{\\Big (}s, x+f(s,x,u){\\Big )}-V(t, x)=g_3(s, x,u), \\\\s\\in [0,T], x\\in {\\mathbb {R}}^d, u\\in {\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace .\\end{array}\\right.$ Since $s, x$ are arbitrary, we have (REF ).",
"Next, we treat sufficiency.",
"By the Itô formula for G-Itô-Lévy processes to $V(t,Y_t)$ , we have (REF ).",
"And then one can apply (REF ) to (REF ) to get (REF ).",
"That is, $F_{s,t}$ is path independent in the sense of (REF ).",
"The proof is complete."
],
[
"Some special cases",
"In the subsection, we analysis some special cases.",
"If $b(t,x)=0, h_{ij}(t,x)=0, \\sigma (t,x)=I_d, f(t,x,u)=u$ , Eq.",
"(REF ) becomes $\\mathrm {d}Y_t=\\mathrm {d}B_t+\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }uL(\\mathrm {d}t, \\mathrm {d}u)=\\mathrm {d}X_t.$ We take $\\alpha =1, \\beta =\\frac{1}{2}, \\gamma =1$ .",
"Thus, by Theorem REF , it holds that $F_{s,t}$ is path independent in the sense of (REF ) if and only if $(V, g_1, g_2, g_3)$ satisfies the partial integral-differential equation $\\left\\lbrace \\begin{array}{ll}\\partial _tV(t,x)=G(\\partial ^2_xV)(t,x)+\\sup \\limits _{\\nu \\in {\\mathcal {V}}} \\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }(V(t, x+u)-V(t, x))\\nu (\\mathrm {d}u),\\\\\\partial ^2_xV(t,x)= g_1(t, x),\\\\\\partial _x V(t, x)=g_2(t, x),\\\\V(t, x+u)-V(t, x)=g_3(t, x,u), \\\\t\\in [0,T], x\\in {\\mathbb {R}}^d, u\\in {\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace .\\end{array}\\right.$ Besides, by Theorem REF , it holds that for $\\phi \\in C_{b, lip}({\\mathbb {R}}^d)$ , $V(t,x)=\\bar{{\\mathbb {E}}}[\\phi (x+X_t)]$ is the unique viscosity solution of the following partial integro-differential equation: $\\partial _t V(t,x)-G(\\partial ^2_xV)(t,x)-\\sup \\limits _{\\nu \\in {\\mathcal {V}}}\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }{\\Big (}V(t, x+u)-V(t, x){\\Big )}\\nu (\\mathrm {d}u)=0$ with the initial condition $V(0,x)=\\phi (x)$ .",
"So, $&&g_1(t, x)=\\partial ^2_x\\bar{{\\mathbb {E}}}[\\phi (x+X_t)], \\\\&&g_2(t, x)=\\partial _x\\bar{{\\mathbb {E}}}[\\phi (x+X_t)], \\\\&&g_3(t, x,u)=\\bar{{\\mathbb {E}}}[\\phi (x+u+X_t)]-\\bar{{\\mathbb {E}}}[\\phi (x+X_t)].$ That is, we can find $g_1, g_2, g_3$ .",
"If $d=1, \\alpha =1, \\beta =0, \\gamma =0$ , it follows from Theorem REF that $F_{s,t}$ is path independent in the sense of (REF ) if and only if $(V, g_1, g_2, g_3)$ satisfies the partial integral-differential equation $\\left\\lbrace \\begin{array}{ll}\\partial _tV(t,x)+\\partial _xV(t,x)b(t,x)=G(g_1)(t,x),\\\\\\partial _xV(t,x)h(t,x)+\\frac{1}{2}\\partial ^2_xV(t,x)\\sigma ^2(t,x)=0,\\\\(\\sigma \\partial _x V)(t, x)=g_2(t, x),\\\\V{\\Big (}t, x+f(t,x,u){\\Big )}-V(t, x)=g_3(t, x,u), \\\\t\\in [0,T], x\\in {\\mathbb {R}}, u\\in {\\mathbb {R}}\\setminus \\lbrace 0\\rbrace .\\end{array}\\right.$ And then if $\\sigma (t,x)\\ne 0$ , the unique solution of the above second equation is $V(t,x)=V(t,0)+\\partial _xV(t,0)\\int _0^xe^{-2\\int _0^z\\frac{h(t,v)}{\\sigma ^2(t,v)}\\mathrm {d}v}\\mathrm {d}z.$ Besides, since ${\\mathcal {Q}}$ is bounded away from 0, $G$ is invertible.",
"Thus, $&&g_1(t,x)=G^{-1}{\\Big (}\\partial _tV(t,x)+b(t,x)\\partial _xV(t,0)e^{-2\\int _0^x\\frac{h(t,v)}{\\sigma ^2(t,v)}\\mathrm {d}v}{\\Big )},\\\\&&g_2(t, x)=\\sigma (t, x)\\partial _xV(t,0)e^{-2\\int _0^x\\frac{h(t,v)}{\\sigma ^2(t,v)}\\mathrm {d}v},\\\\&&g_3(t, x,u)=\\partial _xV(t,0)\\int _x^{x+f(t,x,u)}e^{-2\\int _0^z\\frac{h(t,v)}{\\sigma ^2(t,v)}\\mathrm {d}v}\\mathrm {d}z.$ That is, we also give out $g_1, g_2, g_3$ in the case.",
"Remark 3.3 In the above special cases, we can describe concretely $g_1, g_2, g_3$ .",
"This is interesting and also is one of our motivations."
],
[
"Comparison with some known results",
"In the subsection, we compare our result with some known results.",
"First, if we take $f(t,x,u)=0$ in Eq.",
"(REF ) and $g_3(t,x,u)=0$ in (REF ), Theorem REF becomes [12].",
"Therefore, our result is more general.",
"Second, we take ${\\mathcal {M}}({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )=\\lbrace \\nu \\rbrace , {\\mathcal {Q}}=\\lbrace I_d\\rbrace $ in Subsection REF .",
"Thus, $B$ is a classical Brownian motion with ${\\langle }B^i, B^j{\\rangle }_t=I_{i=j}t$ and $L(\\mathrm {d}t, \\mathrm {d}u)$ is a classical Poisson random measure.",
"And then Eq.",
"(REF ) goes into $\\mathrm {d}Y_t=b(t,Y_t)\\mathrm {d}t +\\sum \\limits _{i=1}^d h_{ii}(t,Y_t)\\mathrm {d}t+\\sigma (t,Y_t)\\mathrm {d}B_t+\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }f(t,Y_t,u)L(\\mathrm {d}t,\\mathrm {d}u).$ Note that $\\nu ({\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace )<\\infty $ .",
"So, by (REF ) (${\\bf H}^1_{b,h,\\sigma ,f}$ )-(${\\bf H}^2_{b,h,\\sigma ,f}$ ), [4] admits us to obtain that $\\int _0^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }f(s,Y_s,u)\\tilde{L}(\\mathrm {d}s,\\mathrm {d}u):=\\int _0^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }f(s,Y_s,u)L(\\mathrm {d}s,\\mathrm {d}u)-\\int _0^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }f(s,Y_s,u)\\nu (\\mathrm {d}u)\\mathrm {d}s$ is a ${B}_t$ -martingale, where ${B}_t:=\\sigma \\lbrace \\omega _s, 0\\leqslant s\\leqslant t\\rbrace , 0\\leqslant t\\leqslant T$ .",
"Therefore, we can rewrite Eq.",
"(REF ) to get that $\\mathrm {d}Y_t&=&b(t,Y_t)\\mathrm {d}t +\\sum \\limits _{i=1}^d h_{ii}(t,Y_t)\\mathrm {d}t+\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }f(t,Y_t,u)\\nu (\\mathrm {d}u)\\mathrm {d}t+\\sigma (t,Y_t)\\mathrm {d}B_t\\\\&&+\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }f(t,Y_t,u)\\tilde{L}(\\mathrm {d}t,\\mathrm {d}u).$ This is a classical stochastic differential equation with jumps.",
"Thus, by [7] it holds that under (${\\bf H}^1_{b,h,\\sigma ,f}$ )-(${\\bf H}^2_{b,h,\\sigma ,f}$ ), the above equation has a unique solution.",
"In the following, note that $G(g_1)=\\frac{1}{2}{\\rm tr}(g_1)=\\frac{1}{2}\\sum \\limits _{i=1}^d g_1^{ii}$ .",
"And then $F_{s,t}$ can be represented as $F_{s,t}&=&\\int _s^t \\left(\\frac{\\alpha }{2}+\\beta \\right)\\sum \\limits _{i=1}^d g_1^{ii}(r,Y_r)\\mathrm {d}r+\\int _s^t {\\langle }g_2(r,Y_r),\\mathrm {d}B_r{\\rangle }+\\int _s^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }g_3(r,Y_r, u)\\tilde{L}(\\mathrm {d}r, \\mathrm {d}u)\\\\&&+\\int _s^t\\int _{{\\mathbb {R}}^d\\setminus \\lbrace 0\\rbrace }(1+\\gamma )g_3(r,Y_r, u)\\nu (\\mathrm {d}u)\\mathrm {d}r.$ This is just right [11] without the distribution of $Y_r$ for $r\\in [s,t]$ .",
"So, in the case Definition REF and Theorem REF are Definition 2.1 and Theorem 3.2 in [11] without the distribution of $Y_r$ for $r\\in [s,t]$ , respectively.",
"Therefore, our result overlaps [11] in some sense.",
"Acknowledgements: The authors are very grateful to Professor Xicheng Zhang for valuable discussions.",
"The first author also thanks Professor Renming Song for providing her an excellent environment to work in the University of Illinois at Urbana-Champaign."
]
] | 2001.03528 |
[
[
"A stable SPH with adaptive B-spline kernel"
],
[
"Abstract Tensile instability, often observed in smoothed particle hydrodynamics (SPH), is a numerical artifact that manifests itself by unphysical clustering or separation of particles.",
"The instability originates in estimating the derivatives of the smoothing functions which, when interact with material constitution may result in negative stiffness in the discretized system.",
"In the present study, a stable formulation of SPH is developed where the kernel function is continuously adapted at every material point depending on its state of stress.",
"Bspline basis function with a variable intermediate knot is used as the kernel function.",
"The shape of the kernel function is then modified by changing the intermediate knot position such that the condition associated with instability does not arise.",
"While implementing the algorithm the simplicity and computational efficiency of SPH are not compromised.",
"One-dimensional dispersion analysis is performed to understand the effect adaptive kernel on the stability.",
"Finally, the efficacy of the algorithm is demonstrated through some benchmark elastic dynamics problems."
],
[
"Introduction",
"The smoothed particle hydrodynamics (SPH) [1], [2] is a particle-based method, evolved as an alternative approach for numerical simulation of many physical processes which are otherwise difficult to model through mesh-based methods.",
"This includes problems in astrophysics [3], [4], fluid dynamics [5], [6], impact mechanics [7], [8], [9], large deformation [10], fracture [11], [12], [13], [14], [15] etc.",
"In SPH, the spatial discretization through particles (instead of elements) provides relief from the numerical difficulty associated with element distortion.",
"Moreover, the absence of explicit connectivity between particles makes SPH better equipped to deal with discontinuity and material separation.",
"However, all these advantages are often overshadowed by some computational pitfalls which may result in inaccurate and sometime unstable computation if not dealt with properly.",
"The tensile instability is perhaps the major concern among them.",
"Such instability in SPH was first witnessed by [16] in gas dynamics problems and subsequently by [17] in magnetohydrodynamics problems.",
"Therein the authors observed a prominent clustering or separation of particles, which were not in the line with the realistic physical behaviour of the system.",
"Almost after a decade since the instability was first reported, [18] performed a stability analysis of SPH.",
"In their analysis, they used the cubic B-spline kernel function which is most common in SPH computations.",
"For such kernel, if smoothing length is taken same as the particle spacing the instability occurs explicitly under tensile state of stress and thus coined as the tensile instability.",
"However, depending on the kernel function and smoothing length (with respect to the inter particle spacing) the instability may also occur under compressive stresses.",
"As per the analysis by [18], the instability is due to the manifestation of a negative stiffness resulting from the interaction of kernel function and the material constitution.",
"Therein, it was also shown that the tensile instability cannot be removed by increasing artificial viscosity, though the artificial viscosity may control, to some extent, the perturbation growth rate.",
"In the literature several corrective measures are suggested to deal with the tensile instability.",
"[19], [20] proposed a stress point approach, where the field variables such as density, stress and energy are evaluated at some prescribed stress points placed in between the actual SPH particles.",
"Velocities are calculated at the particles and mapped to the stress points using the moving least square (MLS) technique.",
"The concept is also extended to higher dimension in [21].",
"The approach estimated better accuracy in one dimensional problems.",
"However, tracking, velocity mapping and subsequent updating of stress points make the approach computationally intensive especially in higher dimension.",
"Through an unified stability analysis, [22] highlighted that the stress points are not effective if Eulerian kernel is used.",
"They also suggested to use a Lagrangian kernel (with or without stress points) for a stable computation.",
"Total Lagrangian formulation is also adopted in [23], [24].",
"Though the total Lagrangian formulation does not suffer any tensile instability, the difficulty arises while modelling large deformation and discontinuities in computational domain due to material fracture.",
"Moreover the particle distribution near the highly deformed region becomes very distorted which may yield poor convergence rate [25].",
"[26], [27] proposed the concept of artificial stress.",
"Therein an artificial stress term is introduced in the momentum Equation, which produces a short range repulsive force between the neighbouring particles in order to keep them in a stable configuration.",
"The parameters which controls the intensity for the artificial stress are determined through a dispersion analysis.",
"Although this approach is widely accepted, the form and associated parameters of artificial stress depend on the physical process to be studied, governing Equations and also the choice of kernel function.",
"Over the years few other approaches are also proposed to overcome the tensile instability, such as conservative smoothing [23], [24], [28], corrected kernel estimate [29], [30] etc.",
"All the corrective measures mentioned above either are computationally intensive or require some artificial penalty force that need to be tuned depending on the problem.",
"Moreover these approaches generally suppress the instability, instead of completely removing them from the system.",
"As the tensile instability is ascribed to the shape of the kernel function, use of adaptive kernel would be more suitable and intrinsic to the root cause of the instability.",
"In the present study an adaptive algorithm is developed where the shape of the kernel function is continuously modified such that the condition associated with the instability does not arise.",
"B-spline basis function constructed over a variable knot vector is taken as the kernel in the proposed formulation.",
"The shape of the kernel is adapted by changing the location of the intermediate knots.",
"The theoretical basis of the algorithm is discussed.",
"One dimensional dispersion analysis is also performed to investigate the effect of adaptive kernel on the stability.",
"Through some benchmark elastic dynamics problem it shown that the algorithm can remove the tensile instability in a computationally simpler and efficient way."
],
[
"SPH and Tensile Instability",
"For a better comprehension of the development to follow, a brief account of SPH (conservation Equations and discretization) and the tensile instability are provided in this section.",
"While revisiting SPH, the basic steps which are followed in the present study are only outlined.",
"Detail information may be found in [31] and the references therein."
],
[
"SPH",
"The conservation Equations (mass, momentum and energy) in a Lagrangian framework are given by, $\\frac{d\\rho }{dt} = -\\rho \\frac{\\partial \\mathbf {v}^{\\beta }}{\\partial \\mathbf {x}^{\\beta }}$ $\\frac{d\\mathbf {v}^{\\alpha }}{dt} = \\frac{1}{\\rho }\\frac{\\partial \\sigma ^{\\alpha \\beta }}{\\partial \\mathbf {x}^{\\beta }}$ $\\frac{de}{dt} = \\frac{\\sigma ^{\\alpha \\beta }}{\\rho }\\frac{\\partial \\mathbf {v}^{\\beta }}{\\partial \\mathbf {x}^{\\beta }}$ where, $\\rho $ is the density; $v^\\alpha $ and $\\sigma ^{\\alpha \\beta }$ are the components of velocity and Cauchy stress tensor respectively; $e$ is the specific energy; $\\alpha $ and $\\beta $ are the spatial coordinates.",
"The computational domain $\\bar{\\Omega }=\\partial \\Omega \\bigcup \\Omega $ is discretized in to a set of particles positioned at ${\\lbrace \\mathbf {x}_i}\\rbrace _{i=1}^N$ , where $N$ is the total number of particles.",
"Let ${\\lbrace m_i}\\rbrace _{i=1}^N$ , ${\\lbrace \\rho _i}\\rbrace _{i=1}^N$ , ${\\lbrace p_i}\\rbrace _{i=1}^N$ ,${\\lbrace v^{\\alpha }_i}\\rbrace _{i=1}^N$ , ${\\lbrace {\\sigma _i}^{\\alpha \\beta }}\\rbrace _{i=1}^N$ be the discrete values of mass, density, pressure, velocity and Cauchy stress respectively.",
"Any continuous field variable is approximated as a functional representation of the discrete values through a smoothed weighting function or kernel function as, $<f(\\mathbf {x})> \\cong \\sum _{j \\in \\mathbb {N}^i} f_j W(\\mathbf {x}-\\mathbf {x}_j,\\kappa h)\\frac{m_j}{\\rho _j},$ where $W\\left(\\mathbf {x},\\kappa h\\right)$ is a bell-shaped compactly supported kernel function with centre at $\\mathbf {x}$ and smoothing length $\\kappa h$ , $\\kappa $ being the support size of the kernel in its parametric space; and $\\mathbb {N}^i$ is the influence domain of $i$ -th particle (i.e., set of indices of other particles which interact with $i$ -th particle) defined as $\\mathbb {N}^i = \\lbrace j \\in \\mathbb {Z}^+ ~|~ |\\mathbf {x}_i-\\mathbf {x}_j| < \\kappa h ~and~ i \\ne j \\rbrace $ .",
"SPH being a particle based method, requires the integral forms of the continuous conservation laws [Eqs.",
"(REF ) - (REF )] to be discretized at each particle position using the kernel approximations [Equation (REF )].",
"The discrete form of the conservation Equations are, $\\frac{d\\rho _i}{dt} = \\sum _{j \\in \\mathbb {N}^i} {m_j ({v_i}^\\beta -{v_j}^\\beta )W_{ij,\\beta }},$ $\\frac{dv_i^{\\alpha }}{dt} = \\sum _{j \\in \\mathbb {N}^i} {m_j\\left(\\frac{\\sigma _i^{\\alpha \\beta }}{\\rho _i^{2}}+\\frac{\\sigma _j^{\\alpha \\beta }}{\\rho _j^{2}}-\\Pi _{ij}\\delta ^{\\alpha \\beta }\\right) W_{ij,\\beta }},$ $\\frac{de_i}{dt} = -\\frac{1}{2} \\sum _{j \\in \\mathbb {N}^i} {m_j({v_i}^\\beta -{v_j}^\\beta )\\left(\\frac{\\sigma _i^{\\alpha \\beta }}{\\rho _i^{2}}+\\frac{\\sigma _j^{\\alpha \\beta }}{\\rho _j^{2}} - \\Pi _{ij} \\delta ^{\\alpha \\beta }\\right) W_{ij,\\beta }},$ where, $W_{ij,\\beta } = \\frac{\\partial W(\\mathbf {x},\\kappa h)}{\\partial \\mathbf {x}^{\\beta }}|_{(\\mathbf {x}_i-\\mathbf {x}_j)}$ .",
"In Equation REF and REF , $\\Pi _{ij}$ is the artificial viscosity [32] which is often required to stabilize the numerical algorithm in the presence of shock.",
"The following form [27] of artificial viscosity is used in the present study.",
"$\\Pi _{ij}={\\left\\lbrace \\begin{array}{ll}\\frac{-\\gamma _1\\overline{C}_{ij}\\mu _{ij} + \\gamma _2\\mu ^2_{ij}}{\\overline{\\rho }_{ij}^2} & \\text{for} \\, \\mathbf {x}_{ij}.\\mathbf {v}_{ij} < 0\\\\0 & \\text{otherwise} \\\\\\end{array}\\right.",
"}$ where, $\\mu _{ij}= \\frac{h\\left(\\mathbf {v}_{ij}.\\mathbf {x}_{ij}\\right)}{X^2_{ij} + \\eta h^2}$ ; $\\overline{C}_{ij} = \\frac{C_i + C_j}{2}$ ; $\\overline{\\rho }_{ij} = \\frac{\\rho _i + \\rho _j}{2}$ ; $\\gamma _1$ and $\\gamma _2$ are parameters which control the intensity of the artificial viscosity; $\\eta $ is a small number to avoid singularity when two interacting particles ($i$ -th and $j$ -th) are close to each other; $C_i = \\sqrt{\\frac{E}{\\rho _i}}$ and $C_j = \\sqrt{\\frac{E}{\\rho _j}}$ are the wave propagation speed evaluated at $i$ -th and $j$ -th particles respectively; and $\\mathbf {v}_{ij} = \\mathbf {v}_i- \\mathbf {v}_j$ and $\\mathbf {x}_{ij} = \\mathbf {x}_i- \\mathbf {x}_j$ indicate the relative velocity and displacement of the $i-j$ particle pair.",
"The particles are moved as per the XSPH Equation [33] as, $\\frac{dx_i^\\alpha }{dt} = v_i^\\alpha - \\epsilon \\sum _{j \\in \\mathbb {N}^i} \\frac{m_j}{\\overline{\\rho }_{ij}} v_{ij}^\\alpha W_{ij},$ where $v_{ij}^\\alpha =v_i^\\alpha -v_j^\\alpha $ and $\\epsilon \\in [0,1]$ .",
"The Cauchy stress component is written in terms of hydrostatic and deviatoric stresses as $\\sigma ^{\\alpha \\beta }= -P\\delta ^{\\alpha \\beta }+S^{\\alpha \\beta }$ , where $P$ and $S^{\\alpha \\beta }$ are respectively the pressure and the components of traceless symmetric deviatoric stress tensor.",
"Pressure is computed from the following Equation of state, $p=K\\left(\\frac{\\rho }{\\rho _0} -1\\right),$ where, $\\rho _0$ is he initial density and $K$ is the bulk modulus.",
"The discretized Equation for the deviatoric stress rate is, $\\begin{split}\\dot{S}_i^{\\alpha \\beta }= - \\sum _{j \\in \\mathbb {N}^i} \\frac{m_j}{\\rho _j} G_i\\Big ( v_{ij}^\\alpha W_{ij,\\beta }+v_{ij}^\\beta W_{ij,\\alpha } - \\frac{2}{3} \\delta ^{\\alpha \\beta } v_{ij}^\\gamma W_{ij,\\gamma } \\Big ) + \\\\\\frac{1}{2}\\sum _j \\frac{m_j}{\\rho _j} \\Big [ S^{\\alpha \\gamma } \\Big ( v_{ij}^\\beta W_{ij,\\gamma }-v_{ij}^\\gamma W_{ij,\\beta } \\Big )+ S^{\\gamma \\beta } \\Big ( v_{ij}^\\alpha W_{ij,\\gamma }-v_{ij}^\\gamma W_{ij,\\alpha } \\Big ) \\Big ]\\end{split}$ where $G$ is the shear modulus.",
"Once the Equations are discretized over space, the resulting ODEs in time are integrated via any appropriate time integration technique.",
"In the present study the two step predictor-corrector method with constant time step is used."
],
[
"Tensile Instability",
"In a continuum if two material points are relatively displaced (approach to or move away from each other) from their equilibrium positions, internal force develop which opposes the motion.",
"This internal force ($F_b$ ) evolves with the internal stresses satisfying the conservation of linear momentum as, $F_b = m\\frac{dv^{\\alpha }}{dt} = m\\frac{1}{\\rho }\\frac{\\partial \\sigma ^{\\alpha \\beta }}{\\partial \\mathbf {x}^{\\beta }}$ In one dimension, the SPH discretization of Equation (REF ) takes the form (without the artificial viscosity term), $m_i\\frac{dv_i}{dt} = - m_i\\sum _{j \\in \\mathbb {N}^i} {m_j\\left(\\frac{\\sigma _i}{\\rho _i^{2}}+\\frac{\\sigma _j}{\\rho _j^{2}}\\right) W^{\\prime }_{ij}}.$ The above Equation gives the cumulative force exerted on particle $i$ from its neighbour $j\\in \\mathbb {N}^i$ .",
"The interaction force between any particle pair (say $i-j$ ) is expressed as, $F_{ij} = - m_i m_j {\\left(\\frac{\\sigma _i}{\\rho _i^{2}}+\\frac{\\sigma _j}{\\rho _j^{2}}\\right) W^{\\prime }_{ij}},$ Figure: Geometric representation of tensile instabilityFrom Equation REF it is evident that if the interacting particles are in constant state of stress (i.e $\\sigma _i = \\sigma _j = \\sigma $ ), the interaction force is guided by the first derivative ($W^{\\prime }$ ) of the kernel function.",
"Figure REF illustrates the shape of a cubic kernel (Equation REF ) function ($W$ ) and its first derivative ($W^{\\prime }$ ) centred at the $i$ -th particle i.e., $x_i$ .",
"Now consider the interaction of $i$ -th paricle with any of its neighbour, say $j$ -th particle.",
"If the $j$ -th particle is shifted from the equilibrium position the interaction force $F_{ij}$ between the pair ($i-j$ ) should increase to retain the particles ($i$ and $j$ ) in equilibrium state.",
"The shape of $W^{\\prime }$ does not always allow such to happen.",
"Initially starting from zero at the $i$ -th particle $W^{\\prime }$ increases (i.e., positive $W^{\\prime \\prime }$ ) with inter-particle distance and after reaching the peak value at M it starts declining (i.e., negative $W^{\\prime \\prime }$ ) and eventually become zero at the boundary of the influence domain.",
"Therefore, initially when $W^{\\prime \\prime }$ is positive the interaction force ($F_{ij}$ ) monotonically increases with increase in inter-particle separation.",
"But when the $j$ -th particle crosses the maxima the magnitude of the interaction force starts decreasing with further increase in inter-particle distance resulting in further separation of the $i-j$ particle pair.",
"This situation is analogous to having a negative stiffness in the system which causes disruption in stability of the computation.",
"Similar argument can be put in case of approaching particles.",
"The instability manifests itself by unphysical clustering or clumping of particles, sometime causing numrical fracture.",
"This is illustrated via a two dimensional example [18] in Figure REF .",
"Herein, a square grid of isotropic linear elastic particles (Figure REF ) with $\\rho =7850 kg/m^3$ , $E = 200GPa$ and $\\nu = 0.3$ is considered.",
"The boundary particles are kept fixed.",
"All interior particles are preassigned with an uniform stress field and the central particle is perturbed with a velocity of $10^{-07}$ m/s.",
"No artificial viscosity and artificial stress corrections are used.",
"Particle configurations after $2.5 \\mu s$ and $5.0 \\mu s$ computed via SPH with standard cubic kernel are shown in Figures REF and REF respectively.",
"The effect of instability is evident in the figures.",
"Figure: Demonstration of tensile instability"
],
[
"Removing Tensile Instability",
"As per the analysis by [18], the tensile instability may be kept away from the SPH computation if the inequaqlities $\\sum _{j \\in \\mathbb {N}^i}^{} W^{^{\\prime \\prime }}_{i,ij} \\le 0$ for tension and $\\sum _{j \\in \\mathbb {N}^i}^{} W^{^{\\prime \\prime }}_{i,ij} \\ge 0$ for compression are preserved.",
"In the standard SPH, the kernel function ($W(x,\\kappa h)$ ) and the smoothing length ($h$ ) are chosen independent of the material constitution.",
"Therefore for a given kernel, the sign of $W^{^{\\prime \\prime }}(x,\\kappa h)$ depends only on the distance between the interacting particles; not on their state of stress.",
"Now the central idea of the present algorithm is to introduce this feature in the SPH formulation i.e., continuously change the sign of $W^{^{\\prime \\prime }}(x,\\kappa h)$ at a material point depending on the state of stress at its neighbourhood such that the above stability condition is satisfied.",
"In other words, the shape of the kernel function is adapted so as to ensure that the neighbour particles lie within the stable region of the kernel derivative (as shown in Figure REF ).",
"This adaptation needs to be done in such a way that, (a) the fundamental properties of kernel function are preserved, (b) no ad hoc parameters are introduced, (c) the computational efficiency of SPH code is not compromised, and (d) definition of neighbour is not changed i.e., for a given particle the size of the influence domain remains same.",
"The B-spline basis functions are the building blocks of the proposed adaptive algorithm.",
"Therefore, to start with, the construction and salient features of B-spline basis are outlined in the next sub-section."
],
[
"B-spline as adaptive kernel in SPH",
"B-Spline basis functions have inherent property of compactness and higher order continuity with minimal number of points [34].",
"These functions also hold partition of unity characteristics.",
"The recursive definition (Cox-eBoor algorithm) of $i$ -th normalized B-spline basis functions of degree $p$ in one dimension is, $N_{i,0}\\left(\\zeta \\right) ={\\left\\lbrace \\begin{array}{ll}1 & for \\zeta _i\\le \\zeta \\le \\zeta _{i+1}\\\\0 & otherwise\\end{array}\\right.",
"}$ $N_{i,p}\\left(\\zeta \\right) = \\frac{\\zeta - \\zeta _i}{\\zeta _{i+p} - \\zeta _i}N_{i,p-1}\\left(\\zeta \\right) + \\frac{\\zeta _{i+p+1} - \\zeta }{\\zeta _{i+p+1} - \\zeta _{i+1}}N_{i+1,p-1}\\left(\\zeta \\right)$ where, $\\Xi =\\lbrace \\zeta _1,\\zeta _2, \\zeta _3,...,\\zeta _{n+p+1} | \\zeta _i \\in \\mathbb {R}\\rbrace $ is the knot vector and $n$ is the number of basis functions.",
"B-spline basis functions are, (a) strictly positive: $N_{i,p} (\\zeta ) \\ge 0 \\forall i, p$ , (b) compactly supported: $N_{i,p} (\\zeta ) = 0 \\forall \\zeta \\in [\\zeta _i, \\zeta _{i+p+1}]$ i.e., $\\left(\\zeta _{i+p+1} - \\zeta _i \\right)$ is the support size of $N_{i,p}$ , (c) partition of unity: $\\sum _{j} N_{i,p}(\\zeta ) = 1$ , and (d) differentiability: $N_{i,p} (\\zeta )$ is $C^p$ continuous everywhere in the domain except at the knot locations where it is $C^{p-1}$ continuous.",
"The above properties qualify the B-spline basis function to be a valid kernel in SPH.",
"The most important feature which constitutes the premise of the present formulation is that for any B-spline basis function, say $N_{i,p}(\\zeta )$ , the shape can be modified by changing the location of intermediate knots $\\lbrace \\zeta _{i+1},...\\zeta _{i+p} \\rbrace $ and still its support (or influence domain) will remain same as $\\left(\\zeta _{i+p+1} - \\zeta _i \\right)$ if the positions of extreme knots $\\lbrace \\zeta _{i},\\zeta _{i+p+1} \\rbrace $ are unchanged.",
"For the use in SPH computation, here $N_{1,p}$ defined over knot vector $\\lbrace \\zeta _1,\\zeta _2, ,...,\\zeta _{1+p+1}\\rbrace $ is taken as the kernel function.",
"Change of shape of cubic B-spline kernel function (i.e., $N_{1,3}$ ) with different intermediate knot positions is demonstrated in Figure REF .",
"The kernel is defined over knot $\\lbrace -b, -a_1, 0, a_2, b\\rbrace $ .",
"Figures are shown for fixed $b=2$ and different $a_1$ and $a_2$ .",
"Though obvious, it is important to take a note (for discussion in subsequent sections) that if the knot positions are not symmetric, the kernel function will also not be symmetric.",
"The expressions (obtained from Equation (REF ) and (REF )) of quadratic (i.e., $N_{1,2}$ ) and cubic (i.e., $N_{1,3}$ ) B-spline kernel with a symmetric knot vectors $\\lbrace -b, -a, a, b\\rbrace $ and $\\lbrace -b, -a, 0, a, b\\rbrace $ respectively are provided in Table REF .",
"This enables to form a more generalised analytical expression, where both shape (varying the value of intermediate knot $a$ ) and support size (varying the value of $b$ ) of the kernel function may be changed as per the requirement.",
"Table: Expressions for parametrised B-Spline functionsFigure: Cubic B-spline kernel and its first derivative constructed over knot -b,-a 1 ,0,a 2 ,b\\left\\lbrace -b, -a_1, 0, a_2, b \\right\\rbrace for fixed b=2b=2 and different a 1 a_1 and a 2 a_2.",
"Knot positions are shown by red dots."
],
[
"Adaptive Algorithm",
"For a better understanding the algorithm is first discussed in one dimension.",
"Consider an arbitrary particle (say $i$ -th), with position $x_i$ and influence domain $\\mathbb {N}^i = \\lbrace j \\in \\mathbb {Z}^+ ~|~ |x_i-x_j| < bh ~and~ i \\ne j \\rbrace $ .",
"Let $\\overline{\\mathbb {N}^i} =\\lbrace i-1, i+1 \\rbrace \\subseteq \\mathbb {N}^i$ be the set of immediate neighbours as represented by red dots in Figure REF .",
"The tensile instability is known to be a short wavelength instability which starts with clumping of pair of immediate neighbours as depicted in Figure REF .",
"Though in the stability condition the summation is over all the particles within the influence domain, stability of the computation is significantly governed by the interaction of immediate neighbours.",
"Therefore, the crucial step in the algorithm is to ensure that the immediate neighbours (particles in $\\overline{\\mathbb {N}^i}$ ) are brought into the stable zone of the kernel function.",
"Towards achieving this, the required shift in the position of the intermediate knot is discussed in this section.",
"For a given particle, the magnitude and the direction of the shift (i.e., whether towards the particle or away from the particle) depend on the relative positions and the state of stress of its neighbour particles.",
"One approach would be to check the interaction of $i$ with all its immediate neighbour $j \\in \\overline{\\mathbb {N}^i}$ separately and adapt the kernel such that all $j \\in \\overline{\\mathbb {N}^i}$ come within the stable zone of $i$ .",
"For instance in the 1D case, with cubic B-spline kernel defined over knot $\\lbrace -b, -a_1, 0, a_2, b\\rbrace $ , first check the interaction between $i-(i-1)$ and depending on whether the particles are approaching (compression) or moving away (tension) fix $a_1$ .",
"Similarly depending on the interaction of $i-(i+1)$ fix $a_2$ .",
"While this seems to be more logical, the problem with this approach is that for $a_1 \\ne a_2$ the kernel no longer remains symmetric (as shown in Figure REF ) and may affect the conservation.",
"Moreover, implementation in higher dimension would be numerically very intensive.",
"Instead of treating the each individual particle pair separately, herein the cumulative effect of all immediate neighbours i.e., $j \\in \\overline{\\mathbb {N}^i}$ on particle $i$ is considered.",
"The kernel function at $x_i$ ($W(x-x_i,h)$ ) is formed with a symmetric knot vector $\\lbrace -b, a, 0, a, b\\rbrace $ .",
"Now, when the neighbouring particles approach towards $i$ , the density of the particle $\\rho _i$ increases and vice versa.",
"Therefore the density ratio $\\rho _i/\\rho _0$ ($\\rho _0$ is the initial density) is taken as the criteria for knot shifting i.e., if $\\rho _i/\\rho _0 >1$ , the intermediate knot is shifted towards particle $i$ and similarly if $\\rho _i/\\rho _0 <1$ , the intermediate knot is moved away from $i$ .",
"Once the direction of the knot shifting (i.e., towards or away with respect to particle $i$ ) is identified, next is to determine the magnitude of the shift.",
"The influence domain (i.e., value of $b$ ) is kept fixed and only the shape of the kernel over the influence domain is adapted by varying $\\chi =a/b$ ratio (depending on the value of $\\rho _i/\\rho _0$ ) in order to obtain a stable region.",
"The two extreme positions of the intermediate knot are $\\chi =\\epsilon $ and $\\chi =1-\\epsilon $ , where $\\epsilon \\in \\mathbb {R}^+$ is a small number used to avoid any repetition of knot at the centre (i.e, $x=0$ ) or at the boundary (i.e., $x=\\pm b$ ) of the kernel support.",
"For the cubic B-spline kernel, $W^{\\prime }$ curve attains maximum at a distance $\\frac{\\chi }{1 + \\chi }b$ .",
"Assuming a quasi-uniform particle distribution (at least locally) in the neighbourhood of $i$ with characteristic spacing $\\Delta p$ , the condition so that the immediate neighbour lies at the peak of $W^{\\prime }$ is, $\\Delta p/h = \\frac{\\chi }{1+\\chi }b \\\\\\Rightarrow \\chi = \\frac{\\Delta p/h}{b-\\Delta p/h}$ Now the following condition is necessary in order to ensure a stable interaction of particle $i$ with its immediate neighbour $j \\in \\overline{\\mathbb {N}^i}$ .",
"$\\chi _i ={\\left\\lbrace \\begin{array}{ll}\\ < \\frac{\\Delta p/h}{b-\\Delta p/h} & \\text{for} \\, \\frac{\\rho _i}{\\rho _0} > 1\\\\> \\frac{\\Delta p/h}{b-\\Delta p/h} & \\text{for} \\, \\frac{\\rho _i}{\\rho _0} < 1 \\\\\\end{array}\\right.",
"}$ Figure: Stable region for different state of stress: a. higher χ\\chi for tension; b. lower χ\\chi for compressionSimilar condition may be obtained for any order B-spline kernel.",
"For instance the condition for quadratic B-spline ( $W^{\\prime }$ is maximum at $\\chi = 1$ ) would be, $\\chi _i ={\\left\\lbrace \\begin{array}{ll}\\ < \\frac{\\Delta p/h}{b} & \\text{for} \\, \\frac{\\rho _i}{\\rho _0} > 1\\\\> \\frac{\\Delta p/h}{b} & \\text{for} \\, \\frac{\\rho _i}{\\rho _0} < 1 \\\\\\end{array}\\right.",
"}$ The first derivative of quadratic and cubic B-spline kernel for different value so $\\chi $ are plotted in Figure REF .",
"The similar concept may be extended to higher dimension.",
"The algorithm in 2D is demonstrated in Figure REF .",
"Cubic B-spline kernel and its first derivative in 2D for different $\\chi $ are shown in Figure REF .",
"Figure: 1st derivative of quadratic and cubic kernel for fixed b=2b=2 and different χ=a/b\\chi =a/b.Figure: Adaptive Kernel pictorial illustrationFigure: Cubic B-spline kernel and its first derivative in 2D for fixed b=2b=2 and different χ=a/b\\chi = a/bAny value of $\\chi _i$ satisfying the above condition ensures a stable interaction between immediate neighbours.",
"However, this may not always guarantee a stable computation as the interaction with other neighbours (i.e., $j \\in \\mathbb {N}^i \\setminus \\overline{\\mathbb {N}^i}$ ) may also contribute to the instability.",
"In the next section a more general condition is obtained through dispersion analysis."
],
[
"Dispersion Analysis",
"Any standard SPH method is associated with its inherent dispersion that arises due to the functional approximation over the support domain.",
"In this current exercise, the additional dispersion introduced in the system for using adaptive kernel approximation is investigated.",
"Let us consider a stable configuration of particles ${\\lbrace \\overline{x}_i}\\rbrace _{i \\in \\mathbb {Z}^+}$ with characteristic spacing $\\Delta p$ in a one dimensional domain $\\overline{\\Omega } \\subset \\mathbb {R}$ .",
"The system is provided with an infinitesimal perturbation from its stable state and subsequently the wave speed of the perturbation is calculated.",
"Here onwords any variable ($f$ ) described in the stable state is noted as $\\overline{f}$ and in the perturbed state as $f$ respectively for the ease of understanding.",
"For the given stable configuration, assuming a uniform density $\\overline{\\rho }$ , the mass associated with any particle may be obtained as $m = \\overline{\\rho } \\Delta p$ .",
"This results in a uniform state of hydrostatic stress $\\overline{\\sigma }_i = \\overline{\\sigma } = K [\\frac{\\overline{\\rho }}{\\rho _0} - 1]$ at each particle (i.e., $\\forall i \\in \\mathbb {Z}^+$ ), where $\\rho _0$ is the initial density and $K$ is the bulk modulus.",
"Ignoring the artificial viscosity term $\\Pi _{ij}$ (as it does not have any effect in removing tensile instability [24]), the governing Equations of continuity and motion at any particle in the stable state are described as per the conservation of mass and the momentum as, $\\frac{d\\overline{\\rho }}{dt} = - \\sum _{j \\in \\mathbb {N}^i } \\overline{\\rho } \\, \\Delta p \\left[{\\overline{v}_j - \\overline{v}_i} \\right] \\triangledown \\overline{W}_{ij}$ $\\frac{d{\\overline{v}}_i}{dt} = - \\sum _{j \\in \\mathbb {N}^i} \\overline{\\rho } \\,\\Delta p \\left[2\\frac{{\\overline{\\sigma }}_i}{\\overline{\\rho }^2} \\right] \\triangledown \\overline{W}_{ij}$ where $\\overline{W}_{ij} = W(\\overline{x}_{ij}) = W(\\overline{x}_i-\\overline{x}_j)$ is the B-spline kernel function described in the stable configuration.",
"The displacement $x_i$ at any $i^{th}$ particle is updated as, $\\frac{d x_i}{dt} = v_i$ Now when the system is provided with an infinitesimal perturbation ($\\delta x$ ) it also induces a similar perturbation $\\delta v$ in velocity and $\\delta \\rho $ in the assigned density at the particles.",
"The configuration such obtained is referred to as the perturbed configuration $\\Omega \\subset \\mathbb {R}$ .",
"Let us describe the position (${{x}_i} = \\overline{{x}}_i + \\delta x_i$ ), velocity (${{v}_i} = \\overline{{v}}_i + \\delta v_i$ ) and density (${{\\rho }_i} = \\overline{{\\rho }} + \\delta \\rho _i$ ) at any arbitrary $i$ -th particle in the perturbed configuration ($\\Omega \\subset \\mathbb {R}$ ) as, ${{x}_i} = \\overline{{x}_i} + Xe^{i(k\\overline{x}_i - \\omega t)}$ ${{v}_i} = \\overline{{v}_i} + Ve^{i(k\\overline{x}_i - \\omega t)}$ ${\\rho _i} = \\overline{{\\rho }_i} + D e^{i(k\\overline{x}_i - \\omega t)}$ where, $X, V, D$ are the respective coefficients for displacement velocity and density variation.",
"$\\omega $ and $k$ are the frequency and wave number of the oscillation.",
"The governing conservation Equations in the perturbed configuration can be written as, $\\frac{d\\rho _i}{dt} = - \\sum _{j \\in \\mathbb {N}^i} \\rho _j \\Delta p \\left( v_j - v_i \\right) \\triangledown W_{ij}$ $\\frac{d{{v}}_i}{dt} = - \\sum _{j \\in \\mathbb {N}^i} \\rho _j \\Delta p \\left(\\frac{{\\sigma }_i}{{\\rho _i}^2} +\\frac{{\\sigma }_j}{{\\rho _j}^2} \\right) \\triangledown W_{ij}$ where $W_{ij} = W(x_i -x_j) = W(\\overline{x}_{ij} + \\delta x_{ij})$ with $\\delta x_{ij} = \\delta x_{i} - \\delta x_{j}$ , is the kernel function described in the perturbed configuration.",
"$W_{ij}$ can be expanded by Taylor series as, $W_{ij} = \\overline{W}_{ij} + \\delta x_{ij} \\triangledown \\overline{W}_{ij}$ Similarly, $\\triangledown W_{ij}$ can be expressed as, $\\triangledown W_{ij} = \\triangledown \\overline{W}_{ij} + \\delta x_{ij} \\triangledown ^2 \\overline{W}_{ij}$ As $\\delta {x_{ij} < < 1}$ , the higher order terms of $\\delta x_{ij}$ are neglected in Equations REF and REF .",
"The state of stress at the $i^{th}$ particle in perturbed configuration may be written as, $\\sigma _i = K[\\frac{\\rho _i}{\\rho _0} -1],$ which further may be expanded as, $\\frac{{\\sigma }_i}{\\rho _i^2} = \\frac{K}{\\left( \\overline{\\rho } + \\delta \\rho _i \\right)^2 }[\\frac{\\overline{\\rho } + \\delta \\rho _i}{\\rho _0} - 1]$ $\\Rightarrow \\frac{{\\sigma }_i}{\\rho _i^2} = \\frac{K}{\\overline{\\rho }^2 }[\\frac{\\overline{\\rho }}{\\rho _0} - 1 + \\frac{\\delta \\rho _i}{\\rho _0}][1 - 2\\frac{\\delta \\rho _i}{\\overline{\\rho }}]$ $\\Rightarrow \\frac{{\\sigma }_i}{\\rho _i^2} = \\frac{K}{\\overline{\\rho }^2}[\\frac{\\overline{\\rho }}{\\rho _0} - 1] + \\frac{K}{\\overline{\\rho }^3 }[2 - \\frac{\\overline{\\rho }}{\\rho _0}]\\delta \\rho _i$ $\\Rightarrow \\frac{{\\sigma }_i}{\\rho _i^2} = \\frac{\\overline{{\\sigma }}}{\\overline{\\rho }^2} + B\\delta \\rho _i$ where $B = \\frac{K}{\\overline{\\rho }^3 }[2 - \\frac{\\overline{\\rho }}{\\rho _0}]$ .",
"Now when Equations REF - REF are replaced in the governing momentum balance Equation REF we can obtain, $\\frac{d{v}_i}{dt} = - \\sum _{j \\in \\mathbb {N}^i} \\overline{\\rho } [1 + \\frac{\\delta \\rho _j}{\\overline{\\rho }}] \\Delta p \\left[ 2\\frac{\\overline{\\sigma }}{\\overline{\\rho }^2} + B \\lbrace \\delta \\rho _i + \\delta \\rho _j \\rbrace \\right] \\left[ \\triangledown \\overline{W}_{ij} + \\delta x_{ij} \\triangledown ^2 \\overline{W}_{ij} \\right]$ Here also the higher order terms are neglected assuming the perturbations to be infinitesimal.",
"The RHS of the Equation REF can be expanded as, $\\begin{split}\\frac{d{\\overline{v}}_i}{dt} + \\frac{d}{dt} \\delta v_i = - \\sum _{j \\in \\mathbb {N}^i} \\Delta p \\left[2 \\frac{\\overline{\\sigma }}{\\overline{\\rho }} \\right] \\triangledown \\overline{W}_{ij} - \\sum _{j \\in \\mathbb {N}^i} 2 \\Delta p \\frac{\\overline{\\sigma }}{\\overline{\\rho }} \\delta x_{ij} \\triangledown ^2 \\overline{W}_{ij} \\\\- \\sum _{j \\in \\mathbb {N}^i} 2 \\Delta p \\frac{\\overline{\\sigma }}{\\overline{\\rho }^2} \\delta \\rho _j \\triangledown \\overline{W}_{ij} - \\sum _{j \\in \\mathbb {N}^i} \\overline{\\rho } \\Delta p B \\lbrace \\delta \\rho _i + \\delta \\rho _j \\rbrace \\triangledown \\overline{W}_{ij}\\end{split}$ The first term in the RHS refers to the acceleration at any $i^{th}$ particle in the stable configuration i.e.",
"$\\frac{d{\\overline{v}}_i}{dt} = - \\sum _{j \\in \\mathbb {N}^i} \\Delta p \\left[2 \\frac{\\overline{\\sigma }}{\\overline{\\rho }} \\right] \\triangledown \\overline{W}_{ij}$ (Equation REF ).",
"Putting the values of $\\delta \\rho _i$ and $\\delta \\rho _j$ the expression can be re framed as, $\\begin{split}-\\omega ^2 \\delta x_{i} = - \\sum _{j \\in \\mathbb {N}^i} 2 \\Delta p \\frac{\\overline{\\sigma }}{\\overline{\\rho }} \\triangledown ^2 \\overline{W}_{ij} \\delta x_{ij} - \\sum _{j \\in \\mathbb {N}^i} 2 \\Delta p \\frac{\\overline{\\sigma }}{\\overline{\\rho }^2} D e^{i(k\\overline{x}_j - \\omega t)} \\triangledown \\overline{W}_{ij} \\\\- \\sum _{j \\in \\mathbb {N}^i} \\overline{\\rho } \\, \\Delta p B D \\left\\lbrace e^{i(k\\overline{x}_i - \\omega t)} + e^{i(k\\overline{x}_j - \\omega t)} \\right\\rbrace \\triangledown \\overline{W}_{ij}\\end{split}$ The expression of the frequency of oscillation ($\\omega $ ) can be obtained by dividing Equation REF by $\\delta x_i$ , where $\\frac{\\delta x_{ij}}{\\delta x_{i}} = 1 - \\frac{\\delta x_j}{\\delta x_i} = 1 - e^{ik(\\overline{x_j} - \\overline{x_i})}$ .",
"As the final expression is valid everywhere within the domain (i.e., $\\forall i \\in \\mathbb {Z}^+$ ), the particle index $i$ is omitted here onwards so that $\\overline{x_{ij}}$ and $\\overline{W_{ij}}$ are written as $\\overline{r_j}$ and $\\overline{W_{j}}$ .",
"$\\overline{r_j}$ is the distance of the neighbourhood particle in the unperturbed state.",
"Replacing $\\frac{D}{X}$ by $\\sum _{j=1}^{\\mathbb {N}^i} \\textit {i} \\, \\overline{\\rho } \\, \\sin (k\\overline{r_{j}}) \\triangledown \\overline{W}_{j}$ (as obtained from conservation of mass) (where $i^2 = -1$ ) the $\\omega ^2$ can be expressed as, $\\omega ^2 = 2\\frac{{\\overline{\\sigma }}}{\\overline{\\rho }} \\Delta p \\sum _{j \\in \\mathbb {N}^i} \\left[1- \\text{cos}(k\\overline{r}_{j}) \\right] \\triangledown ^2 \\overline{W}_{j} + \\frac{K}{{\\rho _0}^2} \\lbrace \\sum _{j \\in \\mathbb {N}^i}\\Delta p \\, \\text{sin}(k\\overline{r}_{j}) \\triangledown \\overline{W}_{j} \\rbrace ^2$ Expressing the kernel function and its derivatives in the parametric form , the frequency of the wave speed can be finally written as, when, $0 < \\frac{\\overline{r_j}}{h} < a$ $\\begin{split}\\omega ^2 = \\frac{2\\overline{{\\sigma }}}{\\overline{\\rho }} \\Delta p \\frac{60}{\\pi a^2 b^2 (a^2 +ab +b^2) h^3}\\sum _{j \\in \\overline{\\mathbb {N}^i}} \\left[1- cos(k\\overline{r_{j}}) \\right] \\lbrace (a+b)\\frac{\\overline{r_{j}}}{h} - ab \\rbrace \\\\+ \\frac{K}{\\overline{\\rho }} (2- \\frac{\\overline{\\rho }}{\\rho _0}) \\left\\lbrace \\sum _{j \\in \\overline{\\mathbb {N}^i}} \\Delta p \\sin (k\\overline{r}_{j}) \\frac{30[(a+b)(\\frac{\\overline{r}_{j}}{h})^2 - 2ab\\frac{\\overline{r}_{j}}{h}]}{\\pi a^2b^2 h^2 (a^2+ab+b^2)} \\right\\rbrace ^2\\end{split}$ when, $a < \\frac{\\overline{r_j}}{h} < b$ $\\begin{split}\\omega ^2 = \\frac{2\\overline{{\\sigma }}}{\\overline{\\rho }} \\Delta p \\frac{60}{\\pi b (b-a) (a^2 +ab +b^2) h^3}\\sum _{j \\in \\overline{\\mathbb {N}^i}} \\left[1- cos(k\\overline{r_{j}}) \\right] \\lbrace b - \\frac{\\overline{r_{j}}}{h} \\rbrace \\\\+ \\frac{K}{\\overline{\\rho }} \\left\\lbrace \\sum _{j \\in \\overline{\\mathbb {N}^i}} \\Delta p \\sin (k\\overline{r}_{j}) \\frac{30\\lbrace b - \\frac{\\overline{r}_{j}}{h} \\rbrace ^2}{\\pi b (b-a) h^2 (a^2+ab+b^2)} \\right\\rbrace ^2\\end{split}$ Figure: Dispersion analysis of a 1D wave propagation problem with adaptive kernelFigure: Stable zone for intermediate knot a for different ρ ¯/ρ\\overline{\\rho }/\\rho and h=1.5Δph=1.5\\Delta pMathematically for any value of $k$ that results in $\\omega < 0$ , the system reflects zero energy modes or instability.",
"So the objective of removing instability clearly points out to ensure $\\omega > 0$ for any wave number($k$ ) of the perturbing wave.",
"Interestingly for any $k\\overline{r_{j}} = n\\pi , \\forall n \\in \\mathbb {Z}^+ \\rightarrow \\omega = 0$ .",
"The smallest wave length that can be captured with a characteristic inter particle distance $\\Delta p$ is $\\lambda = 2\\Delta p$ , which in other form suffices to a wave number $k = \\pi /\\Delta p$ , i.e.",
"$n = 1$ .",
"The dispersions of the numerical solution with cubic B-spline kernel for different values of $a$ are plotted in Figure REF .",
"For this exercise $\\Delta p = 1$ , $h = 1.5\\Delta p$ , $\\overline{\\rho } = 0.96 \\rho $ for tension and $\\overline{\\rho } = 1.05 \\rho $ for compression are taken.",
"As can be seen, appropriate values of $a$ ensure positive shift in the $\\omega $ vs $k$ curve near the zero energy modes, which demonstrates the ability of the method, to remove the instability form the system.",
"However, it is to be noted that for tension (Figure REF ) any value of $a$ satisfying Equation REF may not remove instability.",
"Rather, there is a range within which the value of $a$ needs to be chosen for a stable computation.",
"For a given kernel function this range depends on $h/\\Delta p$ and $\\overline{\\rho } /\\rho $ ratios.",
"This is illustrated in Figure REF .",
"Continuous update of the intermediate knot a provides a way out of tensile instability in SPH computation.",
"The construction of B-spline basis function (section REF ) ensures that for any value of $a \\in \\left(0,b\\right)$ the resulting basis satisfies all the essential properties (such as differentiability, compactness, positivity, symmetry, unique maxima, partition of unity etc.)",
"of a kernel and therefore does not compromise the premise of the kernel approximation in Equation REF .",
"In the next section, the efficacy of the algorithm is demonstrated through some numerical example."
],
[
"Numerical Examples",
"Tensile instability in SPH has been identified and discussed in the literature through various examples.",
"Some test cases among them are revisited in this section and the efficacy of the proposed adaptive formulation in removing tensile instability is demonstrated.",
"In the example problems instability may occur either in tension or compression or both.",
"The adaptive algorithm is implemented with the cubic B-spline kernel as given in Table REF .",
"For a better compression of the relative advantage of the proposed algorithm, simulations are also performed with standard cubic kernel given in Equation REF .",
"$W(q, h)=\\alpha _d{\\left\\lbrace \\begin{array}{ll}1-\\frac{3}{2} q^2 +\\frac{3}{4} q^3,& \\text{if } 0\\le q\\le 1\\\\\\frac{1}{4}(2-q)^3, & \\text{if } 1\\le q\\le 2\\\\0 & \\text{otherwise}\\end{array}\\right.",
"}$ where, $\\alpha _d=\\frac{2}{3h}$ in 1D, $\\alpha _d=\\frac{10}{7\\pi h^2}$ in 2D, $\\alpha _d=\\frac{1}{\\pi h^3}$ in 3D and $q=\\frac{|\\mathbf {x}_i-\\mathbf {x}_j|}{h}$ is the normalised position vector associated with a particle pair.",
"In order to achieve the first order consistency [35], [36] the first derivative of the kernel function is modified as $\\hat{W}_{ij,\\beta }=B^{\\beta \\alpha }W_{ij,\\alpha }$ where $\\textbf {B}$ is a symmetric re-normalization matrix obtained as $\\mathbf {B}^{-1}_i = -\\sum _{j\\in N^i} \\frac{m_j}{\\rho _j} \\mathbf {x}_{ij} \\otimes \\mathbf {\\bigtriangledown } W_{ij}$ .",
"In all the simulations artificial viscosity (Equation REF ) and XSPH correction (Equation REF ) are used.",
"This is to be noted that the purpose of the present study is to remove the tensile instability only by changing the shape of the kernel function and therefore no artificial stress correction is used.",
"In all the example, the smoothing length ($h$ ) is taken as 1.5$\\Delta p$ .",
"The adaptive algorithm is implemented with $b = 2$ , and the intermediate knot $a_i$ updated as, $a_i={\\left\\lbrace \\begin{array}{ll}1.1\\frac{r_d}{h} ,& \\text{if } \\frac{\\rho _i}{\\rho _0} < 1\\\\0.2 & \\text{otherwise}\\end{array}\\right.",
"}$ where, for $i$ -th particle, $r_d=max\\left\\lbrace r_{ij}, j\\in \\overline{\\mathbb {N}^i} \\right\\rbrace $ as defined in Figure REF .",
"The above values of $a_i$ are taken as per the dispersion analysis given in section .",
"For $b=2$ and $h=1.5\\Delta p$ , $a_i = 1.1r_d/h$ yields a stable computation in tension (Figure REF ).",
"For compression, Figure REF suggests that any value of $a_i<1$ (for $b=2$ ) ensures stability and thus $a_i = 0.2$ is uniformly taken in compression.",
"In order to preserve the symmetry in the formulation, when two particles (say $i$ -th and $j$ -th) interact the kernel function is constructed with an average intermediate knot $a_{ij}=(a_j+a_j)/2$ ."
],
[
"2D Stability Test",
"To start with, the 2D stability test considered in section REF to illustrate the tensile instability is considered once again.",
"Geometry, material parameters and initial conditions are kept same as before.",
"Simulations are performed with adaptive kernel and the computed configurations at 100 $\\mu $ s are compared in Figure REF .",
"It can be readily seen that the instability is fairly removed from the solution.",
"In order to ensure that the computation remains stable, the simulation is continued for a long time.",
"As evident in Figure REF , there is no indication of instability even after 1 $ms$ .",
"Figure: 2D stability test: comparison of particle configurations at different time instants"
],
[
"Wave propagation in a bar",
"Next an elastic cantilever rod of length 20cm and depth 1cm subjected to an uniform compressive velocity of $v_0 = 1 m/s$ throughout its length as depicted in Figure REF is considered.",
"Material properties and other computational data are provided in Table REF .",
"As the bar undergoes compression, the compressive wave traverses towards the support and is reflected back as tensile wave.",
"With further progression of the tensile wave, instability that starts to generate in the standard SPH computation, manifests itself in the form of unphysical numerical fracture as shown in Figure REF .",
"However when the computation is performed with the adaptive kernel no instability is observed (Figure REF ).",
"The analytical expression for the the longitudinal displacement at a given point is, $u(x,t) = \\sum _{j=1}^{\\infty } \\frac{8(-1)^{k+1} v_0 L}{\\pi ^2 (2k-1)^2 c} \\text{sin} \\left( (2k-1) \\frac{c\\pi t}{2L} \\right) \\text{cos} \\left( (2k-1) \\frac{\\pi x}{2L} \\right),$ where, $c=\\sqrt{\\frac{E}{\\rho }}$ is the speed of elastic wave in the medium.",
"Figure: Schematic diagram of an axial bar under compression, L = 20 cm , B = 1 cm, V = 1 m/sTable: Parameters for wave propagation in a barThe computed longitudinal displacement time history at the tip of the bar is compared with that of obtained from Equation REF in Figure REF .",
"Displacement time history obtained via the standard SPH is also shown in the same figure.",
"The effect of instability in the standard SPH is evident.",
"On the other hand, the predicted response via SPH with adaptive kernel shows very good agreement with the analytical solution.",
"Figure: Deformed configurations at different time instants obtained via standard SPHFigure: Deformed configurations at different time instants SPH with adaptive kernelFigure: Comparision of displacement time histories at the tip of the bar"
],
[
"Transverse oscillation of a plate",
"Next a thin linear elastic cantilever plate of length $L$ and thickness $B$ , oscillating in two dimensional mode as shown in Figure REF is considered.",
"The frequency of the oscillation is given by, $\\omega ^2 = \\frac{EB^2k^4}{12\\rho (1-\\nu ^2)},$ where, $\\rho , E, \\nu $ are the density, Elastic modulus and Poison ratio respectively.",
"$k$ is the wave number determined from the condition $\\text{cos}(kL)\\text{cosh}(kL) = -1$ .",
"For the first mode of vibration $kL = 1.875$ .",
"The plate is set to motion by giving initial transverse velocity $v_y$ according to the first mode as, $\\frac{v_y}{c} = V_f\\frac{M\\left[\\text{cos}(kx) - \\text{cosh}(kx)\\right] - N\\left[\\text{sin}(kx) - \\text{sinh}(kx)\\right]}{Q}$ where, $c$ is the speed of sound wave in the medium.",
"$M, ~ N$ and $Q$ are constants given as, $M = \\text{sin}(kL) + \\text{sinh}(kL)$ $N = \\text{cos}(kL) + \\text{cosh}(kL)$ $Q = 2 \\left[\\text{cos}(kL)\\text{sinh}(kL) - \\text{sin}(kL)\\text{cosh}(kL) \\right]$ Figure: Cantilever plate under oscillationIn equation REF $V_f$ is the transverse velocity at the free end.",
"The material properties are taken as $\\rho = 7850 kg/m^3, E = 210 GPa, \\nu =0.3$ .",
"Length($L$ ) and thickness ($B$ ) of the plate are considered as 200mm and 20mm respectively, and the initial tip velocity $V_f$ is given as 0.02 m/s.",
"In order to check the effect of adaptive kernel on the convergence behaviour, simulations are performed with different inter particle spacing ($\\Delta p$ ).",
"For all the cases the artificial viscosity parameters are taken as $\\gamma _1 = 1.0, \\gamma _2 = 1.0$ .",
"The computed time periods for different $\\Delta p$ are given in table REF .",
"For $\\Delta p$ = 0.5mm the computed time period ($T_{c}$ ) is very close to its theoretical value ($T_{t} = 2\\pi /\\omega $ , $\\omega $ from Equation REF ) and therefore results for $\\Delta p$ = 0.5mm are considered for subsequent demonstration.",
"Time step ($\\Delta t$ ) is taken as $5\\times 10^{-8}$ sec.",
"Table: Period of Vibration at the tip of the plateDeformed configurations of the plate at different time instants obtained via the standard cubic kernel and the adaptive kernel are shown in Figure REF and REF respectively.",
"As evident from the figures, while standard cubic kernel (Figure REF ) shows instability near the fixed end, no prominent instability is observed in case of adaptive kernel (Figure REF ).",
"Figure: Displacement profiles at different time instants using standard SPH, T = 2.35 msFigure: Displacement profiles at different time instants obtained via SPH with adaptive kernel, T = 2.35 ms; (e) - (h) are the enlarged view A at different time instantsIn order to investigate the performance of the adaptive kernel in a situation where structure undergoes excessive deformation, next a similar plate problem but with different L/B ratio (L = 200mm and B = 10mm) and initial condition $V_f$ = 0.05 m/s is considered.",
"When simulation is performed with the standard cubic kernel, computation suffers an unphysical numerical fracture (Figure REF ).",
"Adaptive kernel, on the other hand, continues to perform in a stable manner as illustrated in Figure REF .",
"The computed time period (T) is 4.62 ms which is very close to the theoretical value of 4.56 ms. Time histories of the transverse displacement at the tip (for both L/B = 10 and L/B = 20) obtained via the adaptive kernel are shown in Figure REF .",
"Figure: Deformed configuration of the plat (L = 200mm , B = 20mm) obtained via standard SPHFigure: Deformed configurations of the plat (L = 200mm , B = 20mm) obtained via SPH with adaptive kernel, T = 4.62 msFigure: Transverse displacement time histories at the tip of the cantilevr plate"
],
[
"Collision of two rubber rings",
"As the last example, elastic collision of two rubber rings with inner diameter 30mm and outer diameter 40mm at a velocity 50m/s is considered.",
"Material properties and other computational data are given in Table REF .",
"The deformed configurations of the rings at different time instants are shown in Figure REF and REF .",
"Instability is clearly visible in SPH with standard Cubic B-spline kernel (Figure REF ).",
"However on implementing the adaptive kernel (Figure REF ) the instability is removed from the system.",
"Table: Parameters for the simulation of rubber collisionFigure: Configurations of the rubber rings at different time instants obtained via standard SPHFigure: Configurations of the rubber rings at different time instants obtained via SPH with adaptive kernel"
],
[
"Closure",
"Like any other computational tool, SPH also has its share of limitations.",
"The tensile instability is perhaps the major stumbling block among them.",
"The cause of the instability lies in the core of the formulation, which uses a bell-shaped compactly supported kernel function for discretizing the strong form of the governing Equations.",
"Swegle's [24] analysis suggests that stability can be enforced by either modifying the stress tensor or choosing a suitable kernel such that their interaction does not induce any spurious negative stiffness.",
"The second approach is adopted in this study.",
"Herein an adaptive algorithm is proposed where the shape of the kernel function is continuously modified depending on the state of stress such that the condition associated with tensile instability does not arise.",
"Towards this, the B-spline basis function constructed over a symmetric knot vector is taken as the kernel function, and its shape is adapted by changing the location of the intermediate knot.",
"Theoretical framework and implementation of the algorithm are discussed in detail.",
"One dimensional dispersion analysis is also performed to have a better insight into the effect of the intermediate knot on the stability.",
"The most distinguishing feature of the proposed algorithm is that it does not require tuning of any user-defined parameters, unlike common practices available in the literature.",
"Moreover, the additional computational effort needed for the algorithm is also insignificant.",
"The efficacy of the algorithm is demonstrated through some benchmark elastic dynamics problems.",
"The proposed approach shows great potential in removing tensile instability at an affordable computational effort.",
"The applicability may further be extended to different class of problems which may constitute the topic for future work."
],
[
"Acknowledgement",
"The authors would like to acknowledge the Naval Research Board, DRDO, India for their support of this work."
]
] | 2001.03416 |
[
[
"Physical conditions and chemical abundances in photoionized nebulae from\n optical spectra"
],
[
"Abstract This chapter presents a review on the latest advances in the computation of physical conditions and chemical abundances of elements present in photoionized gas H II regions and planetary nebulae).",
"The arrival of highly sensitive spectrographs attached to large telescopes and the development of more sophisticated and detailed atomic data calculations and ionization correction factors have helped to raise the number of ionic species studied in photoionized nebulae in the last years, as well as to reduce the uncertainties in the computed abundances.",
"Special attention will be given to the detection of very faint lines such as heavy-element recombination lines of C, N and O in H II regions and planetary nebulae, and collisionally excited lines of neutron-capture elements (Z >30) in planetary nebulae."
],
[
"A very brief introduction on emission line spectra of photoionized nebulae",
"Photoionized nebulae (i. e. H ii regions and planetary nebulae) are among the most “photogenic” objects in the sky.",
"Given their relatively high surface-brightness they are easily accessible, even for non-professional telescopes.",
"This allowed earliest visual spectroscopic observations by William Huggins and William A. Miller ([1]) who obtained the first spectrum of a planetary nebula (The Cat's Eye Nebula), where they detected a bright emission line coming from a mysterious element that Margaret L. Huggins ([2]) called “nebulium”.",
"Several decades later, Ira S. Bowen ([3]) showed that this emission was produced by doubly ionized oxygen (O$^{2+}$ ) at extremely low densities.",
"An historical review on the early steps of the study of the physics of gaseous nebulae was provided by Donald E. Osterbrock ([4]) who used a seminal paper by Bowen ([5]) as the starting point for a review of nebular astrophysics.",
"Photoionized nebulae are excited by the strong ultraviolet (UV) radiation of hot stars ($T_{eff}\\ge 25-30 kK$ ) which produce photons with energy that could be above the ionization threshold of the gas particles and hence, ionize them releasing a free electron.",
"The probability of occurrence of this phenomenon depends on the photoionization cross-section which, in turn, depends on the energy of the photon and the target being considered.",
"Once ionized, the gas particles tend to recombine with the free electrons, and eventually an equilibrium stage is established in which the rate of ionization equals the rate of recombination for each species (see [6]).",
"The optical spectra of photoionized nebulae are dominated by emission lines, which are formed when atoms or ions make a transition from one bound electronic state to another bound state at a lower energy via spontaneous emission.",
"These bound electrons can be excited either by free electrons colliding with the atom/ion, or by absorption of a photon.",
"However, the background radiation field in the interstellar medium in generally not strong enough for excitation by photon absorption to be significant (see chapter 5 of [7]) and therefore, the only way of having a bound electron in an excited state is by collisional excitation from a lower state, which subsequent radiative decays to lower levels originating the collisionally excited lines (hereinafter CELs), or owing to a recombination between a free electron and an ion, which is the mechanism behind the emission of recombination lines (hereinafter RLs).",
"Given that the abundance of H and He ions are several orders of magnitude higher than that of heavier elements, one can instinctively assume that the emission spectra will be dominated by H and He lines, which is not the case.",
"In photoionized nebulae the peak of the energy distribution of free electrons is of the order of 1 eV.",
"Ions of heavy atoms like N, O, Ne, S, Cl, Ar, etc.",
"have electronic structures with low-lying electronic states in the range of fractions to few eV from the ground state and can therefore be effectively excited by collisions.",
"On the other hand, for H and He ions, the gap between the ground state and the first excited state is very large and cannot be excited by collisions, but by recombination.",
"Fig.",
"REF shows a typical optical spectrum of a photoionized nebulae (in this case the planetary nebula Hb 4); remarkably bright H and He RLs and CELs of different ionic species of N, O, Ne, S, Cl and Ar are labelled.",
"Figure: Section of the very deep spectrum of the PN Hb 4 analysed in , showing bright RLs of H i and He i and CELs of different ions of O, N, S and Ne.Therefore, the spectrum of a photoionized nebula is dominated by the emission of RLs of H and He (the most abundant elements) and CELs of heavier elements.",
"The combination of narrow-band images taken in the brightest emission lines allows to construct the beautiful coloured images of photoionized nebulae (see Fig.",
"REF ) from which we can have a first sketch of the ionization structure of the photoionized region.",
"In Fig.",
"REF we show a 3 narrow-band filter combined image of a star-forming region in the Large Magellanic Cloud where is clear that the emission of [O iii] is more internally located than the emission from [S ii].",
"Figure: “Bubbles of Brand New Stars” Composite image of a star-forming region in the Large Magellanic Cloud (LMC) captured by the Multi Unit Spectroscopic Explorer (MUSE) instrument on ESO's Very Large Telescope (VLT).",
"The following colour code was used: [O iii] λ\\lambda 5007 (blue).Hα\\alpha (yellow), [S ii] λ\\lambda 6731 (red).",
"The field-of-view of the image is 7.82 ×\\times 8.00 arcminutes 2 ^2.",
"North is 180.2deg\\deg left of vertical.",
"Credit: ESO, A. McLeod et al.Although the emission line spectra from H ii regions and planetary nebulae (hereinafter, PNe) are roughly similar, there are some remarkable differences between them.",
"H ii regions are large (tens of parsecs), massive (generally between $10^2-10^3$ M$_{\\odot }$ ) regions of gas that are ionized by the ultraviolet (UV) radiation emitted by recently formed OB-type massive stars with typical effective temperatures between 25-50 kK; in general, these stars are not hot enough to ionize nebular He ii, whose ionization potential is $h\\nu =54.4$ eV.",
"However, there are exceptions to this rule, especially in the integrated spectra of blue compact dwarf galaxies (BCDs), Wolf-Rayet (WR) galaxies and a couple of nebulae in the Local Group, associated to WR stars.",
"On the other hand, PNe are much smaller ($10^{-1}$ pc) and less massive ($\\sim $ 10$^{-1}$ M$_{\\odot }$ ) nebulae that are excited by central stars which are generally hotter (central stars can reach temperatures as high as 250 kK); therefore, there will be ionizing photons with enough energy to ionize high-excitation species and hence, producing qualitatively different spectra than that of H ii regions, showing emission lines of He ii, [Ne v], [Ar v], [Fe v], and even more excited species.",
"The analysis of emission line spectra of photoionized nebulae allows us to determine the chemical composition of the interstellar medium (ISM) from the solar neighbourhood to the high redshift star-forming galaxies.",
"It stands as an essential tool for our knowledge of stellar nucleosynthesis and the cosmic chemical evolution.",
"Since the early achievements in spectrophotometry of photoionized nebulae, the quality of deep optical and near-infrared spectrophotometric data of PNe has increased significantly mainly thanks to both the development of more efficient instruments and to the advent of large aperture (8m-10m-class) ground-based telescopes.",
"In this sense, the future installation of giant-class ones (diameters 30-50m) opens new horizons in the field of nebular spectroscopy.",
"The detection of very faint emission lines in ionized nebulae as auroral CELs in faint, distant or high-metallicity objects; optical recombination lines (hereinafter, ORLs) of heavy-element ions or CELs of trans-iron neutron-capture elements are becoming a routine fact and provide new information of paramount interest in many different areas of astrophysics.",
"H ii regions can be observed at considerable distances in the Universe and hence, are crucial to determine the chemical composition of the interestellar medium (ISM) in the extragalactic domain.",
"Since H ii regions lie where star formation is occurring, chemical abundances computed in H ii regions are probes to trace the present-day chemical composition of the ISM.",
"In particular, the study of radial variations of chemical abundances along galactic discs in spiral galaxies are essential observational constraints for chemical evolution models, and precise determinations of chemical abundances in low-metallicity dwarf galaxies, can permit to determine the primordial abundance of helium owing to Big Bang nucleosynthesis (see [10] and references therein).",
"The global picture of abundances in PNe is more complicated because for elements that are supposed to be not modified, such as O and $\\alpha $ -elements, the computed abundances reflect the chemical conditions in the cloud where the progenitor star was formed, while the chemical abundances of N, C, or neutron-capture elements, that could be modified during the cycle of life of low-to-intermediate mass stars allow us to constrain the nucleosynthetic processes in these stars."
],
[
"Recent reviews on chemical abundance determinations",
"Recently, two tutorials focused on the determination of ionized gaseous nebulae abundances have been released ([10] and [11]) although with different points of view.",
"In the former, [10] give a brief review on the physics basics of abundance determinations, like local ionization and local thermal equilibrium, emission line mechanisms and on the calculation of physical conditions and ionic and elemental abundance determinations from observations; these authors also review recent results in abundance determinations in both H ii regions and PNe.",
"However the review is quite focused to the abundance discrepancy problem from the point of view of temperature fluctuations (see Section REF ).",
"In [11], the focus is on the determination of abundances in extragalactic H ii regions from the direct method (when electron temperature, $T_e$ , and electron density, $n_e$ , diagnostic lines are available) and in the use of some strong-line methods calibrated using the direct method (see Section REF ).",
"Further comprehensive tutorials on abundance determinations are those by [12] and [13] where the theoretical background of photoionized nebulae is treated in more detail, and particular emphasis is given to the description of line formation mechanisms, transfer of radiation, as well as to the use of empirical diagnostics based on emission lines and determination of chemical abundances using photoionization models.",
"It is not the scope of this chapter to repeat the basic concepts of the physics of photoionized nebulae, which have been described in different detail in the aforementioned tutorials.",
"Moreover, for a much more detailed description of such processes, we refer the reader to the canonical book of photoionized nebulae: “Astrophysics of gaseous nebulae and active Galactic nuclei” ([6]).",
"In the following sections I will focus on recent advances in chemical abundances determinations in photoionized nebulae from the analysis of deep optical and near-infrared spectra, from an observational point of view.",
"Due to space limitations, I refer the reader to [12], [13] for an overview on abundance determinations based on photoionization model fitting.",
"Similarly, the strong line methods to determine abundances in the extragalactic domain (from giant H ii regions to high-redshift galaxies) will be only briefly discussed in Section REF .",
"In the last years, the number of deep high-quality spectra of photoionized nebulae has increased significantly, allowing the detection of very faint emission lines (see e. g. [14], [15], [16], [17], [18], [19], [20], [8], [21], [22], [23]) and the computation of, in principle, very reliable chemical abundances.",
"The advantages of obtaining deep and high resolution spectra of photoionized regions are clear because one can easily isolate faint lines that in lower resolution spectra would be blended and go unnoticed.",
"As an illustration, in Fig.",
"REF we show an excerpt of the spectrum of the high-excitation PN H 1-50 analysed in [24] with the same spectrum downgraded to a lower resolution overplotted in red.",
"Several permitted lines of O, N, and C would have remained hidden in the low-resolution spectra and ad-hoc atomic physics would have been needed to estimate their fluxes.",
"In the last years, several groups have provided a large sample of deep, high-resolution spectra of both Galactic and extragalactic H ii regions and PNe (see e. g. [25], [26], [24] and the compilation made by [27]).",
"Figure: Portion of a high-resolution (R∼\\sim 15000) spectra of the high-excitation PN H 1-50 showing the zone where the multiplet 1 O ii lines lie.",
"Overplotted in red is the same spectra degraded to a resolution of R∼\\sim 3500.",
"As it can be shown the high-resolution of the original spectra allows one to deblend several very close permitted emission lines of C, O, and N that would have remained hidden in the low-resolution spectra.",
"Data originally published in .However, deep, high signal-to-noise, high-resolution spectra are not the panacea.",
"[28] has recently shown that the effects of observational uncertainties can be very important even making use of high quality spectra, owing to the high number of sources of uncertainty that are acting in the process, which include: assumptions in the nebular structure, atomic data (see section REF ), atmospheric differential refraction, telluric absorption and emission, flux calibration, extinction correction, blends with unknown lines, etc.",
"Therefore, a careful data reduction procedure should be carried out to obtain reliable results.",
"Additionally, an homogeneous analysis determining physical conditions and chemical abundances from the same set of spectra is mandatory if one want to compute precise abundances.",
"For instance, many studies devoted to study the radial abundance gradients have made use of physical conditions derived from radio recombination lines combined with and optical or infrared lines to compute abundances; these approach has been used in several seminal papers on the Galactic abundance gradient (see section REF ; however, it can introduce systematic uncertainties owing to the different areas of the nebula covered in the different wavelength ranges.",
"Additionally, we should use a set of appropriate lines to compute the abundances; as an example, computing O$^+$ /H$^+$ ratios from the trans-auroral [O ii] $\\lambda \\lambda $ 7320+30 lines could introduce undesired uncertainties because these lines could be strongly affected by telluric emission, and are also very sensitive to electron density and temperature.",
"To illustrate these effects, in Fig.",
"REF we show an adaptation of Fig.",
"5 of [29] where the radial oxygen abundance gradient making a consistent analysis of several data sets is presented.",
"Physical conditions have been derived from both radio and optical diagnostics, and abundances have been derived using optical CELs of oxygen (blue points) or far-IR fine-structure CELs of oxygen (red points).",
"As can be seen, both data sets show significant scatter in the oxygen abundance at a given Galactocentric distance, which can be interpreted as an “intrinsic scatter” owing to the gas not being well mixed ([30], [29], [31]).",
"However, high-quality observations seem to rule out this interpretation.",
"In Fig.",
"REF the abundances computed by [32] from an homogeneous analysis of optical spectrophotometric data of 35 H ii regions with direct determinations of the electron temperature have been overplotted on the [29] sample.",
"As it is clearly shown, the scatter in the oxygen abundance is reduced significantly and is not substantially larger than the observational uncertainties, indicating that oxygen seems to be well mixed in the ISM at a given distance along the Galactic disc.",
"Moreover, [33] showed from the analysis of high-quality spectra with high signal-to-noise auroral [O iii] line detections in H ii regions in the inner parts of M 33, a much lower scatter than that found by [31]; this author also found no evidence for significant azimuthal variations in the H ii region metallicity distributions, ruling out large anomalies in the mixing of the gas.",
"Figure: Radial oxygen abundance gradient in the MW using abundances derived from optical lines (blue points) or from far-IR lines (red points) compiled by .",
"The data from obtained through deep spectra taken in 8m-10m class telescopes are overplotted (black squares).",
"The fits for each set of data are represented by lines with the same colour than the data points.",
"References of the original data compiled by are shown in the legend.Finally, one has to take into account some biases that the direct method can have.",
"[34] discussed about the limitations of the direct method to determine O abundances in giant H ii regions at metallicities larger than solar.",
"This author used ab-initio photoionization models of giant H ii regions, and applied to the models the same methods as used for real objects to test the direct method.",
"The global result of this study was that for log(O/H)+12 larger than 8.7 (i.e.",
"larger than the solar value), the computed O/H values were below the ones implied by the photoionization models owing to strong temperature gradients present in giant H ii regions.",
"Finally, [34] propose that PNe, which are not affected by these biases, could be potential probes of the metallicity of the interstellar medium in the internal parts of spiral galaxies as well as in metal-rich elliptical galaxies.",
"However, in Section REF we will discuss that this idea should be taken with some caution."
],
[
"The direct method",
"The most popular way to compute the chemical abundances of the elements that are present in a photoionized gas is the so-called direct method.",
"This method makes use of CELs intensities of different ionic species of elements like N, O, S, Ne, Cl, Ar, Ne, Fe, etc., and involves the determination of the physical conditions (temperature and electron density) in the emitting plasma.",
"In the conditions prevailing in photoionized nebulae like H ii regions and PNe, most of the observed emission lines are optically thinAn emission line is said to be optically thick if on average a photon emitted cannot pass through the ISM without absorption.",
"Conversely, an emission line is said to be optically thin if we can see the radiation coming from behind the nebula (i. e. it is not absorbed).",
"with the exception of some resonance UV lines and some fine-structure IR lines (see [12]) making their use for abundance determinations very robust.",
"In the analysis of photoionized nebulae it is usually assumed that the physical conditions are homogeneous in the photoionized region.",
"Under these assumptions one can compute electron temperature and density by using sensitive line ratios.",
"Electron temperature ($T_e$ ) and density ($n_e$ ) in nebulae are represented by the kinetic energy and density of the free electrons in the photoionized gas.",
"Some CEL intensity ratios of ions of common elements like O, N, S or Ar depend on the physical conditions of the gas, and are useful to calculate $T_e$ and $n_e$ (see Section 3.5 of [10] for more details).",
"In particular, the intensity ratios of emission lines of a given ion that originates in very different energy levels, are sensitive to $T_e$ and almost independent on $n_e$ , since the populations of the different atomic levels are strongly dependent on the kinetic energy of the colliding free electrons.",
"Typical optical electron temperature diagnostics are: [N ii] $\\lambda $ 5754/$\\lambda $ 6583, [O ii] $\\lambda \\lambda $ 7320+30/$\\lambda \\lambda $ 3726+29, [O iii] $\\lambda $ 4363/$\\lambda $ 5007, [Ar iii] $\\lambda $ 5191/$\\lambda $ 7531 or [S iii] $\\lambda $ 6312/$\\lambda $ 9531.",
"Therefore, determination of chemical abundances making use of the direct method in optical spectra requires the detection of faint auroral lines, which correspond to transitions from the state $^1$ S to $^1$ D and are very $T_e$ -sensitive.",
"The detection of such lines is a relatively easy task in Galactic H ii regions and PNe.",
"However, their emissivity decreases rapidly with metallicity and with decreasing surface brightness of the objects so, detecting them is a challenging task in the extragalactic domain, especially in objects beyond the Local Group.",
"However, the combination of high-sensitivity spectrographs with large aperture (10m type) telescopes have allowed the detection of the auroral [O iii] $\\lambda $ 4363 Å line in at least 18 star-forming galaxies at $z>1$ (see [35] and references therein).",
"On the other hand, line ratios sensitive to $n_e$ come from levels with very similar energy, so that the ratio of their populations does not depend on $T_e$ .",
"These levels show different transition probabilities or different collisionally de-excitation rates, such that the ratio between the emission lines generated is strongly dependent on the electron density of the photoionized gas.",
"Typical optical density diagnostics are: [O ii] $\\lambda $ 3726/$\\lambda $ 3729, [S ii] $\\lambda $ 6716/$\\lambda $ 6731, [Cl iii] $\\lambda $ 5517/$\\lambda $ 5537 and [Ar iv $\\lambda $ 4711/$\\lambda $ 4741.",
"A precise determination of the physical conditions is crucial to derive reliable abundances from CELs.",
"As the abundances are computed relative to H by using the relative intensities of CELs or ORLs relative to a H i ORL (usually H$\\beta $ ), and given the very different dependence of the emissivity of CELs and ORLs (see [10]) the abundances from CELs show a strong (exponential) dependence on $T_e$ , while abundances computed from faint metallic ORLs are almost $T_e$ dependent.",
"This has important implications for the so-called abundance discrepancy problem (see Section REF ).",
"Once physical conditions are computed one has to decide the temperature and density structure that is going to be assumed in the nebula.",
"The most common approach, is to assume a two-zone scheme, where the high ionization zone is characterized by $T_e$ (high) (usually $T_e$ ([O iii]), the low-ionization zone is characterized by $T_e$ (low) (usually $T_e$ ([N ii]) and the density is considered homogeneous in the whole nebula and is characterized by $n_e$ ([S ii]).",
"Then, each temperature is applied to compute ionic abundances of species with similar ionization potentials than the ion used in the $T_e$ diagnostic.",
"In a typical spectra, $T_e$ (low) is applied to compute abundances of N$^+$ , O$^+$ , S$^+$ , Cl$^+$ and Fe$^+$ , while $T_e$ (high) is used for the remaining ionic species observed in the optical spectra.",
"However, recent results from [36] have shown that this scheme can be erroneous.",
"These authors, from deep, high-resolution spectra of H ii regions in the Magellanic Clouds, have proposed that for some ions, it is better to adopt other scheme in order to avoid trends with metallicity.",
"In particular, they propose to use $T_e$ ([N ii]) to calculate Cl$^{2+}$ and the mean of $T_e$ ([N ii]) and $T_e$ ([O iii]) for S$^{2+}$ and Ar$^{2+}$ , finding that, in such cases, Cl/O, S/O and Ar/O are approximately constant with metallicity (see their Fig.",
"3) as expected for $\\alpha $ -elements.",
"In deep spectra, covering the whole optical (or even up to 1$\\mu $ m) wavelength range more electron temperature and density diagnostics will be available.",
"In such cases, $T_e$ (low) and $T_e$ (high) can be computed as the average of the values obtained from different diagnostics, which are generally in reasonable good agreement within the uncertainties (see e. g. [15], [37], [19]).",
"In some cases, particularly in relatively high-density PNe ($n_e>10^4$ cm$^{-3}$ ), density stratification can be observed, with the [Ar iv] densities being larger than those computed with the other three diagnostics (see [38]).",
"In such cases it is better to consider also a two-zone density model (see [9]).",
"In some extreme cases of extremely young and dense PNe, with densities higher than the critical densities of the upper levels of the transitions producing the [Ar iv] lines, all the classical electron density diagnostics will be saturated and can provide inaccurate densities.",
"An alternative density indicator is based on the analysis of [Fe iii] emission lines, which are robust density diagnostics when collisional de-excitation dominates over collisional excitation.",
"Indeed, if inappropriate density diagnostics are used, then physical conditions deduced from commonly used line ratios will be in error, leading to unreliable chemical abundances for these objects.",
"(see [39]).",
"The first public code for the computation of physical conditions and ionic abundances was fivel [40], an interactive FORTRAN program which used a basic five-level atom approximation, which considers that only the five low-lying levels (i. e. at energies $\\le $ 5 eV above the ground state are physically relevant for computing the observed emission line spectrum.",
"Later, [41] developed nebular, a set of software tools (based in the fivel program, but extending it to an N-level atom) in the iraf/stsdasiraf is distributed by National Optical Astronomy Observatories, which is operated by AURA (Association of Universities for Research in Astronomy), under cooperative agreement with NSF (National Science Foundation).",
"environment that allow the user to compute diagnostic for a variety of ground-state electron configurations, and compute ionic abundances separately for up to 3 zones of ionization.",
"The main advantage of nebular is that it can be scripted.",
"However, changes of atomic data sets is not trivial and computations of elemental abundances are not included.",
"([42]) developed the Nebular Empirical Analysis Tool (neathttps://www.nebulousresearch.org/codes/neat/), a very simple to use code written in FORTRAN90 which requires little or no user input to return robust results, trying to provide abundance determinations as objective as possible.",
"One of the main advantages of this code is that it can evaluate uncertainties of the computed physical conditions and abundances by using a Monte Carlo approach.",
"Another advantage of this code is that it also accounts for the effect of upward biasing on measurements of lines with low signal-to-noise ratios, allowing to reduce uncertainties of abundance determinations based on these lines.",
"Finally, as atomic data for heavier elements than helium are stored externally in plain text files, the user can easily change the atomic data.",
"The last package to be offered in the field has been PyNebhttps://github.com/Morisset/PyNeb_devel ([43]) which is completely written in python and is designed to be easily scripted, and is more flexible and therefore, powerful than its predecessors.",
"This package allow the user to easily change and update atomic data as well as providing tools to plot and compare atomic data from different publications."
],
[
"Abundances in distant photoionized nebulae: the strong line methods",
"In the absence of reliable plasma diagnostics (a common fact in extragalactic objects) in giant H ii regions or integrated spectra of galaxies, one needs to use alternative methods to derive accurate chemical abundances.",
"This is especially important to estimate the metallicities of giant extragalactic H ii regions as well as of local and high-redshift emission-line galaxies and hence, it has a relevant influence on the study of the chemical evolution of the Universe.",
"The first mention of the strong-line methods was 40 years ago, when [44] and [45] proposed a method to compute the oxygen abundance using strong lines only: the $R_{23}$ method, in which oxygen abundance is a one dimensional function of the $R_{23}$ parameter, defined as: $R_{23}=\\frac{[{\\rm O II}]\\lambda 3727+[{\\rm O III}]\\lambda 4959+5007}{H\\beta }$ This method was calibrated using the few relevant photoionization models available at that time.",
"The problem with dealing with $R_{23}$ is that it is double valued with respect to metallicity.",
"In fact, at low oxygen abundances –12+log(O/H) $\\lesssim $ 8.0– the $R_{23}$ index increases with the abundance, while for high oxygen abundances –12 + log(O/H) $\\ge $ 8.25– the efficiency of the cooling caused by metals make $R_{23}$ to drop with rising abundance.",
"There is also a transition zone between 8.0 and 8.25 (see e. g. [46] for a detailed description of the high and low metallicity branches).",
"This method has been refined multiple times since then and several calibrations, using data sets with abundances from the direct method (e. g. [46], [47]), using photoionization model grids (e. g. [48], [49]), or a combination of both, are now available in the literature.",
"An overview of the most popular calibrations of strong-line methods can be found in [50].",
"A comprehensive critical evaluation of the different semi-empirical strong-line methods has been done by [51] who also develop a method for reducing systematics in the techniques to compute chemical abundances by using electron temperatures and ionization correction factors.",
"In the last years, mainly thanks to the increasingly easy access to super-computing resources, new approaches have been proposed.",
"Bayesian methods have been used by several authors to determine chemical abundances in extragalactic targets (e. g. [52]) although the priors should be selected cautiously to avoid unreliable results.",
"On the other hand, as in most of astronomy fields, machine learning techniques are also being used to infer chemical abundances (see e. g. [53]).",
"However, as has been pointed out by [54], making use of an illustrative example, the use of these techniques ignoring the underlying physics can lead to unphysical inferences.",
"[54] have argued in a comprehensive review that although strong-line methods are routinely used to estimate metallicities owing to their apparent simplicity, the users need to have a solid background on the physics of H ii regions to understand the approximations made on the different approaches, and the limitations each calibration has, to avoid biases, misinterpretations and mistakes.",
"Even taking into account the drawbacks mentioned above, strong-line methods have been widely used for studying giant H ii regions and emission line galaxies in large long-slit spectroscopic surveys as the Sloan Digital Sky Survey (SDSS) [55], or 2D spectroscopic surveys as MANGA (e. g. [56], [57]), CALIFA (e. g. [58]), and AMUSING (e. g. [59])."
],
[
"Advances in abundances determinations in photoionized nebulae",
"In this section I will focus on the latest advances that have been reached in the field of photoionized nebulae.",
"I will pay special attention to atomic data, ionization correction factors and the abundance discrepancy problem, that have been traditionally claimed as potential sources of uncertainty in chemical abundance determinations."
],
[
"Atomic data",
"The atomic data used for computing abundances in photoionized nebulae are ususally considered as a black box by the users.",
"Most users consider the default atomic data sets used by their favourite analysis tools or directly use the last available atomic data in the literature for each ion.",
"In the last years large compilations of atomic data have been done in the chiantihttp://www.chiantidatabase.org and nisthttp://physics.nist.gov databases, although the available atomic data in each database do not always match for a given ion.",
"[60], [61] discussed how to ensure that atomic data are correctly understood and used, as well as on the typical uncertainties in atomic data.",
"High-quality observations of photoionized nebulae are a powerful tool to check the reliability of atomic data.",
"[62] and [38] found, using a large data set of PNe spectra and comparing electron density estimates for PNe based on different density diagnostics, that the [O ii] transition probabilities calculated by [63] yielded systematically lower electron densities than those computed using the [S ii] diagnostic, and that such discrepancies were caused by errors in the computed transition probabilities.",
"Moreover, [38] found that the transition probabilities of [64] and the collision strengths of [65] were completely inconsistent with observations at the high and low density limits, respectively, and should be ruled out.",
"[66] determined chemical abundances of O, N, S, Ne, Cl and Ar for a sample of PNe and H ii regions and evaluated the impact of using different sets of atomic data on the computed physical conditions and abundances.",
"These authors used all the possible combinations of 52 different sets of transition probabilities and collision strengths to calculate physical conditions and chemical abundances, finding that different combinations of atomic data introduce differences in the derived abundances that can reach or surpass 0.6-0.8 dex at higher densities ($n_e>10^{-4}$ cm$^{-3}$ in several abundance ratios like O/H and N/O.",
"Removing the data sets that introduce the largest differences can reduce the total uncertainties, although they can still remain in high-density objects.",
"Additionally, they have pointed out that special attention should be paid to the transition probabilities of the S$^+$ , O$^+$ , Cl$^{++}$ and Ar$^{3+}$ density diagnostic lines, and to the collision strengths of Ar$^{3+}$ which, if incorrectly selected, can lead to unreliable chemical abundances in high-density nebulae.",
"Finally, [54] has pointed out that the role of atomic data in strong-line method calibrations cannot be ignored.",
"Recent changes in routinely used atomic data have revealed that they play a crucial role in direct abundance determinations and in photoionization models."
],
[
"Ionization correction factors",
"The elemental abundance of a particular element is computed by adding up the ionic abundances of all the ions present in a nebula.",
"However, it is usually found that not all the ions of a given element are observed, whether because they are emitted in a different spectral range than that observed or because the spectra is not deep enough to detect them.",
"Therefore, the contribution of these unobserved ions should be estimated in someway.",
"With this aim, the use of Ionization Correction Factors (ICFs) was proposed by [67].",
"These authors proposed to use similarities between ionization potentials of different ions to construct ICFs.",
"This approach has been used by several authors since then (see e. g. [68]).",
"However, [12] argued that these ICFs should be treated with caution because the ionization structure in a photoionized nebula does not depend only on the ionization potential.",
"Moreover, it has been shown that using recent photoionization models, these simple expressions are not always valid and new ICFs are needed to obtain more reliable abundances (see e. g. [69]) The alternative is to compute ICFs using photoionization models, where the physics involved in ionized nebulae is treated with much more detail.",
"Photoionization models allow to compute the detailed ionization structure of the various elements present in a nebula, by taking into account all the processes that govern ionization and recombination (i.e.",
"mostly photoionization, radiative and di-electronic recombination, and charge exchange), as well as all the heating and cooling processes that determine the electron temperature ([70]).",
"Traditionally, different ICFs have been computed for H ii regions and PNe, given the differences in the hardness of the radiation field and the different ionic species detected in each type of object (see Section ).",
"Several authors have derived ICFs from photoionization models for H ii regions ([71], [72], [73], [74], [75], [76], [77]) and for PNe ([78], [79], [70]).",
"It is not the scope of this text to show the details of the different approaches used to compute ICFs from photoionization models, but I think it is worth mentioning some of the most widely used ICF schemes.",
"[75] re-evaluated empirical expressions for the abundance determination of N, O, Ne, S, Cl, Ar and Fe to compute abundances of emission-line galaxies from the Data Release 3 of the Sloan Digital Sky Survey (SDSS).",
"They took special care in the selection of atomic data and constructed an appropriate grid of photoionization models with state-of-the art model atmospheres.",
"In particular, these authors take care of a problem that should not be ignored in the computation of photoionization models, which is the uncertain rate of the dielectronic recombination for sulfur, chlorine and argon ions.",
"They compared the abundances of these elements calculated with different assumed dielectronic recombination rates and could put some constraints on these rates.",
"Additionally, following an approach that was defined by [80] these authors proposed different ICFs depending on the metallicity range of the nebulae.",
"Regarding PNe, [70] constructed ICFs for He, N, O, C, Ne, S, Cl, and Ar using a large grid of photoionization models that are representative of most of the observed PNe.",
"Besides the obvious advantage of covering a wide range of physical parameters with a large photoionization model grid, the main advantage of this work is the provision of analytical expressions to estimate the uncertainties arising from their computed ICFs.",
"Finally, a third scheme to compute ICFs is to derive analytical expressions obtained from observational fittings to large sets of high-quality data (see e. g. [81] for Cl, and [82] for C).",
"In section REF I will come back to the ICFs mentioning some works devoted to the computation of ICFs for neutron-capture elements in PNe."
],
[
"Oxygen enrichment in PNe",
"Oxygen is the element for which more reliable abundances can be obtained and, therefore, it has been traditionally used as a proxy for metallicity.",
"In H ii regions, oxygen reflects the current abundance in the ISM, while in PNe, it is supposed to reflect the chemical composition of the environment where the star was born because its abundance remain unchanged during the life of the star ([83]).",
"However, AGB stars can modify the oxygen abundance by two mechanisms: the third dredge-up (TDU) and the hot bottom burning (HBB), although only nucleosynthesis models which include extra-mixing processes like diffusive convective overshooting (e. g. [84], [85], [86]) predict a significant production of oxygen.",
"Figure: Values of O/Cl as a function of O/H for a sample of Galactic PNe and H ii regions (see ).",
"The red circles represent PNe with oxygen-rich dust, the green diamonds PNe with carbon-rich dust, and the blue stars the H ii regions.",
"The protosolar abundances of are overplotted with the solar symbol.",
"Plot made with data gently provided by Gloria Delgado-Inglada.Until recently, the only observational probes of oxygen production in AGB stars have been restricted to low-metallicity PNe (see e. g. [88]).",
"However, using deep, high-quality optical spectra (with spectral resolution better than 1 Å) [83] recomputed accurate abundances of He, O, N, Ne, C, Ar, and Cl in 20 PNe and 7 H ii regions in our Galaxy at near-solar metallicities.",
"These authors found that all but one of the Galactic PNe with C-rich dust (the one with the highest metallicity according to Cl/H) show higher O/Cl values than the PNe with O-rich dust and the H ii regions (see Fig.",
"REF ), and interpret that result as O is enriched in C-rich PNe due to an efficient third dredge-up in their progenitor stars.",
"These results have been confirmed later by nucleosynthesis models including convective overshooting by [85], [89].",
"These findings confirm that oxygen is not always a good proxy of the original ISM metallicity and other chemical elements such as chlorine or argon, the abundance of which is unaltered in the evolution of low- and intermediate-mass stars, should be used instead.",
"Additionally, as has been pointed out by [89], the production of oxygen by low-mass stars should be thus considered in galactic-evolution models."
],
[
"Abundance gradients in the Milky Way and in nearby spiral galaxies from direct abundance determinations",
"The study of the radial distribution of the gas phase metallicity in a Galaxy (usually using oxygen as a proxy for the metallicity) is fundamental for our understanding of the evolution of Galaxies.",
"The pioneering studies on the gradient of abundances in spiral galaxies were those of [90] and [91], which were based on the spectral differences found by [92] between the H ii regions in the spiral galaxy M33.",
"[93] were the first in carrying out an homogeneous study of abundance gradients in the Milky Way (hereinafter, MW) with a relatively large sample of H ii regions.",
"However, these authors rely on electron temperatures determined from radio RLs and abundances from optical lines, obtaining a relatively large scatter at a given Galactocentric distance.",
"Since these pioneering works, several authors have computed radial abundance gradients using the direct method in our Galaxy (e. g. [94], [95], [29], [96], [97]) and in external galaxies (see e. g. [98], [99], [100], [101], [25] and references therein).",
"Regarding the MW, the relatively large scatter at a given Galactocentric distance found in several works has been claimed as a possible indication that the gas is not as well mixed as commonly thought (see e. g. [29]).",
"Moreover, [102] found significant differences in the radial gradient of O in the MW depending on the Galactic azimuth region considered, strengthening the idea that metals are not well mixed at a given radius.",
"However, [32] made an homogeneous analysis using a set of deep optical spectra of 35 H ii regions, from which they computed accurate physical conditions and ionic and elemental abundances, finding that the scatter of the N and O abundances of H ii regions is of the order of the observational uncertainties, indicating that both chemical elements seem to be well mixed in the ISM at a given Galactocentric distance (see the comparison between radial O abundance from this work and that of [29] in Fig.",
"REF ).",
"In the extragalactic domain, it is worth mentioning the existence of the CHemical Abundances of Spirals (CHAOS) project, which is devoted to surveying several spiral galaxies to determine precise “direct” abundances in large samples of H ii regions in spiral galaxies (see [101], [100], [103]).",
"This project has increased by more than an order of magnitude the number of H ii regions with direct measurements of the chemical abundances in nearby disk galaxies (see [101]).",
"There are many open problems with the abundance gradients of the MW and nearby spiral Galaxies such as a possible temporal evolution ([104], [105]) based on the differences found in the gradients using different populations of PNe and H ii regions; the existence or not of a flattening of the gradient in the outer disc of spiral galaxies, including the MW ([106], [96], [59]) or in the inner disc ([96], [59]); distance determinations uncertainties, particularly for PNe (see [105]) or, as mentioned in section REF , the applicability of oxygen as a reliable element to trace the metallicity in PNe ([83]).",
"Some of the limitations that, in my opinion should be taken into account have been summarized in [107].",
"The determination of precise radial metallicity gradients are precious constraints for chemical evolution models of the MW in particular and of spiral galaxies in general.",
"The presence of a negative gradient agrees with the the stellar mass growth of galaxies being inside-out (see e. g. [108]).",
"However, additional information, such as the possible temporal evolution of the gradients (see e. g. [104], [105]) can give information about physical processes that can modify gradients, and that should be considered by chemical evolution models (see discussion by [105] and references therein).",
"As H ii regions reflect the young stellar populations and, on the other hand PNe, reflect older stellar populations (with a relatively large spread in ages) a careful comparison between gradients obtained from different objects is very useful to constrain the temporal evolution of the gradient predicted by chemical evolution models ([109])."
],
[
"The abundance discrepancy problem",
"The abundance discrepancy problem is one of the major unresolved problems in nebular astrophysics and it has been around for more than seventy years ([110]).",
"It consists in the fact that in photoionized nebulae $-$ both H ii regions and PNe$-$ ORLs provide abundance values that are systematically larger than those obtained using CELs.",
"Solving this problem has obvious implications for the measurement of the chemical content of nearby and distant galaxies, because this task is most often done using CELs from their ionized ISM.",
"For a given ion, the abundance discrepancy factor (ADF) is defined as the ratio between the abundances obtained from ORLs and CELs, i. e., $ADF(X^{i+}) = (X^{i+}/H^+)_{ORLs}/(X^{i+}/H^+)_{CELs},$ and is usually between 1.5 and 3, with a mean value of about 2.0 in H ii regions and the bulk of PNe (see e.g.",
"[111], [27], but in PNe it has a significant tail extending to much larger values, up to 2–3 orders of magnitudeAn updated record of the distribution of values of the ADF in both H ii regions and PNe can be found in Roger Wesson's webpage: http://nebulousresearch.org/adfs.",
"It is important to remark that the ADF is most easily determined for O$^{2+}$ owing to both CELs and RLs are straightforward to detect in the optical.",
"ADFs can be also determined for other ions, such as C$^{2+}$ , N$^{2+}$ , and Ne$^{2+}$ , although the obtained values are more uncertain because CELs and RLs are detected in different wavelength ranges (in the case of C$^{2+}$ and N$^{2+}$ ) or because RLs are intrinsically very faint (in the case of Ne$^{2+}$ ).",
"The possible origin of this discrepancy has been discussed for many years and three main scenarios have been proposed: [112] was the first proposing the presence of temperature fluctuations in the gas to explain the discrepancy between $T_e$ ([O iii]) and $T_e$ (H I) derived from the Balmer jump.",
"After that seminal work, [67] developed a scheme to correct the abundances computed from CELs for the presence of temperature inhomogeneities.",
"Later, [113] suggested that the discrepancy between ORLs and CELs abundances could be explained if spatial temperature variations over the observed volume were considered.",
"[10] have recently summarized the mechanisms proposed to explain and maintain the presence of temperature fluctuations in photoionized nebulae.",
"[114] were the first in proposing the existence of chemical inhomogeneities in the gas as a plausible mechanism to explain the abundance discrepancy.",
"This scenario was later expanded by [14], who claimed that metal-rich (i. e. H-poor) inclusions could be the clue to resolve the abundance discrepancy problem; in this scenario, metal ORLs would be emitted in the metal-rich inclusions, where cooling has been enhanced, while CELs would be emitted in the “normal” metallicity (H-rich) zones.",
"This model was tested by several authors by constructing two-phase photoionization models that, in several cases, successfully simultaneously reproduced the ORLs and CELs emissions in H ii regions [115] and PNe [116].",
"However, at the present time, the origin of such metal-rich inclusions remains elusive, although some scenarios have been proposed for PNe ([117]) and H ii regions ([118]).",
"In the last years, increasing evidence has been found of a link between the presence of a close central binary star at the heart of PNe and very high ($>$ 10) ADFs (see Section REF ).",
"A third scenario was brought into play by [119], who proposed that the departure of the free electron energy distribution from the Maxwellian distribution ($\\kappa $ -distribution) could explain the abundance discrepancies owing to the presence of a long tail of supra-thermal electrons that contribute to an increase in the intensity of the CELs at a given value of the kinetic temperature.",
"However, in the last years little theoretical ([120], [121], [122]) or observational ([123]) support has been presented for this scenario in photoionized nebulae.",
"[121] have shown that the heating or cooling timescales are much longer than the timescale needed to thermalize supra-thermal electrons because they can only travel over distances that are much shorter than the distances over which heating rates change, implying that the electron velocity distribution will be close to a Maxwellian one long before the supra-thermal electrons can affect the emission of CELs and RLs.",
"Moreover, [122] demonstrated analytically that the electron energy distribution relaxes rapidly to a steady-state distribution that is very close to a Maxwellian, having negligible effects on line ratios.",
"One of the most active groups in the study of the abundance discrepancy in PNe in the last two decades has been the University College London/U.",
"Beijing group, who have developed deep medium-resolution spectrophotometry of dozens of PNe to compute the physical and chemical properties of these objects from ORLs.",
"In one of the most detailed and comprehensive studies of this group, [124] showed that the values of the ADF deduced for the four most abundant second-row heavy elements (C, N, O and Ne) are comparable (see their Fig. 18).",
"However, they also computed abundances from ORLs from a third-row element (Mg) and they found that no enhancement of ORL abundances relative to CEL ones is obvious for Mg: the average Mg abundances from ORLs for disk PNe remained in a range compatible to the solar photospheric value, even taking into account the small depletion expected for this element onto dust grains (less than 30%).",
"Finally, these authors also showed that, regardless of the value of the ADF, both CEL and ORL abundances yield similar relative abundance ratios of heavy elements such as C/O, N/O and Ne/O .",
"This has important implications, especially in the case of the C/O ratio, given the difficulties of obtaining this ratio from UV CELs (see Sect.",
"REF ).",
"Several authors have strongly argued in favour of the inhomogeneous composition of PNe and against pure temperature fluctuations (see e. g. [125] and references therein); some of the reasoning that has been presented supporting this model are: i) far-IR [O iii] CELs, which in principle, have a much lower dependence on electron temperature than optical CELs, provide abundances that are consistent with those derived from optical CELs (see e. g. [14]); ii) the analysis of the physical conditions using H, He, O and N ORLs yields electron temperatures that are much lower than those computed from classical CEL diagnostic ratios (see [126], [127], [128], [20]); additionally, ORL density diagnostics provide densities that are higher than those derived from CEL diagnostics; iii) chemically homogeneous photoionization models do not reproduce the required temperature fluctuations to match CEL and ORL abundances, while bi-abundance photoionization models including an H-poor (i. e. metal-rich) component of the gas successfully reproduce the observed intensities of both CELs and ORLs (e. g. [116]).",
"All these arguments strongly favour the presence of a low-mass component of the gas that is much colder and denser than the “normal” gas, and that is responsible for the bulk of the ORL emission.",
"However, we cannot rule out the possibility that different physical phenomena can contribute simultaneously to the abundance discrepancy in PNe.",
"Some physical phenomena have been proposed to explain the abundance discrepancy in the framework of temperature fluctuations or chemical inhomogeneities scenarios ([117], [10]).",
"However, until very recently, there was no observational proof that demonstrated a single physical process to be responsible for the abundance discrepancy.",
"Some recent works on the Orion nebula have observationally linked the abundance discrepancy to the presence of high velocity flows ([129]) or to the presence of high density clumps, such as proto-planetary disks ([130], [39].",
"On the other hand, [125] found a very extreme value of the ADF for the PN Hf 2–2 (ADF$\\sim $ 70) and, for the first time, suggested the possibility that this large ADF could be related to the fact that the central star of the PN, which is a close-binary star, has gone through a common-envelope phase."
],
[
"The link between close binary central stars and large abundance discrepancy factors",
"Several papers in recent years have confirmed the hypothesis proposed by [125] that the largest abundance discrepancies are reached in PNe with close-binary central stars.",
"[131] found that three PNe with known close-binary central stars showed high ADFs, with the PN Abell 46, with an ADF(O$^{2+}$ )$\\sim $ 120, and as high as 300 in its inner regions, being the most extreme object.",
"Their spectroscopic analysis supports the previous interpretation that, in addition to “standard” hot (T$_e$$\\sim $ 10$^4$ K) gas, a colder (T$_e$$\\sim $ 10$^3$ K), metal-rich, ionized component also exists in these nebulae.",
"Both the origin of the metal-rich component and how the two gas phases are mixed in the nebulae are basically unknown.",
"Moreover, this dual nature is not predicted by mass-loss theories.",
"However, it seems clear that the large-ADF phenomena in PNe is linked to the presence of a close-binary central star.",
"In fact, [23] have recently completed a survey of the ADFs in seven PNe with known close-binary central stars and they found ADFs larger than 10 for all of them, confirming the strong link between large ADFs and close-binary central stars.",
"On the other hand, several spectroscopic studies have shown that the ORL emitting plasma is generally concentrated in the central parts of the PNe.",
"This occurs in PNe with known close-binary central stars and large ADFs (e. g. [131], [132]), in PNe with low-to-moderate ADFs and no indication of binarity (e. g. [133], [134]) and in PNe with relatively large ADFs but no known close-binary central star (e. g. M 1–42, see [135]).",
"Figure: MUSE Emission line maps of several lines in the PN NGC 6778, ordered by ionization potential of the parent ion: from left to right and from top to bottm: [O i] λ\\lambda 6300 Å, [S ii] λ\\lambda 6731 Å, Hβ\\beta λ\\lambda 4861 Å, [N ii] λ\\lambda 6548 Å, He i λ\\lambda 5876 Å, O ii λλ\\lambda \\lambda 4649+50 ÅORLs, [O iii] λ\\lambda 4959 ÅCEL, [Ar iv] λ\\lambda 4740 Åand He ii λ\\lambda 4686 Å.",
"It is worth re-emphasising that the O ii and the [O iii] emission comes from the same ion: O 2+ ^{2+} .",
"The “x” marks a reference spaxel[136] recently obtained the first direct image of the PN NGC 6778 (a PN with ADF$\\sim $ 20) in O ii recombination lines, taking advantage of the tunable filters available at the OSIRIS instrument in the 10.4m Gran Telescopio Canarias (GTC).",
"They found that in NGC 6778, the spatial distribution of the O ii $\\lambda \\lambda $ 4649+50 ORL emission does not match that of the [O iii] $\\lambda $ 5007 CEL.",
"[135] found the same behaviour in Abell 46 using direct tunable filter images centred at $\\lambda \\lambda $ 4649+51 Å.",
"Moreover, [135] presented preliminary results obtained from deep 2D spectroscopic observations with MUSE at the 8.2m Very Large Telescope (VLT) of five southern large-ADF PNe, and they confirmed this behaviour in at least the PNe Hf 2-2 (ADF$\\sim $ 84), M 1-42 and NGC 6778 (both with ADF$\\sim $ 20).",
"In Fig REF we show the MUSE emission line maps of several emission lines in the PN NGC 6778.",
"The emission maps are ordered by increasing ionization potential of the parent ion from left to right and from top to bottom.",
"It is clear that O ii $\\lambda $ 4649+50 ORLs emission is more centrally concentrated that [O iii] $\\lambda $ 4959 CEL emission, and seems to be emitted in a zone that correspond to a higher ionization specie.",
"A similar result has been found by [137] from a kinematical analysis of several heavy metal ORLs and CELs in NGC 7009.",
"These authors found that the kinematics of ORLs and CELs were discrepant and incompatible with the ionization structure of the nebula, unless there is an additional plasma component to the CEL emission that arises from a different volume from that giving rise to the RL emission from the parent ions within NGC 7009.",
"Similarly, [138] found that the kinematics of the C ii $\\lambda $ 6578 line is not what expected if this line arises from the recombination of C$^{2+}$ ions or the fluorescence of C$^+$ ions in ionization equilibrium in a chemically homogeneous nebular plasma, but instead its kinematics are those appropriate for a volume more internal than expected.",
"These results clearly support the hypothesis of the existence of two separate plasmas, at least in these large-ADF PNe, with the additional indication that they are not well mixed, perhaps because they were produced in distinct ejection events related to the binary nature of the PN central star.",
"[23] propose that a nova-like outburst from the close-binary central star could be responsible for ejecting H-deficient material into the nebulae soon after the formation of the main nebula.",
"This material would be depleted in H, and enhanced in C,N, O, and Ne, but not in third row elements.",
"It is worth mentioning the similarity of these plasma component with some well-known old nova shells as CP Pup and DQ Her that show low $T_e$ and strong ORLs ([139], [140])."
],
[
"The C/O ratio from recombination lines",
"The determination of accurate C/H and C/O ratios in H ii regions is of paramount importance to constrain chemical evolution models of galaxies ([141]).",
"Furthermore, C is an important source of opacity in stars and one of the main elements found in interstellar dust and organic molecules, making it one of the biogenic elements ([142]).",
"Despite its importance, C abundances in H ii regions have been poorly explored as they are traditionally derived from UV observations of the semi-forbidden C iii] $\\lambda $ 1909 and C ii] $\\lambda $ 2326 CELs from space ([143], [82]).",
"For PNe the situation is better, owing to several successful $IUE$ and $HST$ programs that have provided reliable measurements of these lines in dozens of objects (see e. g. [144], [145], [146], [147], [148] and references therein).",
"However, the determination of reliable fluxes from these lines is difficult as they are severely affected by interstellar reddening.",
"Moreover, the emissivities of these lines are also very dependent on the electron temperature.",
"Finally, aperture effects owing to the different areas covered by UV and optical observations must be taken into account to guarantee the observation of the same volume of the nebula in both ranges.",
"Alternatively, scanning techniques have been used to match UV International Ultraviolet Explorer (IUE) observations and optical spectroscopy (see e. g. [14], [133]).",
"The determination of C and O abundances from ORLs skips these difficulties and takes advantage of the fact that using the same type of lines (CELs or ORLs) the C/O ratios remain the same in most of the objects were both types of ratio have been computed ([124], [149].",
"Thanks to the new CCDs with improved efficiency in the blue and the use of large telescopes, several high-quality observations of the C ii $\\lambda $ 4267 ORL in PNe have been achieved in the last years (e. g. [150], [144], [19], [151], [124], [8], [20], [24]."
],
[
"C/O ratios in H ",
"[95] computed for the first time the C/H and C/O radial gradients of the ionized gas from ORLs in the MW.",
"Later, [152] derived these gradients in M101, and [25] in M33 and NGC 300.",
"The general conclusion of these works is that C abundance gradients are always steeper than those of O, producing negative C/O gradients across the galactic disks which reflect the non-primary behavior of C enrichment.",
"Furthermore, the comparison between the C/H and C/O gradients obtained in the MW with state-of-the-art chemical evolution models revealed that the obtained C gradients can only been explained if the C produced by massive and low-intermediate mass stars depends strongly on time and on the Galactocentric distance ([141]).",
"From the C/O ratios computed from ORLs in several low-metallicity star-forming galaxies compiled by [153], [142] it has been found that H ii regions in star-forming dwarf galaxies have different chemical evolution histories than the inner discs of spiral galaxies and that the bulk of C in the most metal-poor extragalactic H ii regions should have the same origin as in halo stars (see Figs.",
"8 and 9 of [142])."
],
[
"C/O ratios in PNe",
"As was pointed out above, there are multiple determinations of C abundance from UV lines in the literature.",
"These determinations, when considered together with detailed analysis of optical spectra covering the same volume of the nebula, provide strong constraints to nucleosynthesis models (see e. g. [147]).",
"However, owing to limitations in space, in this text I am going to focus on the determination of C/O ratios from ORLs.",
"Thanks to the new CCDs with improved efficiency in the blue and the use of large telescopes, several high-quality observations of the C ii $\\lambda $ 4267 RL in PNe have been achieved in the last years (e. g. [150], [144], [19], [151], [124], [8], [20], [24].",
"C/O ratios derived from ORLs combined with other abundance ratios such as N/O or He/H, can set strong constraints to the initial mass of PNe progenitors.",
"This is because different processes occurring at the interior of AGB stars (third dredge-up episodes, hot bottom burning process) activate at different masses and can strongly modify C/O and N/O ratios (see e. g. [154] and references therein).",
"These ratios can also provide strong constraints to the physics assumed by nucleosynthesis models, i. e. the assumption of convective overshooting into the core during the main sequence and the He-burning phases can diminish the mass limit at which He-flashes can occur at a given metallicity ([89]) compared to those models not considering it ([154]); this has implications for the mass limit at which hot bottom burning can be activated.",
"Additionally, C/O ratios can be used to obtain information about the efficiency of dust formation in C-rich or O-rich environments (see below) and to learn about different dust-formation mechanisms (see [85] and [24])."
],
[
"Abundances of heavy elements in PNe from faint emission lines",
"As I have emphasized several times in this chapter, the increasing efficiency of astronomical detectors as well as the advent of large (8m-10m type) telescopes have boosted the detection of very faint emission lines in photoionized regions.",
"In this section, I will focus on the detection of extremely faint lines in deep spectra of PNe."
],
[
"Faint emission lines of refractory elements",
"The determination of elemental abundances of refractory elements in the spectra of Galactic PNe is not an easy task because the available lines of these elements (mainly iron and nickel) are relatively faint.",
"Additionally, these elements are constituents of dust grains in the ISM and their abundance in photoionized nebulae do not reflect their actual abundances.",
"Several authors have computed abundances of Fe ions in both PNe and H ii regions ([155], [156], [157], [158]) [157] computed detailed Fe abundances for a sample of 28 PNe and found that more than 90% of Fe atoms are condensed on dust grains.",
"These authors did not find differences between the iron abundances in C-rich and O-rich PNe, suggesting similar depletion efficiencies in both environments.",
"[149] combined C/O ratios derived from both UV CELs and optical ORLs (as we comment in Sect.",
"REF , they seem to be equivalent) with information obtained from Spitzer mid-infrared spectra.",
"They also computed Fe depletions onto dust grains, and found that the highest depletion factors are found in C-rich objects with SiC or the 30 $\\mu $ m feature in their infrared spectra, while the lowest depletion factors were found for some of the O-rich objects showing silicates in their infrared spectra.",
"[159] compiled detections of very faint [Ni ii] and [Ni iii] lines in deep spectra of Galactic PNe and H ii regions.",
"They determined the nickel abundance from the [Ni iii] lines using an extensive grid of photoionization models to determine a reliable ionization correction factor (ICF).",
"From the comparison of Fe/Ni ratios with the depletion factor obtained from both [Fe/H] and [Ni/H], they conclude that nickel atoms adhere to dust grains more efficiently than iron atoms in environments where dust formation or growth is more important."
],
[
"Neutron-capture elements",
"Nebular spectroscopy of neutron(n)-capture elements (atomic number Z $>$ 30) is a recent field that has seen rapid development in the last 10 years, and holds promise to significantly advance our understanding of AGB n-capture nucleosynthesis.",
"Nebular spectroscopy can reveal unique and complementary information to stellar spectroscopy.",
"Observations of PNe provide the first opportunity to study the production of the lightest n-capture elements (Z $\\le $ 36) and noble gases (Kr and Xe) in one of their sites of origin.",
"Unlike the case of AGB stars, nucleosynthesis and convective dredge-up are complete in PNe, whose envelopes contain material from the last 2$-$ 3 thermal pulses.",
"Accurate computations of n-capture elements would shed light on the different scenarios proposed for the production of these elements and would constrain the chemical yields of low- and intermediate-mass stars for these elements.",
"n-capture elements were not recognized in any astrophysical nebula until [160] identified emission lines of Br, Kr, Rb, Xe, Ba, and possibly other heavy species in the bright PN NGC 7027.",
"Since then, a breathtaking number of n-capture element emission lines have been identified for the first time in near-infrared ([161], [162], [163], [164], [165]), UV ([166]) and optical ([151], [21], [22]) spectra of PNe.",
"The new detections have led to a dedicated effort to produce atomic data needed for abundance determinations (e. g., see [167], [163], [165], and references therein).",
"The new photoionization cross-sections and recombination coefficients have been incorporated in photoionization calculations to compute reliable ICFs (e. g., [168]).",
"The new collisional strengths have been used for abundance determinations of newly detected ions (see [163], for [Rb iv], [164], for [Se iii] and [Kr vi], and [165] for [Te iii] and [Br v]).",
"Thanks to the fast advances in observations, atomic data determinations and numerical modelling, this field has grown from just 3 PNe with n-capture element abundances in 2001 to more than 100 Galactic PNe in 2019 ([151], [162], [163], [8], [21], [169], [164], [22], [165]).",
"The importance of deep, high-resolution optical spectrophotometry of PNe to detect faint n-capture elements can be understood when comparing the works by [151] and [21].",
"In the first case, several n-capture emission lines were discovered in the spectra of 5 PNe, but even at a resolution of $\\sim $ 22000, many features were not unambiguously detected.",
"[21] took advantage of the very high-resolution (R$\\sim $ 40000) spectrum of NGC 3918 to clearly identify several ions of Kr, Xe, Rb and Se, testing for the first time the complete set of ICFs for Kr created by [168].",
"The combination of deep optical and near-infrared spectra of PNe has been proved to be a powerful tool to test the predictions of modern AGB nucleosynthesis models as well as to test the accuracy of new atomic data computations and ICF prescriptions.",
"[22] have combined a deep optical spectrum and a near-infrared spectrum of NGC 5315, testing for the first time the complete set of ICFs for Se created by [168].",
"Finally, the determination of precise elemental abundances ratios of n-capture elements like Kr/Xe or Te/Se from optical and near-infrared spectra are potential indicators of the time integrated neutron flux in the intershell region between the H- and He-burning shells, giving strong constraints to nucleosynthesis models in the thermally pulsing AGB phase [170]."
],
[
"Acknowledgements",
"First of all, I want to acknowledge support of the Erasmus+ programme of the European Union under grant number 2017-1-CZ01-KA203-035562.",
"I also acknowledge support from an Advanced Fellowship from the Severo Ochoa excellence program (SEV-2015-0548) and support from the State Research Agency (AEI) of the Spanish Ministry of Science, Innovation and Universities (MCIU) and the European Regional Development Fund (FEDER) under grant AYA2017-83383-P.",
"I am grateful to Dr. Gloria Delgado-Inglada, who kindly provided me the data and the script to produce Fig.",
"REF for this chapter.",
"I thank the referee of this chapter, Dr. Roger Wesson, for a detailed report that help to improve the scientific content of the chapter.",
"tocsectionAppendix"
]
] | 2001.03388 |
[
[
"Measuring Female Representation and Impact in Films over Time"
],
[
"Abstract Women have always been underrepresented in movies and not until recently has the representation of women in movies improved.",
"To investigate the improvement of female representation and its relationship with a movie's success, we propose a new measure, the female cast ratio, and compare it to the commonly used Bechdel test result.",
"We employ generalized linear regression with $L_1$ penalty and a Random Forest model to identify the predictors that influence female representation, and evaluate the relationship between female representation and a movie's success in three aspects: revenue/budget ratio, rating, and popularity.",
"Three important findings in our study have highlighted the difficulties women in the film industry face both upstream and downstream.",
"First, female filmmakers, especially female screenplay writers, are instrumental for movies to have better female representation, but the percentage of female filmmakers has been very low.",
"Second, movies that have the potential to tell insightful stories about women are often provided with lower budgets, and this usually causes the films to in turn receive more criticism.",
"Finally, the demand for better female representation from moviegoers has also not been strong enough to compel the film industry to change, as movies that have poor female representation can still be very popular and successful in the box office."
],
[
"Introduction",
"Film is a common entertainment form that fulfills the audience's desire to make emotional connections with characters and learn about their social world.",
"However, it was very rare for women to see inspiring counterparts on the big screen in the past.",
"Many key roles in film-making, such as directors and cinematographers, were for many decades dominated almost entirely by men [8], and women did not have enough power to make demands in the film industry.",
"Consequently, women have been constantly underrepresented in movies.",
"Even when they are present, women are often portrayed in circumscribed and subordinated ways in traditionally feminine (i.e., stereotyped) roles, such as nonprofessionals, homemakers, wives or parents, and sexual gatekeepers [3].",
"Lacking a role model on the big screen is detrimental for young girls.",
"They are discouraged from pursuing ambitions and participating actively in social affairs [4].",
"Therefore, female under-representation is a critical issue that must be addressed.",
"In more recent times, women have made inroads into various fields and films have started to respond to female viewers with strong and well-rounded female characters [12], [5].",
"There are many studies and projects devoted to studying evolving feminism in films, centering on both the upstream effects, in which content is structured through the actions of major filmmakers in gendered organizations who presume the public's preferences, as well as the downstream effects, where audiences respond to content and attitudes are formed and reinforced.",
"In our work of examining gender representation in films, we ask the following two questions: What factors in the film-making process have a significant impact on female representation in films, and do films that feature more female representation outperform those that do not commercially?",
"To answer these questions, we need to define a proper measure for female representation in films.",
"The Bechdel test is a popular measure for examining how well-rounded and complete the representations of women in media are.",
"The test asks only three questions: Are there at least two women in the film who have names?",
"Do those women talk to each other?",
"Do they talk to each other about something other than a man?",
"The test is simple and has been widely used in many film gender studies to measure female representation[9], [10].",
"However, one major criticism of the test is that it fails to reveal the hidden gender imbalance structure[11].",
"A movie can pass the Bechdel test yet still portray women as auxiliary characters with minimal screen time.",
"For example, both Wonder Woman and The Martian passed the test even though the importance of female characters is very different between the two movies.",
"Therefore, we propose a new measure of female representation in films: the percentage of female cast members in the whole cast.",
"This proposal is based on the intuition that if more women are cast for a film, then women will have more representation on the big screen.",
"In addition to a new measure of female representation, we would also like to consider different aspects of movie success.",
"Box office return is a popular measure of movie success because it is directly linked to profit.",
"However, movie ratings and popularity are also important measures of a movie's success.",
"Public acclamation and popularity brings confidence to the studios and producers to inspire similar films or sequels even when the box office return is not substantially high.",
"Our analyses are completed in two steps.",
"Drawing on a sample of the most widely distributed films, we first combine a content analysis using the Bechdel Test with film-making data such as the budget and gender of crews to examine what factors are influential on encouraging more female representation during the film-making process.",
"Next, including all film characteristics such as female representation, budget, genre, and many others, we further examine whether better female representation can increase the chance of success for films while adjusting for other possible confounding variables.",
"In doing so, we contribute to sociological theories about the reciprocal impact of feminism.",
"The remainder of the paper is organized as follows.",
"In section 2, we review the background literature that studies the gender gap in films and analyzes box office performance.",
"In section 3, we describe the data features and the data collection process.",
"In section 4, we present the methods and models used throughout this study.",
"We present our experiments and the corresponding results in Section 4.",
"In section 5, we present and discuss our conclusions and future directions."
],
[
"Related work",
"Most of the studies that center on gender inequality in media are content analysis [12], [5], [11], which means they study how women are portrayed in media.",
"While content analysis is beyond the scope of our study, the related works have shown that the roles of women are evolving on the big screen.",
"There are also many studies about female underrepresentation, albeit on a small scale and with simple analysis.",
"Thomas has shown that in the top 10 worldwide highest grossing films of 2016, women speak only 27% of the lines [15].",
"The finding is novel but the sample size is too small.",
"A team at the Rhodes Information Initiative at Duke University published a report that movies which pass the Bechdel Test have a statistically significant higher return on investment compared to films that do not [2].",
"The tech company Shift72 published a similar report which states that all films that pass the Bechdel Test surpassed the box office returns of films that fail this test https://shift7.com/media-research/#arrow-jump.",
"However, these reports only presented statistics of mere comparisons between movies that pass the test and movies that do not, which are too simple to conclude that “more women in the film means more success (movies are more profitable).\"",
"Such analyses failed to adjust for confounding sources such as the genres.",
"Linder et al.",
"have shown that after adjusting for confounding factors such as production budgets, movies with more female representation (passing the Bechdel test) tend to have smaller production budgets and consequently earn less money.",
"However, given the same budget, female representation does not boost the box office return [10].",
"They have conducted a similar analysis for movie critics using the same set of movies and obtained a similar conclusion that female representation has little effect on critics [9].",
"The analysis in these papers is the closest to ours.",
"However, this study only drew samples of 974 films from the 2000-2009 decade and considered only a few confounding variables such as the budget and genres.",
"Our analysis include a larger sample from a longer time span with a larger set of confounding variables to provide deeper insights over time."
],
[
"Data",
"Three data sources are used in this project: the Bechdel Test Movie List https://https://bechdeltest.com, the Internet Movie Database (IMDb) https://www.imdb.com/interfaces/, and the Movie Database (TMDb) https://developers.themoviedb.org.",
"The Bechdel Test Movie List contains over 8000 movies crowd-sourced through the Internet with a flag showing if each of them passes the Bechdel Test or not.",
"IMDb has an up-to-date movie dataset, which is widely used among many movie related projects.",
"However, since IMDb does not provide an official API, we have limited access to some data, such as the full list of the cast and crew members of each movie.",
"TMDb on the other hand is a crowd-sourced online movie database similar to IMDb.",
"It is used less often, but with the advantage of having an API, we can easily access a variety of movie-related data of interest to us."
],
[
"Data Acquisition",
"The Bechdel Test Movie List is downloaded using its API.",
"It contains 8190 movies at the time, with 7 fields.",
"The relevant features include imdbid (can be used to join with other datasets), title, year, and rating (0 means no two women, 1 means no talking, 2 means talking about a man, 3 means it passes the test).",
"The IMDb dataset is downloaded from its website, where only the title.basic table is used.",
"It contains 9 fields, including tconst (imdbid), titleType, primaryTitle, originalTitle, isAdult, startYear, endYear, runtimeMinutes, and genres.",
"Getting the TMDb dataset requires more work with its API.",
"The first step is to obtain the tmdbids that are associated with the imdbids within the Bechdel Movie List.",
"Then the APIs are called iteratively to access the detail and credits of each movie.",
"By rearranging the tables, we are finally able to get our desired features including: budget, revenue, vote average, vote count, popularity (a comprehensive measure calculated from release data, number of votes, number of views, etc.",
"from TMDb), production companies, number of cast members, and gender of the cast and crew members."
],
[
"Data Preprocessing",
"As mentioned from previous sections, we have three data sources: Bechdel Test Movie List, IMDB, and TMDB.",
"The previous two are well established as tabular forms and require minimal processing.",
"For the TMDb data, new features are created to better represent the movie measurables of interest.",
"First, the revenue-budget ratio is calculated through simple division to represent the profitability of each movie.",
"There are some exceptions: a few non-commercial movies have budgets of 0, so we removed these rare cases from our dataset.",
"Some movies have a long list of production companies, but only those at the top of the list provide major contributions, so we excise the list and set an upper limit of 5 production companies for each movie.",
"In addition, the movie credits features (number, gender, and title of cast and crew members) are aggregated to extract new features such as female cast ratio, number of core crew members (directors, producers and screenplays), and the female core crew ratios.",
"Finally, the Bechdel Movie List, IMDb, and TMDb datasets are joined according to the imdbid.",
"The categorical variables are then one-hot encoded and the numerical variables are min-max normalized in preparation for modeling.",
"The final table contains 4232 observations and 67 variables."
],
[
"Methodology",
"Our primary goal is to evaluate what factors have an effect on female representation in films and the impact of female representation on films' success.",
"Given this purpose, the primary method of our analysis is an explanatory model which focuses on modeling the true generation of sample data rather than making the best prediction.",
"Many believe that models with high explanatory power are inherently of high predictive power as well.",
"Conflation between explanation and prediction is common, however the type of uncertainty associated with explanation is of a different nature than that associated with prediction .",
"Therefore, we add a prediction model as the secondary task to compare the predictive performance with our primary model.",
"For our primary model, we consider Generalized Linear Regression with a penalty term for model selection.",
"Generalized linear regression is one of the most commonly used statistical models for explanatory modeling, and the addition of the penalty term puts constraints on regression coefficient estimation to keep only the relevant variables from the model.",
"It is ideal for our goal of explanatory modeling with variable selection.",
"For our secondary model, we consider Random Forest.",
"Random Forest is known for being robust in predictive modeling with feature ranking by importance.",
"We expect Random Forest to have very satisfactory performance as a predictive model for our data and to use its performance to compare with the predictive performance of our primary model.",
"In order to compare the coefficient estimates of different variables from the regression model, variables are normalized by min-max normalization to $[0,1]$ so that they are on the same scale, and the same dataset is fitted to both the regression model and Random Forest model.",
"After obtaining the variable selection and predictors that rank results from the two methods, we compare the set of selected variables and the top ranked predictors to see if they are consistent.",
"For both methods, we split the samples by a 7:3 ratio into the training and test sets and compare the prediction accuracy to see if they have similar performance."
],
[
"Generalized Linear Regression with $L_1$ Penalty",
"Generalized linear regression is a broad class of models that includes regular regression for continuous response as well as models for discrete responses.",
"The relationship between a dependent variable and one or more independent variables can be explained by the magnitude and sign of the regression coefficient estimates.",
"Depending on the types of response variables, we include the regular linear regression for continuous responses and logistic regression for binary responses in our analysis.",
"In addition, we also include a $L_1$ regularization term to shrink the coefficient estimates toward 0 so that only highly correlated variables remain in the model [16].",
"A tuning parameter controls the degree of shrinkage such that varying the tuning parameter value results in different estimates.",
"We employ the Bayesian Information Criterion (BIC) to choose the optimal model, which focuses on finding the true model that generates the data.",
"If using an error-based or accuracy-based criterion such as cross-validation for variable selection, when correlated variables enter the model, any one of the correlated variables being selected can give a good predictive performance; however BIC selects the variables that yield the best likelihood for the model and removes the rest, thus serving our purpose of explanatory modeling perfectly."
],
[
"Random Forest",
"The Random Forest is one of the most effective methods for predictive analysis as either a classification algorithm or regression model [6].",
"By selecting the nodes randomly and aggregating the trees for pooled results, Random Forest is known to be robust to noise and outliers, as well as the value changes of parameters on a fine scale, including the number of trees aggregated and the number of nodes being selected at each node split.",
"Therefore, we do not manually vary the parameter values on a fine scale in order to achieve the optimal results.",
"The model selection, which is the ranking of predictors by their changes in purity, is done internally through computing the Out-of-bag (OOB) error from aggregating the trees."
],
[
"Experiments",
"In this section, we present the basic features of female representation (i.e.",
"the Bechdel test result and female cast ratio), movie success (i.e.",
"the revenue/budget ratio, movie rating and movie popularity), and our modeling results of their associations."
],
[
"Preliminary",
"We examine the two most important measures of female representation in movies, the Bechdel test result and the female cast ratio.",
"About 55.01% of the movies in our dataset pass the Bechdel test.",
"As Fig.REF shows, in the 21st century, the movies that pass the test have outnumbered the ones that do not.",
"The average female cast ratio is 0.26, and Fig.REF shows that the movies in latter time have higher female cast ratios compared to the former time.",
"Using a simple univariate linear regression, we conclude that as time progresses, the number of movies that pass the Bechdel test increases (coefficient estimate = 1.61, p-value$<0.0001$ ) and more females are cast in the movies (coefficient estimate = 0.11, p-value$<$ 0.0001).",
"These two measures are highly correlated (t test, p-value$<$ 0.0001), as movies that have passed the Bechdel test have a higher female cast ratio (0.31) than those which did not (0.20).",
"Among all the variables, we are particularly interested in budget because we want to see whether movies with better female representation were better supported.",
"The results are disappointing in that higher female cast ratio is significantly associated with lower budget (Pearson correlation coefficient = -0.099, p-value$<$ 0.0001).",
"Similarly, movies that pass the Bechdel test had a significantly lower budget than movies that did not (t test, p-value = 0.001).",
"Figure: Women representation in movies over time.",
"(a) The distribution of the Bechdel test results over the last 12 decades; (b) The distribution of the female cast ratio over the last 12 decades.Exploring their association with movie success, we found that the revenue/budget ratio is not significantly associated with the female cast ratio (Pearson correlation coefficient = 0.016, p-value=0.28).",
"Meanwhile, the revenue/budget ratio does not differ significantly between the movies with different Bechdel test results (t test, p-value=0.14).",
"On the other hand, higher movie rating is significantly correlated with lower female cast ratio (Pearson correlation coefficient = -0.15, p-value$<$ 0.0001).",
"Movie ratings in the group of movies that pass the Bechdel test are significantly lower than the group of movies that do not (t test, p-value$<$ 0.0001); Similarly, higher movie popularity is also significantly correlated with lower female cast ratio (Pearson correlation coefficient = -0.091, p-value$<$ 0.0001).",
"Movies that pass the Bechdel test are significantly less popular than the movies that do not.",
"The statistical analyses have shown that female representation seems to have a negative impact on movie success.",
"However, we cannot conclude the relationship between movie success and female representation.",
"We have found that female representation is improving over time, and audience reaction to movies is also changing over time.",
"From Fig.REF , overall, movie rating decreases significantly as time progresses (Pearson correlation coefficient = -0.25, p-value$<$ 0.0001) and popularity increases significantly as time progresses (Pearson correlation coefficient = 0.097, p-value$<$ 0.0001).",
"The revenue/budget ratio does not change significantly over time (Pearson correlation coefficient = -0.012, p-value=0.43) because both budget (Pearson correlation coefficient = 0.29, p-value$<$ 0.0001) and revenue (Pearson correlation coefficient = 0.19, p-value$<$ 0.0001) increase significantly over time.",
"We further investigate the relationship between movie success and female representation in different time periods.",
"The majority of the movies were produced in the United States, and the feminism movement in the United States reached a peak around 1970.",
"From Fig.REF , movies that did not pass the Bechdel test outnumber the movies that did dramatically before 1970, but afterwards there were more movies that passed the test.",
"Before 1970, the relationship between the female cast ratio and movie success is the same as for the overall data.",
"However, all three aspects did not differ significantly between movies that passed the Bechdel test and movies that did not.",
"Going into 1970-2015, the relationship is the same as for the overall data.",
"After 2015, movie rating is no longer significantly correlated with the female cast ratio (Pearson correlation coefficient = -0.078, p-value=0.053), and all three aspects do not differ significantly between movies that passed the Bechdel test and movies that did not.",
"We can see that before 1970 and after 2015, movies that passed the Bechdel test achieved success at a similar level as movies that did not pass the test, while between these two time points movies with better female representation suffered from a lower rating and lower popularity.",
"From the preliminary analysis, we can see that there exists significant relationship between female representation and movie success.",
"However, they are also correlated with time as female representation is improving, movie popularity is increasing, and ratings are decreasing.",
"We rely on the regression model and Random Forest regression to adjust for any confounding effects throughout the analyses.",
"Figure: Movie budget, revenue, rating and popularity over the last 12 decades."
],
[
"The Improvement in Women Representation",
"We first evaluate the factors that have strong associations with the Bechdel test result.",
"Setting the Bechdel test result as the response variable of a logistic regression with penalty and a Random Forest, we include the variables of year, adult content, run minutes, budget, number of cast members, female director ratio, female producer ratio, female screenplay writer ratio, 26 production company indicators, and 25 genre indicators as the predictors.",
"The same set of predictors is also fitted to the models with the female cast ratio being the response variable.",
"The variable selection results are shown in Table REF .",
"For both response variables, the regression model and the Random Forest model have very similar predictive performance.",
"The female screenplay writer ratio is selected in both regression models to have positive association with the Bechdel test result (coefficient estimate = 0.79) and female cast ratio (coefficient estimate = 0.042) has very high rankings in both Random Forest models (Bechdel test=6, female cast ratio=5), implying that women participation in story-writing is key to determining how women are represented on screen, outperforming the female director ratio and female producer ratio.",
"These two variables are also selected with relatively smaller magnitude and decent ranking in the Random Forest model, due to the fact that these three variables are correlated and their effects on the response variables are adjusted for the presence of the female screenplay writer ratio.",
"Year is also a very strong predictor of female representation, which is consistent with Fig.REF in that as time progresses women become better represented.",
"For the female cast ratio, the number of cast members is the most influential factor (coefficient estimate=-0.19, ranking=1) as a higher number of cast members leads to a lower female cast ratio.",
"Such an association implies that male actors are more likely to be hired than females.",
"Many genres were identified to have better female representation, such as Romance, selected by both regression models (Bechdel test: 0.18, female cast ratio: 0.042) with decent rankings in both Random Forest models (Bechdel test: 14, female cast ratio: 6), and Horror, selected by the Bechdel test regression (coefficient estimate = 0.20) with decent rankings in both Random Forest models (Bechdel test: 11, female cast ratio: 14).",
"Here, Horror was paid special attention due to its association with movie success in future analysis.",
"Some genres, like Action and Crime were identified to have less female representation.",
"Although significant correlations of budget with the female cast ratio and Bechdel test results are found in the preliminary statistical analysis, budget is not selected in either of the regression models.",
"However, this fact can be explained by the gender preference of genres.",
"From Fig.REF , we can see that Action and Crime movies, which are identified in our models to be negatively associated with female representation, are likely to receive a much higher budget than movies that are not associated with these genres.",
"Meanwhile, they also tend to feature more male cast members than female cast members.",
"Romance and Horror movies (surprisingly) feature more females than males on the average.",
"However, they also tend to receive a lower budget.",
"The imbalance between the budgets of genres and gender preference of genres leads to the imbalance of budgets on movies with different levels of female representation.",
"Table: Regression coefficient estimates and Random Forest predictor importance ranking for the variables selected by the regularized regression models of the Bechdel test and female cast ratio.",
"The importance of predictors in the Random Forest model is defined as the change in the Gini-impurity for classification and MSE for regression, where larger changes indicate greater importance.Figure: The female cast ratio and budget distribution in the genres selected by the regression model."
],
[
"Women Representation and Movie Success",
"Setting the movie revenue/budget ratio (R/B ratio), rating, and popularity each as the response variable of an individual regression model with penalty and a Random Forest model, we include the variables of year, adult content, run time minutes, number of cast members, female cast ratio, female core crew ratio, the Bechdel test, 26 production company indicators, and 25 genre indicators as the predictors.",
"The female cast ratio is selected to have strong positive association with R/B ratio (coefficient estimate=$1.19\\times 10^{-3}$ ) and ranks as the most important predictor in the Random Forest model (ranking=1).",
"Meanwhile, three similar genres have been identified to have high positive association with the R/B ratio: Horror (coefficient estimate=$1.69\\times 10^{-3}$ , ranking=7), Mystery (coefficient estimate=$2.07\\times 10^{-3}$ , ranking=5), and Thriller (coefficient estimate=$1.04\\times 10^{-3}$ , ranking=6).",
"As we have acquired the knowledge from previous analysis that the female cast ratio is positively associated with the Horror genre, we are confident to conclude that with the presence of Horror and the other two correlated genres in the model, the female cast ratio has a positive effect on a movie's profitability after considering the effects of genres.",
"This same set of variables is also selected for movie rating, however their associations with rating are in the opposite direction in contrast to the R/B ratio.",
"The female cast ratio has a negative association with rating (coefficient estimate=$-6.75\\times 10^{-3}$ ) and a very high ranking in the Random Forest model (rank=4).",
"Horror (coefficient estimate=$-0.014$ , ranking=6) and Mystery (coefficient estimate=$-0.19$ , ranking=16) are identified to be negatively associated with rating.",
"It seems that the movies featuring more women, horror, and mystery elements are likely to make more money, but receive more criticism.",
"Neither of the female representation measures, the female cast ratio and Bechdel test result, are selected for movie popularity.",
"However, the female cast ratio still receives a high ranking in the Random Forest model (ranking=4), possibly due to its correlation with other variables.",
"Due to the fact that the revenue/budget ratio is calculated directly from budget, and that in the previous analysis we have learned that budget does not have a direct impact on female representation, we exclude budget from our model since its strong correlation with the response variable would overwhelm the effects of other variables.",
"However, our models still demonstrate the relationship between budget and movie success.",
"The run-time in minutes and the number of cast members are identified to have very strong association with all three aspects of movie success and receive very high ranking in the Random Forest models, and they are a proxy of a movie's budget as a higher budget leads to a larger number of cast members (Spearman correlation coefficient=0.41, p-value$<$ 0.0001) and a longer run time in minutes (Pearson correlation coefficient=0.26, p-value$<$ 0.0001).",
"Meanwhile, our preliminary analysis shows that the female cast ratio is not significantly correlated with the R/B ratio, and our regression model identifies it to have quite a strong association possibly due to the fact that the female cast ratio is negatively associated with the budget.",
"Movies featuring a higher female cast ratio tend to have a lower budget, which is likely to lead to a higher R/B ratio even though the revenue is not particularly high.",
"However, such an effect is not strong enough to be detected by the correlation test.",
"Our modeling results are consistent with the preliminary analysis for ratings in that a higher female cast ratio leads to a lower movie ratings.",
"For movie popularity, the female cast ratio is not selected in the regression model despite the fact that the Pearson correlation test identifies them being highly correlated.",
"This again can be explained by the confounding effect of genres and production companies.",
"The regression model identifies some genres being positively associated with popularity, such as Action (coefficient estimate=$2.67\\times 10^{-3}$ , ranking=6) and Adventure (coefficient estimate=$3.19\\times 10^{-3}$ , ranking=5), as well as one production company Disney (coefficient estimate=$6.23\\times 10^{-3}$ , ranking=8).",
"Meanwhile, it also identifies one genre being negatively associated with popularity, Drama (coefficient estimate=$-9.4\\times 10^{-4}$ , ranking=10).",
"Fig.REF shows that Action, Adventure, and Sci-Fi movies, which tend to be very popular, also tend to feature fewer female cast members; Drama movies, on the other hand, feature slightly more female cast members but also tend to be less popular than other genres.",
"Disney, known for creating strong and well-rounded female characters, in fact still features more male cast members (or male voices in animation) than female cast members in their productions.",
"Therefore, the high popularity of Disney movies does not help associate better female representation with high movie popularity.",
"Table: Regression coefficient estimates and Random Forest predictors importance ranking for variables selected by the regularized regression models of movie revenue/budget ratio, rating and popularity.Figure: The female cast ratio and movie popularity distribution in the genres of Action, Adventure, Drama, Sci-Fi and production company of Disney, selected by the regression model."
],
[
"Conclusions",
"Overall, our findings regarding female representation indicate that female representation is critically influenced by the work of female crew members, especially female screenplay writers, in the film-making process.",
"In addition, it is also evolving throughout time as a result of other factors outside movies and thus not included in the dataset (i.e.",
"the variable year).",
"Our findings also reflect some difficulties actresses face, for example, male actors are more likely to be hired than females, genres that tend to receive higher budgets also prefer male actors to portray the stories, and so on.",
"From our investigation into the relationship between female representation and movies' success, we discover that compared to the Bechdel test result, the proposed female cast ratio is more directly linked to a movie's success as it is often selected multiple times.",
"In comparison, the Bechdel test result is not selected once.",
"Moreover, movies featuring more women tend to have a lower budget which leads to a higher revenue/budget return.",
"However, they also suffer from more criticism, possibly due to the low budget invested.",
"Considering that the number of cast members and run-time minutes are both selected to have a very high positive impact on a movie's rating and that they are both a proxy of the budget, this is a very plausible explanation.",
"The female cast ratio is not directly linked to a movie's popularity.",
"However, we have also discovered that genres likely to be popular, such as adventure and action, also tend to feature fewer female cast members.",
"Disney has made contributions to better female representation on the big screen and its productions are often very popular.",
"However, their productions still feature more males than females.",
"Our findings have demonstrated that the difficulties women in the film industry face are both upstream and downstream: female filmmakers, especially female screenplay writers, are instrumental for movies to have better female representation, but the percentages of female filmmakers are very low; only 6.0% of directors, 9.7% of producers, and 12.2% of screenplay writers are female.",
"Meanwhile, lower budgets are provided to support the movies that could tell good stories about women, thus causing the films to receive more criticism.",
"The demand of better female representation from viewers is also not strong enough to press the film industry for change, as movies that have poor female representation can still be very profitable."
],
[
"Discussions",
"Our study provides a larger picture of female representation in movies and how it is perceived by the audience over time.",
"Unfortunately, underrepresentation of women in movies is not the only difficulty women are facing.",
"Apart from the low ratio of female cast members, portrayal of women in stereotypical ways that reflect and sustain socially endorsed views of genders, and depictions of relationships between men and women that emphasize traditional roles and normalize violence against women, are another two important themes of how media portray gender [19].",
"Although our newly-defined measure of female representation, the female cast ratio, has shown to be more successful in evaluating movie success than the Bechdel test, it also has limitations that fail to further explain how women are portrayed in movies.",
"For example, our findings indicate that horror movies like to feature more women than men, and the reason behind this phenomenon is that the audience enjoys the victimization of women more than the victimization of men [14].",
"In other words, the root of \"favoritism\" toward women in horror movies is still that people want to see women being helpless and passive rather than men.",
"Another genre identified by our analysis that favors women, Romance, is also likely to portray women in a stereotypical way.",
"A linguistics study of the movie \"The Best of Me\" shows some differences in the expressions between men and women, in that men prefer to use commanding directives while women use directives for suggesting or requesting, and that men tend to use swear words to express anger and women tend to use swear words to express bad feelings [13].",
"Even in a movie that targeted the female audience, women were portrayed as docile and submissive compared to men.",
"Meanwhile, some studies have used dialogue speaking time to evaluate how much women talk in movies and discovered that even in some female-led movies (such as Disney princess movies), the lead female's speaking time could be outnumbered by male cast members [1].",
"Whether they speak to express their ideas, feelings, or make commands, men speak more than women and their influence on the audience is ultimately stronger.",
"We will also consider many other factors that can influence the image of a character in movies in our future studies to help us better understand how gender works in shaping a character's image and how they influence the audience's perception of movies.",
"For example, race and ethnicity are both important factors that impact the audience's expectation for a character.",
"For example, studies have shown that African American music videos were significantly more likely to portray sexual content and sexualized female characters than White videos [18].",
"In addition, the first African American performer to win the Academy Award was a female, and she won much earlier than her male counterparts.",
"The reason behind the phenomenon that minority women are more popular than minority men is possibly that women and minorities were traditionally both considered to be submissive to white males, thus a minority woman portrayed in such a way is more acceptable to the mainstream than a minority man.",
"Besides race and ethnicity, age is another important factor.",
"While in movies both women and men in their 60s and older are dramatically underrepresented compared to their representation in the U.S. population [7], older actresses experience greater difficulty in finding a job than older actors even for celebrities [17].",
"The majority of male characters are in their 30s and 40s, and the majority of female characters are in their 20s and 30s.",
"For male characters, leadership and occupational power increase with age; however, as female characters age, they are less likely to have goals [7].",
"This observation coincides with the mainstream expectation of genders in that men hold authority and leadership which usually increases with age, while women are in more subordinate roles and often sexualized, which is why younger women are preferred.",
"In conclusion, we hope our findings send a message to the film-making industry that movies with better female representation are more likely to succeed.",
"Improving female representation in movies requires both upstream effort and downstream effort: female crew and cast members contribute to better female representation in movies, and positive feedback from the audience encourages investment into movies to hire more female crew and cast members.",
"Therefore, we encourage the film-making industry to employ more women as a start.",
"Men have been the leading voices of story-telling in movies for decades, so we believe that stories from women's perspectives are worth exploring and would attract larger audiences."
],
[
"Acknowledgement",
"The authors would like to thank Joyce Luo and William Artman for their help with writing."
]
] | 2001.03513 |
[
[
"Multi-Agent Interactions Modeling with Correlated Policies"
],
[
"Abstract In multi-agent systems, complex interacting behaviors arise due to the high correlations among agents.",
"However, previous work on modeling multi-agent interactions from demonstrations is primarily constrained by assuming the independence among policies and their reward structures.",
"In this paper, we cast the multi-agent interactions modeling problem into a multi-agent imitation learning framework with explicit modeling of correlated policies by approximating opponents' policies, which can recover agents' policies that can regenerate similar interactions.",
"Consequently, we develop a Decentralized Adversarial Imitation Learning algorithm with Correlated policies (CoDAIL), which allows for decentralized training and execution.",
"Various experiments demonstrate that CoDAIL can better regenerate complex interactions close to the demonstrators and outperforms state-of-the-art multi-agent imitation learning methods.",
"Our code is available at \\url{https://github.com/apexrl/CoDAIL}."
],
[
"Introduction",
"Modeling complex interactions among intelligent agents from the real world is essential for understanding and creating intelligent multi-agent behaviors, which is typically formulated as a multi-agent learning (MAL) problem in multi-agent systems.",
"When the system dynamics are agnostic and non-stationary due to the adaptive agents with implicit goals, multi-agent reinforcement learning (MARL) is the most commonly used technique for MAL.",
"MARL has recently drawn much attention and achieved impressive progress on various non-trivial tasks, such as multi-player strategy games [32], [19], traffic light control [8], taxi-order dispatching [23] etc.",
"A central challenge in MARL is to specify a good learning goal, as the agents' rewards are correlated and thus cannot be maximized independently [7].",
"Without explicit access to the reward signals, imitation learning could be the most intuitive solution for learning good policies directly from demonstrations.",
"Conventional solutions such as behavior cloning (BC) [33] learn the policy in a supervised manner by requiring numerous data while suffering from compounding error [36], [37].",
"Inverse reinforcement learning (IRL) [31], [38] alleviates these shortcomings by recovering a reward function but is always expensive to obtain the optimal policy due to the forward reinforcement learning procedure in an inner loop.",
"Generative adversarial imitation learning (GAIL) [18] leaves a better candidate for its model-free structure without compounding error, which is highly effective and scalable.",
"However, real-world multi-agent interactions could be much challenging to imitate because of the strong correlations among adaptive agents' policies and rewards.",
"Consider if a football coach wants to win the league, he must make targeted tactics against various opponents, in addition to the situation of his team.",
"Moreover, the multi-agent environment tends to give rise to more severe compounding errors with more expensive running costs.",
"Motivated by these challenges, we investigate the problem of modeling complicated multi-agent interactions from a pile of off-line demonstrations and recover their on-line policies, which can regenerate analogous multi-agent behaviors.",
"Prior studies for multi-agent imitation learning typically limit the complexity in demonstrated interactions by assuming isolated reward structures [2], [22], [25], [42] and independence in per-agent policies that overlook the high correlations among agents [39], [46].",
"In this paper, we cast the multi-agent interactions modeling problem into a multi-agent imitation learning framework with correlated policies by approximating opponents’ policies, in order to reach inaccessible opponents' actions due to concurrently execution of actions among agents when making decisions.",
"Consequently, with approximated opponents model, we develop a Decentralized Adversarial Imitation Learning algorithm with Correlated policies (CoDAIL) suitable for learning correlated policies under our proposed framework, which allows for decentralized training and execution.",
"We prove that our framework treats the demonstrator interactions as one of $\\epsilon $ -Nash Equilibrium ($\\epsilon $ -NE) solutions under the recovered reward.",
"In experiments, we conduct multi-dimensional comparisons for both the reward gap between learned agents and demonstrators, along with the distribution divergence between demonstrations and regenerated interacted trajectories from learned policies.",
"Furthermore, the results reveal that CoDAIL can better recover correlated multi-agent policy interactions than other state-of-the-art multi-agent imitation learning methods in several multi-agent scenarios.",
"We further illustrate the distributions of regenerated interactions, which indicates that CoDAIL yields the closest interaction behaviors to the demonstrators."
],
[
"Markov Game and $\\epsilon $ -Nash Equilibrium",
"Markov game (MG), or stochastic game [27], can be regarded as an extension of Markov Decision Process (MDP).",
"Formally, we define an MG with $N$ agents as a tuple $\\langle N, \\mathcal {S}, {\\mathcal {A}}^{(1)},\\dots ,{\\mathcal {A}}^{(N)}, P, {r}^{(1)}, \\dots ,{r}^{(N)}, \\rho _0, \\gamma \\rangle $ , where $\\mathcal {S}$ is the set of states, ${\\mathcal {A}}^{(i)}$ represents the action space of agent $i$ , where $i \\in \\lbrace 1,2,\\dots ,N\\rbrace $ , $P: \\mathcal {S}\\times {\\mathcal {A}}^{(1)} \\times {\\mathcal {A}}^{(2)}\\times \\cdots \\times {\\mathcal {A}}^{(N)} \\times \\mathcal {S}\\rightarrow [0,1]$ is the state transition probability distribution, $\\rho _0: \\mathcal {S}\\rightarrow [0,1]$ is the distribution of the initial state $s^0$ , and $\\gamma \\in [0,1]$ is the discounted factor.",
"Each agent $i$ holds its policy ${\\pi ^{(i)}}({a}^{(i)}|s): \\mathcal {S}\\times {\\mathcal {A}}^{(i)} \\rightarrow [0, 1]$ to make decisions and receive rewards defined as $r^{(i)}: \\mathcal {S}\\times {\\mathcal {A}}^{(1)} \\times {\\mathcal {A}}^{(2)} \\times \\cdots \\times {\\mathcal {A}}^{(N)} \\rightarrow \\mathbb {R}$ .",
"We use $-i$ to represent the set of agents except $i$ , and variables without superscript $i$ to denote the concatenation of all variables for all agents (e.g., $\\pi $ represents the joint policy and $a$ denotes actions of all agents).",
"For an arbitrary function $f: \\langle s, a \\rangle \\rightarrow \\mathbb {R}$ , there is a fact that $\\mathbb {E}_\\pi [f(s,a)]=\\mathbb {E}_{s\\sim P, a\\sim \\pi }[f(s,a)]\\triangleq \\mathbb {E}\\left[\\sum _{t=0}^\\infty \\gamma ^t f(s_t,a_t) \\right]$ , where $s^{0}\\sim \\rho _0$ , $a_t\\sim \\pi $ , $s_{t+1}\\sim P(s_{t+1}|a_t,s_t)$ .",
"The objective of agent $i$ is to maximize its own total expected return ${R}^{(i)} \\triangleq \\mathbb {E}_\\pi [{r}^{(i)}(s,a)] = \\mathbb {E}\\left[\\sum _{t=0}^\\infty \\gamma ^t {r}^{(i)}(s_t, a_t) \\right]$ .",
"In Markov games, however, the reward function for each agent depends on the joint agent actions.",
"Such a fact implies that one's optimal policy must also depend on others' policies.",
"For the solution to the Markov games, $\\epsilon $ -Nash equilibrium ($\\epsilon $ -NE) is a commonly used concept that extends Nash equilibrium (NE) [30].",
"Definition 1 An $\\epsilon $ -NE is a strategy profile $(\\pi _{*}^{(i)}, \\pi _{*}^{(-i)})$ such that $\\exists \\epsilon >0$ : $\\begin{aligned}\\vspace{-20.0pt}{v}^{(i)}(s, \\pi _{*}^{(i)}, \\pi _{*}^{(-i)})\\ge {v}^{(i)}(s, {\\pi ^{(i)}}, \\pi _{*}^{(-i)}) - \\epsilon , \\forall {\\pi ^{(i)}}\\in {\\Pi }^{(i)}~,\\end{aligned}$ where ${v}^{(i)}(s, {\\pi ^{(i)}}, {\\pi ^{(-i)}})= \\mathbb {E}_{{\\pi ^{(i)}}, {\\pi ^{(-i)}}, s_0=s} \\left[{r}^{(i)}(s_t, a_t^{(i)}, a_t^{(-i)})\\right]$ is the value function of agent $i$ under state $s$ , and ${\\Pi }^{(i)}$ is the set of policies available to agent $i$ .",
"$\\epsilon $ -NE is weaker than NE, which can be seen as sub-optimal NE.",
"Every NE is equivalent to an $\\epsilon $ -NE where $\\epsilon =0$ ."
],
[
"Generative Adversarial Imitation Learning",
"Imitation learning aims to learn the policy directly from expert demonstrations without any access to the reward signals.",
"In single-agent settings, such demonstrations come from behavior trajectories sampled with the expert policy, denoted as $\\tau _E=\\lbrace (s_t, {a}^{(i)}_t) \\rbrace _{t=0}^{\\infty }$ .",
"However, in multi-agent settings, demonstrations are often interrelated trajectories, that is, which are sampled from the interactions of policies among all agents, denoted as $\\Omega _E=\\lbrace (s_t, {a}^{(1)}_t, ..., {a}^{(N)}_t)\\rbrace _{t=0}^{\\infty }$ .",
"For simplicity, we will use the term interactions directly as the concept of interrelated trajectories, and we refer to trajectories for a single agent.",
"Typically, behavior cloning (BC) and inverse reinforcement learning (IRL) are two main approaches for imitation learning.",
"Although IRL theoretically alleviates compounding error and outperforms to BC, it is less efficient since it requires resolving an RL problem inside the learning loop.",
"Recent proposed work aims to learn the policy without estimating the reward function directly, notably, GAIL [18], which takes advantage of Generative Adversarial Networks (GAN [13]), showing that IRL is the dual problem of occupancy measure matching.",
"GAIL regards the environment as a black-box, which is non-differentiable but can be leveraged through Monte-Carlo estimation of policy gradients.",
"Formally, its objective can be expressed as $\\begin{aligned}\\min _{\\pi }\\max _{D} \\mathbb {E}_{\\pi _E}\\left[\\log {D(s,a)}\\right] +\\mathbb {E}_{\\pi } \\left[\\log {(1-D(s,a))}\\right] -\\lambda H(\\pi )~,\\end{aligned}$ where $D$ is a discriminator that identifies the expert trajectories with agents' sampled from policy $\\pi $ , which tries to maximize its evaluation from $D$ ; $H$ is the causal entropy for the policy and $\\lambda $ is the hyperparameter."
],
[
"Correlated Policy",
"In multi-agent learning tasks, each agent $i$ makes decisions independently while the resulting reward ${r}^{(i)}(s_t, a_t^{(i)}, a_t^{(-i)})$ depends on others’ actions, which makes its cumulative return subjected to the joint policy $\\pi $ .",
"One common joint policy modeling method is to decouple the $\\pi $ with assuming conditional independence of actions from different agents [1]: $\\begin{aligned}\\pi ({a}^{(i)}, {a}^{(-i)} | s) \\approx {\\pi ^{(i)}}({a}^{(i)} | s) {\\pi ^{(-i)}}({a}^{(-i)} | s)~.\\end{aligned}$ However, such a non-correlated factorization on the joint policy is a vulnerable simplification which ignores the influence of opponents [44].",
"And the learning process of agent $i$ lacks stability since the environment dynamics depends on not only the current state but also the joint actions of all agents [41].",
"To solve this, recent work has taken opponents into consideration by decoupling the joint policy as a correlated policy conditioned on state $s$ and ${a}^{(-i)}$ as $\\begin{aligned}\\pi ({a}^{(i)}, {a}^{(-i)} | s) = {\\pi ^{(i)}}({a}^{(i)} | s, {a}^{(-i)}) {\\pi ^{(-i)}}({a}^{(-i)} | s)~,\\end{aligned}$ where ${\\pi ^{(i)}}({a}^{(i)} | s, {a}^{(-i)})$ is the conditional policy, with which agent $i$ regards all potential actions from its opponent policies ${\\pi ^{(-i)}}({a}^{(-i)} | s)$ , and makes decisions through the marginal policy ${\\pi ^{(i)}}({a}^{(i)} | s) = \\int _{{a}^{(-i)}} {\\pi ^{(i)}}({a}^{(i)} | s, {a}^{(-i)}) {\\pi ^{(-i)}}({a}^{(-i)} | s) \\mathop {}\\!\\mathrm {d}{{a}^{(-i)}} = \\mathbb {E}_{{a}^{(-i)}} {\\pi ^{(i)}}({a}^{(i)} | s, {a}^{(-i)})$ ."
],
[
"Generalize Correlated Policies to Multi-Agent Imitation Learning",
"In multi-agent settings, for agent $i$ with policy ${\\pi }^{(i)}$ , it seeks to maximize its cumulative reward against demonstrator opponents who equip with demonstrated policies ${\\pi }^{(-i)}_E$ via reinforcement learning: $\\begin{aligned}\\vspace{-20.0pt}{\\operatorname{RL}}^{(i)}({r}^{(i)}) = \\operatornamewithlimits{arg\\,max}_{{\\pi ^{(i)}}} \\lambda H({\\pi ^{(i)}}) + \\mathbb {E}_{{\\pi ^{(i)}}, {\\pi _E^{(-i)}}}[r^{(i)}(s, a^{(i)}, a^{(-i)})]~,\\end{aligned}$ where $H({\\pi ^{(i)}})$ is the $\\gamma $ -discounted entropy [5], [16] of policy ${\\pi ^{(i)}}$ and $\\lambda $ is the hyperparameter.",
"By coupling with Eq.",
"(REF ), we define an IRL procedure to find a reward function ${r}^{(i)}$ such that the demonstrated joint policy outperforms all other policies, with the regularizer $\\psi : \\mathbb {R}^{\\mathcal {S}\\times {\\mathcal {A}}^{(1)}\\times \\cdots \\times {\\mathcal {A}}^{(N)}} \\rightarrow {\\overline{\\mathbb {R}}}$ : $\\begin{aligned}\\vspace{-20.0pt}\\operatorname{IRL}_\\psi ^{(i)}({\\pi _E^{(i)}}) = &\\operatornamewithlimits{arg\\,max}_{{r}^{(i)}} -\\psi ({r}^{(i)})- \\max _{{\\pi ^{(i)}}} (\\lambda H({\\pi ^{(i)}}) + \\mathbb {E}_{{\\pi ^{(i)}}, {\\pi _E^{(-i)}}}[r^{(i)}(s, a^{(i)}, a^{(-i)})])\\\\&+\\mathbb {E}_{\\pi _E} [r^{(i)}(s, a^{(i)}, a^{(-i)})]~.\\end{aligned}$ It is worth noting that we cannot obtain the demonstrated policies from the demonstrations directly.",
"To solve this problem, we first introduce the occupancy measure, namely, the unnormalized distribution of $\\langle s, a\\rangle $ pairs correspond to the agent interactions navigated by joint policy $\\pi $ : $\\begin{aligned}\\vspace{-20.0pt}\\rho _{\\pi }(s, a) = \\pi (a|s)\\sum _{t=0}^{\\infty }\\gamma ^t P(s_t=s|\\pi )~.\\end{aligned}$ With the definition in Eq.",
"(REF ), we can further formulate $\\rho _\\pi $ from agent $i$ 's perspective as $\\begin{aligned}\\vspace{-20.0pt}\\rho _{\\pi }(s, {a}^{(i)}, {a}^{(-i)}) &= \\pi ({a}^{(i)}, {a}^{(-i)}|s)\\sum _{t=0}^{\\infty }\\gamma ^t P(s_t=s|{\\pi ^{(i)}}, {\\pi ^{(-i)}}) \\\\&= \\rho _{{\\pi ^{(i)}}, {\\pi ^{(-i)}}}(s, {a}^{(i)}, {a}^{(-i)})\\\\&={\\left\\lbrace \\begin{array}{ll}\\underbrace{{\\pi ^{(i)}}({a}^{(i)}| s){\\pi ^{(-i)}}({a}^{(-i)}| s)}_{\\text{non-correlated form}}\\sum _{t=0}^{\\infty } \\gamma ^t P(s_t=s|{\\pi ^{(i)}}, {\\pi ^{(-i)}}) \\\\~\\\\\\underbrace{{\\pi ^{(i)}}({a}^{(i)}| s, {a}^{(-i)}){\\pi ^{(-i)}}({a}^{(-i)}| s)}_{\\text{correlated form}}\\sum _{t=0}^{\\infty } \\gamma ^t P(s_t=s|{\\pi ^{(i)}}, {\\pi ^{(-i)}})\\end{array}\\right.",
"}~,\\end{aligned}$ where ${a}^{(i)}\\sim {\\pi ^{(i)}}$ and ${a}^{(-i)}\\sim {\\pi ^{(-i)}}$ .",
"Furthermore, with the support of Eq.",
"(REF ), we have $\\begin{aligned}\\vspace{-20.0pt}\\mathbb {E}_{{\\pi ^{(i)}}, {\\pi ^{(-i)}}}[\\cdot ]&= \\mathbb {E}_{s\\sim P, {a}^{(i)}\\sim {\\pi ^{(i)}}}[ \\mathbb {E}_{{a}^{(-i)}\\sim {\\pi ^{(-i)}}}[\\cdot ]] \\\\&=\\sum _{s,{a}^{(i)},{a}^{(-i)}}\\rho _{{\\pi ^{(i)}}, {\\pi ^{(-i)}}}(s,{a}^{(i)},{a}^{(-i)})[\\ \\cdot \\ ]~.\\end{aligned}$ In analogy to the definition of occupancy measure of that in a single-agent environment, we follow the derivation from [18] and state the conclusion directlyNote that [18] proved the conclusion under the goal to minimize the cost instead of maximizing the reward of an agent..",
"Proposition 1 The IRL regarding demonstrator opponents is a dual form of a occupancy measure matching problem with regularizer $\\psi $ , and the induced optimal policy is the primal optimum, specifically, the policy learned by RL on the reward recovered by IRL can be characterize by the following equation: $\\begin{aligned}\\vspace{-20.0pt}{\\operatorname{RL}}^{(i)} \\circ {\\operatorname{IRL}}^{(i)} = \\operatornamewithlimits{arg\\,min}_{{\\pi ^{(i)}}}-\\lambda H({\\pi ^{(i)}})+\\psi ^{*}(\\rho _{{\\pi ^{(i)}}, \\pi ^{(-i)}_E} - \\rho _{\\pi _E})~.\\end{aligned}$ With setting the regularizer $\\psi =\\psi _{GA}$ similar to [18], we can obtain a GAIL-like imitation algorithm to learn ${\\pi _E^{(i)}}$ from $\\pi _E$ given demonstrator counterparts ${\\pi _E^{(-i)}}$ by introducing the adversarial training procedures of GANs which lead to a saddle point $({\\pi ^{(i)}}, {D}^{(i)})$ : $\\begin{aligned}\\vspace{-20.0pt}\\min _{{\\pi ^{(i)}}}\\max _{{D}^{(i)}} -\\lambda H({\\pi ^{(i)}}) + \\mathbb {E}_{\\pi _E}\\left[\\log {{D}^{(i)}(s,{a}^{(i)}, {a}^{(-i)})}\\right]+\\mathbb {E}_{{\\pi ^{(i)}}, {\\pi _E^{(-i)}}} \\left[\\log {(1-{D}^{(i)}(s, {a}^{(i)}, {a}^{(-i)}))} \\right]~,\\end{aligned}$ where ${D}^{(i)}$ denotes the discriminator for agent $i$ , which plays a role of surrogate cost function and guides the policy learning.",
"However, such an algorithm is not practical, since we are unable to access the policies of demonstrator opponents ${\\pi _E^{(-i)}}$ because the demonstrated policies are always given through sets of interactions data.",
"To alleviate this deficiency, it is necessary to deal with accessible counterparts.",
"Thereby we propose Proposition REF .",
"Proposition 2 Let $\\mu $ be an arbitrary function such that $\\mu $ holds a similar form as ${\\pi ^{(-i)}}$ , then $\\mathbb {E}_{{\\pi ^{(i)}}, {\\pi ^{(-i)}}} [\\cdot ] =\\mathbb {E}_{{\\pi ^{(i)}}, \\mu } \\Big [\\frac{\\rho _{{\\pi ^{(i)}}, {\\pi ^{(-i)}}}(s, {a}^{(i)}, {a}^{(-i)})}{\\rho _{{\\pi ^{(i)}}, \\mu }(s, {a}^{(i)}, {a}^{(-i)})}\\ \\cdot \\ \\Big ]$ .",
"Substituting ${\\pi ^{(-i)}}$ with $\\mu $ in Eq.",
"(REF ) by importance sampling.",
"Proposition REF raises an important point that a term of importance weight can quantify the demonstrator opponents.",
"By replacing ${\\pi _E^{(-i)}}$ with ${\\pi ^{(-i)}}$ , Eq.",
"(REF ) is equivalent with $\\begin{small}\\begin{aligned}\\vspace{-20.0pt}\\min _{{\\pi ^{(i)}}}\\max _{{D}^{(i)}} -\\lambda H({\\pi ^{(i)}}) + \\mathbb {E}_{\\pi _E}\\left[ \\log {{D}^{(i)}(s,{a}^{(i)}, {a}^{(-i)})}\\right]+\\mathbb {E}_{{\\pi ^{(i)}}, {\\pi ^{(-i)}}} \\left[\\alpha \\log {(1-{D}^{(i)}(s,{a}^{(i)}, {a}^{(-i)}))}\\right]~,\\end{aligned}\\end{small}$ where $\\alpha = \\frac{\\rho _{{\\pi ^{(i)}}, \\pi ^{(-i)}_E}(s, {a}^{(i)}, {a}^{(-i)})}{\\rho _{{\\pi ^{(i)}}, {\\pi ^{(-i)}}}(s, {a}^{(i)}, {a}^{(-i)})}$ is the importance sampling weight.",
"In practice, it is challenging to estimate the densities and the learning methods might suffer from large variance.",
"Thus, we fix $\\alpha =1$ in our implementation, and as the experimental results have shown, it has no significant influences on performance.",
"Besides, a similar approach can be found in [20].",
"So far, we've built a multi-agent imitation learning framework, which can be easily generalized to correlated or non-correlated policy settings.",
"No prior has to be considered in advance since the discriminator is able to learn the implicit goal for each agent."
],
[
"Learn with the Opponents Model",
"With the objective shown in Eq.",
"(REF ), demonstrated interactions can be imitated by updating discriminators to offer surrogate rewards and learning their policies alternately.",
"Formally, the update of discriminator for each agent $i$ can be expressed as: $\\begin{aligned}\\vspace{-20.0pt}\\nabla _{\\omega }J_D(\\omega ) =& \\mathbb {E}_{s \\sim P, {a}^{(-i)}\\sim {\\pi ^{(-i)}}}\\left[\\int _{{a}^{(i)}} \\pi _{\\theta }^{(i)}({a}^{(i)} | s, {a}^{(-i)}) \\nabla _{\\omega }\\log {(1-D_{\\omega }^{(i)}(s,{a}^{(i)},{a}^{(-i)}))} \\mathop {}\\!\\mathrm {d}{{a}^{(i)}}\\right] \\\\&+\\mathbb {E}_{(s,{a}^{(i)},{a}^{(-i)}) \\sim \\Omega _E}\\left[\\nabla _{\\omega } \\log {D_{\\omega }^{(i)}(s,{a}^{(i)},{a}^{(-i)})}\\right]~,\\end{aligned}$ and the update of policy is: $\\begin{aligned}\\vspace{-20.0pt}\\nabla _{\\theta }J_{\\pi }(\\theta ) = \\mathbb {E}_{s \\sim P, {a}^{(-i)}\\sim {\\pi ^{(-i)}}}\\left[\\nabla _{{\\theta }^{(i)}}\\int _{{a}^{(i)}} \\pi _{\\theta }^{(i)}({a}^{(i)} | s, {a}^{(-i)}) {A}^{(i)}(s, {a}^{(i)}, {a}^{(-i)})\\mathop {}\\!\\mathrm {d}{{a}^{(i)}}\\right]-\\lambda \\nabla _{{\\theta }^{(i)}}H(\\pi _{\\theta }^{(i)})~,\\end{aligned}$ where discriminator ${D}^{(i)}$ is parametrized by $\\omega $ , and the policy ${\\pi }^{(i)}$ is parametrized by $\\theta $ .",
"It is worth noting that the agent $i$ considers opponents' action ${a}^{(-i)}$ while updating its policy and discriminator, with integrating all its possible decisions to find the optimal response.",
"However, it is unrealistic to have the access to opponent joint policy $\\pi ({a}^{(-i)} | s)$ for agent $i$ .",
"Thus, it is possible to estimate opponents' actions via approximating ${\\pi ^{(-i)}}({a}^{(-i)} | s)$ using opponent modeling.",
"To that end, we construct a function ${\\sigma }^{(i)}({a}^{(-i)} | s): \\mathcal {S}\\times {\\mathcal {A}}^{(1)} \\times \\cdots \\times {\\mathcal {A}}^{(i-1)} \\times {\\mathcal {A}}^{(i+1)} \\times \\cdots \\times {\\mathcal {A}}^{(N)} \\rightarrow {[0, 1]}^{N-1}$ , as the approximation of opponents for each agent $i$ .",
"Then we rewrite Eq.",
"(REF ) and Eq.",
"(REF ) as: $\\begin{aligned}\\vspace{-20.0pt}\\nabla _{\\omega }J_D(\\omega ) &\\approx \\mathbb {E}_{s \\sim P, {\\hat{a}}^{(-i)}\\sim {\\sigma }^{(i)}, {a}^{(i)}\\sim \\pi _{\\theta }^{(i)}}\\left[ \\nabla _{{\\omega }^{(i)}}\\log (1-D_{\\omega }^{(i)}(s,{a}^{(i)},{\\hat{a}}^{(-i)})) \\right]\\\\&+\\mathbb {E}_{(s,{a}^{(i)},{a}^{(-i)}) \\sim \\Omega _E}\\left[\\nabla _{\\omega } \\log {D_{\\omega }^{(i)}(s,{a}^{(i)},{a}^{(-i)})}\\right]\\end{aligned}$ and $\\begin{aligned}\\vspace{-20.0pt}\\nabla _{\\theta }J_{\\pi }(\\theta ) \\approx \\mathbb {E}_{s \\sim P, {\\hat{a}}^{(-i)}\\sim {\\sigma }^{(i)}, {a}^{(i)}\\sim \\pi _{\\theta }^{(i)}}\\left[\\nabla _{{\\theta }^{(i)}}\\log {\\pi _{\\theta }^{(i)}({a}^{(i)} | s, {\\hat{a}}^{(-i)})} {A}^{(i)}(s, {a}^{(i)}, {\\hat{a}}^{(-i)}) \\right]-\\lambda \\nabla _{{\\theta }^{(i)}}H(\\pi _{\\theta }^{(i)})~\\end{aligned}$ respectively.",
"Therefore, each agent $i$ must infer the opponents model ${\\sigma }^{(i)}$ to approximate the unobservable policies ${\\pi ^{(-i)}}$ , which can be achieved via supervised learning.",
"Specifically, we learn in discrete action space by minimizing a cross-entropy (CE) loss, and a mean-square-error (MSE) loss in continuous action space: $\\begin{aligned}\\vspace{-20.0pt}L ={\\left\\lbrace \\begin{array}{ll}\\frac{1}{2}\\mathbb {E}_{s\\sim p}\\left[ \\left\\Vert {\\sigma }^{(i)}({a}^{(-i)} | s) - {\\pi ^{(-i)}}({a}^{(-i)} | s) \\right\\Vert ^2\\right],~&\\text{continuous action space} \\\\\\mathbb {E}_{s\\sim p}\\left[{\\pi ^{(-i)}}({a}^{(-i)} | s)\\log {{\\sigma }^{(i)}({a}^{(-i)} | s)}\\right],~&\\text{discrete action space}.\\end{array}\\right.",
"}\\end{aligned}$ With opponents modeling, agents are able to be trained in a fully decentralized manner.",
"We name our algorithm as Decentralized Adversarial Imitation Learning with Correlated policies (Correlated DAIL, a.k.a.",
"CoDAIL) and present the training procedure in Appendix Algo.",
"REF , which can be easily scaled to a distributed algorithm.",
"As a comparison, we also present a non-correlated DAIL algorithm with non-correlated policy assumption in Appendix Algo.",
"REF ."
],
[
"Theoretical Analysis",
"In this section, we prove that the reinforcement learning objective against demonstrator counterparts shown in the last section is essentially equivalent to reaching an $\\epsilon $ -NE.",
"Since we fix the policies of agents $-i$ as ${\\pi _E^{(-i)}}$ , the RL procedure mentioned in Eq.",
"(REF ) can be regarded as a single-agent RL problem.",
"Similarly, with a fixed ${\\pi _E^{(-i)}}$ , the IRL process of Eq.",
"(REF ) is cast to a single-agent IRL problem, which recovers an optimal reward function $r_*^{(i)}$ which achieves the best performance following the joint action $\\pi _E$ .",
"Thus we have $\\begin{aligned}\\vspace{-20.0pt}{\\operatorname{RL}}^{(i)}(r_*^{(i)}) &= \\operatornamewithlimits{arg\\,max}_{{\\pi ^{(i)}}} \\lambda H({\\pi ^{(i)}}) + \\mathbb {E}_{{\\pi ^{(i)}}, {\\pi _E^{(-i)}}}[r^{(i)}(s, a^{(i)}, a^{(-i)})] \\\\&= {\\pi _E^{(i)}}~.\\end{aligned}$ We can also rewrite Eq.",
"(REF ) as $\\begin{aligned}\\vspace{-20.0pt}\\lambda H({\\pi _E^{(i)}}) + \\mathbb {E}_{{\\pi _E^{(i)}}, {\\pi _E^{(-i)}}}[r^{(i)}(s, a^{(i)}, a^{(-i)})] \\ge \\lambda H({\\pi ^{(i)}}) + \\mathbb {E}_{{\\pi ^{(i)}}, {\\pi _E^{(-i)}}}[r^{(i)}(s, a^{(i)}, a^{(-i)})]\\end{aligned}$ for all ${\\pi ^{(i)}}\\in {\\Pi }^{(i)}$ , which is equivalent to $\\vspace{-20.0pt}&\\mathbb {E}_{a_t^{(i)}\\sim {\\pi _E^{(i)}}, a_t^{(-i)}\\sim {\\pi _E^{(-i)}}, s_0=s} \\left[ \\sum _{t=0}^\\infty \\gamma ^t r_*^{(i)}(s_t, a_t^{(i)}, a_t^{(-i)})\\right] \\ge \\\\&\\mathbb {E}_{a_t^{(i)}\\sim {\\pi ^{(i)}}, a_t^{(-i)}\\sim {\\pi _E^{(-i)}}, s_0=s} \\left[ \\sum _{t=0}^\\infty \\gamma ^t r_*^{(i)}(s_t, a_t^{(i)}, a_t^{(-i)})\\right]+ \\lambda (H({\\pi ^{(i)}}) - H({\\pi _E^{(i)}})), \\forall {\\pi ^{(i)}}\\in {\\Pi }^{(i)}~.",
"\\nonumber $ Given the value function defined in Eq.",
"(REF ) for each agent $i$ , for $H({\\pi ^{(i)}})-H({\\pi _E^{(i)}}) < 0$ , $\\forall {\\pi ^{(i)}}\\in {\\Pi }^{(i)}$ , we have $\\begin{aligned}\\vspace{-20.0pt}{v}^{(i)}(s, {\\pi _E^{(i)}}, {\\pi _E^{(-i)}}) & \\ge {v}^{(i)}(s, {\\pi ^{(i)}}, {\\pi _E^{(-i)}})- \\lambda (H({\\pi _E^{(i)}})-H({\\pi ^{(i)}}))~.\\end{aligned}$ For $H({\\pi ^{(i)}})-H({\\pi _E^{(i)}}) \\ge 0$ , $\\forall {\\pi ^{(i)}}\\in {\\Pi }^{(i)}$ we have $\\begin{aligned}\\vspace{-20.0pt}{v}^{(i)}(s, {\\pi _E^{(i)}}, {\\pi _E^{(-i)}})& \\ge {v}^{(i)}(s, {\\pi ^{(i)}}, {\\pi _E^{(-i)}}) + \\lambda (H({\\pi ^{(i)}})-H({\\pi _E^{(i)}}))\\\\& \\ge {v}^{(i)}(s, {\\pi ^{(i)}}, {\\pi _E^{(-i)}}) - \\lambda (H({\\pi ^{(i)}})-H({\\pi _E^{(i)}}))~.\\end{aligned}$ Let $\\epsilon =\\lambda \\max \\left\\lbrace \\big | H({\\pi ^{(i)}})-H({\\pi _E^{(i)}}) \\big |, \\forall {\\pi ^{(i)}}\\in {\\Pi }^{(i)} \\right\\rbrace $ , then we finally obtain $\\begin{aligned}\\vspace{-20.0pt}{v}^{(i)}(s, {\\pi _E^{(i)}}, {\\pi _E^{(-i)}})\\ge {v}^{(i)}(s, {\\pi ^{(i)}}, {\\pi _E^{(-i)}}) - \\epsilon , \\forall {\\pi ^{(i)}}\\in {\\Pi }^{(i)}~,\\end{aligned}$ which is exactly the $\\epsilon $ -NE defined in Definition REF .",
"We can always prove that $\\epsilon $ is bounded in small values such that the $\\epsilon $ -NE solution concept is meaningful.",
"Generally, random policies that keep vast entropy are not always considered as sub-optimal solutions or demonstrated policies ${\\pi _E^{(i)}}$ in most reinforcement learning environments.",
"As we do not require those random policies, we can remove them from the candidate policy set ${\\Pi }^{(i)}$ , which indicates that $H({\\pi ^{(i)}})$ is bounded in small values, so as $\\epsilon $ .",
"Empirically, we adopt a small $\\lambda $ , and attain the demonstrator policy $\\pi _E$ with an efficient learning algorithm to become a close-to-optimal solution.",
"Thus, we conclude that the objective of our CoDAIL assumes that demonstrated policies institute an $\\epsilon $ -NE solution concept (but not necessarily unique) that can be controlled the hyperparameter $\\lambda $ under some specific reward function, from which the agent learns a policy.",
"It is worth noting that [46] claimed that NE is incompatible with maximum entropy inverse reinforcement learning (MaxEnt IRL) because NE assumes that the agent never takes sub-optimal actions.",
"Nevertheless, we prove that given demonstrator opponents, the multi-agent MaxEnt IRL defined in Eq.",
"(REF ) is equivalent to finding an $\\epsilon $ -NE."
],
[
"Related Work",
"Albeit non-correlated policy learning guided by a centralized critic has shown excellent properties in couple of methods, including MADDPG [28], COMA [10], MA Soft-Q [43], it lacks in modeling complex interactions because its decisions making relies on the independent policy assumption which only considers private observations while ignores the impact of opponent behaviors.",
"To behave more rational, agents must take other agents into consideration, which leads to the studies of opponent modeling [1] where an agent models how its opponents behave based on the interaction history when making decisions [9], [14], [44], [41].",
"For multi-agent imitation learning, however, prior works fail to learn from complicated demonstrations, and many of them are bounded with particular reward assumptions.",
"For instance, [3] proposed Parameter Sharing Generative Adversarial Imitation Learning (PS-GAIL) that adopts parameter sharing trick to extend GAIL to handle multi-agent problems directly, but it does not utilize the properties of Markov games with strong constraints on the action space and the reward function.",
"Besides, there are many works built-in Markov games that are restricted under tabular representation and known dynamics but with specific prior of reward structures, as fully cooperative games [2], [22], [40], [6], two-player zero-sum games [25], two-player general-sum games [26], and linear combinations of specific features [34], [42].",
"Recently, some researchers take advantage of GAIL to solve Markov games.",
"Inspired by a specific choice of Lagrange multipliers for a constraint optimization problem [46], [39] derived a performance gap for multi-agent from NE.",
"It proposed multi-agent GAIL (MA-GAIL), where they formulated the reward function for each agent using private actions and observations.",
"As an improvement, [46] presented a multi-agent adversarial inverse reinforcement learning (MA-AIRL) based on logistic stochastic best response equilibrium and MaxEnt IRL.",
"However, both of them are inadequate to model agent interactions with correlated policies with independent discriminators.",
"By contrast, our approach can generalize correlated policies to model the interactions from demonstrations and employ a fully decentralized training procedure without to get access to know the specific opponent policies.",
"Except for the way of modeling multi-agent interactions as recovering agents' policies from demonstrations, which can regenerate similar interacted data, some other works consider different effects of interactions.",
"[15] proposed to learn a policy representation function of the agents based on their interactions and sets of generalization tasks using the learned policy embeddings.",
"They regarded interactions as the episodes that contain only $k$ (in the paper they used 2 agents), which constructs an agent-interaction graph.",
"Different from us, they focused on the potential relationships among agents to help characterize agent behaviors.",
"Besides, [21] and [12] proposed to use the Dynamic Bayesian Model that describes physical relationships among vehicles and driving behaviors to model interaction-dependent behaviors in autonomous driving scenario.",
"Correlated policy structures that can help agents consider the influence of other agents usually need opponents modeling [1] to infer others' actions.",
"Opponent modeling has a rich history in MAL [4], [11], and lots of researches have recently worked out various useful approaches for different settings in deep MARL, e.g., DRON [17] and ROMMEO [41].",
"In this paper, we focus on imitation learning with correlated policies, and we choose a natural and straightforward idea of opponent modeling that learning opponents' policies in the way of supervised learning with historical trajectories.",
"Opponent models are used both in the training and the execution stages."
],
[
"Environment Description",
"We test our method on the Particle World Environments [28], which is a popular benchmark for evaluating multi-agent algorithms, including several cooperative and competitive tasks.",
"Specifically, we consider two cooperative scenarios and two competitive ones as follows: 1) Cooperative-communication, with 2 agents and 3 landmarks, where an unmovable speaker knowing the goal, cooperates with a listener to reach a particular landmarks who achieves the goal only through the message from the speaker; 2) Cooperative-navigation, with 3 agents and 3 landmarks, where agents must cooperate via physical actions and it requires each agent to reach one landmark while avoiding collisions; 3) Keep-away, with 1 agent, 1 adversary and 1 landmark, where the agent has to get close to the landmark, while the adversary is rewarded by pushing away the agent from the landmark without knowing the target; 4) Predator-prey, with 1 prey agent with 3 adversary predators, where the slower predactor agents must cooperate to chase the prey agent that moves faster and try to run away from the adversaries."
],
[
"Experimental Details",
"We aim to compare the quality of interactions modeling in different aspects.",
"To obtain the interacted demonstrations sampled from correlated policies, we train the demonstrator agent via a MARL learning algorithm with opponents modeling to regard others' policies into one's decision making, since the ground-truth reward in those simulated environments is accessible.",
"Specifically, we modify the multi-agent version ACKTR [45], [39], an efficient model-free policy gradient algorithm, by keeping an auxiliary opponents model and a conditioned policy for each agent, which can transform the original centralized on-policy learning algorithm to be decentralized.",
"Note that we do not necessarily need experts that can do well in our designated environments.",
"Instead, any demonstrator will be treated as it is from an $\\epsilon $ -NE strategy concept under some unknown reward functions, which will be recovered by the discriminator.",
"In our training procedure, we first obtain demonstrator policies induced by the ground-truth rewards and then generate demonstrations, i.e., the interactions data for imitation training.",
"Then we train the agents through the surrogate rewards from discriminators.",
"We compare CoDAIL with MA-AIRL, MA-GAIL, non-correlated DAIL (NC-DAIL) (the only difference between MA-GAIL and NC-DAIL is whether the reward function depends on joint actions or individual action) and a random agent.",
"We do not apply any prior to the reward structure for all tasks to let the discriminator learn the implicit goals.",
"All training procedures are pre-trained via behavior cloning to reduce the sample complexity, and we use 200 episodes of demonstrations, each with a maximum of 50 timesteps.",
"Table: Average reward gaps between demonstrators and learned agents in 2 cooperative tasks.",
"Means and standard deviations are taken across different random seeds.Table: Average reward gaps between demonstrators and learned agents in 2 competitive tasks, where `agent+' and `agent-' represent 2 teams of agents and `total' is their sum.",
"Means and standard deviations are taken across different random seeds.Table: KL divergence of learned agents position distribution and demonstrators position distribution from an individual perspective in different scenarios.",
"`Total' is the KL divergence for state-action pairs of all agents, and `Per' is the averaged KL divergence of each agent.",
"Experiments are conducted under the same random seed.",
"Note that unmovable agents are not recorded since they never move from the start point, and there is only one movable agent in Cooperative-communication."
],
[
"Reward Gap",
"Tab.",
"REF and Tab.",
"REF show the averaged absolute differences of reward for learned agents compared to the demonstrators in cooperative and competitive tasks, respectively.",
"The learned interactions are considered superior if there are smaller reward gaps.",
"Since cooperative tasks are reward-sharing, we show only a group reward for each task in Tab.",
"REF .",
"Compared to the baselines, CoDAIL achieves smaller gaps in both cooperative and competitive tasks, which suggests that our algorithm has a robust imitation learning capability of modeling the demonstrated interactions.",
"It is also worth noting that CoDAIL achieves higher performance gaps in competitive tasks than cooperative ones, for which we think that conflict goals motivate more complicated interactions than a shared goal.",
"Besides, MA-GAIL and NC-DAIL are about the same, indicating that less important is the surrogate reward structure on these multi-agent scenarios.",
"To our surprise, MA-AIRL does not perform well in some environments, and even fails in Predator-prey.",
"We list the raw obtained rewards in Appendix , and we provide more hyperparameter sensitivity results in Appendix .",
"Figure: The density and marginal distribution of agent positions, in 100 repeated episodes with different initialized states, generated from different learned policies upon Keep-away.",
"The top row of each sub-figure is drawn from state-action pairs of all agents.",
"Meanwhile, the bottom row explains for each individual (KL means the KL divergence between generated interactions shown in the top row and the demonstrators)."
],
[
"Divergence over Interactions",
"Since we aim to recover the interactions of agents generated by the learned policies, it is proper to evaluate the relevance between distributions of regenerated interactions and demonstration data.",
"Specifically, we collect positions of agents over hundreds of state-action tuples, which can be regarded as the low-dimension projection of the state-action interactions.",
"We start each episode from a different initial state but the same for each algorithm in one episode.",
"We run all the experiments under the same random seed, and collect positions of each agent in the total 100 episodes, each with a maximum of 50 timesteps.",
"We first estimate the distribution of position $(x,y)$ via Kernel Density Estimation (KDE) [35] with Gaussian kernel to compute the Kullback-Leibler (KL) divergence between the generated interactions with the demonstrated ones, shown in Tab.",
"REF .",
"It is evident that in terms of the KL divergence between regenerated interactions with demonstrator interactions, CoDAIL generates the interaction data that obtains the minimum gap with the demonstration interaction, and highly outperforms other baseline methods.",
"Besides, MA-GAIL and NC-DAIL reflect about-the-same performance to model complex interactions, while MA-AIRL behaves the worst, even worse than random agents on Predator-prey."
],
[
"Visualizations of Interactions",
"To further understand the interactions generated by learned policies compared with the demonstrators, we visualize the interactions for demonstrator policies and all learned ones.",
"We plot the density distribution of positions, $(x,y)$ and marginal distributions of $x$ -position and $y$ -position.",
"We illustrate the results conducted on Keep-away in Fig.",
"REF , other scenarios can be found in the Appendix .",
"Higher frequency positions in collected data are colored darker in the plane, and higher the value with respect to its marginal distributions.",
"As shown in Fig.",
"REF , the interaction densities of demonstrators and CoDAIL agents are highly similar (and with the smallest KL divergence), which tend to walk in the right-down side.",
"In contrast, other learned agents fail to recover the demonstrator interactions.",
"It is worth noting that even different policies can interact to earn similar rewards, but still keep vast differences among their generated interactions.",
"Furthermore, such a result reminds us that the real reward is not the best metric to evaluate the quality of modeling the demonstrated interactions or imitation learning [24]."
],
[
"Conclusion",
"In this paper, we focus on modeling complex multi-agent interactions via imitation learning on demonstration data.",
"We develop a decentralized adversarial imitation learning algorithm with correlated policies (CoDAIL) with approximated opponents modeling.",
"CoDAIL allows for decentralized training and execution and is more capable of modeling correlated interactions from demonstrations shown by multi-dimensional comparisons against other state-of-the-art multi-agent imitation learning methods on several experiment scenarios.",
"In the future, we will consider covering more imitation learning tasks and modeling the latent variables of policies for diverse multi-agent imitation learning."
],
[
"Acknowledgement",
"We sincerely thank Yaodong Yang for helpful discussion.",
"The corresponding author Weinan Zhang is supported by NSFC (61702327, 61772333, 61632017).",
"The author Minghuan Liu is supported by Wu Wen Jun Honorary Doctoral Scholarship, AI Institute, Shanghai Jiao Tong University."
],
[
"CoDAIL Algorithm",
"Algo.",
"REF demonstrates the outline for our CoDAIL algorithm with non-correlated policy structure defined in Eq.",
"(REF ), where we approximate the opponents model ${\\sigma }^{(i)}({a}^{(-i)} | s)$ , improve the discriminator ${D}^{(i)}$ and the policy ${\\pi ^{(i)}}$ iteratively.",
"[] CoDAIL Algorithm [1] Input: Expert interactive demonstrations $\\Omega _E \\sim \\pi _E$ , $N$ policy parameters ${\\theta }^{(1)}, ..., {\\theta }^{(N)}$ , $N$ value parameters ${\\phi }^{(1)}, ..., {\\phi }^{(N)}$ , $N$ opponents models parameters ${\\psi }^{(1)}, ..., {\\psi }^{(N)}$ and $N$ discriminator parameters ${\\omega }^{(1)}, ..., {\\omega }^{(N)}$ ; $k = 0, 1, 2, \\cdots $ Sample interactions among $N$ agents $\\Omega _k \\sim \\pi $ , where $\\pi $ is in correlated policy structure and each agent $i$ utilizes its opponents model ${\\sigma }^{(i)}$ to help making decisions.",
"agent $i = 1, 2, \\cdots , N$ Use state-action pairs $(s, {a}^{(-i)}) \\in \\Omega _k$ to update ${\\psi }^{(i)}$ to minimize the objective as shown in Eq.",
"(REF ).",
"For every state-action pair $(s, {a}^{(i)}) \\in \\Omega _k$ , sample estimated opponent policies from opponents model: ${\\hat{a}}^{(-i)} \\sim {\\sigma }^{(i)}({a}^{(i)}|s)$ , and update ${\\omega }^{(i)}$ with the gradient as shown in Eq.",
"(REF ).",
"Compute advantage estimation ${A}^{(i)}$ for each tuple $(s,{a}^{(i)}, ,{\\hat{a}}^{(-i)})$ with surrogate reward function ${r}^{(i)}(s,{a}^{(i)}, {\\hat{a}}^{(-i)}) = \\log (D^{(i)}_{{\\omega }^{(i)}}(s,{a}^{(i)},{\\hat{a}}^{(-i)}))-\\log (1-D^{(i)}_{{\\omega }^{(i)}}(s,{a}^{(i)},{\\hat{a}}^{(-i)}))$ ${A}^{(i)}(s_t,a_{t}^{(i)},\\hat{a}_{t}^{(-i)}) &= \\sum _{k=0}^{T-1}(\\gamma ^t {r}^{(i)}(s_{t+k},a_{t+k}^{(i)}, \\hat{a}_{t+k}^{(-i)})) + \\gamma ^{T}V_{{\\phi }^{(i)}}^{(i)}(s, a_{T-1}^{(-i)}) \\\\&- V_{{\\phi }^{(i)}}^{(i)}(s, a_{t-1}^{(-i)})$ Update ${\\phi }^{(i)}$ to minimize the objective: $L({\\phi }^{(i)}) = \\left\\Vert \\sum _{t=0}^{T}\\gamma ^t {r}^{(i)}(s,a_t^{(i)}, a_t^{(-i)}) - {\\hat{V}}^{(i)}(s_t,a_{t-1}^{(-i)}) \\right\\Vert ^2$ Update ${\\theta }^{(i)}$ following the gradient shown in Eq.",
"(REF ): $\\mathbb {E}_{s \\sim P, {\\hat{a}}^{(-i)}\\sim {\\sigma }^{(i)}, {a}^{(i)}\\sim \\pi _{\\theta }^{(i)}}\\left[\\nabla _{{\\theta }^{(i)}}\\log {\\pi _{\\theta }^{(i)}({a}^{(i)} | s, {\\hat{a}}^{(-i)})} {A}^{(i)}(s, {a}^{(i)}, {\\hat{a}}^{(-i)}) \\right]-\\lambda \\nabla _{{\\theta }^{(i)}}H(\\pi _{\\theta }^{(i)})$"
],
[
"NC-DAIL Algorithm",
"We outline the step by step NC-DAIL algorithm with non-correlated decomposition of joint policy defined in Eq.",
"(REF ) in Algo.",
"REF .",
"[] NC-DAIL Algorithm [1] Input: Expert interactive demonstrations $\\Omega _E \\sim \\pi _E$ , $N$ policy parameters ${\\theta }^{(1)}, ..., {\\theta }^{(N)}$ , $N$ value parameters ${\\phi }^{(1)}, ..., {\\phi }^{(N)}$ and $N$ discriminator parameters ${\\omega }^{(1)}, ..., {\\omega }^{(N)}$ ; $k = 0, 1, 2, \\cdots $ Sample interactions between $N$ agents $\\Omega _k \\sim \\pi $ , where $\\pi $ is in non-correlated policy structure.",
"agent $i = 1, 2, \\cdots , N$ Use $(s,{a}^{(i)},{a}^{(-i)}) \\in \\Omega _k$ to update ${\\omega }^{(i)}$ with the gradient: $\\hat{\\mathbb {E}}_{\\Omega _k}\\left[\\nabla _{\\omega }\\log (D^{(i)}_{{\\omega }^{(i)}}(s,{a}^{(i)},{a}^{(-i)}))\\right] + \\hat{\\mathbb {E}}_{\\Omega _E}[\\nabla _{{\\omega }^{(i)}} \\log (D^{(i)}_{{\\omega }^{(i)}}(s,{a}^{(i)},{a}^{(-i)}))]~.$ Compute advantage estimation ${A}^{(i)}$ for $(s,{a}^{(i)},{a}^{(-i)}) \\in \\Omega _k$ with surrogate reward function ${r}^{(i)}(s,{a}^{(i)}, {a}^{(-i)}) = \\log (D^{(i)}_{{\\omega }^{(i)}}(s,{a}^{(i)},{a}^{(-i)}))-\\log (1-D^{(i)}_{{\\omega }^{(i)}}(s,{a}^{(i)},{a}^{(-i)}))$ ${A}^{(i)}(s_t,a_{t}^{(i)},a_{t}^{(-i)}) &= \\sum _{k=0}^{T-1}(\\gamma ^t {r}^{(i)}(s_{t+k},a_{t+k}^{(i)}, a_{t+k}^{(-i)})) + \\gamma ^{T}V_{{\\phi }^{(i)}}^{(i)}(s, a_{T-1}^{(-i)}) \\\\&- V_{{\\phi }^{(i)}}^{(i)}(s, a_{t-1}^{(-i)})$ Update ${\\phi }^{(i)}$ to minimize the objective: $L({\\phi }^{(i)}) = \\left\\Vert \\sum _{t=0}^{T}\\gamma ^t {r}^{(i)}(s,a_t^{(i)}, a_t^{(-i)}) - {\\hat{V}}^{(i)}(s_t,a_{t-1}^{(-i)}) \\right\\Vert ^2$ Update ${\\theta }^{(i)}$ by taking a gradient step with: $\\hat{\\mathbb {E}}_{\\Omega _k}\\left[ \\nabla _{{\\theta }^{(i)}}\\log {\\pi ^{(i)}}({a}^{(i)},s){A}^{(i)}(s,a) \\right] - \\lambda \\nabla _{{\\theta }^{(i)}}H(\\pi _{\\theta }^{(i)})~.$"
],
[
"Model Architectures",
"During our experiments, we use two layer MLPs with 128 cells in each layer, for policy networks, value networks, discriminator networks and opponents model networks on all scenarios.",
"Specifically for opponents models, we utilize a multi-head-structure network, where each head predicts each opponent's action separately, and we get the overall opponents joint action ${a}^{(-i)}$ by concatenating all actions.",
"The batch size is set to 1000.",
"The policy is trained using K-FAC optimizer [29] with learning rate of 0.1 and with a small $\\lambda $ of 0.05.",
"All other parameters for K-FAC optimizer are the same in [45].",
"We train each algorithm for 55000 epochs with 5 random seeds to gain its average performance on all environments."
],
[
"Raw Results",
"We list the raw obtained rewards of all algorithms in each scenarios.",
"Table: Raw average total rewards in 2 comparative tasks.",
"Means and standard deviations are taken across different random seeds.Table: Raw average rewards of each agent in 2 competitive tasks, where agent+ and agent- represent 2 teams of agents and total is their sum.",
"Means and standard deviations are taken across different random seeds."
],
[
"Hyperparameter Sensitivity",
"We evaluate how the stability of our algorithm when the hyperparameters change during our experiments on Communication-navigation.",
"Tab.",
"REF shows the total reward difference between learned agents and demonstrators when we modify the training frequency of $D$ and $G$ (i.e., the policy), which indicates that the frequencies of $D$ and $G$ are more stable when $D$ is trained slower than $G$ , and the result reaches a relative better performance when the frequency is 1:2 or 1:1.",
"Fig.",
"REF illustrates that the choice of $\\lambda $ has little effect on the total performance.",
"The reason may be derived from the discrete action space in this environment, where the policy entropy changes gently."
],
[
"Interaction Visualizations upon Other Scenarios",
"We show the density of interactions for different methods along with demonstrator policies conducted upon Cooperative-communication in Fig.",
"REF .",
"Figure: The density and marginal distribution of agents’ positions, (x, y), in 100 repeated episodes with different initialized states, generated from different learned policies upon Cooperative-communication.",
"Experiments are done under the same random seed, and we only consider one movable agent.",
"KL is the KL divergence between generated interactions (top figure) with the demonstrators.We show the density of interactions for different methods along with demonstrator policies conducted upon Cooperative-navigation in Fig.",
"REF .",
"Figure: The density and marginal distribution of agents’ positions, (x, y), in 100 repeated episodes with different initialized states, generated from different learned policies upon Cooperative-navigation.",
"Experiments are done under the same random seed.",
"The top of each sub-figure is drawn from state-action pairs of all agents while the below explain for each one.",
"KL is the KL divergence between generated interactions (top figure) with the demonstrators.We show the density of interactions for different methods along with demonstrator policies conducted upon Predator-prey in Fig.",
"REF .",
"Figure: The density and marginal distributions of agents’ positions, (x, y), in 100 repeated episodes with different initialized states, generated from different learned policies upon Predator-prey.",
"Experiments are conducted under the same random seed.",
"The top of each sub-figure is drawn from state-action pairs of all agents while the below explains for each one.",
"The KL term means the KL divergence between generated interactions (top figure) with the demonstrators."
]
] | 2001.03415 |
[
[
"Detrending Exoplanetary Transit Light Curves with Long Short-Term Memory\n Networks"
],
[
"Abstract The precise derivation of transit depths from transit light curves is a key component for measuring exoplanet transit spectra, and henceforth for the study of exoplanet atmospheres.",
"However, it is still deeply affected by various kinds of systematic errors and noise.",
"In this paper we propose a new detrending method by reconstructing the stellar flux baseline during transit time.",
"We train a probabilistic Long Short-Term Memory (LSTM) network to predict the next data point of the light curve during the out-of-transit, and use this model to reconstruct a transit-free light curve - i.e.",
"including only the systematics - during the in-transit.",
"By making no assumption about the instrument, and using only the transit ephemeris, this provides a general way to correct the systematics and perform a subsequent transit fit.",
"The name of the proposed model is TLCD-LSTM, standing for Transit Light Curve Detrending LSTM.",
"Here we present the first results on data from six transit observations of HD 189733b with the IRAC camera on board the Spitzer Space Telescope, and discuss some of its possible further applications."
],
[
"Introduction",
"Since the first exoplanet atmosphere observation twenty years ago [5], more than 3000 transiting extrasolar planets have been discovered.",
"Transit spectroscopy - i.e.",
"multi-wavelength transit observations - has opened the way for the characterization of atmospheric content and properties of exoplanets.",
"In effect, this can be done by first reconstructing the transmission or emission spectrum from the transit depth measurements at various wavelengths, and at a typical precision level of just a few parts-per-million (ppm) for hot gaseous planets.",
"This is to be contrasted with the imprints left in the stellar light curve by various instrumental and astrophysical effects which make the measurement of the transit depths extremely challenging.",
"Given the shift of the field towards increasingly smaller planets, the need for efficient detrending methods is thus ever growing.",
"Here we present a long- short term memory (LSTM) neural network approach to effectively model and detrend instrument and astrophysical systematics in transit light curves.",
"The total flux $F(t)$ received by a detector at time $t$ can be broken down as follows: Star flux: $F_s(t)$ Planetary signal: $\\delta (t) = (R_P(t)/R_S)^2$ in the case of primary transit obstruction with no limb darkening, where $R_S$ is the stellar radius and $R_P$ the apparent planetary radius Background stars and transient events: $F_b(t)$ Noise and instrumental systematics: $G(.",
")$ The total flux received by each pixel of the detector can then be written as $F(t) = G\\big ( (1-\\delta (t))F_s(t) + F_b(t)\\big )$ , where $F_s$ and $F_b$ may vary depending on the position on the detector and are then subject to instrumental systematics.",
"We will refer to individual pixel time series as pixel light curves, and to the summed contribution of pixels over time as a raw light curve.",
"Essentially, the main instrumental systematics trend observed both with the Hubble WFC3 and the Spitzer IRAC cameras are the so-called ramp effect [16], hypothesized to be due to the charge trapping in the detector [1], and intra-pixel and inter-pixel variations which are correlated with the position of the source on the detector which shows variations in quantum efficiency across different pixelsThis effect has been described in the IRAC instrument handbook: http://irsa.ipac.caltech.edu/data/SPITZER/docs/irac.",
"Footprints of these entangled variability sources can be found in additional instrumental data collected besides the detector raw flux.",
"In particular, the center and scale of the stellar point spread function (PSF) can be processed to give valuable information on the systematics while being mostly uncorrelated with the planetary signal itself.",
"Considering the analysis of time-correlated light curves with the end goal of detrending transit light curves and extracting the transit parameters as precisely as possible, one can approach the problem in several ways.",
"Indeed, the disentanglement of various independent signals might naturally guide one toward blind source separation techniques, which have been applied on this problem ([28], [19], [18]) using the pixel light curves as correlated components.",
"In a complementary way, signal processing analysis techniques have also been used to denoise the raw or pixel light curves, with Gaussian processes [8], pixel level decorrelation [7] or wavelet analysis ([3], [26], [18]).",
"Here we choose the angle of interpolation, i.e.",
"we want to provide predictions for the raw light curve during the transit time provided the out-of transit parts of the light curves.",
"The interpolation method we propose is non-linear and thus capable of capturing complex long term dependencies in the light curve.",
"The use of artificial neural networks (ANNs) is burgeoning in various fields including Astronomy.",
"In particular, [6] presented one of the first use of recurrent neural networks (RNNs) in astronomy for supernovae classification.",
"Yet, in the subfield of exoplanetary sciences, only a few studies have been using ANNs so far, with namely [13] who predicted stellar and planetary parameters from Kepler light curves using RNNs and representation learning, [30] on exoplanetary spectra retrieval, [24], [2], [21] for the supervised classification of transit candidates and [9], [29] for planet detection in direct imaging.",
"Here we make use of a long short-term memory (LSTM) neural network [14] to interpolate the flux of a raw light curve during the transit, given additional time-series data coming from the PSF centroid.",
"The LSTM network learns to predict the next value of the light curve at each time step.",
"The predictions of future time steps are then performed in a probabilistic manner using ancestral sampling, i.e.",
"by injecting the current prediction as input to the subsequent prediction and so on.",
"We thus assume that the pre-transit and post-transit information, along with additional data such as centroid time-series, are sufficient to predict the flux that the detector would have received in the absence of a planet transit.",
"This paper is organised as follows: Section contains background information about neural networks, Section presents the interpolating model and how it can be used for transit light curve fitting, and finally Section is dedicated to an application on Spitzer data."
],
[
"Recurrent Networks and LSTMs",
"In a typical supervised statistical learning task, the goal is to learn a model $h(x) \\simeq y$ that maps an input $x$ to an output $y$Note that $x$ and $y$ can be scalars, or more generally $n$ -dimensional vectors.",
"given examples of pairs $(x,y)$ in such a way that the expected error of future predictions is minimized.",
"Feed-forward neural networks or multi-layer perceptrons (MLPs) represent the simplest architecture of deep neural networks.",
"An example of this type of architecture is shown on Figure REF .",
"No feedback connections exist in these models.",
"Every layer consists of a set of neurons and the neurons of the input layer represent each of the original input variables $x$ .",
"The output of each neuron is a scalar value and is used as input for the neurons of the next layer.",
"Each subsequent layer transforms a linear co mbination of the outputs of the neurons of the previous layer using an activation function $\\sigma $ : $h_{l+1} = \\sigma (W_lh_{l} +b_l)$ where $W_l$ is a matrix of multiplicative weights, $b_l$ the bias vector, $h_{l}$ the vector of units and $\\sigma _l$ the activation function, all at layer $l$ .",
"If we interchangeably write $h_l$ for the function represented at layer $l$ as well as its output, the full function represented by a feed-forward network can then be written: $y = h_D(h_{D-1}(...h_1(X)))$ where $D$ is the depth of the network.",
"Note that the non-linearity of at least one of the layers activation functions is key to obtaining a non-linear predictor.",
"Figure: An example of a feed-forward neural network with 2 hidden layers.",
"When evaluating the underlying function hh, the information is flowing from the input on the left towards the output layer on the right.The main characteristic of Recurrent Neural Networks is that they allow for recurrent connectionsThis means that – unlike in feed-forward neural networks – in RNNs the output of neurons from one layer can be used as input for neurons of the same or a previous layer..",
"If we consider an input sequence $\\lbrace x_1, x_2...\\rbrace $ of vectors, a recurrent hidden layer will thus process it sequentially, receiving at step $t$ both the input $x_t$ as well as other previous hidden state(s) in order to compute the current state $h_t$ .",
"A typical example is shown on Figure REF , where the recurrence occurs between the hidden units of the same layer: $h_t=h_t(x_t,h_{t-1})$ .",
"Figure: An example of a recurrent neural network with no output from .Compared to MLPs, RNNs allow us to reduce the number of parameters of the network by sharing weights between time-steps while seeking temporal patterns in the data.",
"In practice, several more sophisticated recurrent architectures are often more effective than basic RNNs, with most being variants of the long short-term memory [14] architecture whose cell is shown in Figure REF .",
"LSTM networks have proven successful in a large range of applications including unconstrained handwriting recognition [11], speech recognition [12], machine translation [25], to cite only a few.",
"An LSTM cell contains four different gates (see Figure REF ), allowing the network to either retain or forget information from the past of the input sequence.",
"This enables the relevant long-term time dependencies to be picked up more easily.",
"The main addition in LSTMs compared to the basic RNNs has been to introduce self-loops, which are conditioned on the context and controlled by the gates.",
"Below we state the detailed update formulae for the gates and states composing each LSTM unit: The input gate: $i_t = W_{ix}x_t + W_{ih}h_{t-1}+b_i $ The forget gate: $f_t = W_{fx}x_t + W_{fh}h_{t-1}+b_f $ The output gate: $ o_t = W_{ox}x_t + W_{oh}h_{t-1}+b_o $ The cell state: $ c_t = \\sigma (f_t) \\odot c_{t-1} + \\sigma (i_t) \\odot \\tanh (j_t)$ The output vector: $ h_t = \\sigma (o_t) \\odot \\tanh (c_t) $ Where $t$ denotes the time step, $W_{ab}$ the matrix of weights relative to the vectors $a$ and $b$ , $b_a$ the bias vector relative to $a$ , $\\odot $ the Hadamart (i.e.",
"entrywise) product and $\\sigma $ is the activation function, typically a logistic sigmoid or $\\tanh $ function.",
"Incidentally, these types of gated RNNs also have the advantage of being easier to train than basic RNNs, by alleviating the well known vanishing or exploding gradient issue Neural networks are trained via gradient-based minimization of a loss function.",
"In each iteration of training, each parameter of the model (weight) receives an update proportional to the partial derivative of the loss w.r.t the current weight.",
"Allowing these gradients to grow vanishingly small or too large can cause numerical instabilities, slow down training or stop it prematurely. [17].",
"Figure: An LSTM cell from , which replaces a usual hidden unit (i.e.",
"neuron) in a feed-forward neural network.",
"The input, forget and output gating units enable the cell to accumulate or shut off respectively the current input, long-term dependencies and output through a sigmoidal activation function.",
"The square here indicates a delay of one-time-step, and operation symbols in the circles indicate the operation involving the gates' outputs."
],
[
"TLCD-LSTM",
"Here we describe the proposed model to interpolate a time-series on a pre-defined prediction range.",
"As the final goal of this paper is to study the transit signal contained in the interpolation range after correction of the systematic errors, we name the method Transit Light Curve Detrending LSTM (TLCD-LSTM).",
"The model is based on the deep auto-regressive neural network model described in [23].",
"It assumes that temporal relations exist in the time-series and learns to predict the next step in the training range of the input time-series.",
"It can also make use of additional data available for prediction contained in the so-called covariate time-series, which is to be distinguished from the main time-series.",
"In general, one can consider both the main and covariate time-series to be multi-variate, i.e.",
"to be composed of several time-series each.",
"TLCD-LSTM is specifically adapted for interpolation within a given range, and therefore differs from [23] mainly in that the values it tries to predict are not in the future (i.e.",
"the end of the time-series) but in timesteps somewhere within the time-series."
],
[
"Model description",
"Let us denote with $\\lbrace x_1, x_2, ..,x_T\\rbrace $ (abbreviated $\\lbrace x_t\\rbrace $ ) the main time-series of length $T$ we ought to interpolate on the prediction range $[t_1 .. t_2]$ with $t_1$ and $t_2$ integers in $[1..T]$ , and $\\lbrace z_1, z_2, ..,z_T\\rbrace $ (abbreviated $\\lbrace z_t\\rbrace $ ) the time-series of covariates, which constitute additional data available for prediction on the whole time range.",
"Finally, let us also denote with $\\lbrace y_1, y_2, ..,y_T\\rbrace $ (abbreviated $\\lbrace y_t\\rbrace $ ) the target time-series, identical to the main time-series in the training range but which may differ in the prediction range.",
"In the case of $\\lbrace x_t\\rbrace $ being a transit light curve, $\\lbrace y_t\\rbrace $ is the hypothetical light curve without any transit signal.",
"As sketched in Figure REF , each value of the input time-series passes through a stack of LSTM layers, the output of which branches into two distinct feed-forward layers outputting two parameters $\\widehat{\\mu _t}$ and $\\widehat{\\sigma _t}$ at each time-step, which are the predicted mean and standard deviation for the distribution of the current value $x_t$ , respectively.",
"The details and hyperparameters of the architecture are presented in Appendix .",
"Figure: Sketch of the interpolating probabilistic LSTM neural network.",
"The main and covariate time-series are processed through three LSTM layers consisting of 256 units each, and then decoded into two outputs for each of the interpolated points: the mean and the standard deviation.The same network is used both for the training and prediction ranges with only the inputs differing in each case.",
"Table: Differences of inputs at each time step tt between the training and prediction ranges.At each timestep $t$ , the network predicts the current value $x_{t}$ from all past timesteps $x_1,..,x_{t-1}$ as well as from the current covariate $z_{t}$ .",
"While the actual previous time-series value $x_{t-1}$ is used as input in the training ranges, in the prediction range the previous prediction $\\mu _{t-1}$ is injected as an input instead of it (see Table REF ).",
"We assume each value $z_t$ is sampled from a normal distribution: $z_t \\sim \\mathcal {N}(\\mu ,\\,\\sigma ^{2})$ The loss function is then computed as the product of individual likelihoods outside the prediction range: $\\mathcal {L}(\\mu _{t_1..t_2}) = \\prod _{t\\in [1..t_1]\\cup [t_2..T]} \\frac{e^{(x_t-\\widehat{\\mu }_t)^2/2\\widehat{\\sigma }_t}}{\\sqrt{2\\pi \\sigma _t}}$ Note that the log-loss is only computed in the training ranges.",
"However, the last output of the prediction range is taken as the first input of the second training range, thus providing a way to link together the outputs in the different ranges."
],
[
"Predicting the time-series",
"There are several ways one can generate predictions in $[t_1..t_2]$ , once a model is trained.",
"Since the inputs of the network consist of parameters of a probability distribution, the simplest one is to directly take the vector of predicted means $\\widehat{y_t} = \\mu _t$ .",
"However, one can also generate a trace by drawing every value from the Gaussian distribution at every timestep in the prediction range: $\\widehat{y_t}\\sim \\mathcal {N}(\\mu _t,\\,\\sigma _t^{2})$ , and injecting each of these predictions as input for the next time step.",
"Multiple traces obtained with this process then represent the joint predicted distribution (of which they are samples) in a more general way than merely using the means vector.",
"To generate a single vector of predictions from multiple traces, one can – for instance – select the median or mean value at every timestep to construct the median trace or mean trace on the prediction range.",
"In Section , we focus on the simplest approach, i.e.",
"selecting the output means and standard deviations."
],
[
"Covariant Features",
"The covariates time-series $\\lbrace z_t\\rbrace $ can consist of single-dimensional or multi-dimensional data available both in the training and prediction ranges.",
"It is used by the network as additional information besides the target time-series.",
"This works merely by concatenating $x_t$ (conversely $\\widehat{x_t}$ in prediction mode) to the covariate data $z_t$ to construct the new input to the network at every timestep.",
"Ideally, one wants $\\lbrace z_t\\rbrace $ to be correlated with the target time-series.",
"Several time-series might be related to the time-correlated noise we intend to correct, and therefore can be used as covariate data in the model.",
"In the application presented in Section we suggest the use of PSF-related time-series, namely the instrument's point spread function (PSF) centers and widths of a 2D Gaussian fit on the images at every time step.",
"One could also think of other potentially relevant information such as simultaneous host star activity, calibration data relative to the detector and estimations of background flux.",
"For ground-based applications, information about airmass, seeing and weather patterns could be included."
],
[
"Application to transit light curves",
"Here we discuss the use of the interpolating model specifically to transit light curves.",
"The transit signal must be contained within the prediction range.",
"This requires either to know beforehand when the transit occurs, or to adapt the prediction range during the first phase of the training.",
"Pre-transit and post-transit data are used for training the network, and are assumed to not be contaminated by any transit event.",
"They can however contain any sort of variability coming from the star, the background or the instrument.",
"In fact, the model aims at picking up variations due to all sources other than a transit event in order to predict the flux due to these sources alone during the transit time.",
"We perform a transit fit at each evaluation step even though our model does not strictly require it for the training.",
"This is done for two main reasons: 1) The transit fit can be used as a proxy to evaluate the quality of the prediction and provide us with a criterion for early-stopping the training of our model.",
"The transit fit is performed on the detrended light curve normalized with respect to the star $(1-\\delta _t)$ .",
"For details, see Appendix .",
"2) We can use the transit fit to adapt the prediction range $[t_1..t_2]$ during training so that it matches better the actual transit range of the data.",
"This can be done by extracting the fitted mid-transit time and transit duration to compute the times for the beginning and end of transit."
],
[
"Application",
"We present an application to 6 transit observations of planet HD 189733 b from the Spitzer/IRAC detector at 8 $\\mu $ m, collected in 2007 and 2008 (PI: E. Agol, program: 40238).",
"This hot-Jupiter planet has been extensively studied and makes a good candidate for bench-marking our method.",
"In this wavelength channel, the ramp effect can be heavily pronounced [1], while the intra-pixel variations due to pointing jitter are less important than at shorter wavelengths.",
"A few preprocessing steps are applied to the dataThe data used are publicly available and were downloaded from: https://sha.ipac.caltech.edu/applications/Spitzer/SHA/ and detailed in Appendix .",
"These include outlier removal, raw light curve extraction and normalization, centroids fitting and background light estimation.",
"In this section, we present predictions on the pre-transit range (on intervals not used in the training of the model) as an initial evaluation of the model, and then show results on the real transit ranges on which we derive the detrended light curve and subsequent transit fit.",
"Figure: (Top) 6 Spitzer/IRAC 8μm\\mu m raw transit light curves of HD 189733 b after preprocessing.",
"(Bottom) X/Y centroid positions of the point spread function."
],
[
"Testing ",
"As the ground truth, i.e.",
"the predicted stellar and instrumental flux, is not available in the transit range, we chose to first test the interpolating model on the pre-transit range instead, where its predictions can be evaluated more directly.",
"In practice, three prediction ranges are selected in the first 250 timesteps of the time-series, where no transit signal is present, and the mean squared error (MSE) metric is used to evaluate and compare various models.",
"An example of prediction is shown on Figure REF , where the prediction is obtained by averaging 50 sampled traces.",
"Figure: Example of interpolations on 3 light curves containing no transit.",
"The prediction range is located inside the vertical dashed lines.",
"The raw light curve is displayed in blue, and the predicted traces in grey and the median prediction in orange.We perform a grid search over different types of inputs and hyperparameters.",
"More specifically, we vary the aperture width of the sub-array used for computing the raw light curve between 5 and 7 pixels; we experiment with including and excluding covariate features, namely: 1) excluding covariate features altogether; 2) including centroid time-series, and 3) including centroid and PSF width time-series.",
"Furthermore, we vary the number of layers (between 1 and 4), units per layer (powers of 2 up to 1024 and dropout rate (between $0\\%$ and $50\\%$ in steps of $1\\%$ ) Dropout is a common regularization technique in deep learning consisting in randomly reinitializing a fraction of the neurons of a given layer.",
"The dropout rate refers to this fraction.",
"values for the LSTM block; and a unidirectional or bidirectional networkBy `unidirectional' network we mean one that uses just past timesteps to infer the current one.",
"With `bidirectional' we mean using timesteps from both past and future to infer the current one.. We train each different model on the 6 light curves and 3 different prediction ranges, monitoring the average MSE for these 18 predictions and using it as a criterion for early stopping and comparison between the different models.",
"From these tests we observe the following: Including the centroids information improves the quality of the prediction by a factor of $\\sim 2$ , and including the PSF widths time-series besides the centroids brings a further increase in MSE.",
"Dropping $3\\%$ of the recurrent units improves slightly the predictions, especially when the number of parameters of the network increases.",
"Using a bidirectional network slightly decreases the quality of predictions.",
"More information on the hyperparameters and model training used is presented in Appendix ."
],
[
"Performance",
"We present in Table REF the results of the best tested model in the explored grid.",
"As a reference for the performance of the interpolation, we include a baseline model, which is a linear composition of the centroid X/Y time-series $\\lbrace z^X_t\\rbrace $ and $\\lbrace z^Y_t\\rbrace $ : $ \\widehat{y_t} = a + bz^X_t + cz^Y_t $ where $a,b,c \\in \\mathbb {R}$ .",
"The model is trained on the training rangesThe model was fitted using scikit-learn's linear regression module: https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html and evaluated in the prediction ranges.",
"The metrics computed for both models include the MSE, the mean-absolute error (MAE) and the mean signal-to-noise (SNR) ratio, defined as: $MSE = \\frac{1}{N}\\sum _{t_1}^{t_2}{(\\widehat{y_t} - y_t)^2.",
"}$ $MAE = \\frac{1}{N}\\sum _{t_1}^{t_2}{|\\widehat{y_t} - y_t|.",
"}$ $SNR = \\frac{1}{N}\\sum _{t_1}^{t_2}{|\\widehat{y_t} - y_t|/\\sigma _{noise}.",
"}$ where N is the number of observations, and $\\sigma _{noise}$ an estimate of the noise level computed by taking the mean value of the running standard deviation of width 15 over each input light curve.",
"Table: Comparison of performance on the 6 light curves for each model.",
"Every value is averaged between prediction and actual value over 3 different ranges of length 60 and starting respectively at timeseteps 80, 100 and 120.",
"The three last lines show the mean performance over all light curves and ranges in terms of MSE, MAE and SNR.Given its simplicity, this baseline model does a remarkably good job at interpolating $\\lbrace x_t\\rbrace $ , and this is why it was chosen here as a reference for the MSE.",
"Furthermore, since the TLCD-LSTM also uses the centroid time-series, the increase in performance seen on Table REF can directly be interpreted as the improvement brought by the LSTM's ability to identify temporal dependencies in the raw light curve.",
"Using the optimised hyperparameters listed in REF and after training the model for 3000 epochs we extract the output of the network for the whole time ranges, shown in red on Figure REF .",
"Note that the decreasing learning rate used guarantees the convergence of the network towards a stable solution.",
"Visually, the model seems to be able to pick up the trends and variability of each time-series, while joining smoothly the pre and post transit ranges where the ground truth is known.",
"Figure: Raw data (blue) and model output, i.e.",
"interpolated light curve in the absence of transit (red) for the light curves.",
"Dashed vertical lines indicate the initial prediction ranges.The last step is to perform the transit fit on the detrended light curve $\\lbrace 1-\\delta _t\\rbrace $ normalized with respect to the stellar flux $F_s(t)$ .",
"Since the limb darkening effect is minor at 8 $\\mu $ m, we chose a transit model with linear limb darkening (bound between $0.05$ and $0.25$ ), and compute the best fit using a Markov Chain Monte Carlo optimization procedureThe transit model was fitted using PylightCurve package: https://github.com/ucl-exoplanets/pylightcurve [27].",
"The fitted parameters are $R_p/R_s$ , the mid-time transit time $t_c$ , a linear limb darkening coefficient $u$ , the orbit inclination $i$ and orbital semi-major axis relatively to the stellar radius $a/R_s$ .",
"The fitted model, residuals and auto-correlated functions (ACF) are shown in Figure REF and the fitted parameters are presented in Table REF .",
"The higher variance present in the residuals of the $5th$ lightcurve is due to a higher noise level in the input data for this light curve.",
"Figure: (Top) Best Transit Fit (red curve) to the detrended light curve (blue points) normalized with respect to the stellar flux.",
"(Center) Fit residuals (blue points) along with the moving average (red curve) and standard deviation ( orange) of the residuals.",
"(Bottom) Auto-Correlated Function of the residualsTable: Fitted physical parameters for each of the 6 transits.We compare the retrieved transit depths with the results published in [1] for the same data set and preprocessing steps (Figure REF ).",
"Although slightly smaller, the scatter of the predictions is still present with a standard deviation of $91.7$ ppm instead of 144ppm.",
"The mean weighted by the standard deviations of the 6 transit depths is also found to be slightly smaller in our case by 94ppm $\\approx 4\\sigma $ .",
"Figure: Comparison of fitted transit depths between this work and , for the six transit observations of HD189733b.",
"The horizontal lines show the means of the observations from both papers weighted by their respective standard deviations.",
"The dotted lines show the standard deviations of these weighted means."
],
[
"Discussion and conclusions",
"We presented a deep learning model suitable for interpolating time-series, and showed how it can be used to predict the variability of stellar light curves for subsequent transit fit.",
"This approach has the advantage of not making any assumption on the types of noise, systematics or transit shape.",
"The presented method is similar to the Gaussian Process (GP) approach ([8], [22]) in that they both construct highly non-linear models, avoid explicit physical modelling of the systematics and provide probabilistic predictions.",
"However, they differ in various aspects: 1) The neural network lightcurve interpolation approach we propose does not need any transit model whereas it is included in the kernel of the GP.",
"This makes the TLCD-LSTM approach more generally applicable as it does not depend on a pre-defined kernel function.",
"2) The GP approach requires fewer parameters to train and provide fully Bayesian predictions compared to our LSTM-based approach.",
"The smaller number of free parameters may make GPs the preferred choice for short time series.",
"However, GPs computation scales more poorly with the number of data points, preventing them to be applicable to datasets of more than $\\approx 1000$ time steps without binning of the time series.",
"The proposed interpolating LSTM can on the other hand be applied to longer or multiple light curves as commonly found in Kepler and TESS time series allowing for even very long period variability to be captured in the predictive LSTM model.",
"This is because the computational complexity in the case of GPs mainly depends on the number of data points, while in the case of the deep neural networks in the architecture chosen (i.e.",
"the number of layers, number of nodes per layer & type of layers in our case).",
"While the current implementation still relies on a few preprocessing steps such as computing the raw light curve or centroids fits, it constitutes a first step towards the ultimate goal of developing an end-to-end detrending pipeline where the input would be the raw pixel light curves or focal plane images.",
"Furthermore, while we trained our network on data from six real light curves only, taking advantage of a large number of light curves, real or simulated, would allow developing a more general detrending approach for each instrument.",
"LSTMs allow for efficient transfer learning between data sets and instruments (e.g.",
"Kepler to TESS).",
"This may become important in modelling common systematics such as stellar noise between planet-star systems observed by multiple instruments.",
"As we have firmly entered the era of `big data' in planet detection (e.g.",
"Kepler, TESS and ground based surveys) and with upcoming characterisation missions and instruments (e.g.",
"JWST, Ariel, CHEOPS and the ELTs), the opportunities for data detrending and modelling with scalable deep learning methods, capable of processing large numbers of high dimensional data will become increasingly prevalent in the future.",
"Software The data and code used in this paper are available on GitHub under a Creative Commons Attribution 4.0 International License (https://github.com/ucl-exoplanets/deepARTransit, archived on Zenodo) and a MIT License (https://github.com/ucl-exoplanets/pylightcurve).",
"Acknowledgements 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 758892, ExoAI), under the European Union's Seventh Framework Programme (FP7/2007-2013)/ ERC grant agreement numbers 617119 (ExoLights) and the European Union's Horizon 2020 COMPET programme (grant agreement No 776403, ExoplANETS A).",
"Furthermore, we acknowledge funding by the Science and Technology Funding Council (STFC) grants: ST/K502406/1, ST/P000282/1, ST/P002153/1 and ST/S002634/1."
],
[
"Transit fit",
"To obtain a light curve normalized with respect to the star, three steps are required: transformation to the original units $ y_t \\rightarrow y^{\\prime }_t$ , subtraction of background flux $F_b(t)$ , and division of the background subtracted raw light curve by the predicted star flux: $ 1-\\delta (t) = \\frac{F_{received}(t) - F_b(t)}{F_s(t)-F_b(t)} $ With the time-series notations where $x^{\\prime }_t$ and $\\hat{y}^{\\prime }_t$ are the input and mean prediction of the neural network in the original units: $ 1-\\delta _t = \\frac{x^{\\prime }_t - F_{b,t}}{\\hat{y}^{\\prime }_t - F_{b,t}}$ .",
"Note that during training, we use a simple piecewise-linear transit model with four parameters described in [4] optimized by least-square fitting and neglect the contribution of the background $F_b \\ll x^{\\prime }_t, \\hat{y}^{\\prime }_t$ ."
],
[
"Pre-processing",
"Here we describe the different preprocessing steps applied to the raw subarray data."
],
[
"Outlier removal",
"Due to a number of causes, such as remaining cosmic rays or bad pixels, the flux on individual pixels can exhibit great fluctuations within short timescales ($\\approx 1sec$ ).",
"These abnormal values are identified by computing the absolute difference of the pixels' flux with their corresponding median within a time window of width 5 ($2\\sec $ exposure).",
"The values of the median-subtracted time-series greater than 4$\\sigma $ are then replaced by the median values, where $\\sigma $ is the standard deviation of the time-series."
],
[
"Raw light curve extraction",
"In order to limit the influence of background light and focus on the brightest pixels of the stellar PSF, $3\\times 3$ , $5\\times 5$ and $7\\times 7$ pixel regions are extracted around the brightest pixel.",
"The raw light curve is then obtained by summing all the individual pixel light curves."
],
[
"Centroid fitting",
"As mentioned earlier the centroid position time-series are highly correlated with the flux received by the detector.",
"In order to compute the centroids, we perform a two-dimensional Gaussian fit with offset to the data at every timestep, and hence extract four useful time-series, two of which are monitoring the position of the center on the detector and two for the width of the Gaussian.",
"As discussed in [1], this method provides by far a better estimate of the centroids over other methods such as the flux-weighted ratio extraction."
],
[
"Background extraction",
"The background flux contribution to the total flux, although minor, increases with the aperture size used for the light curve extraction.",
"We estimate it here by taking the median flux value of the pixels located in the four corners of each frame, corners delimited by the complement of a circular aperture of radius 16.",
"It accounts for $0.67\\%$ to $1.2\\%$ in our analysis, and should therefore be taken into account.",
"However, as the background estimation is necessarily approximate, we advocate to still interpolate on the raw light curve directly, and only correct for it before the transit fit."
],
[
"Normalization",
"The raw light curve and centroid time-series are all locally standardized, i.e.",
"individually centered around a mean value of zero and rescaled to have their standard deviation equal to one.",
"The preprocessed raw light curves and centroid X/Y positions are shown on Figure REF .",
"Note the diversity of effects among them, showing more or less stochastic noise, ramps or jitter."
],
[
"Training parameters",
"Training was performed using the ADAM optimizer [15] with parameter values $\\beta _1 =0.9$ , $\\beta _2 =0.99$ , $\\epsilon =10−8$ .",
"The learning rate was decreased from 0.01 to 0.0001 using a polynomial decay law with exponent 20.",
"We train the model using a batch size of 6 (all the lightcurves) for faster training.",
"Table: Table of the network hyperparameters"
]
] | 2001.03370 |
[
[
"Co-evolution of language and agents in referential games"
],
[
"Abstract Referential games offer a grounded learning environment for neural agents which accounts for the fact that language is functionally used to communicate.",
"However, they do not take into account a second constraint considered to be fundamental for the shape of human language: that it must be learnable by new language learners.",
"Cogswell et al.",
"(2019) introduced cultural transmission within referential games through a changing population of agents to constrain the emerging language to be learnable.",
"However, the resulting languages remain inherently biased by the agents' underlying capabilities.",
"In this work, we introduce Language Transmission Engine to model both cultural and architectural evolution in a population of agents.",
"As our core contribution, we empirically show that the optimal situation is to take into account also the learning biases of the language learners and thus let language and agents co-evolve.",
"When we allow the agent population to evolve through architectural evolution, we achieve across the board improvements on all considered metrics and surpass the gains made with cultural transmission.",
"These results stress the importance of studying the underlying agent architecture and pave the way to investigate the co-evolution of language and agent in language emergence studies."
],
[
"Introduction",
"Human languages show a remarkable degree of structure and complexity.",
"In the evolution of this complex structure, several different intertwined pressures are assumed to have played a role.",
"The first of these pressures concerns the function of language: as language is to communicate, it should allow effective communication between proficient language users [34].",
"This pressure is strongly intertwined with the nature of the proficient user: what features of language allow effective communication depends on the abilities of the user to use the language.",
"A second pressure on the shape of human language stems from the fact that language must be learnable.",
"Unlike animal languages, which are taken to be mostly innate, human languages must be re-acquired by each individual [31], [12].",
"A language can only survive if it can successfully be transmitted to a next generation of learners.",
"In the field of language evolution, this transmission process is referred to as cultural transmission, while the process of change that occurs as a consequence is called cultural evolution.The importance of cultural evolution for the emergence of structure is supported by a number of artificial language learning studies [33] and computational studies using the Iterated Learning paradigm, in which agents learn a language by observing the output produced by another agent from the previous `generation' [15], [18], [20].",
"Like the pressures arising from the function of language, the way that cultural evolution shapes language also depends on the language users: what is learnable depends on the inductive biases of the learner.",
"Computationally, the emergence of language can be studied through simulation with artificial agents and by investigating the resulting languages for structure, level of compositionality, and morphosyntactic properties [17], [19].",
"Originally based on logic and symbolic representations [17], [5], with the advent of modern deep learning methods, there has been a renewed interest in simulating the emergence of language through neural network agents [23], [10].",
"Such work typically involves the use of referential games [25], in which two or more agents have to emerge a language to obtain a shared reward.",
"These studies are motivated by the first pressure: language as a tool for effective communication.",
"However, they fail to consider the second pressure: language must be learnable by new agents.",
"They also fail to study the impact of the learning biases of the artificial agents themselves, which underlies both pressures.",
"In a recent study, [6] proposed a method to include cultural evolution in a language emergence game.",
"Their approach is more naturally aligned with pressures in humans language evolution than single agent referential games [36], but fails to account for the fact that cultural evolution and the learning biases of the artificial agents are two sides of the same coin: what language is learnable depends on the learning biases of the learner.",
"In this paper, we will therefore integrate the three components described above – communication, cultural evolution and learning biases – and setup a framework in which their interaction can be studied.",
"This framework, which we refer to with the term Language Transmission Simulator, consists of a referential game, played by a changing population of agents – simulating cultural evolution – which are subject to architectural evolution – simulating the learning biases of the learners and allowing them to co-evolve with the language.",
"Our contributions are three-fold fold: We introduce the Language Transmission Simulator, that allows to model both cultural and architectural evolution in a population of agents; We collect a large number of tests from previous work and combine them into an extensive test suits for language emergence games; We demonstrate that emerging languages benefit from including cultural transmission as well as architectural evolution, but the best results are achieved when languages and agents can co-evolve."
],
[
"Related Work",
"Much work has been done on the emergence of language in artificial agents and investigating its subsequent structure, compositionality and morphosyntax [17], [19].",
"Originally, such work was based on logic and symbolic representations [17], [5], but with the advent of modern deep learning [24], there has been a renewed interest in simulating the emergence of language through neural network agents [8], [21], [4], [23], [10], [28].",
"In the exploration of language emergence, different training approaches and tasks have been proposed to encourage agents to learn and develop communication.",
"In a typical setup, two players aim to develop a communication protocol in which one agent must communicate information it has access to (typically an image), while the other must guess it out of a line-up [7], [23].",
"[21] show that `natural' language does not arise naturally in these communication games and it has to be incentivised by imposing specific restrictions on games and agents.",
"[10] first demonstrated that using straight-through estimators were more effective than reinforcement learning in a collaborative task, and that optimizing rewards can lead to structured protocols (i.e.",
"strings of symbols) to be induced from scratch.",
"[28] find that syntactic structure emerges in the stream of symbol uttered by agents, where symbols and syntax can be mapped to specific meanings or instructions.",
"[4] use qualitative analysis, visualization and a zero-shot test, to show that a language with compositional properties can emerge from environmental pressures.",
"[3] find that emerged languages, unlike human languages, do not naturally prefer non-redundant encodings.",
"[2] further find that while generalization capabilities can be found in the languages, compositionality itself does not arise from simple generalization pressures.",
"[32] encourage desirable properties of human languages, such as compositionality, to emerge through well-crafted auxiliary pressures.",
"Finally, [9] demonstrate with experiments on populations of agents that language can evolve and simplify through the interactions of different communities.",
"[6] build upon the emergent language research by introducing cultural transmission as a pressure in referential games.",
"They use a pool of agents with a resetting mechanism and show that this further encourages the emerging language to display compositional properties and structure allowing it to generalize better.",
"Pairing agents with one another in a larger population setting introduces cultural evolution, but it is the pressure introduced by the partial resetting which forces remaining agents to emerge a language that is quickly learnable by a new agent.",
"While [6] is the most related work to ours, an important difference is that they focus on cultural evolution only, without taking into account the learning biases of the agents via modelling architectural evolution."
],
[
"Approach",
"In this paper, we introduce architectural evolution in language emergence games and study the interaction between cultural and architectural evolution with a range of different metrics.",
"Below, we first give a definition of the referential game we consider (Subsection REF ).",
"We then briefly explain our Language Transmission Simulator (Section REF and how we model cultural and architectural evolution within it (Section REF and REF respectively)."
],
[
"Referential games",
"We study language emergence in a referential game inspired by the signalling games proposed by [25].",
"In this game, one agent (the sender) observes an image and generates a discrete message.",
"The other agent (the receiver) uses the message to select the right image from a set of images containing both the sender image and several distractor images.",
"Since the information shown to the sender agent is crucial to the receivers success, this setup urges the two agents to come up with a communication protocol that conveys the right information.",
"Formally, our referential game is similar to [10]: The meaning space of the game consists of a collection $D$ of $K$ images $\\lbrace d_0,d_1, ..., d_K\\rbrace $ , represented by $z$ -dimensional feature vectors.",
"In each round $i$ of the game, a target item $d_i$ is randomly sampled from $D$ , along with a set $C$ of $n$ distractor items.",
"The sender agent $s$ of the game, parametrised by a neural network, is given item $d_i$ , and generates a discrete message $m_i$ from a vocabulary $V$ .",
"The message is capped to a max message length of $L$ .",
"The receiver agent $r$ , also parametrised by a neural network, receives message $m_i$ and uses it to identify $d_i$ in the union of $d_i$ and $C$ .",
"We use $z=512$ , and $n=3$ and train agents with Gumbel-Softmax [13] based on task-success."
],
[
"Language Transmission Simulator",
"In Language Transmission Simulator, depicted in Figure REF , we simulate a population of communicating agents.",
"In every training iteration, two random agents are sampled to play the game.",
"This forces the agents to adopt a simpler language naturally: to succeed they must be able to communicate or understand all opposing agents.",
"In our setup, agents are either sender or receiver, they do not switch roles during their lifetime."
],
[
"Cultural evolution in referential games",
"Following [6], we simulate cultural evolution by periodically replacing agents in the population with newly initialised agents.",
"Cultural evolution is implicitly modelled in this setup, as new agents have to learn to communicate with agents that already master the task.",
"Following [6], we experiment with three different methods to select the agents that are replaced: randomly (no selection pressure), replacing the oldest agents or replacing the agents with the lowest fitness (as defined in Section REF ).",
"We call these setups cu-random, cu-age and cu-best, respectively.",
"Figure: The Language Transmission Simulator: Agent pairs are randomly sampled from each population and trained.",
"After ll training steps, a portion α\\alpha of the population is culled."
],
[
"Architectural evolution in referential games",
"To model architectural evolution, rather than periodically replacing agents with randomly initialised new agents, we instead mutate the most successful agents and replace the worst agents with variations of the best agents, as outlined in Section REF .",
"Note that cultural evolution is still implicitly modelled in this setup, as new agents still have to learn to communicate with older agents.",
"Therefore, we call this setup with the term co-evolution."
],
[
"Culling",
"We refer to the selection process and subsequent mutation or re-initialisation step as culling.",
"In biology, culling is the process of artificially removing organisms from a group to promote certain characteristics, so, in this case, culling consists of removing a subset of the worst agents and replacing them with variations of the best architecture.",
"The proportion of agents from each population selected to be mutated is determined by the culling rate $\\alpha $ , where $\\alpha \\in [0, 1)$ .",
"The culling interval $l$ defines the number of iterations between culling steps.",
"A formalisation of the LTE can be found in appendix REF ."
],
[
"Mutation Algorithm",
"Our mutation algorithm is an intentionally simple implementation of a Neural Architectural Search (NAS).",
"NAS focuses on searching the architecture space of networks, unlike many traditional evolutionary techniques which often include parameter weights in their search space.",
"We opted to use the DARTS (Differentiable Architecture Search) RNN search space defined by [27], which has obtained state-of-the-art performance on benchmark natural language tasks [26].",
"The DARTS search space includes recurrent cells with up to $N$ nodes, where each node $n_1, n_2,...,n_N$ can take the output of any preceding nodes including $n_0$ , which represents the cell's input.",
"All potential connections are modulated by an activation function, which can be the identity function, Tanh, Sigmoid or ReLU.",
"Following [27] and [30], we enhance each operation with a highway bypass [37] and the average of all intermediate nodes is treated as the cell output.",
"To sample the initial model, we sample a random cell with a single node ($N=1$ ).",
"As this node must necessarily be connected to the input, the only variation stems from the possible activation functions applied to the output of $n_1$ , resulting in four possible starting configurations.",
"We set a node cap of $N=8$ .",
"We mutate cells by randomly sampling an architecture which is one edit step away from the previous architecture.",
"Edit steps are uniformly sampled from i) changing an incoming connection, ii) changing an output operation or iii) adding a new node; the mutation location is uniformly sample from all possible mutations.For a formal description of the mutation process, we refer to Appendix REF ."
],
[
"Fitness Criterion",
"The fitness criterion that we use in both the cu-best and co-evolution setup is based on task performance.",
"However, rather than considering agents' performance right before the culling step, we consider the age of the youngest agent in the population (defined in terms of number of batches that it was trained) and for every agent compute their performance up until when they had that age.",
"For any agent $a_j$ in population $\\mathbf {A}$ this is defined as: $\\text{fitness}(a_j)= \\frac{1}{\\mathcal {T}_A}\\sum _{t=0}^{\\mathcal {T}_A} \\mathcal {L}(a_j^t)$ where $\\mathcal {T}_A = \\min _{a \\in A}\\mathcal {T}(a)$ is the age $\\mathcal {T}(a)$ of the youngest agent in the population, and $\\mathcal {L}(a_j^t)$ is the loss of agent $a_j$ at time step $t$ .",
"This fitness criterion is not biased towards older agents, that have seem already more data and have simply converged more.",
"It is thus not only considering task performance but also the speed at which this performance is reached."
],
[
"Experiments",
"We test the LTE framework on a compositionally defined image dataset, using a range of different selection mechanisms."
],
[
"Dataset",
"In all our experiments, we use a modified version of the Shapes dataset [1], which consists of 30 by 30 pixel images of 2D objects, characterised by shape (circle, square, triangle), colour (red, green, blue), and size (small, big).",
"While every image has a unique symbolic description – consisting of the shape, colour and size of the object and its horizontal and vertical position in a 3x3 grid – one symbolic representation maps to multiple images, that differ in terms of exact pixels and object location.",
"We use 80k, 8k, 40k images for train, validation and test sets, respectively.",
"Some example images are depicted in Figure REF .",
"Figure: The modified Shapes task consists of showing an image the sender, and then letting the receiver deduce from the sender's message which image out of the target and kk distractors is the correct one.We pre-train a CNN feature extractor for the images in a two-agent setting of the task (see Appendix REF for more details)."
],
[
"Architecture and Training",
"For our co-evolution experiments, we use the DARTS search space as described above.",
"For all cultural evolution approaches, we use an LSTM [11] for both the sender and receiver architecture (see Appendix REF for more details).",
"The sender and receiver models have a hidden size of 64 for the recurrent layer and an embedding layer of size 64.",
"Further, we use a vocabulary size $V$ of 4, with an additional bound token serving as the indicator for beginning and end-of-sequence.",
"We limit the maximum length of a sentence $L$ to 5.",
"The values for $V$ and $L$ were picked to provide a strong communication bottleneck to promote the emergence of structured and compressed languages, following the intuitions from [21] that natural language patterns do not emerge `naturally'.",
"We back-propagate gradients through the discrete step outputs (message) of the sender by using the Straight-Through (ST) Gumbel-Softmax Estimator [14].",
"We run all experiments with a fixed temperature $\\tau =1.2$ .",
"We use the default Pytorch [29] Adam [16] optimiser with a learning rate of 0.001 and a batch-size of 1024.",
"Note that the optimiser is reset for every batch.",
"For all multi-agent experiments we use a population size of 16 senders and 16 receivers.",
"The culling rate $\\alpha $ is set to $0.25$ or four agents, and we cull (re-initialise or mutate) every $l=5k$ iterations.",
"We run the experiments for a total of $I=500k$ iterations, and evaluate the populations before each culling step."
],
[
"Evaluation",
"We use an range of metrics to evaluate both the population of agents and the emerging languages."
],
[
"Jaccard Similarity",
"We measure the consistency of the emerged languages throughout the population using Jaccard Similarity, which is defined as the ratio between the size of the intersection and the union of two sets.",
"We sample 200 messages per input image for each possible sender-receiver pair and average the Jaccard Similarity of the samples over the population.",
"A high Jaccard Similarity between two messages is an indication that the same tokens are used in both messages."
],
[
"Proportion of Unique Matches",
"We compute how similar the messages that different agents emit for the same inputs by looking at all possible (sender, message) pairs for one input and assess whether they are the same.",
"This metric is 1 when all agents always emit the same messages for the same inputs."
],
[
"Number of Unique Messages",
"We compute the average number of unique messages generated by each sender in the population.",
"An intuitive reference point for this metric is the number of images with distinct symbolic representations.",
"If agents generate more messages than expected by this reference point, this demonstrates that they use multiple messages for the images that are – from a task perspective – identical."
],
[
"Topographic Similarity",
"Topographic similarity, used in a similar context by [22], represents the similarity between the meaning space (defined by the symbolic representations) and the signal space (the messages sent by an agent).",
"It is defined as the correlation between the distances between pairs in meaning space and the distances between the corresponding messages in the signal space.",
"We compute the topographic similarity for an agent by sampling 5,000 pairs of symbolic inputs and corresponding messages and compute the Pearson's $\\rho $ correlation between the cosine similarity of the one-hot encoded symbolic input pairs and the cosine similarity of the one-hot encoded message pairs."
],
[
"Average Population Convergence",
"To estimate the speed of learning of the agents in the population, estimate the average population convergence.",
"For each agent, at each point in time, this is defined as the agents average performance from the time it was born until it had the age of the current youngest agent in the population (analogous to the fitness criterion defined in Section REF ).",
"To get the average population convergence, we take we average those values for all agents in the population."
],
[
"Average Agent Entropy",
"We compute the average certainty of sender agents in their generation process by computing and averaging their entropy during generation."
],
[
"Results",
"We now present a detailed comparison of our cultural and co-evolution setups.",
"For each approach, we averaged over four random seeds, the error bars in all plots represent the standard deviation across these four runs.",
"To analyse the evolution of both agents and languages, we consider the development of all previously outlined metrics over time.",
"We then test the best converged languages and architectures in a single sender-receiver setup, to assess the impact of cultural and genetic evolution more independently.",
"In these experiments, we compare also directly to a single sender-receiver baseline, which is impossible for most of the metrics we consider in this paper.",
"Finally, we briefly consider the emerged architectures from a qualitative perspective."
],
[
"Task performance",
"We first confirm that all setups in fact converge to a solution to the task.",
"As can be seen in Figure REF , all populations converge to a (close to perfect) solution to the game.",
"The cu-age approach slightly outperforms the other approaches, with a accuracy that surpasses the $95\\%$ accuracy mark.",
"Note that, due to the ever changing population, the accuracy at any point in time is an average of both `children' and `adults', that communicate with different members of the population.",
"Figure: Average Population Accuracy of final populations."
],
[
"Analysis",
"In this section we analyse the resulting behaviour and success of agents in Language Transmission Simulator.",
"We first use standard approaches such as average agent entropy and loss (convergence) to measure the success of agents with respect to their language and the task.",
"Secondly, we use other metrics to analyse the emergent language itself in terms of consistency and diversity by using Jaccard Similarity, the proportion of unique matches, the number of unique messages, and the topographic similarity.",
"Thirdly, we perform a qualitative analysis of the architecture that emerge from our Language Transmission Simulator.",
"Finally, we design Frozen Experiments, in which we test the emerged languages and architectures in a 1v1 setting with a fresh agent.",
"This allows us to compare and measure the the improvement gains made by the architecture and those made by the language which emerged.",
"We show through these experiments that the co-evolution setting leads to a language that is both more successful and easier to learn for a given new agent."
],
[
"Agent behaviour",
"To assess the behaviour of the agents over time, we monitor their average message entropy convergence speed.",
"As can be seen in Figure REF , the co-evolution setup results in the lowest average entropy scores, the messages that they assign to one particular image will thus have lower variation than in the other setups.",
"Of the cultural evolution setups, the lowest entropy score is achieved in the cu-best setup.",
"Figure: Average agent entropy over time.Figure REF shows the average population convergence over time.",
"Also in this case, we observe a clear difference between cultural evolution only and co-evolution, with an immediately much lower convergence time for co-evolution and a slightly downward trending curve.",
"Figure: Average convergence for all cultural transmission modes and evolution."
],
[
"Language Analysis",
"To check the consistencies of languages within a population, we compare the Jaccard Similarity and the Average Proportion of Unique Matches, which we plot in Figure REF .",
"This shows that, compared to cultural evolution only, not only are the messages in co-evolution more similar across agents (higher Jaccard Similarity), but also that agents are considerably more aligned with respect to the same inputs (less unique matches).",
"Figure: Average Number of Unique Messages and Topographic Similarity for all cultural evolution modes and co-evolution.For comparison, we also plot the number of unique messages for a symbolic solution that fully encodes all relevant features of the image (since we have three possible shapes and colours, two possible sizes, and a 3×33 \\times 3 grid of possible positions, this symbolic reference solution has 3×3×2×9=1623 \\times 3 \\times 2 \\times 9 = 162 distinct messages.To assess the level of structure of the emerged languages, we plot the average Topographic Similarity and the Average Number of Unique Messages generated by all senders (Figure REF ).",
"The co-evolution condition again outperforms all cultural only conditions, with a simpler language (the number of the unique messages closer to the symbolic reference point) that is structurally more similar to the symbolic representation of the input (higher Topographical Similarity)."
],
[
"Architecture Analysis",
"In Figure REF we show the co-evolution of an agent and a sample of its language during three selected iterations in the co-evolution setup.",
"Strikingly, the best sender architecture does not evolve from its original form, which could point towards the limitations of of our search strategy and space.",
"On the contrary, the receiver goes through quite some evolution steps and converges into a significantly more complex architecture than its original form.",
"We observe a unification of language throughout evolution in Figure REF , which is also supported by Figure REF .",
"The population of senders starts out 11 different unique messages and ends with only two to describe the same input image.",
"We will leave more detailed analysis of the evolved architectures for future work."
],
[
"Frozen Experiments",
"With a series of experiments we test the a priori suitability of the evolved languages and agents for the task at hand, by monitoring the accuracy of new agents that are paired with converged agents and train them from scratch.",
"We focus, in particular, on training receivers with a frozen sender from different setups, which allows us to assess 1) whether cultural evolution made languages evolve to be more easily picked up by new agents 2) whether the genetic evolution made architectures converge more quickly when faced with this task.",
"We compare the accuracy development of: An LSTM receiver trained with a frozen sender taken from cu-best; An evolved receiver trained with a frozen evolved sender.",
"For both these experiments, we compare with two baselines: The performance of a receiver agent trained from scratch along with a receiver agent that has either the cu architecture or the evolved co architecture (cu-baseline and co-baseline, respectively); The performance of an agent trained with an agent that is pretrained in the single agent setup, with either the cu architecture or an evolved architecture (cu-baseline-pretrained and co-baseline-pretrained).",
"Each experiment is run 10 times, keeping the same frozen agent.",
"The results confirm cultural evolution contributes to the learnability and suitability of emerging languages: the cu-best accuracy (green line) converges substantially quicker and is substantially higher than the cu-baseline-pretrained accuracy (orange line).",
"Selective pressure on the language appears to be important: the resulting languages are only easier to learn in the cu-best setup.cu-age and cu-random are ommitted from the plot for clarity reasons.",
"In addition, they show that the agents benefit also from the genetic evolution: the best accuracies are achieved in the co-evolution setup (red line).",
"The difference between the cu-baseline (blue) and the co-baseline (brown) further shows that even if the evolved architectures are trained from scratch, they perform much better than a baseline model trained from scratch.",
"The difference between the co-baseline-pretrained (only genetic evolution, purple line) and the co-evolution of agents and language line (red line) illustrates that genetic evolution alone is not enough: while a new evolved receiver certainly benefits from learning from a (from scratch) pretrained evolved sender, without the cultural transmission pressure, it's performance is still substantially below a receiver that learns from an evolved sender whose language was evolved as well.",
"Figure: Receiver accuracies trained with different types of frozen senders."
],
[
"Conclusion",
"In this paper, we introduced a language transmission bottleneck in a referential game, where new agents have to learn the language by playing with more experienced agents.",
"To overcome such bottleneck, we enabled both the cultural evolution of language and the architectural evolution of agents, using a new Language Transmission Simulator.",
"Using a battery of metrics, we monitored their respective impact on communication efficiency, degree of linguistic structure and intra-population language homogeneity.",
"While we could find important differences in between cultural evolution strategies, it is when we included architectural evolution that agents scored best.",
"In a second experiment, we paired new agents with evolved languages and agents and again confirmed that, while cultural evolution makes a language easier to learn, co-evolution leads to the best communication.",
"In future research, we would like to apply the Language Transmission Simulator on new, more complex tasks and further increase our understanding of the properties of the emerged languages and architectures.",
"Recent research has also found that relaxing the vocabulary size $V$ and sequence length $L$ constraints can lead to greater syntactic structure in emergent languages [35].",
"We thus hope to investigate further relaxation of hyper-parameters and other neuro-evolution techniques in future work."
],
[
"Acknowledgements",
"We would like to thank Angeliki Lazaridou for her helpful discussions and feedback on previous iterations of this work."
],
[
"Language Transmission Engine",
"We formalise our Language Transmission process in the pseudo code shown in Algorithm REF .",
"We select hyper-parameters $l$ as the number of iterations or batches shown between culling steps, and $I$ as the total number of iterations.",
"Language Transmission Engine $S \\leftarrow \\lbrace s_0, s_1 ... ,s_N \\rbrace $ $R \\leftarrow \\lbrace r_0, r_1 ... ,r_N \\rbrace $ $i \\leftarrow 1$ $i \\le I$ batch $b$ in $D$ Sample $\\hat{s}$ from $S$ Sample $\\hat{r}$ from $R$ $\\mathbf {train}(\\hat{s}, \\hat{r}, b)$ $i \\bmod l = 0 $ $\\mathbf {cull}(S, R)$ $i \\leftarrow i+1$"
],
[
"Mutation Algorithms Pseudo-code",
"Genotype-level Mutation $\\mathbf {mg}$$genotype$ $g \\leftarrow copy(genotype)$ $a \\leftarrow \\mathcal {U}(1, 3)$ $n \\leftarrow \\mathcal {U}(1, len(g))$ $a = 1$ $p \\leftarrow \\mathcal {U}[ReLU, I, \\tanh , \\sigma ]$ $n.activation \\leftarrow p$ $a = 2$ $r \\leftarrow \\mathcal {U}(1, n)$ $n .connection \\leftarrow r$ $a = 3$ $n^\\prime \\leftarrow new\\_node()$ $p \\leftarrow \\mathcal {U}[ReLU, I, \\tanh , \\sigma ]$ $r \\leftarrow \\mathcal {U}(1, len(g))$ $n^\\prime .activation \\leftarrow p$ $n^\\prime .connection \\leftarrow r$ $g.append(n^\\prime )$ $g$ The genotype mutation is described in pseudo-code by algorithm REF , and takes as input a genotype containing nodes describing the cell.",
"The genotype is mutated by either changing the input connection or primitive (output activation function) for a randomly sampled node $n$ , or adding a new node altogether.",
"See section REF for explanations on the workings of the DARTS cell structure.",
"Population-level Mutation $\\mathbf {mutate}$$\\mathbf {P}$ $p^\\prime \\leftarrow \\operatornamewithlimits{arg\\,min}_{convergence}(P)$ $\\mathbf {p} \\leftarrow \\pi (P)$ $p_i$ in $\\mathbf {p}$ $p_i.genotype \\leftarrow \\mathbf {mg}(p^\\prime .genotype)$ In order to mutate a population $\\mathbf {P}$ using $\\pi $ as a replacement policy, we use the process outlined in algorithm REF ."
],
[
"Sender Architecture",
"The sender architecture comprises of a linear layer input mapping the input feature size (512) to the hidden size.",
"The image feature vector is therefore mapped to the same dimension as the RNN layer, where it is used as the initial hidden state.",
"When training, for each step of the sender RNN we apply the cell and use the straight-through Gumbel-Softmax trick to be able back-propagate gradients through the discrete message output.",
"During evaluation however, we sample the categorical distribution at each step to produce each token in the sentence."
],
[
"Receiver Architecture",
"The receiver architecture is simpler and takes as an input the message outputted by the sender and outputs a vector of input feature size (512).",
"A single embedding matrix is used to encode the sender's message.",
"During training the message is linearly transformed using the embedding matrix, while during the evaluation pass the discrete message outputs of the sender are used to map to the specific embedding dimensions.",
"The embedded message is then passed to the RNN layer, and the final state of the RNN is linearly mapped back to the feature size.",
"Doing so allows us to obtain a prediction for each image feature (distractors and true image), by comparing the alignment between the receiver output and the respective feature vectors."
],
[
"Feature Extraction",
"In order to obtain image features, we pre-trained a convolutional model on the task using the raw image as input.",
"Due to the input size requirements of the convolutional model, we resize the images linearly to be 128 by 128 (height, width) by 3 (RGB channels).",
"We used early stopping conditions on the validation accuracy, an embedding size of 256, and hidden size of 512.",
"The two agents are otherwise trained with the same parameters as other experiments: vocab size and max sentence length of 5, Adam optimizer with learning rate of 0.001.",
"For the visual module itself, we used a similar architecture to that in [4] albeit smaller.",
"We used a five-layer convolution network with 20 filters, and a kernel size and stride of 3 for all layers.",
"For every convolutional layer, ReLU activation was applied on the output, after a Batch normalization step with no bias parameter.",
"The linear layer which followed the convolutional layers had output dimensions of 512 and a ReLU activation function.",
"This allows us to obtain image features of size 512, which we then used for all experiments."
]
] | 2001.03361 |
[
[
"AutoDNNchip: An Automated DNN Chip Predictor and Builder for Both FPGAs\n and ASICs"
],
[
"Abstract Recent breakthroughs in Deep Neural Networks (DNNs) have fueled a growing demand for DNN chips.",
"However, designing DNN chips is non-trivial because: (1) mainstream DNNs have millions of parameters and operations; (2) the large design space due to the numerous design choices of dataflows, processing elements, memory hierarchy, etc.",
"; and (3) an algorithm/hardware co-design is needed to allow the same DNN functionality to have a different decomposition, which would require different hardware IPs to meet the application specifications.",
"Therefore, DNN chips take a long time to design and require cross-disciplinary experts.",
"To enable fast and effective DNN chip design, we propose AutoDNNchip - a DNN chip generator that can automatically generate both FPGA- and ASIC-based DNN chip implementation given DNNs from machine learning frameworks (e.g., PyTorch) for a designated application and dataset.",
"Specifically, AutoDNNchip consists of two integrated enablers: (1) a Chip Predictor, built on top of a graph-based accelerator representation, which can accurately and efficiently predict a DNN accelerator's energy, throughput, and area based on the DNN model parameters, hardware configuration, technology-based IPs, and platform constraints; and (2) a Chip Builder, which can automatically explore the design space of DNN chips (including IP selection, block configuration, resource balancing, etc.",
"), optimize chip design via the Chip Predictor, and then generate optimized synthesizable RTL to achieve the target design metrics.",
"Experimental results show that our Chip Predictor's predicted performance differs from real-measured ones by < 10% when validated using 15 DNN models and 4 platforms (edge-FPGA/TPU/GPU and ASIC).",
"Furthermore, accelerators generated by our AutoDNNchip can achieve better (up to 3.86X improvement) performance than that of expert-crafted state-of-the-art accelerators."
],
[
"Introduction",
" We have seen the rapid adoption of Deep Neural Networks (DNNs) for solving real-life problems, such as image classification , , object detection , natural language processing , etc.",
"Although DNNs enable high-quality inferences, they also require a large amount of computation and memory demand during deployment due to their inherently immense complexity , , , , .",
"Moreover, DNN-based applications often require not only high inference accuracy, but also aggressive hardware performance, including high throughput, low end-to-end latency, and limited energy consumption.",
"Recently, we have seen intensive studies on DNN accelerators in hardware, which attempt to take advantage of different hardware design styles, such as GPUs, FPGAs, and ASICs, to improve the speed and efficiency of DNN inference and training , , , , , , , , , , , , .",
"However, developing customized DNN accelerators presents significant challenges as it asks for cross-disciplinary knowledge in machine learning, micro-architecture, and physical chip design.",
"Specifically, to build accelerators on FPGAs or ASICs, it is inevitable to include (1) customized architectures for running DNN workloads, (2) RTL programming for implementing accelerator prototypes, and (3) reiterative verifications for validating the functionality correctness.",
"The whole task requires designers to have a deep understanding of both DNN algorithms and hardware design.",
"In response to the intense demands and challenges of designing DNN accelerators, we have seen rapid development of high-level synthesis (HLS) design flow , , , and DNN design automation frameworks , , , , , that improve the hardware design efficiency by allowing DNN accelerator design from high-level algorithmic descriptions and using pre-defined high-quality hardware IPs.",
"Still, they either rely on hardware experts to trim down the large design space (e.g., use pre-defined/fixed architecture templates and explore other factors , ) or conduct merely limited design exploration and optimization, hindering the development of optimal DNN accelerators that can be deployed into various platforms.",
"To address the challenges above, we propose AutoDNNchip, an end-to-end automation tool for generating optimized FPGA- and ASIC-based accelerators from machine learning frameworks (e.g., Pytorch/Tensorflow) and providing fast and accurate performance estimations of hardware accelerators implemented on various tar- geted devices.",
"The main contributions of this paper are as follows: [leftmargin=*] One-for-all Design Space Description.",
"We make use of a graph-based representation that can unify design factors in all of the three design abstraction levels (including IP, architecture, and hardware-mapping levels) of DNN accelerator design, allowing highly flexible architecture configuration, scalable architecture/IP/mapping co-optimization, and algorithm-adaptive accelerator design.",
"Chip Predictor.",
"Built on top of the above design space description, we propose a DNN Chip Predictor, a multi-grained performance estimation/simulation tool, which includes a coarse-grained, analytical-model based mode and a fine-grained, run-time-simulation based mode.",
"Experiments using 15 DNN models and 4 platforms (edge-FPGA/TPU/GPU and ASIC) show that our Chip Predictor's predicted error is within $10\\%$ of real-measured energy/latency/resource-consumption.",
"Chip Builder.",
"We further propose a DNN Chip Builder, which features a two-stage Design Space Exploration (DSE) methodology.",
"Specifically, our Chip Builder realizes: (1) an architecture/IP design based on the Chip Predictor's coarse-grained, analytical-model based prediction for a 1st-stage fast exploration and optimization, and (2) an IP/pipeline design based on the Chip Predictor's fine-grained, run-time-simulation based prediction as a 2nd-stage IP-pipeline co-optimization.",
"Experiments show that the Chip Builder's 1st-stage DSE can efficiently rule out infeasible choices, while its 2nd-stage co-optimization can effectively boost the performance of remaining design candidates, e.g., 36.46% throughput improvement and 2.4$\\times $ idle cycles reduction.",
"AutoDNNchip.",
"Integrating the aforementioned two enablers (i.e., Chip Predictor and Chip Builder), we develop AutoDNNchip, which can automatically generate optimized DNN accelerator implementation (i.e., synthesizable RTL implementation) given the user-defined DNN models from machine learning frameworks (e.g., Pytorch), application-driven specifications (e.g., energy and latency), and resource budget (e.g., size of the processing array and memories).",
"Experiments demonstrate that the optimized FPGA- and ASIC-based DNN accelerators generated by AutoDNNchip outperform the recent award-winning design by 11% and a state-of-the-art accelerator by up to 3.86$\\times $ .",
"As an automated DNN accelerator design tool, AutoDNNchip is the first to highlight all of the following features: (1) efficient and accurate performance prediction of DNN accelerators on 4 platforms, enabling fast optimal algorithm-to-accelerator mapping design and algorithm/accelerator co-design/co-optimization; (2) a design space description that unifies the descriptions of design factors from all of the three design abstraction levels in DNN accelerators into one directed graph, supporting arbitrary accelerator architectures (e.g., both homogeneous and heterogeneous IPs and their inter-connections), and (3) can automatically generate both FPGA- and ASIC-based DNN accelerator implementation that outperforms expert-crafted state-of-the-art designs for various applications."
],
[
"Background and related works",
" Figure: Overview of the proposed AutoDNNchip framework, which accepts user-defined DNN models/datasets and application-driven specifications to automatically generate optimized FPGA- or ASIC-based DNN accelerator designs.FPGA- and ASIC-based DNN Accelerators.",
"There has been intensive study in customized FPGA- and ASIC-based DNN accelerators.",
"The accelerator in uses loop tiling for accelerating convolutional layers on FPGAs.",
"The DNNBuilder accelerator applies an optimal resource allocation strategy, fine-grained layer-based pipeline, and column-based cache to deliver high-quality FPGA-based DNN accelerators.",
"The work in proposes a throughput-oriented accelerator with multiple levels (i.e., task, layer, loop, and operator levels) of parallelisms.",
"The recent designs in , introduce a hardware-efficient DNN and accelerator co-design strategy by considering both algorithm and hardware optimizations, using DNN building blocks (called Bundles) to capture hardware constraints.",
"For ASIC-based DNN accelerators, efforts have been made in both industry and academia, where representative ones include TPU , , ShiDianNao , and Eyeriss , and different accelerators exploit different optimizations for various applications.",
"DNN Accelerator Performance Prediction.",
"For designing FPGA-based DNN accelerators, current practice usually relies on roofline models or customized analytical tools , to estimate the achievable performance.",
"For ASIC-based accelerators, recently published designs , , introduce various performance prediction methods.",
"Eyeriss proposes an energy model for capturing the energy overhead of the customized memory and computation units and a delay model that simplifies the latency calculation.",
"Similarly, MAESTRO develops an energy estimation model that considers hardware design configurations and memory access behaviors, while Timeloop adopts a loop-based description of targeted workloads and analyzes the data movement and memory access for latency estimation.",
"DNN Accelerator Generation.",
"The tremendous need for developing FPGA-/ASIC-based DNN accelerators motivates the development of automated DNN accelerator generation.",
"For example, DeepBurning is a design automation tool for building FPGA-based DNN accelerators with customized design parameters using a pre-constructed RTL module library.",
"DNNBuilder and FP-DNN propose end-to-end tools that can automatically generate optimized FPGA-based accelerators from high-level DNN symbolic descriptions in Caffe/Tensorflow frameworks.",
"Caffeine is another automation tool that provides guidelines for choosing FPGA hardware parameters, such as the number of processing elements (PEs), bit precision of variables, and parallel data factors.",
"By using these automation tools, it is easier to bridge the gap between fast DNN construction in popular machine learning frameworks and slow implementation of targeted hardware accelerators."
],
[
"Overview of ",
" Figure: AutoDNNchip's three-step design flow for the design space exploration, optimization, and DNN-to-RTL generation.Fig.",
"REF shows an overview of the proposed AutoDNNchip, which can automatically generate optimized FPGA- or ASIC-based DNN accelerators as well as an optimal algorithm-to-hardware mapping (i.e., dataflow), according to three customized inputs: (1) the high-level DNN descriptions trained in desired datasets, (2) the application-driven specifications regarding DNN inference quality and hardware performance, and (3) the available resources of targeted platforms.",
"The realization of AutoDNNchip is achieved by the proposed One-for-all Design Space Description (see Section ), Chip Predictor (see Section ), and Chip Builder (see Section ).",
"Table: Conclusions"
]
] | 2001.03535 |
[
[
"Higher-dimensional generalizations of the Berry curvature"
],
[
"Abstract A family of finite-dimensional quantum systems with a non-degenerate ground state gives rise to a closed 2-form on the parameter space: the curvature of the Berry connection.",
"Its cohomology class is a topological invariant of the family.",
"We seek generalizations of the Berry curvature to families of gapped many-body systems in D spatial dimensions.",
"Field theory predicts that in spatial dimension D the analog of the Berry curvature is a closed (D+2)-form on the parameter space (the Wess-Zumino-Witten form).",
"We construct such closed forms for arbitrary families of interacting lattice systems in all dimensions.",
"In the special case of systems of free fermions in one dimension, we show that these forms can be expressed in terms of the Bloch-Berry connection on the product of the Brillouin zone and the parameter space.",
"In the case of families of Short-Range Entangled systems, we argue that integrals of our forms over spherical cycles are quantized."
],
[
"Introduction",
"Consider a quantum-mechanical system with a Hamiltonian depending on parameters, a unique ground state for all values of the parameters, and an energy gap to the lowest excited state.",
"To these data one can associate a 2-form $\\Omega $ on the parameter space called the Berry curvature [1].",
"This 2-form is closed and quantized: its periods (integrals over closed surfaces in the parameter space ${\\mathsf {M}}$ ) are integral multiples of $2\\pi $ .",
"Quantization of the periods of the Berry curvature can be explained as follows.",
"On the one hand, the ground states of the system fit into a rank-one complex vector bundle ${\\mathcal {B}}$ over ${\\mathsf {M}}$ .",
"The Berry curvature $\\Omega $ is the curvature of a certain unitary connection on this vector bundle (the Berry connection).",
"On the other hand, it is well-known that any rank-one vector bundle ${\\mathcal {L}}$ over a manifold ${\\mathsf {M}}$ has a topological invariant called the 1st Chern class taking values in $H^2({\\mathsf {M}},{\\mathbb {Z}})$ .",
"Its image in the de Rham cohomology $H^2({\\mathsf {M}},{\\mathbb {R}})$ can be represented by $F/2\\pi $ , where $F$ is the curvature 2-form of any unitary connection on ${\\mathcal {L}}$ .",
"These two facts imply that periods of $\\Omega $ are $2\\pi $ times an integer.",
"While definitions of the Berry connection and the Berry curvature do not make an explicit reference to the dimension of space, they are not directly applicable to models of quantum statistical mechanics in spatial dimension $D\\ge 1$ , or to models of Quantum Field Theory.",
"The reason is that the Hilbert space of these models is infinite-dimensional, and consequently the formulas defining the Berry connection and Berry curvature may be afflicted with divergences.",
"In the case of models of statistical mechanics these are volume divergences, while in the case of QFT there can be both volume and short-distance divergences.",
"The presence of volume divergences is easy to see in the case of an infinite system of identical decoupled spins coupled to a magnetic field.",
"Each spin contributes additively to the Berry curvature 2-form, therefore the total Berry curvature is proportional to the volume of the system.",
"One could try to define Berry curvature per unit volume, but its existence and quantization for more general extended systems is far from clear.",
"Even if the Berry curvature could somehow be defined for infinitely extended systems, its physical and mathematical interpretation would be unclear.",
"For such systems, only the algebra of observables is independent of the parameters of the Hamiltonian.",
"The Hilbert space is not specified from the outset.",
"It depends on the choice of a suitable state on the algebra of observables via the Gelfand-Naimark-Segal construction.",
"This state is a ground state of the Hamiltonian and thus itself depends on the parameters.",
"There is no natural way to identify Hilbert spaces for different values of the parameters, and thus ground states do not form a well-defined vector bundle.",
"Therefore one would not be able to interpret the Berry curvature as the curvature of a connection on a rank-one vector bundle.",
"A few years ago A. Kitaev proposed that for a family of Short-Range Entangled (SRE) gapped systems in spatial dimensions $D$ one can define a closed $(D+2)$ -form on the parameter space.",
"This form is a higher-dimensional generalization of the Berry curvature.",
"The cohomology class of this $(D+2)$ -form would serve as a topological invariant of the family.",
"One difficulty in making this proposal concrete is that currently there is no useful definition of Short-Range Entangled systems, beyond the \"negative\" statement that these are systems which exhibit neither spontaneous symmetry-breaking nor topological order.",
"Unlike the notion of a Short-Range Entangled system, the notion of a gapped system is straightforward to define.",
"In this paper we define and study higher-dimensional generalizations of the Berry curvature for gapped lattice systems on ${\\mathbb {R}}^D$ .",
"For any family of such systems we define a closed $(D+2)$ -form $\\Omega ^{(D+2)}$ on the parameter space ${\\mathsf {M}}$ .",
"The form depends on some choices (essentially, a choice of a differential $D$ -form on ${\\mathbb {R}}^D$ whose integral over ${\\mathbb {R}}^D$ is 1).",
"The cohomology class of $\\Omega ^{(D+2)}$ is independent of any choices and is a topological invariant of the family.",
"It is an obstruction to continuously deforming the family to a constant family of gapped systems.",
"It also can be viewed as an obstruction to having a gapped boundary which varies continuously with parameters.",
"We also argue that when all systems in the family are Short-Range Entangled, the integral of $\\Omega ^{(D+2)}$ over any $(D+2)$ -dimensional sphere is an integral multiple of $2\\pi $ .",
"In this special case our set-up matches that in Kitaev's proposal.",
"For families of Euclidean lattice systems in $D+1$ dimensions with exponentially decaying correlations, A. Kitaev outlined a construction of a closed $(D+2)$ -form on the parameter space [2].",
"Our results can be viewed as a Hamiltonian version of this construction.",
"In the case of translationally-invariant tight-binding free fermion Hamiltonians in 1d we show that the cohomology class of $\\Omega ^{(D+2)}$ is determined by the curvature of the Berry-Bloch connection.",
"We conjecture that this is true in any dimension.",
"Free fermion systems provide examples of families whose topological invariants are non-trivial.",
"Recently Cordova, Freed, Lam, and Seiberg studied field theories with \"anomalies in the space of couplings\" [3], [4].",
"Via the bulk-boundary correspondence, this subject is closely related to topologically-nontrivial families of gapped field theories in one dimension higher.",
"It is natural to conjecture that there is a 1-1 correspondence between topological invariants of families of gapped field theories in $(D+1)$ space-time dimensions and topological invariants of families of gapped lattice models in $D$ spatial dimensions some of which we study here.",
"The content of the paper as follows.",
"In Section we interpret higher Berry curvature forms in the language of Quantum Field Theory, specifically as Wess-Zumino-Witten terms in the effective action for the parameters.",
"This serves as a motivation for subsequent discussion.",
"In Section we show how to associate a closed 3-form to a family of gapped 1d lattice systems.",
"In Section we extend our construction to families of gapped lattice systems in arbitrary spatial dimension.",
"This requires some mathematical machinery which we review.",
"We discuss our results in Section .",
"In Appendix A we argue that if all systems in the family are Short-Range Entangled, the integral of $\\Omega ^{(D+2)}$ over any spherical cycle in the parameter space is an integral multiple of $2\\pi $ .",
"We also explain the interpretation of the period of $\\Omega ^{(D+2)}$ on $S^{D+2}$ as an obstruction to having a gapped boundary condition defined globally on $S^{D+2}$ .",
"In Appendix B we compute the 3-form $\\Omega ^{(3)}$ for families of tight-binding free fermion 1d systems of class A and express its cohomology class in terms of the Berry-Bloch connection.",
"This allows us to give examples of families of systems where our higher Berry curvature is topologically non-trivial (lies in nonzero cohomology classes).",
"A. K. would like to thank the other members of the gang of the seven (D. Freed, M. Freedman, M. Hopkins, A. Kitaev.",
"G. Moore, and C. Teleman) for discussions of family invariants of gapped systems and related issues.",
"We are especially grateful to A. Kitaev for reading a preliminary draft of the paper and pointing out an error.",
"This research was supported in part by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632.",
"A.K.",
"was also supported by the Simons Investigator Award."
],
[
"Effective action considerations",
"The purpose of this section is to motivate the constructions in subsequent sections.",
"It is not essential for understanding the rest of the paper.",
"Readers not familiar with topological aspects of QFT are advised to skip it on first reading.",
"If a gapped system in $D$ spatial dimensions is described by a trivial topological field theory at long distances, then its low-energy effective action is a well-defined function of background fields, such as the metric and the gauge fields which couple to global symmetries.",
"If one deals with a family of such systems parameterized by a manifold ${\\mathsf {M}}$ , one can let the parameters vary slowly from point to point, and the effective action is still a well-defined function of the background fields.",
"The variation of the parameters can be described by a map $\\phi : X\\rightarrow {\\mathsf {M}}$ , where $X$ is the space-time.",
"The effective action depends on $\\phi $ as well as other background fields.",
"Loosely speaking, topological terms in the action are those terms which survive when one re-scales the metric $g_{\\mu \\nu }\\mapsto e^{\\sigma } g_{\\mu \\nu }$ and takes the limit $\\sigma \\rightarrow +\\infty $ .",
"The simplest such terms are those which depend only on $\\phi $ and not on other background fields.",
"For example, for $D=0$ (ordinary quantum mechanics) such a topological term schematically has the form $S_{top}(X,\\phi )=\\int _X \\omega ^{(1)}_j \\partial _t\\phi ^j dt=\\int _X\\phi ^*\\left(\\omega ^{(1)}\\right),$ where $X$ is a one-dimensional manifold ($S^1$ or ${\\mathbb {R}}$ ) and $\\omega ^{(1)}$ is the 1-form on ${\\mathsf {M}}$ representing the Berry connection.",
"This formula is only schematic because in general the Berry connection on the parameter space can be represented by a 1-form $\\omega ^{(1)}$ only locally on ${\\mathsf {M}}$ .",
"If the cohomology class of the Berry curvature $\\Omega ^{(2)}$ is non-trivial, then one cannot write $\\Omega ^{(2)}=d\\omega ^{(1)}$ for a globally-defined 1-form $\\omega ^{(1)}$ .",
"Rather, one needs to cover the parameter space with charts, in each of which the connection is represented by a 1-form.",
"On the overlaps of the charts these 1-forms are related by gauge transformations.",
"To define $S_{top}(X,\\phi )$ properly, one needs to know both the locally-defined 1-forms and the gauge transformations connecting them.",
"Another important point is that only $\\exp (i S_{top}(X,\\phi ))$ can be defined unambiguously, while $S_{top}(X,\\phi )$ is defined only up to an integer multiple of $2\\pi $ .",
"To see this, let us pick an oriented two-dimensional manifold $Y$ such that $\\partial Y=X$ .",
"If the map $\\phi :X\\rightarrow {\\mathsf {M}}$ extends to a continuous map $\\tilde{\\phi }: Y\\rightarrow {\\mathsf {M}},$ then one can write a more precise formula for the topological action as follows: $S_{top}=\\int _Y \\tilde{\\phi }^*\\left(\\Omega ^{(2)}\\right).$ This expression depends on the choice of $\\tilde{\\phi }$ .",
"But $\\exp (iS_{top}(X,\\phi ))$ is unambiguously defined since periods of $\\Omega ^{(2)}$ are \"quantized\": the integral of $\\Omega ^{(2)}$ over any 2-cycle on ${\\mathsf {M}}$ is $2\\pi $ times an integer.",
"If $\\phi $ does not extend to $Y$ , one can still define $\\exp (iS_{top}(X,\\phi ))$ as the holonomy of the Berry connection, but to compute it one needs to use choose local trivializations, as sketched above.",
"Although the cohomology class of $\\Omega ^{(2)}$ does not completely determine the Berry connection, it does determine it up to an addition of a globally defined 1-form.",
"Since any 1-form can be deformed to zero, this means that the cohomology class of the Berry curvature determines $\\exp (iS_{top}(X,\\phi ))$ up to a continuous deformation.",
"This cohomology class is often easier to compute than the Berry curvature itself, because the Berry curvature is not a deformation invariant and depends on dynamical details.",
"For $D>0$ the story is similar.",
"A topological action which does not depend on fields other than $\\phi $ schematically has the form $S_{top}(X,\\phi )=\\int _X \\phi ^*\\left(\\omega ^{(D+1)}\\right)=\\frac{1}{(D+1)!",
"}\\int _X \\omega ^{(D+1)}_{i_0\\ldots i_{D}} \\left(\\partial _0 \\phi ^{i_0}\\right)\\ldots \\left(\\partial _{D}\\phi ^{i_{D}}\\right)dx^0\\ldots dx^{D},$ where $\\omega ^{(D+1)}$ is an $(D+1)$ -form on ${\\mathsf {M}}$ and $X$ is a closed oriented $(D+1)$ -manifold.",
"If one takes this formula literally, then all such actions can be deformed to zero, since any $(D+1)$ -form can be deformed to zero.",
"But if one interprets $\\omega ^{(D+1)}$ more creatively, as a sort of \"higher connection\", one can get more interesting actions which cannot be deformed to the trivial one.",
"One way to find such a generalization is to note that the r.h.s.",
"of the above equation does not change under $\\omega ^{(D+1)}\\mapsto \\omega ^{(D+1)}+d\\lambda ^{(D)}$ , where $\\lambda ^{(D)}$ is an arbitrary $D$ -form.",
"Then it is natural to consider an object specified by locally-defined $(D+1)$ -forms $\\omega ^{(D+1)}_\\alpha $ , where $\\alpha $ labels the charts.",
"On the overlaps of charts these $(D+1)$ -forms are related by $D$ -form gauge transformations.",
"The full story is rather complicated, since in order to be able to define \"higher holonomy\" along a $(D+1)$ -dimensional submanifold one needs compatibility conditions for the gauge transformations which involve $(D-1)$ -forms on triple overlaps, etc.",
"An alternative approach (first appearing in a mathematical paper by Cheeger and Simons [5]) is to postulate the following natural property.",
"If $X=\\partial Y$ for some $(D+2)$ -manifold $Y$ , and if $\\phi $ extends to a map $\\tilde{\\phi }:Y\\rightarrow {\\mathsf {M}}$ , then one must have $\\exp \\left(i S_{top}(X,\\phi )\\right)=\\exp \\left(i\\int _Y\\tilde{\\phi }^*\\left(\\Omega ^{(D+2)}\\right)\\right),$ where $\\Omega ^{(D+2)}$ is a $(D+2)$ -form on ${\\mathsf {M}}$ .",
"For this formula to make sense, $\\Omega ^{(D+2)}$ must be closed and its periods must be integer multiples of $2\\pi $ .",
"For example, to see that $\\Omega ^{(D+2)}$ must be closed, one can vary $\\tilde{\\phi }$ infinitesimally while keeping its boundary value $\\phi $ fixed.",
"It is easy to see that the r.h.s.",
"will be unchanged only if $d\\Omega ^{(D+2)}=0$ .",
"To see that $\\Omega ^{(D+2)}$ must have periods which are integral multiples of $2\\pi $ , take $X$ to be the empty manifold, and take $Y$ to be any closed $(D+2)$ -manifold.",
"Locally on ${\\mathsf {M}}$ one can write $\\Omega ^{(D+2)}=d\\omega ^{(D+1)}$ .",
"If the cohomology class of $\\Omega ^{(D+2)}$ is trivial, one can do it globally, and then $S_{top}(X,\\phi )$ can be defined by the simple formula (REF ).",
"In general, one can show that given a closed $(D+2)$ -form $\\Omega ^{(D+2)}$ with \"quantized\" periods there exists an exponentiated action $\\exp (i S_{top}(X,\\phi ))$ satisfying the above equation.",
"It is unique up to a factor $\\exp (i\\int _X\\phi ^*(\\alpha ))$ , where $\\alpha $ is a closed $(D+1)$ -form on ${\\mathsf {M}}$ .",
"As in the case $D=0$ , this implies that the cohomology class of $\\Omega ^{(D+2)}$ determines $\\exp (i S_{top}(X,\\phi ))$ up to a factor which can be deformed to 1.",
"Thus one can say that deformation classes of such topological actions (known as Wess-Zumino-Witten terms) are classified by \"quantized\" cohomology classes of degree $D+2.$ There is also an interpretation of Wess-Zumino-Witten terms as holonomies of \"higher connections\" on \"higher bundles\" on ${\\mathsf {M}}$ .",
"Then the cohomology class of $\\Omega ^{(D+2)}$ determines the topology of the corresponding \"higher bundle\".",
"But since such an interpretation is quite abstract, we will not use it in this paper.",
"The conclusion is that given a family of trivial gapped systems in spatial dimension $D$ , one should be able to obtain a closed $(D+2)$ -form on the parameter space with \"quantized\" periods.",
"While the form itself depends on the dynamical details, its cohomology class is a topological invariant.",
"It classifies possible deformation classes of Wess-Zumino-Witten terms on the parameter space.",
"The statement about quantization of periods needs some qualification in the case of fermionic systems.",
"A fermionic path-integral depends on spin structure on $X$ .",
"For fermionic systems it is unreasonable to restrict attention to topological terms which depend only on the map $\\phi $ , one needs to study topological terms which depend both on $\\phi $ and the spin structure.",
"Then one needs to generalize the Cheeger-Simons approach by requiring the manifolds $X$ and $Y$ to be spin manifolds.",
"Such spin-structure-dependent Wess-Zumino-Witten terms were first considered in [6].",
"Alternatively, if one limits oneself to the case of systems on $X={\\mathbb {R}}^{D+1}$ or its one-point compactification $S^{D+1}$ , then one can always take $Y=B^{D+2}$ ($(D+2)$ -dimensional ball).",
"Then the quantization condition is relaxed: only integrals of the form $\\int _{S^{D+2}} h^*\\left( \\Omega ^{(D+2)}\\right)$ need to be integral multiples of $2\\pi $ .",
"Here $h:S^{D+2}\\rightarrow {\\mathsf {M}}$ is any smooth map.",
"We will call such an $h$ a spherical cycle.",
"Thus for fermionic systems only integrals of $\\Omega ^{(D+2)}$ over spherical cycles are quantized.",
"Of course, not all topological terms which are consistent on ${\\mathbb {R}}^{D+1}$ or $S^{D+1}$ will remain consistent when considered on a general space-time.",
"That is, quantization on spherical cycles is not enough to make the Wess-Zumino-Witten action well-defined on arbitrary spin manifolds."
],
[
"Higher Berry curvature for gapped 1d systems",
"As explained in the previous section, given a family of trivial gapped theories on a $D$ -dimensional lattice and assuming that the field theory description applies at each point in the parameter space ${\\mathsf {M}}$ , there should be a way to construct a closed $(D+2)$ -form on ${\\mathsf {M}}$ whose integrals over spherical cycles are quantized.",
"The cohomology class of the form is a topological invariant of the family (cannot change under deformations).",
"In this section we construct such a closed form $\\Omega ^{(D+2)}$ on ${\\mathsf {M}}$ for the case of gapped spin chains, that is, gapped lattice $D=1$ systems.",
"We do not use the existence of the field theory limit.",
"In Appendix we argue that integrals of $\\Omega ^{(D+2)}$ over spherical 3-cycles are quantized.",
"That is, integrals of the form $\\int _{S^3}h^*\\Omega ^{(D+2)}$ , where $h$ is a map from $S^3$ to ${\\mathsf {M}}$ , are integer multiples of $2\\pi $ .",
"To begin with, let us recall how the Berry 2-form is defined for gapped 0d systems and why this definition does not work for $D>0$ .",
"Let $G=1/(z-H)$ be the Green's function for a positive bounded Hamiltonian $H$ which depends on some parameters.",
"Assume that 0 is an isolated eigenvalue of $H$ for all values of the parameters.",
"Let $\\Omega ^{(2)}=\\frac{i}{2}\\oint \\frac{dz}{2\\pi i}{\\rm Tr}(G dH G^2 dH),$ where $\\oint $ is the counterclockwise contour integral around $z=0$ and $d$ denotes the exterior derivative on the paramter space ${\\mathsf {M}}$ .",
"That is, $d = \\sum _{\\ell }^{} d\\lambda ^\\ell \\frac{\\partial }{\\partial \\lambda ^\\ell }$ where $\\lambda ^\\ell $ are parameters.",
"The wedge product of forms $\\wedge $ is implicit in Eq.",
"(REF ).",
"$\\Omega ^{(2)}$ is a closed 2-form on ${\\mathsf {M}}$ .",
"Indeed, since $dG=G dH G$ , we compute $d\\Omega ^{(2)}=\\frac{i}{2}\\oint \\frac{dz}{2\\pi i} {\\rm Tr}(GdHGdHG^2dH-GdHG^2dHGdH-GdHGdHG^2dH)=\\\\=-\\frac{i}{2}\\oint \\frac{dz}{2\\pi i}{\\rm Tr}(GdHG^2dHGdH)=\\frac{i}{6}\\oint \\frac{dz}{2\\pi i}\\frac{\\partial }{\\partial z} {\\rm Tr}(GdHGdHGdH) =0.$ $\\Omega ^{(2)}$ is the usual Berry curvature, as one can verify by inserting a complete set of states.",
"Suppose now $H$ is a many-body Hamiltonian for an infinite 1d lattice system with an energy gap.",
"More explicitly, we assume that $H=\\sum _{p\\in \\Lambda } H_p$ where $H_p$ is bounded and finite-range and $\\Lambda \\subset {\\mathbb {R}}$ is a discrete subset of real numbers without accumulation points.",
"Then $H$ is unbounded, but one can still define a bounded operator $G=1/(z-H)$ for $z$ which are away from the spectrum of $H$ .",
"We assume again that $H$ is positive and that 0 is an isolated eigenvalue for all values of the parameters.",
"Fixing $p,q\\in \\Lambda $ , we can define a non-closed 2-form on the parameter space $\\Omega ^{(2)}_{pq}=\\frac{i}{2}\\oint \\frac{dz}{2\\pi i}{\\rm Tr}(G dH_p G^2 dH_q).$ If the Hamiltonian $H$ is gapped, $\\Omega ^{(2)}_{pq}$ decays exponentially away from $p=q$ (see [7]).",
"The Berry curvature is formally given by $\\Omega ^{(2)}=\\sum _{p,q\\in \\Lambda } \\Omega ^{(2)}_{pq},$ but the contribution of the points near the diagonal, $p\\simeq q$ , is divergent for infinite-volume systems.",
"Instead of the ill-defined Berry curvature 2-form, consider the following 2-form depending on a site $p$ : $F^{(2)}_p=\\frac{i}{2}\\oint \\frac{dz}{2\\pi i}{\\rm Tr}(GdHG^2 dH_p).$ It is well-defined, but not closed.",
"Instead one has an identity $dF^{(2)}_q=\\sum _{p\\in \\Lambda } F^{(3)}_{pq},$ where the 3-form $F^{(3)}_{pq}$ is given by $F^{(3)}_{pq}=\\frac{i}{6}\\oint \\frac{dz}{2\\pi i}{\\rm Tr}(G^2dHGdH_p GdH_q-GdHG^2dH_p GdH_q) -(p\\leftrightarrow q).$ The identity (REF ) can be verified by a straightforward computation.",
"Note that $F^{(3)}_{pq}$ decays exponentially away from the diagonal $p=q$ thanks to the results of [7].",
"The identity (REF ) and other similar identities are key for defining topological invariants of families of gapped systems in one and higher dimensions.",
"In the context of Euclidean lattice systems, analogous identities were first observed by A. Kitaev who used them to define invariants of families of such systems [2].",
"In this paper we essentially derive Hamiltonian analogs of Kitaev's formulas.",
"Let $f:\\Lambda \\rightarrow {\\mathbb {R}}$ be a function which is 0 for $p \\ll 0$ and 1 for $p\\gg 0$ .",
"For example, it could be simply 0 for $p<a$ and 1 for $p\\ge a$ .",
"Then we define a 3-form on the parameter space by $\\Omega ^{(3)}(f)=\\frac{1}{2} \\sum _{p,q\\in \\Lambda } F^{(3)}_{pq} (f(q)-f(p)).$ It is well-defined because on the one hand $F^{(3)}_{pq}$ decays exponentially for large $|p-q|$ , and on the other hand $f(q)-f(p)$ is non-zero only when $p>a$ and $q<a$ , or the other way around.",
"For the specific choice of $f(p)$ equal 0 for $p<a$ and 1 for $p\\ge a$ , the equation (REF ) takes a simple form $\\Omega ^{(3)}(f)= \\sum _{p<a \\atop q>a} F^{(3)}_{pq},$ which makes its convergence more transparent.",
"Later in this paper we will show that $dF^{(3)}_{qr}= \\sum _{p\\in \\Lambda } F^{(4)}_{pqr},$ where $F^{(4)}_{pqr}$ is a function which is anti-symmetric in $p,q,r$ and decays exponentially away from the diagonal $p=q=r$ .",
"We find $\\begin{split}d\\Omega ^{(3)}(f)= \\frac{1}{2} \\sum _{q,r\\in \\Lambda }(f(r)-f(q)) dF^{(3)}_{qr}=\\frac{1}{2} \\sum _{p,q,r\\in \\Lambda } (f(r)-f(q)) F^{(4)}_{pqr}\\\\=\\frac{1}{6} \\sum _{p,q,r\\in \\Lambda } (f(r)-f(q)+f(p)-f(r)+f(q)-f(p)) F^{(4)}_{pqr}=0,\\end{split}$ where we have used the anti-symmetry of $F^{(4)}_{pqr}$ .",
"Therefore the 3-form $\\Omega ^{(3)}(f)$ is closed.",
"Closedness of $\\Omega ^{(3)}(f)$ implies that its cohomology class is a topological invariant of the family of gapped systems.",
"Indeed, let us regard ${\\mathsf {M}}$ as a submanifold in the space ${\\mathfrak {M}}_D$ of all gapped systems in dimension $D$ .",
"Obviously, the form $\\Omega ^{(D+2)}$ is a restriction of a closed form on ${\\mathfrak {M}}_D$ defined in exactly the same way.",
"Deforming ${\\mathsf {M}}$ within ${\\mathfrak {M}}_D$ can be thought of as a flow along a vector field on ${\\mathfrak {M}}_D$ .",
"Since the Lie derivative of a closed form along any vector field is exact, deforming ${\\mathsf {M}}$ cannot change the cohomology class of $\\Omega ^{(D+2)}$ .",
"The cohomology class of the 3-form $\\Omega ^{(3)}(f)$ is independent of the choice of the function $f$ as long as $f(p)=0$ for $p\\ll 0$ and $f(p)=1$ for $p\\gg 0$ .",
"Indeed, any two such functions differ by a function $g$ which is compactly supported, and for such a function we can write $\\begin{split}\\Omega ^{(3)}(g)=\\frac{1}{2} \\sum _{p,q\\in \\Lambda } (g(q)-g(p))F^{(3)}_{pq}= \\sum _{q\\in \\Lambda } g(q) \\sum _{p \\in \\Lambda } F^{(3)}_{pq} = \\sum _{q\\in \\Lambda } g(q) d F^{(2)}_q=d\\sum _{q\\in \\Lambda } g(q) F^{(2)}_q.\\end{split}$ This means that $\\Omega ^{(3)}(f+g)$ and $\\Omega ^{(3)}(f)$ differ by a total derivative of a well-defined 2-form on ${\\mathsf {M}}$ and therefore are in the same cohomology class.",
"We note the following obvious properties of the 3-form $\\Omega ^{(3)}(f)$ .",
"It vanishes for constant families (i.e.",
"families where the Hamiltonian is independent of parameters), and it is additive under stacking of families (with the same parameter space)."
],
[
"Higher Berry curvature for gapped systems in any dimension",
"To construct analogs of Berry curvature in higher dimensions, the language of chains and cochains is very useful.",
"Let $\\Lambda $ be a discrete subset of $\\mathbb {R}^D$ without accumulation points.",
"For $n\\ge 0$ , an $n$ -chain is a quantity $A_{p_0 \\dots p_n}$ which depends on $n+1$ points $p_0,\\ldots ,p_n \\in \\Lambda $ , is skew-symmetric under permutations of $p_0,\\ldots ,p_n$ , and decays exponentially away from the diagonal $p_0=p_1=\\ldots =p_n$ .",
"The space of $n$ -chains will be denoted $n(\\Lambda )$ .",
"The boundary operator $\\partial :n(\\Lambda )\\rightarrow {n-1}(\\Lambda )$ is defined as follows: $(\\partial A)_{p_1 \\dots p_n}=\\sum _{p_0\\in \\Lambda } A_{p_0 \\dots p_n}.$ It is easy to see that $\\partial ^2=0.$ Thus $\\oplus _{n\\ge 0}\\, n(\\Lambda )$ is a chain complex.",
"Dually, an $n$ -cochain (with values in reals) is a real-valued function $\\alpha (p_0,\\ldots ,p_n)$ which depends on $p_0,\\ldots ,p_n\\in \\Lambda $ , is bounded, skew-symmetric under permutations, and obeys the following condition: when restricted to any $\\delta $ -neighborhood of the diagonal, it vanishes when any of the points is outside some finite set.",
"Let $n(\\Lambda )$ be the space of $n$ -cochains.",
"There is a pairing between $n(\\Lambda )$ and $n(\\Lambda )$ defined by $\\langle A,\\alpha \\rangle =\\frac{1}{(n+1)!",
"}\\sum _{p_0,\\ldots ,p_n} A_{p_0,\\ldots , p_n}\\alpha (p_0,\\ldots ,p_n)$ There is also an operator $\\delta : n(\\Lambda )\\rightarrow {n+1}(\\Lambda )$ satisfying $\\delta ^2=0$ and uniquely defined by the condition $\\langle A, \\delta \\alpha \\rangle =\\langle \\partial A,\\alpha \\rangle $ for any $(n+1)$ -chain $A$ and an $n$ -cochain $\\alpha $ and $n\\ge 0$ .",
"One can regard (REF ) as a version of Stokes' theorem.",
"Explicitly, the operator $\\delta $ is given by $(\\delta \\alpha )(p_0,\\ldots ,p_{n+1})=\\sum _{j=0}^{n+1} (-1)^j \\alpha (p_0,\\ldots ,p_{j-1},p_{j+1},\\ldots ,p_{n+1}).$ In particular, if $\\Lambda \\subset {\\mathbb {R}}$ is a 1d lattice, and $f:\\Lambda \\rightarrow {\\mathbb {R}}$ is a function such that $f(p)=1$ for $p\\gg 0$ and $f(p)=0$ for $p\\ll 0$ , then $(\\delta f)(p,q)=f(q)-f(p)$ is a closed 1-cochain on $\\Lambda $ .",
"It is not exact, since $f$ does not have a finite support.",
"One can define the product $\\alpha \\cup \\gamma $ of an $n$ -cochain $\\alpha $ and an $m$ -cochain $\\gamma $ as an $n+m$ -cochain given by $(\\alpha \\cup \\gamma )(p_0,\\ldots ,p_{n+m})=\\frac{1}{(n+m+1)!",
"}\\sum _{\\sigma \\in {\\mathcal {S}}_{n+m+1}} (-1)^{{\\rm sgn}\\, \\sigma } \\alpha (p_{\\sigma (0)},\\ldots ,p_{\\sigma (n)})\\gamma (p_{\\sigma (n)},\\ldots ,p_{\\sigma (n+m)}),$ It satisfies $\\alpha \\cup \\gamma =(-1)^{nm}\\gamma \\cup \\alpha ,\\quad \\delta (\\alpha \\cup \\gamma )=\\delta \\alpha \\cup \\gamma +(-1)^n\\alpha \\cup \\delta \\gamma .$ Using this notation, we see that $\\Omega ^{(3)}(f)=\\langle F^{(3)},\\delta f\\rangle $ , where $F^{(3)}$ is a 1-chain with values in 3-forms on ${\\mathsf {M}}$ with components $F^{(3)}_{pq}.$ Furthermore, eq.",
"(REF ) can be written as a relation between a 1-chain $F^{(3)}$ valued in 3-forms and a 2-chain $F^{(4)}$ valued in 4-forms: $dF^{(3)}=\\partial F^{(4)}.$ Then the computation leading to (REF ) can be shortened to $d\\Omega ^{(3)}(f)=\\langle d F^{(3)},\\delta f\\rangle =\\langle \\partial F^{(4)},\\delta f\\rangle =\\langle F^{(4)},\\delta \\delta f\\rangle =0.$ Similarly, the computation leading to (REF ) can be shortened to $\\Omega ^{(3)}(g)=\\langle F^{(3)},\\delta g\\rangle =\\langle \\partial F^{(3)}, g\\rangle =d\\langle F^{(2)}, g\\rangle .$ Here $g:\\Lambda \\rightarrow {\\mathbb {R}}$ is supported on a finite set, therefore the application of the Stokes' theorem is legitimate.",
"Now we will generalize the construction of the previous section to arbitrary dimensions and define a closed $(D+2)$ -form $\\Omega ^{(D+2)}$ on the parameter space of a family of gapped lattice systems in $D$ spatial dimensions.",
"We define a family of $D$ -dimensional gapped lattice systems in the same way as for $D=1$ , the only difference being that the lattice $\\Lambda $ is a subset of $\\mathbb {R}^D$ instead of $\\mathbb {R}$ .",
"For $D>1$ not all gapped systems are can be continuously connected to the trivial one, thanks thanks to the possibility of topological order.",
"Therefore we do not expect our $(D+2)$ -form to have quantized periods even on spherical cycles.",
"Nevertheless we will argue in Appendix A that for families of systems in an SRE phase its periods are quantized on spherical $(D+2)$ -cycles, as expected from the field theory analysis.",
"We will define higher Berry curvatures recurrently via the following \"descent equation\": $dF^{(n)}=\\partial F^{(n+1)},$ where $F^{(n)}$ is $(n-2)$ -chain with values in $n$ -forms on the parameter space.",
"Analogous equations for families of Euclidean lattice systems were used in [2].",
"Starting from $F^{(2)}$ defined in (REF ), we can find all its descendants.",
"The result is $\\begin{split}F^{(n)}_{p_0 \\dots p_{n-2}} &= \\frac{i (-1)^n}{ n(n-1)} \\sum _{\\sigma \\in \\mathcal {S}_{n-1}}{ {\\rm sgn}( \\sigma )} \\oint \\frac{dz}{2\\pi i} \\\\ &\\sum _{j=0}^{n-2} (n-j-1) {\\rm Tr}\\left( GdH G dH_{p_{\\sigma (0)}}GdH_{p_{\\sigma (1)}}\\dots G^2 dH_{p_{\\sigma (j)}} \\dots G dH_{p_{\\sigma (n-2)}} \\right).\\end{split}$ For this to be a well-defined chain, it must decay exponentially when any two of the points $p_0,\\ldots ,p_{n-2}$ are separated by a large distance.",
"For $n=3$ this was proved in [7], and we expect that the proof can be generalized to arbitrary $n$ .",
"Heuristically, exponential decay follows from the physical interpretation of the above correlators in terms of generalized local susceptibilities.",
"For $n=2$ the correlator is a variation of the expectation value of a local operator $dH_{p_0}$ with respect to an arbitrary infinitesimal variation of the Hamiltonian.",
"That is, it is a local susceptibility.",
"For $n=3$ it can be interpreted as a variation of a local susceptibility with respect to a variation of the Hamiltonian elsewhere.",
"For $n=4$ it can be interpreted as a variation of a variation, etc.",
"We expect all such quantities to decay exponentially for large spatial separations because the correlation length is finite for a gapped system at zero temperature.",
"In order to find a topological invariant of a family of gapped systems we need to contract this $(n-2)$ -chain with an $(n-2)$ -cochain.",
"Let $\\alpha $ be an $(n-2)$ -cochain, then $\\langle F^{(n)},\\alpha \\rangle $ is an $n$ -form on the parameter space.",
"But in general it is not closed: $d\\langle F^{(n)},\\alpha \\rangle = \\langle dF^{(n)},\\alpha \\rangle = \\langle \\partial F^{(n+1)}, \\alpha \\rangle =\\langle F^{(n+1)}, \\delta \\alpha \\rangle .$ In order for the integral of the $n$ -form $\\int _{C_n}\\langle F^{(n)},\\alpha \\rangle $ to be independent of the deformation of the cycle $C_n$ , the cochain $\\alpha $ must be closed, $\\delta \\alpha =0.$ On the other hand, if the cochain $\\alpha $ is exact, $\\alpha =\\delta \\gamma $ , we find $\\langle F^{(n)},\\alpha \\rangle = \\langle F^{(n)},\\delta \\gamma \\rangle = \\langle \\partial F^{(n)},\\gamma \\rangle = d\\langle F^{(n-1)},\\gamma \\rangle ,$ and all integrals $\\int _{C_n}\\langle F^{(n)},\\alpha \\rangle $ over cycles $C_n$ will be zero.",
"We see that in order to get a non-trivial invariant of a family we need to contract the chain $F^{(n)}$ with a cochain which is closed but not exact.",
"Moreover, adding to such a cochain an exact cochain will not change the invariant.",
"Thus we need to understand the space of closed cochains modulo the subspace of exact cochains, that is, the cohomology of the cochain complex $(n(\\Lambda ),\\delta )$ .",
"If we omit the word \"bounded\" from the definition of cochains, then the cohomology of the corresponding complex is known in the mathematical literature as the coarse cohomology of $\\Lambda $ [10].",
"For physical applications, one may assume that $\\Lambda \\subset \\mathbb {R}^D$ uniformly fills the whole ${\\mathbb {R}}^D$ , in the sense that there exists $\\delta >0$ such that each point of ${\\mathbb {R}}^D$ is within distance $\\delta $ of some point of $\\Lambda $ , and that $\\Lambda $ has no accumulation points.",
"Then the $n$ -th coarse cohomology group of $\\Lambda $ is isomorphic to the $n$ -th cohomology group of ${\\mathbb {R}}^D$ with compact support [10].",
"The latter is non-trivial only for $n=D$ and is one-dimensional.",
"The generator of $D$ -th coarse cohomology group can be taken to be $\\delta f_1\\cup \\dots \\cup \\delta f_D$ , where $f_\\mu (p)=\\theta (x^\\mu (p))$ and $x^\\mu (p)$ is the $\\mu $ -coordinate of $p$ and $\\theta (x)$ is theta function.",
"More generally, one can choose $f_\\mu $ to be any function which depends only on $x^\\mu (p)$ and is 0 for $x^\\mu (p) \\ll 0$ and 1 for $x^\\mu (p)\\gg 0$ .",
"Note that such cochains are bounded and thus also define a nontrivial cohomology class in the sense that we need.",
"For a family of $D$ -dimensional systems parameterized by ${\\mathsf {M}}$ we therefore define a $(D+2)$ -form on ${\\mathsf {M}}$ : $\\Omega ^{(D+2)}(f_1,\\dots ,f_D)= \\langle F^{(D+2)}, \\delta f_1\\cup \\dots \\cup \\delta f_D\\rangle .$ This $(D+2)$ -form is closed: $d\\Omega ^{(D+2)}(f_1,\\dots ,f_D)=\\langle dF^{(D+2)}, \\delta f_1\\cup \\dots \\cup \\delta f_D\\rangle =\\langle \\partial F^{(D+3)}, \\delta f_1\\cup \\dots \\cup \\delta f_D\\rangle \\\\=\\langle F^{(D+3)},\\delta ( \\delta f_1\\cup \\dots \\cup \\delta f_D)\\rangle =0.$ Its cohomology class is unchanged under the shift $f_1\\rightarrow f_1+g$ by a compactly-supported function $g$ since $\\Omega ^{(D+2)}(g,f_2,\\dots ,f_D)= \\langle F^{(D+2)}, \\delta g\\cup \\delta f_2\\cup \\dots \\cup \\delta f_D\\rangle =\\langle F^{(D+2)}, \\delta (g\\cup \\delta f_2\\cup \\dots \\cup \\delta f_D)\\rangle \\\\=\\langle \\partial F^{(D+2)}, g\\cup \\delta f_2\\cup \\dots \\cup \\delta f_D\\rangle =d\\langle F^{(D+1)}, g\\cup \\delta f_2\\cup \\dots \\cup \\delta f_D\\rangle $ and analogously for other shifts $f_\\mu \\rightarrow f_\\mu +g$ .",
"In general, periods of $\\Omega ^{(D+2)}(f_1,\\ldots ,f_D)$ are not subject to quantization.",
"In Appendix we argue that integrals over spherical cycles in the parameter space are quantized for families of SRE systems."
],
[
"Discussion",
"In this paper, we have constructed higher-dimensional generalizations of the Berry curvature starting from the ordinary Berry curvature for quantum-mechanical systems and solving the descent equation (REF ) .",
"In fact, this procedure of constructing higher-dimensional generalization of topological invariants from lower dimensional ones via descent equations is rather general.",
"For example, the Thouless charge pump for 1d systems [11] and its higher-dimensional generalizations can be constructed from the ground-state charge of a quantum-mechanical system with a $U(1)$ symmetry.",
"This will be discussed in a separate publication.",
"The cohomology class of the Berry curvature is an obstruction to having a continuously varying family of ground states.",
"Similarly one can show that the cohomology class of the form $\\Omega ^{(D+2)}$ restricted to a $(D+2)$ -sphere in the parameter space is an obstruction to having a smoothly varying gapped boundary defined everywhere on this sphere.",
"See Appendix A for details.",
"An example of a three-parameter family of 1d lattice systems whose cohomology class is non-trivial is given in Appendix , where we relate the 3-form $\\Omega ^{(3)}(f)$ to the Berry-Bloch connection over the Brillouin zone.",
"For $D=0$ the cohomology class of the Berry curvature (regarded as an integral class) is the only topological invariant of the family.",
"It is trivial if and only if the family can be deformed to a constant family without closing the gap.",
"One can ask if the same is true for $D>0$ or if there are additional independent invariants.",
"The existence of topological order for $D>1$ means that the answer will probably depend on which topological phase one considers.",
"The case $D=1$ is special since all gapped 1d systems are Short-Range Entangled.",
"Moreover, for $D=1$ it has been conjectured by A. Kitaev that a properly defined space of all gapped bosonic systems has the homotopy type $K({\\mathbb {Z}},3)$ .",
"That is, its only non-trivial homotopy group is in degree 3 and is isomorphic to ${\\mathbb {Z}}$ .",
"If this is true, then all cohomology classes on the space of gapped bosonic 1d systems can be expressed as some complicated functions of the basic class which sits in degree 3.",
"That is, for $D=1$ bosonic families there are no further independent invariants beyond the one we constructed."
],
[
"Quantization of higher Berry curvatures",
"Consider a family of gapped systems in spatial dimension $D$ .",
"In the body of the paper we showed how to define a closed form $\\Omega ^{(D+2)}$ on the parameter space ${\\mathsf {M}}$ .",
"It depends on some additional data ($D$ functions on $\\Lambda $ ), but the cohomology class was shown to be independent of these data.",
"Thus periods of $\\Omega ^{(D+2)}$ are also independent of these additional data.",
"In this appendix we argue that if all systems in the family are Short-Range Entangled (SRE), and if $h$ is a spherical cycle in ${\\mathsf {M}}$ (i.e.",
"a map $h:S^{D+2}\\rightarrow {\\mathsf {M}}$ ), then the integral of $\\Omega ^{(D+2)}$ over such a cycle is \"quantized\": $\\frac{1}{2\\pi }\\int _{S^{D+2}} h^* (\\Omega ^{(D+2)})\\in {\\mathbb {Z}}.$ We begin with the 1d case, where there is no topological order, and thus all gapped systems without spontaneous symmetry breaking are SRE.",
"Thus all systems in the family belong to the same SRE phase.",
"In the bosonic case, this means that they can all be deformed to a trivial system whose Hamiltonian is a sum of one-site operators and the ground state is a product state.",
"In the fermionic case, there is a unique non-trivial SRE phase corresponding to Kitaev's Majorana chain.",
"So there are two options: either all systems in the family are in the trivial phase, or they can all be deformed to the Majorana chain.",
"In the latter case we can stack the whole family with the \"constant\" Majorana chain and get a family of fermionic systems in the trivial phase.",
"Since $\\Omega ^3(f)$ is unchanged under stacking the family with a system independent of parameters, this reduces the problem to studying a family of systems in the trivial phase.",
"Let $f(p)=\\theta (p)$ (a step-function on $\\Lambda \\subset {\\mathbb {R}}$ ).",
"Recall that we denote the space of all gapped 1d system by ${\\mathfrak {M}}_1$ .",
"(Our argument will be the same for bosonic and fermionic systems, so we do not need to distinguish the two possibilities).",
"This is an infinite-dimensional space which can be thought of as a union of an infinite number of finite-dimensional manifolds.",
"The parameter space ${\\mathsf {M}}$ is a submanifold in this infinite-dimensional space, and the 3-form $\\Omega ^{(3)}$ on ${\\mathsf {M}}$ is a restriction of the 3-form on ${\\mathfrak {M}}_1$ defined in exactly the same way.",
"Let us fix a particular trivial system ${\\mathbf {m}}_0\\in {\\mathfrak {M}}_1$ .",
"Each point in ${\\mathsf {M}}$ can be connected to ${\\mathbf {m}}_0$ by a continuous path in ${\\mathfrak {M}}_1$ .",
"This applies to all points in the image of the spherical cycle $h$ .",
"If this could be done continuously over the whole $S^3$ , it would mean that the cycle is contractible to a point ${\\mathbf {m}}_0$ in ${\\mathfrak {M}}_1$ , and the corresponding integral $\\int _{S^3}h^*(\\Omega ^{(3)}(f))$ would be zero.",
"While in general it is not possible to contract the whole spherical cycle, it is always possible to contract $S^3$ with a point removed.",
"In particular, it is possible to contract $S^3$ without either north or south pole.",
"Let $S^3_S$ and $S^3_N$ be $S^3$ with the north and south poles removed, respectively.",
"Let us denote the contractions in the space of the gapped Hamiltonians by $\\mathcal {P}_S$ and $\\mathcal {P}_N$ .",
"These are continuous maps from $[0,1]\\times S^3_S$ to ${\\mathfrak {M}}_1$ and from $[0,1]\\times S^3_N$ to ${\\mathfrak {M}}_1$ , respectively.",
"Let us parameterize $[0,1]$ by $t$ .",
"For $t=0$ they are just restrictions of $h$ to $S^3_S$ and $S^3_N$ .",
"For $t=1$ they are constant maps to ${\\mathbf {m}}_0$ .",
"Let the Hamiltonian corresponding to a point ${\\mathbf {m}}\\in {\\mathfrak {M}}_1$ be $H({\\mathbf {m}})=\\sum _p H_p({\\mathbf {m}})$ .",
"The family of Hamiltonians corresponding to the spherical cycle $h$ is $H[{\\mathbf {s}}]=\\sum _p H_p(h({\\mathbf {s}}))$ , where ${\\mathbf {s}}\\in S^3$ .",
"For ${\\mathbf {s}}\\in S^3_N$ we define another Hamiltonian $H^+[{\\mathbf {s}}]$ which is the same as $H[{\\mathbf {s}}]$ except that on the far right part of the lattice $p\\gg 0$ it adiabatically interpolates to $H({\\mathbf {m}}_0)$ .",
"More precisely, $H^+[{\\mathbf {s}}]=\\sum _{p\\in \\Lambda } H^+_p[{\\mathbf {s}}]$ is sum of on-site Hamiltonians $H^+_p[{\\mathbf {s}}] = H_p({\\mathbf {m}}({\\mathbf {s}},p))$ where we let the parameters of the Hamiltonian depend slowly on $p$ as ${\\mathbf {m}}({\\mathbf {s}},p)=\\mathcal {P}_N(t_N(p),{\\mathbf {s}})$ .",
"The function $t_N:{\\mathbb {R}}\\rightarrow {\\mathbb {R}}$ is equal to 1 for $p\\in [2L,+\\infty )$ , smoothly interpolates from 1 to 0 in the region $p\\in [L,2L]$ , and is 0 for $p\\in (-\\infty ,L]$ .",
"Similarly, we define a local Hamiltonian $H^-[{\\mathbf {s}}]$ for all ${\\mathbf {s}}\\in S^3_S$ via $H^-[{\\mathbf {s}}]=\\sum _{p\\in \\Lambda } H_p(P_S(t_S(p),{\\mathbf {s}}))$ where the function $t_S:{\\mathbb {R}}\\rightarrow {\\mathbb {R}}$ is 1 for $p\\in (-\\infty ,-2L],$ smoothly interpolates from 1 to 0 in the region $p\\in [-2L,-L],$ and is 0 for $p\\in [-L,+\\infty )$ .",
"Lastly, we define $H^{+-}_p[{\\mathbf {s}}]$ for all ${\\mathbf {s}}\\in S^3_N\\bigcap S^3_S$ as a Hamiltonian which coincides with $H_p[{\\mathbf {s}}]$ in the region $p\\in [-L,L]$ , coincides with $H_p({\\mathbf {m}}_0)$ for $p\\notin [-2L,2L]$ , and smoothly interpolates between these regions using the paths $\\mathcal {P}_S$ and $\\mathcal {P}_N$ .",
"Our main assumption is that all these families of Hamiltonains are gapped for sufficiently large $L$ .",
"This seems reasonable since for a fixed $t$ and ${\\mathbf {s}}$ all Hamiltonians $H(P_N(t,{\\mathbf {s}}))$ and $H(P_S(t,{\\mathbf {s}}))$ are gapped and there should be an upper bound on the correlation length.",
"However, a proof of this would be very desirable.",
"We denote by $\\Omega ^{(3)}_+(f),\\Omega ^{(3)}_-(f)$ and $\\Omega ^{(3)}_{+-}(f)$ the 3-forms corresponding to the families $H^+$ , $H^-$ and $H^{+-}$ .",
"They are defined on $S^3_N$ , $S^3_S$ and $S^3_N\\bigcap S^3_S$ , respectively.",
"We write an integral over $S^3$ as a sum of integrals over its lower and upper hemispheres which we call $B_-$ and $B_+$ : $\\int _{S^3} h^*(\\Omega ^{(3)}(f) )= \\int _{B_+} h^*(\\Omega ^{(3)}(f))+ \\int _{B_-} h^*(\\Omega ^{(3)}(f)) =\\int _{B_+} \\Omega ^{(3)}_+(f) +\\int _{B_-} \\Omega ^{(3)}_-(f) + O(L^{-\\infty }).$ In the last step we replaced $h^*(\\Omega ^{(3)})$ with $\\Omega ^{(3)}_\\pm $ on $B_\\pm $ .",
"Since by our assumption $H[{\\mathbf {s}}]$ , $H^+[{\\mathbf {s}}],$ and $H^-[{\\mathbf {s}}]$ are all gapped, the 3-form $h^*(\\Omega ^{(3)})$ is only sensitive to the Hamiltonian of the system in the neighborhood of the point $p=0$ where the function $f(p)=\\theta (p)$ has a discontinuity.",
"Since all these Hamiltonians coincide near the point $p=0$ , for large $L$ the error introduced by this replacement is of order $L^{-\\infty }$ .",
"Let us now define $f_+(p)=\\theta (p-3L)$ and $f_-(p)= \\theta (p+3L)$ and write $\\int _{B_+} \\Omega ^{(3)}_+(f) +\\int _{B_-} \\Omega ^{(3)}_-(f)=\\int _{B_+} \\Omega ^{(3)}_+(f_+) +\\int _{B_-} \\Omega ^{(3)}_-(f_-)+\\int _{B_+} \\Omega ^{(3)}_+(f-f_+) +\\int _{B_-} \\Omega ^{(3)}_-(f-f_-),$ The on-site Hamiltonian $H_p^+[{\\mathbf {s}}]$ coincides with the constant Hamiltonian $H_p({\\mathbf {m}}_0)$ near $p=3L$ .",
"Therefore the form $\\Omega ^{(3)}_+(f_+)$ is of order $L^{-\\infty }$ , and so is its integral over $B_+$ .",
"Similarly, $\\int _{B_-} \\Omega ^{(3)}_-(f_-) = O(L^{-\\infty })$ .",
"The remaining terms in the above equation contain functions $f_\\pm -f$ which have compact support.",
"For any such function $g:\\Lambda \\rightarrow {\\mathbb {R}}$ we can write $\\Omega ^{(3)}_\\pm (g) =\\langle F^{(3)}_\\pm ,\\delta g\\rangle = d \\langle F^{(2)}_\\pm , g\\rangle $ .",
"Therefore we get $\\int _{B_+} \\Omega ^{(3)}_+(f-f_+) +\\int _{B_-} \\Omega ^{(3)}_-(f-f_-) = \\int _{S^2} \\langle F^{(2)}_+,f-f_+\\rangle -\\int _{S^2} \\langle F^{(2)}_-,f-f_-\\rangle $ where $S^2$ is the equator of $S^3$ and the common boundary of $B_-$ and $B_+$ .",
"The minus sign arises because the orientation on $S^2$ induced by $B_-$ is opposite to the one induced by $B_+$ .",
"We can now replace $F^{(2)}_+$ and $F^{(2)}_-$ with $F^{(2)}_{+-}$ in both integrals, since the integrands are only sensitive to the Hamiltonian of the system in the region where $H^+_p[{\\mathbf {s}}]=H^{+-}_p[{\\mathbf {s}}]$ and $H^-_p[{\\mathbf {s}}]=H^{+-}_p[{\\mathbf {s}}]$ .",
"Such a replacement introduces an error of order $L^{-\\infty }$ .",
"Therefore the above expression becomes $\\begin{split}\\int _{S^2} \\langle F^{(2)}_+,f-f_+\\rangle -\\int _{S^2} \\langle F^{(2)}_-,f-f_-\\rangle = \\int _{S^2} \\langle F^{(2)}_{+-}, f-f_+\\rangle -\\int _{S^2} \\langle F^{(2)}_{+-}, f-f_-\\rangle +O(L^{-\\infty }) \\\\= - \\int _{S^2} \\langle F^{(2)}_{+-},f_+-f_-\\rangle +O(L^{-\\infty }).\\end{split}$ By construction $H^{+-}_p[{\\mathbf {s}}]=H_p[{\\mathbf {s}}]$ for $p\\in [-L,L]$ , while $H^{+-}_p[{\\mathbf {s}}]=H({\\mathbf {m}}_0)$ for $p\\notin [-2L,2L]$ .",
"Since outside $[-2L,2L]$ the Hamiltonian is constant, that part of the system does not contribute to $F^{(2)}$ and can be discarded.",
"What remains is a system with a finite-dimensional Hilbert space.",
"Since $f_+-f_-= \\theta (p-3L)-\\theta (p+3L)$ and thus is equal $-1$ in the region $[-2L,2L]$ , we have $-\\langle F^{(2)}_{+-},f_+-f_-\\rangle =\\sum _{p\\in [-2L,2L]} F^{(2)}_{+- p}+O(L^{-\\infty }).$ This is simply the Berry curvature of this finite-dimensional system.",
"Therefore its integral over $S^2$ is an integer multiple of $2\\pi $ .",
"We conclude that $\\int _{S^3} h^*(\\Omega ^{(3)}(f))=2\\pi n+O(L^{-\\infty }), \\quad n\\in {\\mathbb {Z}}.$ Taking the limit $L\\rightarrow \\infty $ we get the desired result.",
"In general we proceed by induction in $D$ .",
"For $D>1$ the restriction to SRE systems is a nontrivial constraint on the kind of families we allow.",
"Other than that, we can proceed in the same way as for $D=1$ .",
"First we tensor with a suitable constant SRE system to reduce to the case of a family of systems in a trivial phase.",
"Then we remove the north and south pole from $S^{D+2}$ and define three families of gapped Hamiltonians $H^+[{\\mathbf {s}}]$ , $H^-[{\\mathbf {s}}],$ and $H^{+-}[{\\mathbf {s}}]$ which are defined on $S^{D+2}_N$ , $S^{D+2}_S$ and $S^{D+2}_N\\bigcap S^{D+2}_S$ , respectively.",
"They approach $H({\\mathbf {m}}_0)$ on the far right, far left, and both far right and far left, respectively.",
"By far right we mean the region $x^D(p)\\gg 0$ , while far left is the region $x^D(p)\\ll 0$ .",
"The same manipulations as before reduce the integral of $\\Omega ^{(D+2)}$ over $S^{D+2}$ to an integral of $\\Omega ^{(D+1)}$ over the equatorial $S^{D+1}$ up to terms of order $L^{-\\infty }$ .",
"This completes the inductive step.",
"An interpolation between $H({\\mathbf {m}})$ and $H({\\mathbf {m}}_0)$ can also be viewed as a gapped boundary condition for $H({\\mathbf {m}})$ .",
"Given a smooth family of gapped boundary conditions for $H[{\\mathbf {s}}]$ defined on some open subset $U\\subset S^3$ (not necessarily arising from a smooth interpolation as above), one can write $\\Omega ^{(D+2)}(f_1,\\ldots ,f_D)\\vert _U$ as an exact form.",
"This is done in exactly the same way as above.",
"Therefore if the cohomology class of $\\Omega ^{(D+2)}$ is non-trivial, it is impossible to find a family of gapped boundary conditions for $H[{\\mathbf {s}}]$ which is defined on the whole $S^3$ and varies smoothly with ${\\mathbf {s}}$ .",
"For $D=0$ the analogous statement is that the cohomology class of the Berry curvature is an obstruction to finding a family of ground states on the whole parameter space which depends continuously on the parameters."
],
[
"Higher Berry curvature for 1d insulators of class A",
"In this appendix we compute the higher Berry curvature 3-form in the case of gapped systems of free fermions in 1d with conserved charge (that is, insulators of class A).",
"Then we specialize to the case of translationally-invariant systems and compare with forms constructed out of the Bloch-Berry connection.",
"We start with the many-body expression for the 3-form $F^{(3)}_{pq}$ divided by $2\\pi $ : $\\frac{F^{(3)}_{pq}}{2\\pi }=-\\frac{i}{12\\pi }\\oint _{z=E_0} \\frac{dz}{2\\pi i} {\\rm Tr}(2GdHG^2 dH_p G dH_q+GdHGdH_p G^2dH_q) -(p\\leftrightarrow q)$ We will consider the following many-body Hamiltonian: $H_p=\\frac{1}{2}\\sum _{m\\in \\Lambda } \\left(a^\\dagger _ph(p,m)a_m+a^\\dagger _m h(m,p) a_p\\right).$ Here $h(p,q)$ is an Hermitian matrix $h(p,q)^*=h(q,p)$ .",
"The fermionic creation-annihilation operators $a_p^\\dagger , a_p$ satisfy canonical anti-commutation relations $\\begin{split}\\lbrace a^\\dagger _p, a_q\\rbrace &= \\delta _{pq},\\\\\\lbrace a_p, a_q\\rbrace &= \\lbrace a^\\dagger _p, a^\\dagger _q\\rbrace =0,\\end{split}$ where $\\delta _{pq}$ is the Kronecker delta.",
"Since all relevant operators are sums of single particle operators, matrix elements $\\langle m |A| n\\rangle $ vanish unless many-body states $n$ and $m$ differ by exactly one single-particle excitation.",
"The above expression can be written in terms of one-particle quantities as follows: $\\frac{F^{(3)}_{pq}}{2\\pi }=-\\frac{i}{12\\pi }\\oint \\frac{dz}{2\\pi i} {\\rm tr}(2gdhg^2 dh_p g dh_q+gdhgdh_p g^2dh_q) -(p\\leftrightarrow q).$ Here the contour of integration encloses all states below Fermi level and all lower case letters denote the corresponding single-particle operators acting on the single-particle Hilbert space $\\ell ^2(\\Lambda )$ .",
"Naively, this integral contains additional contributions compared to (REF ) where a fermion jumps from an empty state or jumps into a filled state.",
"But these contributions cancel each other and the result coincides with (REF ).",
"Hamiltonian density at a point $p$ can be written as $h_p= \\frac{1}{2} (\\delta _p h + h\\delta _p)$ , where $\\delta _p$ is Kronecker's delta (equal 1 on $p$ and 0 on other cites) and functions are understood as operators on the one-particle Hilbert space acting by multiplication.",
"Contracting $F^{(3)}_{pq}$ with the cochain $f(q)-f(p)$ we find $\\begin{split}\\frac{1}{2\\pi }\\langle F^{(3)},\\delta f\\rangle = -\\frac{i}{24\\pi }\\oint \\frac{dz}{2\\pi i} {\\rm tr}\\Big (&[dh,f](gdhgdhg^2-g^2dhgdhg)\\\\&-2[h,f](gdhg^2dhgdhg-gdhgdhg^2dhg) \\Big ).\\end{split}$ Note that multiplication by $f$ is not a trace class operator, since it acts on infinitely many sites.",
"Therefore traces containing them are not guaranteed to exist.",
"On the other hand, commutators like $[dh,f]$ are supported only on a finite number of sites and traces containing them are well-defined.",
"On the other hand, given a gapped 1d system of free fermions with translational symmetry which depends on three parameters $\\lambda _1,\\lambda _2,\\lambda _3$ , one may consider the Bloch bundle of filled states over the product of the Brillouin zone $S^1$ and the parameter space $\\Sigma $ .",
"It carries the non-Abelian Bloch-Berry connection, and one can consider various Chern-Weil forms on $S^1\\times \\Sigma $ constructed out of this connection.",
"In particular, one can consider the degree-4 component of the Chern character of the Berry-Bloch connection and its integral over $S^1\\times \\Sigma $ : $\\int _{S^1\\times \\Sigma } {\\rm Ch}({\\mathcal {F}})=-\\frac{1}{8\\pi ^2}\\int _{S^1\\times \\Sigma } {\\rm Tr}({\\mathcal {F}}\\wedge {\\mathcal {F}}).$ Here ${\\mathcal {F}}$ is the non-Abelian curvature 2-form of the Bloch-Berry connection and trace is taken over filled bands.",
"It can be shown (see Sec.",
"IIIA in [8]) that this quantity can be expressed in terms of the one-particle Green's function as follows: $\\begin{split}&-\\frac{1}{8\\pi ^2}\\int _{S^1\\times \\Sigma } {\\rm Tr}({\\mathcal {F}}\\wedge {\\mathcal {F}})\\\\ &= \\frac{\\pi ^2}{15} \\epsilon ^{\\mu \\nu \\rho \\sigma \\tau } \\oint \\frac{dz}{2\\pi i} \\int _{S_1} \\frac{dk}{2\\pi }\\int _{\\Sigma } \\frac{d^3\\lambda }{(2\\pi )^3} {\\rm tr^{\\prime }}\\left[ \\left(g \\frac{\\partial g^{-1}}{\\partial q^\\mu }\\right)\\left(g \\frac{\\partial g^{-1}}{\\partial q^\\nu }\\right)\\left(g \\frac{\\partial g^{-1}}{\\partial q^\\rho }\\right)\\left(g \\frac{\\partial g^{-1}}{\\partial q^\\sigma }\\right)\\left(g \\frac{\\partial g^{-1}}{\\partial q^\\tau }\\right)\\right],\\end{split}$ where $q^\\mu =(z,k,\\lambda _1 ,\\lambda _2,\\lambda _3)$ .",
"The first integral encloses filled levels, the second integral is over the Brillouin zone, and the last integral is over the parameter space $\\Sigma $ .",
"The trace ${\\rm tr^{\\prime }}$ is taken over subspace with fixed momentum $k$ .",
"In translationally invariant system we can interpret $ \\int _{S_1} \\frac{dk}{2\\pi }$ as part of the trace ${\\rm tr}$ over the whole one-particle Hilbert space and substitute $\\frac{\\partial g^{-1}}{\\partial k} = -\\frac{\\partial h}{\\partial k} =- i[h,f]$ .",
"Expanding the derivatives $\\partial /\\partial q^\\mu $ and combining parameter derivatives into forms, $\\sum _i\\dfrac{\\partial h}{\\partial \\lambda ^i}d\\lambda ^i = dh $ , we find $-\\frac{1}{8\\pi ^2}\\int _{S^1\\times \\Sigma } {\\rm Tr}({\\mathcal {F}}\\wedge {\\mathcal {F}})=\\frac{i}{24\\pi } \\oint \\frac{dz}{2\\pi i} \\int _{\\Sigma } {\\rm tr}\\Big ( g^2[h,f]gdhgdhgdh\\\\-g^2dhg[h,f]gdhgdh+g^2dhgdhg[h,f]gdh-g^2dhgdhgdhg[h,f]\\Big ).$ One can see that the integrand of this expression differs from (REF ) by a total derivative proportional to $d \\left( \\oint \\frac{dz}{2\\pi i} {\\rm tr}\\Big ([h,f](gdhgdhg^2-g^2dhgdhg) \\Big ) \\right).$ Since $\\Sigma $ was an arbitrary three-dimensional submanifold of the parameter space, we have shown that the first higher Berry 3-form divided by $2\\pi $ is in the same cohomology class as $\\int _{S^1\\times \\Sigma } {\\rm Ch}({\\mathcal {F}})$ .",
"We conjecture that more generally for class A insulators in $D$ dimensions the form $\\Omega ^{(D+2)}$ is in the same cohomology class as the integral of the degree $2D+2$ component of the Chern character of the Bloch-Berry connection over the Brillouin zone.",
"An example of a free 1d fermion system with a non-trivial integral $\\int _{S^1\\times \\Sigma } {\\rm Ch}({\\mathcal {F}})$ can be constructed using the 4d Chern insulator (see sec.",
"IIIB of [8]).",
"The Hamiltonian is $H=\\sum _{k_x}\\psi ^\\dagger _{k_x} d_a(k_x,{\\vec{\\lambda }}) \\Gamma ^a \\psi _{k_x},$ where $\\Gamma ^a$ are five Dirac matrices generating a Clifford algebra, and $d_a(k_x,{\\vec{\\lambda }}) = \\left[(m+c+\\cos k_x+c\\sum _{i=1}^3 \\cos \\lambda _i), \\sin k_x, \\sin \\lambda _1,\\sin \\lambda _2,\\sin \\lambda _3 \\right].$ It was shown in Ref.",
"[8] that if we chose $\\Sigma $ to be 3-torus $S^1\\times S^1\\times S^1$ defined by identification $\\lambda _i\\sim \\lambda _i +2\\pi $ this model has a non-zero integer value of the integral $\\int _{S^1\\times S^1\\times S^1\\times S^1} {\\rm Ch}({\\mathcal {F}})$ for a particular choice of $m$ and $c$ .",
"One can think about this family of 1d models as a \"dimensional reduction\" of the 4d Chern insulator where we treat three out of four components of momentum as parameters.",
"Note that the Atiyah-Singer index theorem [12] implies that the integral of the Chern character of a vector bundle over a four-torus is an integer.",
"Therefore the integral of $\\Omega ^{(3)}$ over the parameter space $T^3$ is $2\\pi $ times an integer, despite the fact that the parameter space is a torus rather than a sphere."
]
] | 2001.03454 |
[
[
"A note on Nicolas-Auguste Tissot: At the origin of quasiconformal\n mappings (to appear in Vol. VII of the Handbook of Teichm{\\\"u}ller Theory)"
],
[
"Abstract Nicolas-Auguste Tissot (1824--1897) was a French mathematician and cartographer.",
"He introduced a tool which became known among geographers under the name ``Tissot indicatrix'', and which was widely used during the first half of the twentieth century in cartography.",
"This is a graphical representation of a field of ellipses, indicating at each point of a geographical map the distorsion of this map, both in direction and in magnitude.",
"Each ellipse represented at a given point is the image of an infinitesimal circle in the domain of the map (generally speaking, a sphere representing the surface of the earth) by the projection that realizes the geographical map.",
"Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane, and he also developed a theory for the distorsion of mappings between general surfaces.",
"His ideas are close to those that are at the origin of the work on quasiconformal mappings that was developed several decades after him by Gr{\\\"o}tzsch, Lavrentieff, Ahlfors and Teichm{\\\"u}ller.",
"Gr{\\\"o}tzsch, in his papers, mentions the work of Tissot, and in some of the drawings he made for his articles, the Tissot indicatrix is represented.",
"Teichm{\\\"u}ller mentions the name Tissot in a historical section in one of his fundamental papers in which he points out that quasiconformal mappings were initially used by geographers.",
"The name Tissot is missing from all the known historical reports on quasiconformal mappings.",
"In the present article, we report on this work of Tissot, showing that the theory of quasiconformal mappings has a practical origin.",
"The final version of this article will appear in Vol.",
"VII of the Handbook of Teichm{\\\"u}ller Theory (European Mathematical Society Publishing House, 2020)."
],
[
"Introduction",
"Darboux, starts his 1908 ICM talk whose title is Les origines, les méthodes et les problèmes de la géométrie infinitésimale (The origins, methods and problems of infinitesimal geometry) with the words: “Like many other branches of human knowledge, infinitesimal geometry was born in the study of practical problems,\" and he goes on explaining how problems that arise in the drawing of geographical maps, that is, the representation of regions of the surface of the Earth on a Euclidean piece of paper, led to the most important developments in geometry made by Lagrange, Euler, Gauss and others.",
"The theory of quasiconformal mappings has its origin in the problems of drawing geographical maps.",
"Teichmüller, in the last part of his paper Extremale quasikonforme Abbildungen und quadratische Differentiale (Extremal quasiconformal mappings and quadratic differentials), published in 1939 [35], which is the main paper in which he develops the theory that became known as Teichmüller theory, makes some comments on this origin, mentioning the work of the French mathematician and geographer Nicolas-Auguste Tissot (1824–1897).",
"Grötzsch, in his paper Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfach zusammenhängender schlichter Bereiche (On the distortion of non-conformal schlicht mappings of multiply-connected schlicht regions), published in 1930 [19], mentions several times the name Tissot, referring to the Tissot indicatrix which he represents in the pictures he drew for his article.",
"The directions of the major and minor and minor axes of this ellipse constitute are important element in some of his results.",
"A geographical map is the image of a mapping—henceforth called a projection—from the surface of the Earth, considered as a sphere or spheroid, onto the Euclidean plane.",
"The Tissot indicatrix is a device introduced by Tissot, who called it the indicating ellipse (ellipse indicatrice, which was used by geographers until the middle of the twentieth century.",
"It is a field of ellipses drawn on the geographical map, each ellipse representing the image by the projection—assumed to be differentiable—of an infinitesimal circleThe expression “infinitesimal circle\" means here, as is usual in the theory of quasiconformal mappings, a circle on the tangent space at a point.",
"In practice, it is a circle on the surface which has a “tiny radius.\"",
"In the art of geographical map drawing, these circles, on the domain surfaces, are all supposed to have the same small size, so that the collection of relative sizes of the image ellipses becomes also a meaningful quantity.",
"at the corresponding point on the sphere (or spheroid) representing the surface of the Earth.",
"Examples of Tissot indicatrices are given in Figure REF .",
"Figure: Four geographical maps on which the field of ellipses (Tissot indicatrix) are drawn.",
"The maps are extracted from the book Album of map projections .",
"These are called, from left to right, top to bottom, the stereographic p. 180, Lagrange p. 180, central cylindrical p. 30 and equidistant conical projections p. 92.",
"The first two projections are conformal and not area-preserving.",
"The last two are neither conformal nor area-preserving.In Figure REF , we have reproduced drawings from a paper of Grötzsch in which he represents the Tissot indicatrix of the maps he uses.",
"Figure: Two figures from Grötzsch's paper Über die Verzerrung bei nichtkonformen schlichten Abbildungen mehrfachzusammenhängender schlichter Bereiche .Grötzsch drew the Tissot indicatrices of his quasiconformal mappings.",
"(In each drawing, the major and minor axes of the ellipses are shown.",
")Although the work of Tissot is closely related to the theory of quasiconformal mappings, his name is never mentioned in the historical surveys of this subject, and the references by Grötzsch and by Teichmüller to his work remained unnoticed.",
"In this note, I will give a few indications on this work.",
"Before surveying the work of Tissot in §, I will give, in §, a short biographical note on him."
],
[
"Biographical note on Tissot",
"Nicolas-Auguste Tissot was born in 1824, in Nancy, which was to become, 26 years later, the birthplace of Henri Poincaré (whom we shall mention soon).",
"Tissot entered the École Polytechnique in 1841.",
"He started by occupying a career in the ArmyThe reader should note that the École Polytechnique was, and is still, is a military school.",
"and defended a doctoral thesis on November 17, 1851; cf.",
"[36].",
"On the cover page of his thesis, he is described as “Ex-Capitaine du Génie.\"",
"Tissot became later a professor at the famous Lycée Saint-Louis in Paris, and at the same time examiner at the École Polytechnique, in particular for the entrance exam.",
"He eventually became an assistant professor (répétiteur) in geodesy at the École Polytechnique.",
"After having published, in the period 1856–1858, several papers and Comptes Rendus notes on cartography, in which he analyzed the distortion of some known geographical maps (see [38], [39], [40]), Tissot started developing his own theory, on which he published three notes, in the years 1859–1860, [41], [42], [43], and then a series of others in the years 1865–1880 [45], [46], [47], [48], [49], [50], [51].",
"He then collected his results in the memoir [52], published in 1881, in which he gives detailed proofs.",
"In a note on p. 2 of this memoir, Tissot declares that after he published his first Comptes Rendus notes on the subject, the statements that he gave there without proof were reproduced by A. Germain in his Traité des projections des cartes géographiques [18] and by U. Dini in his memoir Sopra alcuni punti della teoria delle superfici [12].",
"He notes that Germain and Dini gave their own proofs of these statements, which are nevertheless more complicated than those he had in mind and which he gives in the memoir [52].",
"He also writes that Dini showed that the whole theory of curvature of surfaces may be deduced from the general theory that he had developed himself.",
"In fact, Dini applied this theory to the representation of a surface on a sphere, using Gauss's methods.",
"Tissot also says that his ideas were used in astronomy, by Hervé Faye, in his Cours d'astronomie de l'École Polytechnique [16].",
"The texts of the two Comptes rendus notes [43] and [40] of Tissot are reproduced in the Germain's treatise [18].",
"Besides his work on geographical maps, Tissot wrote several papers on elementary geometry.",
"We mention incidentally that several preeminent French mathematicians of the nineteenth and the beginning of the twentieth century published papers on this topics.",
"We mention Serret, Catalan, Laguerre, Darboux, Hadamard and Lebesgue; see e.g.",
"[44], [37], [53].",
"On the title page of Tissot's memoir [52] (1881), the expression Examinateur à l'École Polytechnique follows his name, as he was in charge of the entrance examination.",
"In his Éloge historique de Henri Poincaré [8], Darboux relates the following episode about Tissot, examining Poincaré:Poincaré entered the École Polytechnique in 1873.",
"In the French system of oral examinations, which is still in use, a student is given a question or a set of questions which he is asked to prepare while another student (who had already been given some time to prepare his questions) is explaining his solutions at the blackboard, in the same room.",
"Thus, it is not unusual that at such an examination, some students listen to the examinations of others.",
"Before asking his questions to Poincaré, Mr. Tissot suspended the exam during 45 minutes: we thought it was the time he needed to prepare a sophisticated question.",
"Mr. Tissot came back with a question of the Second Book of Geometry.",
"Poincaré drew a formless circle, he marked the lines and the points indicated by the examiner, then, after wandering long enough in front of the blackboard, with his eyes fixed on the ground, he concluded loudly: “It all comes down to proving the equality $AB=CD$ .",
"This is a consequence of the theory of mutual polars, applied to the two lines.\"",
"Mr. Tissot interrupted him: “Very good, Sir, but I want a more elementary solution.\"",
"Poincaré started wandering again, this time not in front of the blackboard, but in front of the table of the examiner, facing him, almost unconscious of his acts; then suddenly he developed a trigonometric solution.",
"Mr. Tissot objected: “I would like you to stay in Elementary Geometry.\"",
"Almost immediately after that, the examiner of Elementary Geometry was given satisfaction.",
"He warmly congratulated the examinee and announced that he deserves the highest grade.Avant d'interroger Poincaré, M. Tissot suspendit l'examen pendant trois quarts d'heure : le temps de préparer une question raffinée, pensions-nous.",
"M. Tissot revint avec une question du deuxième Livre de Géométrie.",
"Poincaré dessina un cercle informe, il marqua les lignes et les points indiqués par l'examinateur; puis, après s'être promené devant le tableau les yeux fixés à terre pendant assez longtemps, conclut à haute voix: Tout revient à démontrer l'égalité $AB = CD$ .",
"Elle est la conséquence de la théorie des polaires réciproques, appliquée aux deux droites.",
"“Fort bien, Monsieur, interrompit M. Tissot; mais je voudrais une solution plus élémentaire.\"",
"Poincaré se mit à repasser, non plus devant le tableau, mais devant la table de l'examinateur, face à lui, presque inconscient de ses actes, puis tout à coup développa une solution trigonométrique.",
"“Je désire que vous ne sortiez pas de la Géométrie élémentaire,\" objecta M. Tissot, et presque aussitôt satisfaction fut donnée à l'examinateur d'élémentaires, qui félicita chaleureusement l'examiné et lui annonça qu'il avait mérité la note maxima.",
"Poincaré kept a positive momory of Tissot's examinations.",
"He expresses this in a letter to his mother sent on May 6, 1874, opposing them to the 10-minute examinations (known as “colles\") that he had to take regularly at the École Polytechnique and which he said are pitiful.",
"He writes:[30], letter No.",
"62.",
"“When I think about the exams of Tissot and others, I can not help but take pity of these 10 minutes little colles where one puts in danger his future with an expression which is more or less exact or a sentence which is more or less well crafted, and where a person is judged upon infinitesimal differences.",
"\"Quand je pense aux exams de Tissot et autres, [...] je ne puis m'empêcher de prendre en pitié ces petites colles de 10 minutes où on joue son avenir dans une expression plus ou moins exacte ou sur une phrase plus ou moins bien tournée et où on juge un individu sur des différences infinitésimales."
],
[
"On the work of Tissot on geographical maps",
"Tissot studied at the École Polytechnique, an engineering school where the students had a high level of mathematical training and at a period where the applications of the techniques of differential geometry to all the domains of science were an integral part of the curriculum.",
"His work is part of a well-established tradition where mathematical tools are applied to the craft of map drawing.",
"This tradition passes through the works of preeminent mathematicians such as Ptolemy [31], [3], Lambert [22], [23], Euler [13], [14], [15] Lagrange [20], [21], Gauss [17], Chebyshev [5], [6], Beltrami [2], Liouville (see the appendices to [26]), Bonnet [4] Darboux [9], [10], [11], and there are others.",
"It was known since antiquity that there exist conformal (that is, angle-preserving) projections from the sphere to the Euclidean plane.Ptolemy, in his Geography, works with the strereographic projection, see [3].",
"See also Ptolemy's work on the Planisphere [32].But it was noticed that these projections distort other quantities (length, area, etc.",
"), and the question was to find projections that realize a compromise between these various distorsions.",
"For instance, one question was to find the closest-to-conformal projection among the maps that are area-preserving.",
"Hence, the idea of “closest-to-conformal\" projection came naturally.",
"Among the mathematicians who worked on such problems, Tissot came closest to the notion of quasiconformality.",
"Let us summarize a few of his results on this subject.",
"An important observation made by Tissot right at the beginning of his memoir [52] (p. 1) is that finding the most appropriate mode of projection depends on the shape of the region—and not only its size, that is, on the properties of its boundary,.",
"Finding maps of small “distorsion\" (where, as we mentioned, this word has several possible meanings) was the aim of theoretical cartography.",
"Tissot discovered that in order for the map to minimize an appropriately defined distortion, a certain function $\\lambda $ , defined by setting $d\\sigma ^2=(1+\\lambda )^2ds^2,$ must be minimized in some appropriate sense, where $ds$ and $d\\sigma $ are the line elements at the source and the target surfaces respectively.",
"The minimality of $\\lambda $ may mean, for example, that the value of the gradient of its square must be the smallest possible.",
"In fact, Tissot studied mappings between surfaces that are more general than those between subsets of the sphere and of the Euclidean plane.",
"He started by noting that for a given mapping between two surfaces, there is, at each point of the domain, a pair of orthogonal directions that are sent to a pair of orthogonal directions on the image surface.",
"Unless the mapping is angle-preserving at the given point, these pairs of orthogonal directions are unique.",
"The orthogonal directions at the various points on the two surfaces define a pair of orthogonal foliations preserved by the mapping.",
"Tissot calls the tangents to these foliations principal tangents at the given point.",
"They correspond to the directions where the ratio of lengths of the corresponding infinitesimal line elements attains its greatest and smallest values.",
"Using the foliations defined by the principal tangents, Tissot gave a method for finding the image of an infinitely small figure drawn in the tangent plane of the first surface.",
"In particular, for a differentiable mapping, the images of infinitesimal circles are ellipses.",
"In this case, he gave a practical way of finding the major and minor axes of these ellipses, and he provided formulae for them.",
"This is the theory of the Tissot indicatrix.",
"From the differential geometric point of view, the Tissot indicatrix gives information on the metric tensor obtained by pushing forward the metric of the sphere (or the spheroid) by the projection mapping.",
"We recall that in modern quasiconformal theory, an important parameter of a map is the quasiconformal dilatation at a point, defined as the ratio of the major axis to the minor axis of the infinitesimal ellipse which is the image of an infinitesimal circle by the map (assumed to be differentiable at the give point, so that its derivative sends circles centered at the origin in the tangent plane to ellipses).",
"The Tissot indicatrix gives much more information than this quasiconformal dilatation, since it keeps track of (1) the direction of the great and small axes of the infinitesimal ellipse, and (2) the size of this ellipse, compared to that of the infinitesimal circle of which it is the image.",
"Darboux got interested in the work of Tissot on geography, and in particular, in a projection described in Chapter 2 of his memoir [52].",
"He wrote a paper on Tissot's work [11] explaining more carefully some of his results.",
"He writes: “[Tissot's] exposition appeared to me a little bit confused, and it seems to me that while we can stay in the same vein, we can follow the following method [...]\"Son exposition m'a paru quelque peu confuse, et il m'a semblé qu'en restant dans le même ordre d'idées on pourrait suivre avec avantage la méthode suivante.",
"Tissot showed then how to construct mappings that have minimal distortion.",
"Tissot's work was considered as very important by cartographers.",
"The American cartographer, in his book Flattening the earth: two thousand years of map projections [34], published in 1997 and which is a reference in the subject, after presenting the existing books on cartography, writes: “Almost all of the detailed treatises presented one or two new projections, they basically discussed those existing previously, albeit with very thorough analysis.",
"One scholar, however, proposed an analysis of distorsion that has had a major impact on the work of many twentieth-century writers on map projections.",
"This was Tissot [...].\"",
"Modern cartographers are still interested in the theoretical work of Tissot, see [24].",
"We mentioned several preeminent mathematicians who before Tissot worked on the theory of geographical maps.",
"From the more recent era, let me mention Milnor's paper titled A problem in cartography[25], published in 1969.",
"The reader interested in the theory of geographical maps developed by mathematicians is referred to the papers [29], [27] and [28] which also contain more on the work of Tissot.",
"geographical map Grötzsch, Herbert (1902–1993) indicatrix Tissot Poincaré, Henri (1854–1912) quasiconformal mapping Tissot indicatrix Tissot, Nicolas-Auguste (1824–1897)"
]
] | 2001.03434 |
[
[
"Sparse Vector Transmission: An Idea Whose Time Has Come"
],
[
"Abstract In recent years, we are witnessing bewildering variety of automated services and applications of vehicles, robots, sensors, and machines powered by the artificial intelligence technologies.",
"Communication mechanism associated with these services is dearly distinct from human-centric communications.",
"One important feature for the machine-centric communications is that the amount of information to be transmitted is tiny.",
"In view of the short packet transmission, relying on today's transmission mechanism would not be efficient due to the waste of resources, large decoding latency, and expensive operational cost.",
"In this article, we present an overview of the sparse vector transmission (SVT), a scheme to transmit a short-sized information after the sparse transformation.",
"We discuss basics of SVT, two distinct SVT strategies, viz., frequency-domain sparse transmission and sparse vector coding with detailed operations, and also demonstrate the effectiveness in realistic wireless environments."
],
[
"Introduction",
"These days, automated things such as vehicles, drones, sensors, machines, and robots, combined with artificial intelligence (AI) technologies, have found their way into almost every industry.",
"Remarkable growth of business models using autonomous machines is accelerating the need for communication between machines as well as machine to human communications [1].",
"One important feature of machine-centric communications over the long-standing human-oriented communications is that the amount of information to be transmitted is tiny.",
"For example, information to be exchanged in the autonomous driving, robot, smart factory, and home appliance is in a form of control and command-type information such as start/stop, turn on/off, move left/right, speed up/slow down, shift, and rotate.",
"Typically, required information bit in these applications is in the range of 10 $\\sim $ 100 bits.",
"Information acquired from the sensors (e.g., temperature, pressure, speed, gas density) is in the order of 10 bits.",
"Also, similar sized packets are used in many feedback or control channels (e.g., ACK/NACK feedback in 4G LTE/5G NR PUCCH [2], [3]).",
"Crucial observation in these applications is that conventional transmission mechanism is unduly complicated and inefficient, resulting in a waste of resources, transmit power, and processing latency.",
"Shannon's channel coding theorem, governing principle of today’s packet transmission, is based on the law of large numbers so that it works properly only when the packet size is sufficiently large.",
"In fact, when the packet length is short, noise introduced by the channel cannot be averaged out properly, degrading the packet reception quality substantially (see, e.g., information theoretic analysis in [4]).",
"Further, in the ultra short-packet transmission regime, size of the non-payload (pilot signals and control data) easily exceeds the payload size so that the cost caused by the non-payload outweighs the cost of payload.",
"In particular, in some applications requiring high reliability (e.g., ultra-reliable and low latency communications (URLLC) in 5G [5]), cost caused by the pilot signaling increases sharply, further degrading the resource utilization efficiency.",
"Without doubt, relying on today's transmission mechanism would not be efficient due to the waste of resources, large decoding latency, and also expensive operational cost.",
"Our intent in this article is to introduce new type of short packet transmission scheme referred to as sparse vector transmission (SVT).",
"Key idea of SVT is to transmit the short-sized information after the sparse vector transformation.",
"Using the principle of compressed sensing (CS), we decode the packet using a small number of resources.",
"SVT has a number of advantages over the conventional transmission strategies; it is simple to implement, reduces the transmission latency as well as the encoding/decoding complexity.",
"When the position of a sparse vector is used to encode the information exclusively, decoding can be done without the channel knowledge, saving the pilot transmission overhead and the channel estimation effort.",
"Further, SVT can inherently improve the user identification quality and security.",
"In a nutshell, SVT is a viable solution for massive machine-type communication (mMTC) and URLLC scenarios having many advantages over the conventional packet transmission mechanism.",
"The rest of this paper is organized as follows.",
"In Section II, we explain the principle of sparse vector transmission.",
"In Section III and IV, we discuss two distinct SVT strategies: the frequency domain sparse transmission and sparse vector coding.",
"We present the future research direction and conclusion in Section V."
],
[
"Principle of Sparse Vector Transmission",
"In this section, we discuss the basic principle of SVT.",
"After the brief review of the CS principle, we explain two types of SVT schemes: frequency-domain sparse transmission and sparse vector coding."
],
[
"Basics of Compressed Sensing",
"A system with $m$ -dimensional measurement vector $\\mathbf {y}$ and $n$ -dimensional input vector $\\mathbf {s}$ given by $\\mathbf {y}=\\mathbf {A}\\mathbf {s}$ where $\\mathbf {A}$ is the system (sensing) matrix relating the input vector $\\mathbf {s}$ and the measurement vector $\\mathbf {y}$ .",
"If $\\mathbf {A}$ is a tall or square matrix, meaning that the dimension of $\\mathbf {y}$ is larger than or equal to the dimension of $\\mathbf {s}$ $(m \\ge n)$ , one can recover $\\mathbf {s}$ using the conventional techniques (e.g., Gaussian elimination) as long as the sensing matrix is a full rank.",
"However, if the matrix $\\mathbf {A}$ is fat $(m < n)$ , meaning that the number of unknowns is larger than the dimension of observation vector, it is in general not possible to find out the unique solution since there exist infinitely many possible solutions.",
"In this pathological scenario where the inverse problem is ill-posed, sparsity of an input vector comes to the rescue.",
"A vector $\\mathbf {s}$ is called sparse if the number of nonzero entries is sufficiently smaller than the dimension of the vector.",
"If a vector $\\mathbf {s}$ is $k$ -sparse, meaning that there are $k$ nonzero elements in $\\mathbf {s}$ , the measurement vector $\\mathbf {y}$ is expressed as a linear combination of $k$ columns of $\\mathbf {A}$ associated with the nonzero entries of $\\mathbf {s}$ .",
"If the support $\\Omega _{\\mathbf {s}}$ (set of nonzero indices in $\\mathbf {s}$ ) is known a priori by any chance, then by removing columns corresponding to the zero entries in $\\mathbf {s}$ , we can convert the underdetermined system into over-determined one and thus can find out the solution using the standard technique.",
"CS theory asserts that as long as the sensing matrix is generated at random, $k$ -sparse vector can be recovered with $m \\approx ck\\log (n/k)$ measurements ($c$ is a constant).",
"In performing the recovery task, $\\ell _{1}$ -norm minimization technique and greedy sparse recovery algorithm (e.g., orthogonal matching pursuit (OMP)) have been popularly used (see, e.g., [6]).",
"It is worth mentioning that underlying assumption in many CS-based studies is that the signal is sparse in nature or can be sparsely represented in a properly chosen basis.",
"Indeed, applications of CS in wireless communications have been mainly on the recovery of naturally sparse signals such as sparse millimetre wave channel estimation in angle and delay domains, angle (DoA/AoD) estimation, and spectrum sensing [6], [7], [8].",
"Intriguing feature of SVT over these is to purposely transmit the sparse vector to achieve the gain in performance, latency, and energy efficiency.",
"Figure: System models for SVT schemes (FDST and SVC)."
],
[
"Two Types of SVT",
"Basically, there are two options in SVT.",
"To ease our exposition, we discuss the orthogonal frequency division multiplexing (OFDM) system, standard systems for 4G, 5G cellular and WiFi systems, as a baseline.",
"Nevertheless, main principle can be readily extended to different transmission schemes.",
"In the first option, referred to as the frequency-domain sparse transmission (FDST), information is embedded into a small number of subcarriers and then transmitted (see Figure 1).",
"In this case, composite of the channel matrix $\\mathbf {H}$ and the IDFT matrix $\\mathbf {F}^{H}$ becomes the sensing matrix so that the time-domain sample vector $\\mathbf {y}=\\mathbf {A}\\mathbf {s}+\\mathbf {v}=\\mathbf {H}\\mathbf {F}^{H}\\mathbf {s}+\\mathbf {v}$ becomes the measurement vector.",
"While the symbol decoding in the conventional OFDM systems is initiated after receiving all time-domain samples, sparse vector $\\mathbf {s}$ in FDST can be recovered with a small number of time-domain samples using the CS technique.",
"Let $\\mathbf {P}$ be the matrix taking early $m$ samples of $\\mathbf {y}$ , then the vector of the first $m$ measurements is expressed as $\\tilde{\\mathbf {y}}=\\mathbf {P}\\mathbf {y}$ (see Figure 1).",
"For instance, if $k=16$ , $n=1024$ , and $c=4$ , then only 11% of samples $(m \\approx 115)$ is needed to decode $\\mathbf {s}$ .",
"In the second option, called the sparse vector coding (SVC), spreading matrix $\\mathbf {C}=[\\mathbf {c}_{1} \\ \\cdots \\ \\mathbf {c}_{n}]$ is applied to the sparse vector $\\mathbf {s}$ before the transmission (see Figure 1).",
"Each position in a vector $\\mathbf {s}$ has its own spreading sequence $\\mathbf {c}_{i}$ so that the transmit vector can be expressed as a linear combination of spreading sequences corresponding to nonzero symbols.",
"Distinctive feature of SVT over the conventional transmission scheme is that positions as well as symbols can be employed to convey the information.",
"By way of analogy, one can imagine a process to generate the sparse vector as placing a few balls into the empty boxes.",
"When we try to put $k$ balls in $n$ boxes $(k \\le n)$ , we have ${n \\atopwithdelims ()k}$ choices, so that we can encode $\\lfloor \\log _{2} {n \\atopwithdelims ()k} \\rfloor $ bits of information into the position of the sparse vector $\\mathbf {s}$ .",
"Suppose the modulation order is the same for all nonzero positions (say $b_{s}$ bit per symbol), then $kb_s$ bits can be encoded to the active symbols (symbols in the nonzero positions) so that one SVT block conveys $kb_s + \\lfloor \\log _{2} {n \\atopwithdelims ()k}\\rfloor $ bits in total.",
"There are various options to encode the information in SVT.",
"One simple option is to use both positions and active symbols in the information transmission.",
"Alternatively, one can map the user ID (UID) to the positions and the rest information to the active symbols to elegantly divide the user identification process and information decoding.",
"Yet another option is to embed the message to the positions and the parity bits for the error detection and correction to the active symbols."
],
[
"Frequency-domain Sparse Transmission",
"In this section, we discuss the FDST scheme in detail.",
"In contrast to the conventional OFDM systems, FDST transmits the information in a form of a sparse vector and then uses the CS technique to decode the input sparse vector.",
"We first discuss the system model and then explain the FDST decoding and environment-aware user identification, an approach to simplify the user identification process using environmental information."
],
[
"System Model",
"As discussed, the system model for FDST is $\\mathbf {y}=\\mathbf {H}\\mathbf {F}^{H}\\mathbf {s}+\\mathbf {v}$ .",
"Due to the addition of the cyclic prefix, $\\mathbf {H}$ is a circulant matrix and thus can be eigen-decomposed by DFT basis.",
"That is, $\\mathbf {H}=\\mathbf {F}^{H}\\Lambda \\mathbf {F}$ where $\\mathbf {F}$ is the DFT matrix and $\\Lambda $ is the diagonal matrix ($\\lambda _{ii}$ represents the channel of $i$ -th subcarrier).",
"The corresponding system model is expressed as $\\mathbf {y}=\\mathbf {F}^{H}\\Lambda \\mathbf {s}+\\mathbf {v}$ .",
"Since the supports of $\\mathbf {s}$ and $\\Lambda \\mathbf {s}$ are the same, by letting $\\mathbf {x}=\\Lambda \\mathbf {s}$ , the system model is converted to $\\mathbf {y}=\\mathbf {F}^{H}\\mathbf {x}+\\mathbf {v}$ .",
"Recalling that CS operates with far fewer measurements than the conventional techniques require, a small part of $\\mathbf {y}$ is enough to recover $\\mathbf {x}$ .",
"From the principle of CS, the recovery performance depends heavily on the quality of sensing matrix.",
"As a metric to evaluate the sensing matrix, mutual coherence defined as the largest magnitude of normalized inner product between two distinct columns of sensing matrix is widely used [6].",
"As long as we choose consecutive samples, the mutual coherence of IDFT submatrix $(\\mathbf {P}\\mathbf {F}^{H})$ remains constant so that the recovery performance would not be affected by the choice of samples in $\\mathbf {y}$ .",
"Hence, one can use early arrived samples to achieve a reduction in transmission and decoding latencies (see Figure 2).",
"Since this is a standard setting for CS, any sparse recovery algorithm can be employed to decode $\\mathbf {x}$ from $\\tilde{\\mathbf {y}}$ ."
],
[
"FDST Decoding",
"Basically, decoding of FDST consists of two steps; in the first step, support of $\\mathbf {x}$ is identified by the sparse recovery algorithm.",
"For example, greedy sparse recovery algorithm identifies one column (position of a vector) of the sensing matrix in each iteration.",
"In our case, a column of $\\mathbf {P}\\mathbf {F}^{H}$ that is maximally correlated with the measurement vector is chosen.",
"Once the support $\\Omega _{\\mathbf {x}}$ of $\\mathbf {x}$ (equivalently the support $\\Omega _{\\mathbf {s}}$ of $\\mathbf {s}$ ) is identified, by removing components associated with the zero entries in $\\mathbf {s}$ , an over-determined system model to decode the symbol $\\mathbf {s}$ can be obtained.",
"In the decoding of $\\mathbf {s}$ , conventional technique such as the linear minimum mean square error (LMMSE) estimator followed by the symbol slicer can be used for the symbol detection.",
"Figure: Illustration of the FDST-based short packet transmission in the TDD systems (k=3)(k=3).In the mapping of the UID to the support, one can consider the environment-aware user identification (EA-UI).",
"Key idea of EA-UI is to use a support $\\Omega _{\\mathbf {s}}$ derived from the environmental information as a UID.",
"By the environmental information, we mean the information obtained from wireless environments such as channel impulse response (CIR), angle (AoD, DoA), location, delay spread, to name just a few.",
"EA-UI is conceptually similar to the biometric user identification.",
"Biometric identifier, such as iris or fingerprint, intrinsically representing the unique identity of individual’s body, can greatly simplify the user identification process.",
"Principle of EA-UI is essentially the same since the environmental information is reflected in the support of the transmit vector.",
"One simple example, illustrated in Figure 2, is to choose positions of $k$ subcarriers having the largest channel gain as a support and use this as a UID.",
"For example, in TDD systems, base station (BS) can acquire the channel information (and thus UID) of all mobile devices from the uplink pilot signals due to the channel reciprocity.",
"In the uplink scenario, therefore, BS can identify which mobile device has sent the packet by checking the support (UID) of the received packet.",
"Similarly, in the downlink scenario, mobile device can easily check whether the packet is for itself by comparing the decoded support $\\hat{\\Omega }_{\\mathbf {x}}$ and its own support $\\Omega _{\\mathbf {x}}$ .",
"EA-UI has several advantages; first, it improves the security since the UID is derived from naturally acquired environmental (channel) information.",
"Second, in most physical channels (e.g., UE-specific channels like PDCCH, PDSCH in 4G LTE/5G NR [3]), BS sends the data together with UID to notify which device the packet is delivered to.",
"Since the EA-UI mechanism separates the user identification and data decoding elegantly, time and effort to decode whole packet just for the user identification purpose can be saved.",
"Indeed, since EA-UI is done by the identification of the support in $\\mathbf {x}$ , not by the accurate recovery of sparse vector $\\mathbf {s}$ , support of $\\mathbf {x}$ can be recovered using $\\tilde{\\mathbf {y}}$ and $\\mathbf {D}=\\mathbf {P}\\mathbf {F}^{H}$ (the submatrix of IDFT).",
"Recalling that $\\mathbf {D}$ is independent of the channel and $\\mathbf {x}=\\Lambda \\mathbf {s}$ , the channel estimation is unnecessary in the support detection.",
"Third, since $k$ is in general very small, sparse recovery algorithm can quickly identify the support.",
"Further, transmission and decoding latency can be greatly reduced since only a small fraction of early arrived (time-domain) samples is used for the packet decoding (see Figure 2).",
"As a final note, one can easily add the error correction capability to EA-UI since the sparsity of subcarriers lends itself to the addition of error correction mechanism.",
"For example, when the correlation between adjacent columns in $\\mathbf {D}$ is large, which is true for the submatrix of IDFT, an index of a column adjacent to the correct one can be chosen as a support element incorrectly.",
"Incorrect support element can also be chosen when the supports of BS and UE are slightly different due to the channel estimation error or imperfect channel reciprocity.",
"Since $k$ is small $(k \\ll n)$ in $\\mathbf {s}$ , by relaxing the success condition in the support identification, an error can be corrected.",
"Basic idea of this strategy is to replace the selected index $\\hat{\\omega }$ with the nearest support element $\\omega \\in \\Omega _{\\mathbf {x}}$ .",
"In other words, as long as the mismatch level $\\hat{\\omega }-\\omega $ is smaller than the properly designed threshold, the error caused by the different supports can be corrected [9].",
"Table: System Setup for FDST SimulationsFigure: BLER of URLLC packet transmission (m=256)(m=256)."
],
[
"Numerical Performance Evaluation",
"In this subsection, we present the numerical results to evaluate the performance of FDST.",
"In our simulations, the OFDM systems (with 512 subcarriers) under the i.i.d.",
"Rayleigh fading channels are used.",
"As performance measures, the block error rate (BLER) and the processing latency (see Table I for detailed setup) are considered.",
"In Figure 3, the BLER of FDST and PDCCH in 4G LTE are compared.",
"In our simulations, sizes of payload and non-payload are set to 144 and 16 bits, respectively.",
"Due to the selective use of good subchannels and also properly designed error correction mechanism, FDST outperforms PDCCH by a large margin, achieving more than 5 dB gain when BLER is $10^{-2}$ .",
"We next evaluate the average processing latency defined as the sum of the buffering latency and decoding latency for one OFDM symbol.",
"The processing latencies of FDST for $m=256$ $(73.4 \\mu s)$ and $m=128$ $(36.7 \\mu s)$ are reduced by the factor of 56% and 78% over the LTE PDCCH $(166.8 \\mu s)$ , respectively.",
"In practice, to decode a packet in 4G LTE/5G NR systems, we need to receive 7 (4G LTE) or 2 (5G NR) OFDM symbols while only one symbol is enough for the proposed FDST so that the latency gain of FDST is pronounced.",
"We note that, since the required number of samples in the receiver is small, the base station does not need to transmit whole samples, resulting in the saving of the transmit power."
],
[
"Sparse Vector Coding",
"In this section, we discuss the SVC transmission scheme.",
"Distinctive feature of SVC over FDST is to transmit the sparse vector after the random spreading.",
"We first discuss the basic SVC operation and then explain the deep neural network based SVC decoding.",
"Figure: The block diagram for the SVC transceiver using the deep neural network."
],
[
"Basics of SVC",
"Key idea behind SVC is to transmit the short-sized packet after the sparse vector transformation and compression.",
"To be specific, an information vector is mapped to the sparse vector and then transmitted after the pseudo random spreading (see Figure 4).",
"In view of this, one can consider SVC as a reminiscent of CDMA.",
"However, major difference is that the length of the spreading sequence $\\mathbf {c}_{i}$ is far shorter than the size of an input vector $\\mathbf {s}$ and only a few entries in $\\mathbf {s}$ are nonzero.",
"Note, in contrast to FDST using the sub-matrix of IDFT as a sensing matrix, the sensing matrix $\\mathbf {C}=[\\mathbf {c}_{1} \\ \\mathbf {c}_{2} \\ \\cdots \\ \\mathbf {c}_{n}]$ of SVC is generated at random so that the vector $\\mathbf {C}$$\\mathbf {s}$ after the spreading contains enough information to recover the sparse vector $\\mathbf {s}$ .",
"The input-output relationship of SVC is $\\mathbf {y}=\\mathbf {H}\\mathbf {C}\\mathbf {s}+\\mathbf {v}$ where $\\mathbf {H}$ is the diagonal matrix where $h_{ii}$ is the channel for $i$ -th subcarrier (e.g., resource element in LTE) and $\\mathbf {v}$ is the additive Gaussian noise.",
"Major benefit of SVC is that the decoding process can be greatly simplified when the channel is approximately constant, which is true for most short packet transmission scenarios.",
"Note that when the channel is approximately constant (i.e., $\\mathbf {H} \\approx h\\mathbf {I}$ ), the system model can be simplified to $\\mathbf {y} \\approx h\\mathbf {C}\\mathbf {s}+\\mathbf {v}=\\mathbf {C}\\mathbf {x}+\\mathbf {v}$ where $\\mathbf {x}=h\\mathbf {s}$ .",
"Validity of this system model can be fortified when the packet is transmitted in a narrowband channel or approximately static environments.",
"In fact, when the packet transmission time $T_{p}$ is smaller than the channel coherence time $T_{c}$ $(T_c \\gg T_p)$ , channel can be readily assumed to be a constant.",
"For example, when the carrier frequency $f_c=1.8$ GHz and the mobile speed is $\\nu =10$ km/h, then $T_c \\approx 11$ ms is much larger than the duration of one LTE OFDM symbol $(\\approx 0.07 \\text{ ms})$ [10], which justifies the validity of this approximation.",
"Suppose an information is embedded only in the positions of $\\mathbf {s}$ .",
"Then the decoding task is to find out the support $\\Omega _{\\mathbf {s}}$ of $\\mathbf {s}$ (equivalently $\\Omega _{\\mathbf {x}}$ ).",
"Fortunately, since the system matrix $\\mathbf {C}$ does not contain the channel components, the support identification can be done without the channel knowledge, resulting in savings of the resources and power for the pilot transmission, not to mention the saving the trouble for the channel estimation.",
"Also, SVC is also good fit for the small cell deployed scenarios or cell-edge environments since the randomly spread sparse vector is robust to the co-channel interference."
],
[
"SVC Decoding with Deep Neural Network",
"In the SVC packet decoding, we can basically use any sparse signal recovery algorithm.",
"In most sparse recovery algorithms, such as OMP, an index of a column in $\\mathbf {C}$ that is maximally correlated to the measurement $\\mathbf {y}$ is chosen as an estimate of the support element [11], [12].",
"Thus, if two columns of $\\mathbf {C}$ are strongly correlated and only one of these is associated with the nonzero values in $\\mathbf {s}$ , then it might not be easy to distinguish the right column from wrong one.",
"Clearly, support identification performance depends heavily on the correlation between columns of $\\mathbf {C}$ .",
"In fact, when a packet is transmitted using a small amount of resources, underdetermined ratio $m/n$ of the system will increase sharply, causing a severe degradation in the decoding performance.",
"To address this problem, we employ a deep neural network (DNN), an outstanding tool to approximate the complicated and nonlinear function.",
"Particularly, in this work, we use a convolutional neural network (CNN) due to its computation efficiency [13].",
"In the training phase, CNN learns the nonlinear mapping function $g$ between the received signal $\\mathbf {y}$ and support $\\Omega $ (see Figure 4).",
"The support identification problem can be expressed as $\\hat{\\Omega }=g(\\mathbf {y};\\Theta )$ where $\\Theta $ is the set of weights and biases.",
"In order to find out the network parameters $\\Theta $ that approximate the mapping function correctly, a received signal $\\mathbf {y}$ passes through multiple hidden layers.",
"In each hidden layer, the relationship between input $\\mathbf {z}^{i}$ and output $\\mathbf {z}^{o}$ can be expressed as $\\mathbf {z}^{o}=f(\\mathbf {W} \\ast \\mathbf {z}^{i} + \\mathbf {b})$ where $\\mathbf {W}$ and $\\mathbf {b}$ are the weight and bias, respectively, $f$ is the nonlinear activation function (e.g., ReLu function), and $\\ast $ is the convolution operator.",
"In order to achieve a reduction in the computational complexity, the max pooling operation taking the maximal value in the filtered window is employed.",
"It is now well-known from the universal approximation theorem that DNN processed by the deeply stacked hidden layers well approximates the desired function.",
"In our context, this means that the trained neural network with multiple hidden layers can handle the whole SVC decoding process, resulting in an accurate support identification.",
"When one tries to use the deep learning techniques to the wireless communication systems (e.g., packet decoding and channel estimation), there are two major problems in the training phase.",
"First, it is very difficult to handle the randomness of the received vector $\\mathbf {y}$ .",
"Indeed, since the channel depends heavily on the environmental factors such as frequency band and geometric objects, DNN needs to learn a huge amount of channel training data.",
"Fortunately, the SVC decoding is essentially the same as the support identification and all channel components are contained in an input sparse vector $\\mathbf {x}=h\\mathbf {s}$ .",
"Thus, the DNN-based SVC decoding only needs to learn the codebook matrix $\\mathbf {C}$ (known in advance), not the channel statistics, which greatly simplifies the learning process and also improves accuracy.",
"Next, in the training phase, abundant amount of dataset is required to train the DNN.",
"Using the real received signals as training data would be a natural option but it requires huge training overhead.",
"For example, when gathering ten million received signals in LTE systems, it will take more than 4 hours (1 ms subframe consisting of 14 symbols).",
"Luckily, since the DNN-based SVC decoding is not so sensitive to the channel variations, the training vectors can be generated synthetically and then used in the offline training phase.",
"In doing so, time and effort to collect huge training data can be prevented.",
"Figure: The block diagram for the SVC transceiver using the deep neural network.Table: Computational complexity analysis for deep-SVC"
],
[
"Numerical Performance Evaluation",
"In this subsection, we examine the BLER performance of SVC-encoded short packet in the downlink of OFDM systems.",
"For comparison, we also test the PDCCH of 4G LTE systems and 5G NR system (using polar code) under AWGN channel condition.",
"In the 4G PDCCH, the convolution code with rate 1/3 with the 16-bit CRC is used.",
"In the SVC scheme, the binary codebook generated from the Bernoulli distribution is employed.",
"Also, we set $m=42$ , $n=96$ , and $k=2$ for 12 bits transmission.",
"In the decoding of the SVC-encoded packet, one-dimensional CNN consisting of 6 hidden layers whose size is 84 (twice as $m$ ) and convolution filter whose size is 3 are employed.",
"In Figure 5, we observe that the DNN-based SVC outperforms the conventional schemes, achieving 0.5 dB gain over the conventional SVC scheme, 3.3 dB gain over the PDCCH with convolutional code, and 1.2 dB gain over the PDCCH with polar code at BLER=$10^{-5}$ .",
"In Table 2, we compare the computational complexities of the proposed Deep-SVC and conventional OMP algorithm under the simulation setup.",
"Here, the number of filters $T$ and pooling size $p$ are set to 32 and 2, respectively.",
"In order to examine the overall behaviour, we compute the required floating point operations (flops) for various input sizes $(m=42,54)$ .",
"We observe that the complexity of Deep-SVC is much smaller than that of OMP.",
"For example, when $m=54$ , the complexity of the proposed scheme is 53% lower than that of the OMP algorithm.",
"It is worth mentioning that the complexity of the proposed scheme depends heavily on the DNN network parameter (e.g., $L$ and $T$ ), not the system parameter ($m$ and $k$ ).",
"For instance, when m increases from 42 to 54, the computational complexity of the Deep-SVC changes slightly but that of OMP algorithm increases significantly."
],
[
"Conclusion and Future Direction",
"In this article, we presented an overview of the sparse vector transmission suitable for the short packet transmission in machine-centric communication scenarios (mMTC and URLLC).",
"We discussed basics of SVT, two distinct SVT approaches (FDST and SVC) with detailed operations, and also demonstrated the effectiveness of SVT in realistic wireless environments.",
"We observed that SVT is an effective means to transmit the short packet having many advantages over the conventional transmission scheme yet much work remains to be done.",
"For example, we did not elaborate the coding schemes in this work.",
"Perhaps simplest option is to combine the channel coding scheme and SVT mechanically.",
"Better option would be to consider the correlation of the sensing matrix and the quality of channel in the sparse vector generation and decoding.",
"In designing the coding scheme, we can recycle the wasted information (i.e., bit loss $\\log _{2}{n \\atopwithdelims ()k} - \\lfloor \\log _{2}{n \\atopwithdelims ()k} \\rfloor $ ).",
"Note that $b$ bits are mapped into $k$ positions of $n$ -dimensional vector so that ${n \\atopwithdelims ()k}-2^{b}$ choices are wasted.",
"By the deliberate mapping of these choices to the information bits, decoding error probability can be reduced.",
"Also, SVT can simplify complicated transmission procedure.",
"For instance, SVT can be used as a grant signal in the user scheduling process.",
"It can also be used as a grant-free uplink transmission of the short packet.",
"As communication between machines proliferates, short packet transmission will be more popular and will eventually be a dominating transmission mode in machine-centric wireless systems.",
"We believe that the proposed SVT would serve as a useful tool in the machine communication era."
]
] | 2001.03366 |
[
[
"Doubling nodal solutions to the Yamabe equation in $\\mathbb{R}^n$ with\n maximal rank"
],
[
"Abstract We construct a new family of entire solutions to the Yamabe equation $$-\\Delta u=\\frac{n(n-2)}{4}|u|^{\\frac{4}{n-2}}u \\mbox{ in }\\mathcal{D}^{1,2}(\\mathbb{R}^n).$$ If $n=3$, our solutions have maximal rank, being the first example in odd dimension.",
"Our construction has analogies with the doubling of the equatorial spheres in the construction of minimal surfaces in $S^3(1)$."
],
[
"Introduction",
"Consider the problem $-\\Delta u=\\gamma |u|^{p-1}u\\hbox{ in }\\mathbb {R}^n,\\qquad \\gamma :=\\frac{n(n-2)}{4},\\qquad u\\in {D}^{1,2}({\\mathbb {R}}^n),$ where $n\\geqslant 3$ , $p:=\\frac{n+2}{n-2}$ and ${D}^{1,2}({\\mathbb {R}}^n)$ is the completion of $C_0^\\infty ({\\mathbb {R}}^n)$ with the norm $\\Vert \\nabla u\\Vert _{L^2({\\mathbb {R}}^n)}$ .",
"Problem (REF ) corresponds to the steady state of the energy-critical focusing nonlinear wave equation $\\partial _t^2 u-\\Delta u- |u|^{\\frac{4}{n-2}} u=0, \\ (t,x)\\in {\\mathbb {R}}\\times {\\mathbb {R}}^n,$ whose study (see for instance [4], [6], [7], [14], [15]) naturally relies on the complete classification of the set of non-zero finite energy solutions to (REF ), which is defined by $\\Sigma := \\left\\lbrace Q \\in {{D}}^{1,2} ({\\mathbb {R}}^n) \\backslash \\lbrace 0\\rbrace : \\ -\\Delta Q= \\frac{n(n-2)}{4} |Q|^{\\frac{4}{n-2}} Q\\right\\rbrace ,$ in particular in connection with the soliton resolution conjecture for which only a few examples have become known [13], [14], [5], [6], [7].",
"Observe that (REF ) is the Euler-Lagrange equation of the functional defined by ${\\bf e} (u) := {1\\over 2} \\int _{{\\mathbb {R}}^n} |\\nabla u |^2 \\, dx - {\\gamma (n-2) \\over 2 n} \\int _{{\\mathbb {R}}^n} |u|^{2n \\over n-2} \\, dx.$ Positive solutions to (REF ) solve the Yamabe problem on the sphere (after a stereographic projection) and are the extremal functions for the Sobolev embedding.",
"Thanks to the classical work of Caffarelli-Gidas-Spruck [2], it is known that all positive solutions to (REF ) are given by the so called bubble and all its possible translations and dilations, that is, $U(y):=\\left(\\frac{2}{1+|y|^2}\\right)^{\\frac{n-2}{2}} \\mbox{ and }\\;\\;U_{\\alpha ,y_0}(y):=\\alpha ^{-\\frac{n-2}{2}}U\\left(\\frac{y-y_0}{\\alpha }\\right),\\;\\; \\alpha >0,\\; y_0\\in \\mathbb {R}^n,$ previously discovered independently by Aubin [1] and Talenti [19].",
"In fact, all radial solutions in $\\Sigma $ have the form (REF ).",
"Sign-changing solutions belonging to $\\Sigma $ have thus to be non radial.",
"Using the classical theory of Ljusternik-Schnirelman category, Ding proved in [3] the existence of infinitely many elements in $\\Sigma $ that are non radial, sign-changing, and with arbitrary large energy.",
"The key idea in [3] is to look for solutions to (REF ) that are invariant under the action of $O(2) \\times O(n-2) \\subset O(n)$ to recover compactness for the functional ${\\bf e} (u)$ .",
"No further information though is known on the solutions found by Ding.",
"Recently, more explicit constructions for sign-changing (non radial) solutions to (REF ) have been obtained by del Pino-Musso-Pacard-Pistoia and Medina-Musso-Wei (see [8], [9], [16]).",
"The solutions obtained in [8] are invariant under the action of $D_k \\times O(n-2)$ , where $D_k$ is the dihedral group of rotations and reflections leaving a regular polygon with $k$ sides invariant.",
"More precisely, for any $k$ large enough, the authors construct a solution to (REF ) looking like the bubble $U$ in (REF ) surrounded by $k$ negative scaled copies of $U$ arranged along the vertices of a $k$ -regular polygon in ${\\mathbb {R}}^2$ .",
"At main order the solution looks like $U(y)-\\sum _{j=1}^k\\lambda ^{-\\frac{n-2}{2}}U\\left(\\lambda ^{-1}(y-\\xi _j)\\right),$ where $\\xi _j:=(e^{\\frac{2\\pi (j-1)i}{k}},0,\\ldots )$ , $\\lambda =O(k^{-2})$ if $n\\geqslant 4$ and $\\lambda =O((k\\ln k)^{-2})$ if $n=3$ , as $k \\rightarrow \\infty $ .",
"Observe that $\\sum _{j=1}^k \\lambda ^{-\\frac{n-2}{2}}U\\left(\\lambda ^{-1}(y-\\xi _j)\\right) \\rightharpoonup c_n \\delta _\\Gamma , \\quad {\\mbox{as}} \\quad k \\rightarrow \\infty ,$ for a positive constant $c_n$ , where $\\delta _\\Gamma $ is the Dirac-delta at the equatorial on the $(y_1 , y_2)$ -plane $\\Gamma =\\lbrace y \\in {\\mathbb {R}}^n \\, : \\, y_1^2 + y_2^2=1 \\rbrace $ in $S^n(1)$ .",
"We thus can think of the solutions obtained in [8] and described at main order in (REF ) as the sum of a positive fixed central bubble surrounded by a negative smooth function that desingularizes a Dirac-delta along the equatorial $\\Gamma $ , in the limit as $k \\rightarrow \\infty $ .",
"We call this construction a desingularization of the equatorial, in analogy with similar desingularization constructions for minimal surfaces in Riemannian three-manifolds [10].",
"We remark that these solutions are not invariant under the action of $O(2) \\times O(n-2)$ , thus they differ from the ones found by Ding.",
"Besides, these solutions are the first example of non degenerate sign-changing solutions to (REF ) as shown in [17].",
"For any $Q \\in \\Sigma $ , consider the linear operator $L_Q:=-\\Delta -\\gamma p |Q|^{p-1}$ and define the null space ${{Z}}_Q := \\lbrace f \\in {{D}}^{1,2} ({\\mathbb {R}}^n) \\, : \\, f \\ne 0, \\, L_Q (f) = 0\\rbrace .$ Duyckaerts-Kenig-Merle [5] introduced the following definition of non-degeneracy for a solution of problem (REF ): $Q\\in \\Sigma $ is said to be non degenerate if ${{Z}}_Q$ coincides with the vector space generated by the elements in ${{Z}}_Q$ related to the group of isometries in ${{D}}^{1,2} ({\\mathbb {R}}^n)$ under which problem (REF ) is invariant, given by translations, scalings, rotations and Kelvin transformation.",
"More precisely, $Q$ is non degenerate if ${{Z}}_Q = \\tilde{{{Z}}}_{Q},$ where $\\tilde{{{Z}}}_{Q}:= \\mbox{span} \\ \\left\\lbrace \\begin{array}{l}(2-n) x_j Q +|x|^2 \\partial _{x_j} Q- 2 x_j x \\cdot \\nabla Q, \\quad \\partial _{x_j} Q, \\quad 1\\leqslant j\\leqslant n, \\\\\\\\(x_j \\partial _{x_k} - x_k \\partial _{x_j})Q, \\quad 1\\leqslant j <k\\leqslant n, \\quad \\frac{n-2}{2} Q + x \\cdot Q \\end{array}\\right\\rbrace .$ The rank of a solution $Q \\in \\Sigma $ is the dimension of the vector space $\\tilde{{Z}}_Q$ , and it cannot exceed the number ${N}:=2n+1+\\frac{n(n-1)}{2},$ being this the largest possible dimension for $\\tilde{{Z}}_Q$ .",
"It is well known for instance that $U$ in (REF ) is non degenerate and its rank is $n+1$ (see [18]).",
"In [17] it is proven that the solutions built in [8], looking at main order as in (REF ), are non degenerate and their rank is $3n$ .",
"A question we address in [16] is about the existence of solutions $Q \\in \\Sigma $ to (REF ) whose rank is maximal.",
"In fact, observe that not the bubbles in (REF ) nor the solutions built in [8] have maximal rank.",
"We partially answer this question building a new family of solutions to (REF ), where another polygon with a large number of sides is replicated for $n\\geqslant 4$ in the third and fourth coordinates, giving rise to a new family of non degenerate solutions to (REF ) that at main order look like $U(y)-\\sum _{j=1}^k\\lambda ^{-\\frac{n-2}{2}}U(\\lambda ^{-1}(y-\\overline{\\xi }_j))-\\sum _{j=1}^h\\mu ^{-\\frac{n-2}{2}}U(\\mu ^{-1}(y-\\underline{\\xi }_j)),$ where $\\overline{\\xi }_j:=(e^{\\frac{2\\pi (j-1)i}{k}},0,\\ldots )$ , $\\underline{\\xi }_j:=(0,0, e^{\\frac{2\\pi (j-1)i}{h}},0,\\ldots )$ , $\\lambda =O(k^{-2})$ and $\\mu =O(h^{-2})$ , for $k$ and $h$ sufficiently large, see [16].",
"Using the terminology we introduced before, this construction is a desingularization of the two equators, one in the $(y_1, y_2)$ -plane and the other in the $(y_3 , y_4)$ -plane.",
"These solutions are non degenerate.",
"Furthermore, their rank is $5 \\, (n-1)$ , which is maximal when $n=4$ .",
"A generalization of this result is to consider non degenerate solutions obtained by gluing a central positive bubble with negative scaled copies of $U$ centered at the vertices of $\\ell $ regular polygons with a large number of sides and lying in consecutive planes, which have maximal rank provided the dimension $n$ is $2\\ell $ .",
"In other words, a desingularization of $\\ell $ equators in consecutive planes would provide an example of non degenerate solutions with maximal rank in any even dimension $2\\ell $ .",
"If the dimension $n$ is odd, the existence of non degenerate solutions for (REF ) with maximal rank remains an open problem and a different construction is required.",
"Roughly speaking, the solution built in [16] breaks the radial behavior of the bubble in the first four coordinates, loosing the related invariances.",
"This fact adds extra terms in the kernel of the linearized operator, being all the possible precisely when $n=4$ .",
"Thus, to prove the analogue in odd dimension one needs to find a solution breaking the radiality in an odd number of coordinates.",
"The aim of this work is to address this question and to provide a new family of sign-changing solutions for problem (REF ) that we claim to have maximal rank in dimension $n=3$ .",
"We prove the following result.",
"Theorem 1.1 Let $n\\geqslant 3$ and let $k$ be a positive integer.",
"Then for any sufficiently large $k$ there is a finite energy solution to (REF ) of the form $u(y)=U(y)-\\sum _{j=1}^k\\lambda ^{-\\frac{n-2}{2}}U(\\lambda ^{-1}(y-\\overline{\\xi }_j))-\\sum _{j=1}^k\\lambda ^{-\\frac{n-2}{2}}U(\\lambda ^{-1}(y-\\underline{\\xi }_j))+o_k(1)(1+\\lambda ^{-\\frac{n-2}{2}}),$ where $\\overline{\\xi }_j:=R(\\sqrt{1-\\tau ^2}e^{\\frac{2\\pi (j-1)}{k}i},\\tau ,0,\\ldots ),\\; \\;\\underline{\\xi }_j:=R(\\sqrt{1-\\tau ^2}e^{\\frac{2\\pi (j-1)}{k}i},-\\tau ,0,\\ldots ), \\;\\;j=1,\\ldots ,k,$ and $\\begin{split}\\lambda := {\\ell ^{2\\over n-2} \\over k^2} , \\; \\tau := {t \\over k^{1-{2\\over n-1}} }, \\;{\\mbox{if}} \\quad n\\geqslant 4, \\qquad \\lambda := {\\ell ^{2} \\over k^2 (\\ln k)^2 } , \\; \\tau := {t \\over \\sqrt{\\ln k} }, \\; {\\mbox{if}} \\quad n=3.\\end{split}$ Here $\\lambda ^2+R^2=1, \\quad {\\mbox{and}} \\quad \\eta < \\ell , \\, t < \\eta ^{-1},$ for some positive fixed number $\\eta $ independent of $k$ .",
"The term $o_k(1)\\rightarrow 0$ uniformly on compact sets of ${\\mathbb {R}}^n$ as $k \\rightarrow \\infty $ .",
"These solutions have maximal rank in dimension 3.",
"Some remarks are in order.",
"Remark 1.2 Let us briefly discuss our construction.",
"The solution predicted by Theorem REF looks at main order as $u_0(y) := U(y)-\\sum _{j=1}^k\\lambda ^{-\\frac{n-2}{2}}U(\\lambda ^{-1}(y-\\overline{\\xi }_j))-\\sum _{j=1}^k\\lambda ^{-\\frac{n-2}{2}}U(\\lambda ^{-1}(y-\\underline{\\xi }_j)).$ The polygonal distribution of the points $\\overline{\\xi }_1, \\ldots , \\overline{\\xi }_k$ and $\\underline{\\xi }_1, \\ldots \\underline{\\xi }_k$ makes $u_0$ a function with several important symmetries: it is invariant under rotation of angle ${2\\pi \\over k}$ in the $(y_1, y_2)$ -plane, and even in the other variables $y_j$ , $j=3, \\ldots , n$ .",
"The assumption that $\\lambda ^2 + R^2=1$ gives that $u_0$ is also invariant under Kelvin transformation.",
"We will take great advantage of these symmetries in many different ways in our proof.",
"For instance they allow us to choose the same scaling factor $\\lambda $ for each one of the negative bubbles centred at the different points $\\overline{\\xi }_j$ and $\\underline{\\xi }_j$ , $j=1, \\ldots , k$ , which in principle may not be the same, reducing substantially the number of scaling parameters to adjust.",
"Taking $\\lambda $ and $\\tau $ small positive parameters as $k \\rightarrow \\infty $ , a formal computation shows that the energy functional defined in (REF ) and evaluated at $u=u_0$ has the following expansion, as $k \\rightarrow \\infty $ , ${\\bf e} (u_0)& \\sim (2k +1) a_n + k \\left( \\lambda ^{n-2 \\over 2} (b_n- c_n \\tau ^2 ) - d_n \\lambda ^{n-2} k^{n-2} - e_n { \\lambda ^{n-2} \\over \\tau ^{n-3}} \\right) ,$ when dimension $n\\geqslant 4$ , for some explicit positive constants $a_n$ , $b_n$ , $c_n$ , $d_n$ and $e_n$ .",
"Our choice for $\\lambda $ and $\\tau $ in terms of $k$ is to get at main order the balance $\\nabla _{\\lambda , \\tau } {\\bf e} (u_0) \\sim 0.$ We will justify this heuristic argument in Section 2.",
"Remark 1.3 The construction obtained in Theorem REF differs from the desingularization of the equatorial in (REF ) obtained in [8] or of two equators in (REF ) in [16].",
"In fact the solutions in Theorem REF can be thought as the sum of a positive fixed central bubble surrounded by a negative smooth function that desingularizes Dirac-deltas located in points on two circles that are collapsing into a Dirac-delta supported along the equatorial $\\Gamma $ , in the limit as $k \\rightarrow \\infty $ .",
"We call this construction a doubling of the equatorial in the $(y_1 , y_2)$ -plane, in analogy with similar doubling constructions for minimal surfaces in Riemannian three-manifolds obtained in [10], [11], [12].",
"Remark 1.4 Let $u$ be the solution predicted in Theorem REF and define the following $4n-2$ functions $z_0(y):=\\frac{n-2}{2}u(y)+\\nabla u(y)\\cdot y,\\quad z_\\alpha (y):=\\frac{\\partial }{\\partial y_\\alpha }u(y),\\;\\alpha =1,\\ldots ,n,$ $z_{n+1}(y):=-2y_1z_0(y)+|y|^2z_1(y),\\quad z_{n+2}(y):=-2y_2z_0(y)+|y|^2z_2(y),$ $ z_{n+3}(y):=-2y_3z_0(y)+|y|^2z_3(y),$ $z_{n+\\alpha +2}(y):=-y_\\alpha z_1(y)+y_1z_\\alpha (y),\\;\\alpha =2,\\ldots , n,$ $z_{2n+\\alpha }(y):=-y_\\alpha z_2(y)+y_2z_\\alpha (y),\\;\\alpha =3,\\ldots , n,$ $z_{3n+\\alpha -3}(y):=-y_\\alpha z_3(y)+y_3z_\\alpha (y),\\;\\alpha =4,\\ldots , n.$ These functions belong to ${D}^{1,2}({\\mathbb {R}}^n)$ and also to the kernel of $L_u:=-\\Delta -\\gamma p |u|^{p-1}$ .",
"Furthermore, if $k$ is even they are linearly independent (see Appendix B).",
"Consider now the case $n=3$ : we have that $4n-2={N}=10$ .",
"Thus the solutions constructed in Theorem REF have maximal rank in dimension 3 when $k$ is even.",
"The non-degeneracy of the solution remains an open problem.",
"Remark 1.5 The doubling of the equatorial in ${\\mathbb {R}}^3$ presumably closes the question about solutions with maximal rank.",
"Indeed, for any odd dimension we could combine a doubling of the equatorial in three coordinates with a desingularization of the equatorial as in [8] in two coordinates or a desingularization of two equators as in [16] in four coordinates, as many times as needed.",
"We conjecture that combining these three structures we can build a maximal solution in any odd dimension.",
"Remark 1.6 This doubling construction also provides an alternative example of solution with maximal rank for even dimensions of the form $n=6\\ell $ , $\\ell \\in \\mathbb {N}$ .",
"For these dimensions, in [16] the authors propose to replicate $3\\ell $ times the desingularization of [8].",
"We conjecture that combining $2\\ell $ structures like the doubling of the equatorial of Theorem REF would provide a different solution with maximal rank in these dimensions.",
"Remark 1.7 The construction described in Theorem REF is not the only possible way to double the equatorial $\\Gamma $ .",
"For any even integer $2m$ , we can construct another sequence of solutions that doubles the equatorial in the form of the sum of a positive fixed central bubble surrounded by a negative smooth function that desingularizes Dirac-deltas located in points on $2m$ circles that are collapsing into a Dirac-delta supported along the equatorial $\\Gamma $ , in the limit as $k \\rightarrow \\infty $ , where $m$ of these circles collapse onto $\\Gamma $ from above and $m$ from below.",
"We can also combine desingularization of the equatorial and doubling of the equatorial.",
"For any odd integer $2m +1$ , we can construct a sequence of solutions with the form of the sum of a positive fixed central bubble surrounded by a negative smooth function that consists of two parts.",
"One part desingularizes Dirac-deltas located in points on $2m$ circles that are collapsing into a Dirac-delta supported along the equatorial $\\Gamma $ , in the limit as $k \\rightarrow \\infty $ .",
"The other part desingularizes Dirac-deltas located at points along the equatorial, desingularizing a Dirac-delta along the equatorial $\\Gamma $ , in the limit as $k \\rightarrow \\infty $ .",
"Since the proofs of these constructions are in the same spirit as the one of Theorem REF , we will briefly describe them in section , explaining the principal differences.",
"The proof of Theorem REF is based on a Lyapunov-Schmidt reduction in the spirit of [8]: we define a first approximation and we look for a solution in a nearby neighborhood, by linearizing around the approximation and, after developing an appropriate linear theory, solving by a fixed point argument.",
"This allows to reduce the original problem to the solvability of a finite dimensional one.",
"However, for the construction in Theorem REF this last step is rather delicate.",
"In fact, the finite dimensional reduction leads to two equations (in the two parameters to adjust, $\\lambda $ and $\\tau $ ) where the sizes of the error and the non linear term play a fundamental role.",
"If one follows the strategy of [8], the nonlinear term cannot be controlled and the reduced problem cannot be solved.",
"For this reason, we need to carry on a much more refined argument.",
"The key point is the following: if one pays attention to the error term near the bubbles, this can be decomposed in a (relatively) large but symmetric part, and a smaller but non symmetric part (see Proposition REF ).",
"In the final argument (the reduction procedure) the symmetric part is orthogonal to the element of the kernel, and therefore not seen.",
"Thus, the part playing a role in the reduction is the non symmetric one, that is significatively smaller.",
"Roughly speaking, this allows to solve the linearized problem in two parts, a symmetric and large but irrelevant part, and a non symmetric but small one.",
"Indeed, if our solution has the form $u_*+\\phi $ , being $u_*$ the approximation, we will split $\\phi $ near each bubble as $\\phi =\\phi ^s+\\phi ^*$ , where $\\phi ^s$ is symmetric with respect to the hyperplane $y_3=\\tau $ (or analogously $y_3=-\\tau $ ).",
"This behavior is also inherited by the nonlinear part of the equation and we will be able to perform the fixed point argument setting the size of $\\phi ^*$ very small (see Proposition REF ), what will allow us to conclude the reduction.",
"This new strategy requires delicate decompositions and estimates of every term of the equations, as well as the development of a sharp linear invertibility theory.",
"The structure of the article is the following.",
"In section we detail the approximation and the error associated, estimating it near and far from the bubbles in different norms, and identifying the symmetric and non symmetric parts and their sizes.",
"Section is devoted to the linear theory, where a refinement of the theory in [8] is developed.",
"Section is the core of the strategy, where the gluing scheme is performed, together with the precise decomposition of the function in their symmetric and non symmetric part.",
"In section we carry on the dimensional reduction, concluding the proof of Theorem REF .",
"The appendix contains fundamental computations concerning the shape of the approximation."
],
[
"Doubling construction: a first approximation",
"Let $n \\geqslant 3$ and $\\tau \\in (0,1)$ .",
"In ${\\mathbb {R}}^n$ we fix the following points $\\overline{P} := (\\sqrt{1-\\tau ^2 } , 0 , \\tau , 0 , \\ldots , 0) , \\quad \\underline{P} := (\\sqrt{1-\\tau ^2 } , 0 , -\\tau , 0 , \\ldots , 0).$ Let $ \\lambda \\in (0,1)$ be a positive number, and define $R$ as $\\lambda ^2 + R^2 = 1.$ Let $k$ be an integer number and $\\textbf {u} [\\lambda , \\tau ] (y) := U(y) -\\sum _{j=1}^k \\underbrace{\\lambda ^{-{n-2 \\over 2}} U\\left( {y-\\overline{\\xi }_j \\over \\lambda } \\right)}_{\\overline{U}_j (y)}-\\sum _{j=1}^k \\underbrace{\\lambda ^{-{n-2 \\over 2}} U\\left( {y- \\underline{\\xi }_j \\over \\lambda } \\right)}_{ \\underline{U}_j (y) }$ where $\\begin{aligned}\\overline{\\xi }_j &:= R (\\sqrt{1-\\tau ^2} \\cos \\theta _j , \\sqrt{1-\\tau ^2} \\sin \\theta _j , \\tau , 0, \\ldots , 0 ), \\\\\\underline{\\xi }_j &:= R (\\sqrt{1-\\tau ^2} \\cos \\theta _j , \\sqrt{1-\\tau ^2} \\sin \\theta _j , -\\tau , 0, \\ldots , 0 ),\\end{aligned} \\quad {\\mbox{with}} \\quad \\theta _j := 2\\pi {j-1 \\over k}.$ Observe that $\\overline{\\xi }_1 = R\\overline{P}$ and $\\underline{\\xi }_1 = R\\underline{P}$ , while $\\overline{\\xi }_j$ and $ \\underline{\\xi }_j$ are obtained respectively from $\\overline{P}$ and $\\underline{P}$ after a rotation in the $(y_1 , y_2)$ -plane of angle $2\\pi \\, {{j-1} \\over k} $ .",
"Thanks to (REF ), the functions $\\overline{U}_j$ , $\\underline{U}_j$ are invariant under Kelvin transform, so that $\\textbf {u}$ is also invariant under this transformation, that is $\\textbf {u} (y) = |y|^{2-n} \\textbf {u} \\left({y\\over |y|^2} \\right).$ A direct observation reflects that $\\textbf {u}$ also shares the following symmetries $\\textbf {u} (y_1, \\ldots , - y_j , \\ldots , y_n) = \\textbf {u} (y_1, \\ldots , y_j , \\ldots , y_n), \\quad j=2, \\ldots , n,$ $\\textbf {u} (e^{2\\pi { (j-1)\\over k} i} \\bar{y} , y_3, \\ldots , y_n) = \\textbf {u} ( \\bar{y} , y_3, \\ldots , y_n), \\quad \\bar{y}= (y_1 , y_2), \\quad j=2, \\ldots , k.$ In our construction, we assume that the integer $k$ is large and the parameters $\\lambda $ and $\\tau $ are given by $\\begin{split} \\lambda &:= {\\ell ^{2\\over n-2} \\over k^2} , \\quad \\tau := {t \\over k^{1-{2\\over n-1}} }, \\quad {\\mbox{if}} \\quad n\\geqslant 4, \\\\\\lambda &:= {\\ell ^{2} \\over k^2 (\\ln k)^2 } , \\quad \\tau := {t \\over \\sqrt{\\ln k} }, \\quad {\\mbox{if}} \\quad n=3,\\end{split} \\quad \\quad {\\mbox{where}} \\quad \\eta < \\ell , \\, t < \\eta ^{-1},$ for some $\\eta $ small and fixed, independent of $k$ , for any $k$ large enough.",
"The error function, defined as $E [\\ell , t] (y) := \\Delta \\textbf {u} + \\gamma |\\textbf {u}|^{p-1} \\textbf {u}, \\quad y \\in {\\mathbb {R}}^n,$ inherits the symmetries (REF ), (REF ).",
"As a consequence, fixing a small $\\delta >0$ independent of $k$ , it is enough to describe the error function in the sets $B(\\overline{\\xi }_1 , {\\delta \\over k})$ and ${\\mathbb {R}}^n \\setminus \\bigcup _{j=1}^k \\left( B(\\overline{\\xi }_j , {\\delta \\over k}) \\cup B( \\underline{\\xi }_j , {\\delta \\over k}) \\right) $ in order to know it in the whole space ${\\mathbb {R}}^n$ .",
"For our purpose it is convenient to measure the error using the following weighted $L^q$ norm $\\Vert h\\Vert _{**} := \\Vert \\, (1+ |y|)^{{n+2} - \\frac{2n}{q}} h \\Vert _{L^q({\\mathbb {R}}^n)}.$ For the moment we request that $q$ is a fixed number with $ {n \\over 2} <q<n$ .",
"Later on, we will need a more restrictive assumption on $q$ , $\\frac{n}{2}<q<\\frac{n}{2-\\frac{2}{n-1}}$ .",
"We will evaluate the $\\Vert \\cdot \\Vert _{**}$ -norm of $E$ in the interior regions $B(\\overline{\\xi }_j, {\\delta \\over k} )$ and $ B( \\underline{\\xi }_j, {\\delta \\over k} )$ , for any $j=1, \\ldots , k$ , and in the exterior region ${\\mathbb {R}}^n \\setminus \\bigcup _{j=1}^k \\left( B(\\overline{\\xi }_j, {\\delta \\over k}) \\cup B( \\underline{\\xi }_j, {\\delta \\over k} ) \\right)$ .",
"The error in the interior regions $B (\\overline{\\xi }_j , {\\delta \\over k} )$ and $ B( \\underline{\\xi }_j, {\\delta \\over k} )$ , $j=1, \\ldots , k$.",
"To describe the error $E$ in each one of the balls $B(\\overline{\\xi }_j, {\\delta \\over k})$ , $ B( \\underline{\\xi }_j, {\\delta \\over k} )$ , it is enough to do it in $B(\\overline{\\xi }_1, {\\delta \\over k})$ , as already observed.",
"In this region the dominant term of the function $\\textbf {u}$ in (REF ) is $\\overline{U}_1$ .",
"Thus, for some $s \\in (0,1)$ , we have $\\begin{split}\\gamma ^{-1} E (x) &= p \\left( \\overline{U}_1 (x) + s [ \\sum _{j\\ne 1} \\overline{U}_j (x) + \\sum _{j=1}^k \\underline{U}_j (x) - U(x) ] \\right)^{p-1} \\left[ -\\sum _{j\\ne 1} \\overline{U}_j (x) - \\sum _{j=1}^k \\underline{U}_j (x) + U(x) \\right] \\\\&+ \\sum _{j\\ne 1} \\overline{U}_j^p (x) + \\sum _{j=1}^k \\underline{U}_j^p (x) - U^p (x),\\end{split}$ for $x \\in B (\\overline{\\xi }_1 , {\\delta \\over k} )$ .",
"Let us introduce the change of variable $\\lambda y = x-\\overline{\\xi }_1$ , so that $\\overline{U}_1 (\\lambda y + \\overline{\\xi }_1 )= \\lambda ^{-{n-2 \\over 2}} U(y)$ .",
"In these expanded variables, the error takes the form $\\begin{split} \\lambda ^{n+2 \\over 2} \\gamma ^{-1} E (\\overline{\\xi }_1 + \\lambda y ) &= p \\left( U (y) + s \\lambda ^{n-2 \\over 2} [ \\sum _{j\\ne 1} \\overline{U}_j (\\lambda y + \\overline{\\xi }_1 ) + \\sum _{j=1}^k \\underline{U}_j (\\lambda y + \\overline{\\xi }_1) - U(\\lambda y + \\overline{\\xi }_1 ) ] \\right)^{p-1} \\times \\\\&\\times \\lambda ^{n-2 \\over 2} \\, \\left[ \\sum _{j\\ne 1} \\overline{U}_j (\\lambda y + \\overline{\\xi }_1 ) + \\sum _{j=1}^k \\underline{U}_j (\\lambda y + \\overline{\\xi }_1 ) - U(\\lambda y + \\overline{\\xi }_1 ) \\right] \\\\&+ \\lambda ^{n+2 \\over 2} \\left[ \\sum _{j\\ne 1} \\overline{U}_j^p (\\lambda y + \\overline{\\xi }_1 ) + \\sum _{j=1}^k \\underline{U}_j^p (\\lambda y + \\overline{\\xi }_1 ) - U^p (\\lambda y + \\overline{\\xi }_1 )\\right],\\end{split}$ for some $s \\in (0,1)$ , and uniformly for $|y|<{\\delta \\over \\lambda k}$ .",
"A direct Taylor expansion gives that $\\begin{split} \\overline{U}_j (\\lambda y + \\overline{\\xi }_1) &= {2^{n-2 \\over 2} \\lambda ^{n-2 \\over 2} \\over (\\lambda ^2 +|\\lambda y + \\overline{\\xi }_1 - \\overline{\\xi }_j |^2)^{n-2 \\over 2} }\\quad {\\mbox{for}} \\quad j\\ne 1 \\\\&= {2^{n-2 \\over 2} \\lambda ^{n-2 \\over 2} \\over |\\overline{\\xi }_1- \\overline{\\xi }_j|^{n-2}} \\left[ 1- (n-2) {(y, \\overline{\\xi }_1-\\overline{\\xi }_j) \\over |\\overline{\\xi }_1 - \\overline{\\xi }_j|^2 } \\lambda +O\\left( {\\lambda ^2 (1+ |y|^2) \\over |\\overline{\\xi }_1 - \\overline{\\xi }_j |^2} \\right) \\right],\\\\\\underline{U}_j (\\lambda y + \\overline{\\xi }_1) &= {2^{n-2 \\over 2} \\lambda ^{n-2 \\over 2} \\over (\\lambda ^2 +|\\lambda y + \\overline{\\xi }_1 - \\underline{\\xi }_j |^2)^{n-2 \\over 2} }\\quad {\\mbox{for}} \\quad j=1 , \\ldots , k \\\\&= {2^{n-2 \\over 2} \\lambda ^{n-2 \\over 2} \\over |\\overline{\\xi }_1- \\underline{\\xi }_j|^{n-2}} \\left[ 1- (n-2) {(y, \\overline{\\xi }_1- \\underline{\\xi }_j) \\over |\\overline{\\xi }_1 - \\underline{\\xi }_j|^2 } \\lambda +O\\left( {\\lambda ^2 (1+ |y|^2) \\over |\\overline{\\xi }_1 - \\underline{\\xi }_j |^2} \\right) \\right],\\\\U(\\lambda y + \\overline{\\xi }_1)&= U(\\overline{\\xi }_1 ) \\left[ 1- (n-2) {(y, \\overline{\\xi }_1 ) \\over 1+ |\\overline{\\xi }_1 |^2} \\lambda + O\\left({\\lambda ^2 |y|^2 \\over 1+ |\\overline{\\xi }_1|^2} \\right) \\right],\\end{split}$ uniformly for $|y| \\leqslant {\\delta \\over \\lambda k}$ .",
"In the Appendix we will show that $\\begin{split} \\sum _{j=2}^k {1\\over |\\overline{\\xi }_1 - \\overline{\\xi }_j|^{n-2}} =\\sum _{j=2}^k {1\\over |\\underline{\\xi }_1 - \\underline{\\xi }_j|^{n-2}} ={\\left\\lbrace \\begin{array}{ll} A_n \\, k^{n-2} \\, \\left(1+ O( \\tau ^2) \\right) \\quad {\\mbox{if}} \\quad n \\geqslant 4,\\\\ A_3 \\, k \\, \\ln k \\, \\left(1+ O(\\tau ^2 ) \\right) \\quad {\\mbox{if}} \\quad n =3,\\end{array}\\right.",
"}\\end{split}$ and $\\begin{split} \\sum _{j=1}^k {1\\over |\\overline{\\xi }_1 - \\underline{\\xi }_j|^{n-2}} ={\\left\\lbrace \\begin{array}{ll} B_n \\, {k \\over \\tau ^{n-3}} \\, \\left(1+ O( (\\tau k)^{-2}) \\right) \\quad {\\mbox{if}} \\quad n \\geqslant 5,\\\\B_n \\, {k \\over \\tau ^{n-3}} \\, \\left(1+ O( \\tau ) \\right) \\quad {\\mbox{if}} \\quad n =4,\\\\A_3 \\, k \\, \\ln \\left(\\frac{\\pi }{\\tau }\\right)\\, \\left(1+ O(|\\ln \\tau |^{-1} ) \\right) \\quad {\\mbox{if}} \\quad n =3,\\end{array}\\right.",
"}\\end{split}$ where $A_3 := \\pi ^{-1}$ and, if $n\\geqslant 4$ , $A_n:= {2\\over (2\\pi )^{n-2}} \\, \\sum _{j=1}^\\infty j^{2-n},\\; \\quad B_n := {2\\over 2^{n-2} \\pi } \\int _0^\\infty {ds \\over (1+ s^2)^{n-2 \\over 2}}.$ From (REF ), (REF ), (REF ) and (REF ) we get that $| \\lambda ^{n+2 \\over 2} \\gamma ^{-1} E (\\overline{\\xi }_1 + \\lambda y ) |\\leqslant C \\left[ \\frac{ \\lambda ^{\\frac{n-2}{2}}}{1+ |y|^4} + \\lambda ^{\\frac{n+2}{2}} \\right]$ for $|y| \\leqslant {\\delta \\over \\lambda k}$ .",
"Notice that, to obtain this estimate, we do not need to use the precise information gathered in (REF ) and (REF ), but only the order in $k$ .",
"Indeed, it is enough to get the upper bound $|\\overline{U}_j (\\lambda y + \\overline{\\xi }_1)|\\leqslant C\\quad \\mbox{ for }|y| \\leqslant {\\delta \\over \\lambda k}, $ and the estimate follows since, as (REF ) reflects, $|\\underline{U}_j (\\lambda y + \\overline{\\xi }_1)|$ is smaller.",
"Direct computations (see [8]) give $\\Vert \\, (1+ |y|)^{{n+2} - \\frac{2n}{q}}\\, \\lambda ^{n+2 \\over 2} \\gamma ^{-1} E (\\overline{\\xi }_1 + \\lambda y ) \\Vert _{L^q( |y| < {\\delta \\over \\lambda k})} \\ \\leqslant \\ {\\left\\lbrace \\begin{array}{ll}Ck^{-{n\\over q}} \\quad {\\mbox{if}} \\quad n\\geqslant 4,\\\\{C \\over k \\ln k}\\quad {\\mbox{if}} \\quad n=3,\\end{array}\\right.",
"}$ for some fixed constant $C>0$ .",
"Therefore, by symmetry we conclude that, for any $j=1, \\ldots , k$ , $\\Vert \\, (1+ |y|)^{{n+2} - \\frac{2n}{q}}\\, \\lambda ^{n+2 \\over 2} \\gamma ^{-1} E (\\overline{\\xi }_j + \\lambda y ) \\Vert _{L^q( |y| < {\\delta \\over \\lambda k})} \\ \\leqslant \\ \\left\\lbrace \\begin{array}{cc}Ck^{- \\frac{n}{q} } \\quad {\\mbox{if}} \\quad n\\geqslant 4,\\\\{C\\over k \\ln k} \\quad {\\mbox{if}} \\quad n=3,\\end{array}\\right.$ and $\\Vert \\, (1+ |y|)^{{n+2} - \\frac{2n}{q}}\\, \\lambda ^{n+2 \\over 2} \\gamma ^{-1} E (\\underline{\\xi }_j + \\lambda y ) \\Vert _{L^q( |y| < {\\delta \\over \\lambda k})} \\ \\leqslant \\ \\left\\lbrace \\begin{array}{cc}Ck^{- \\frac{n}{q} } \\quad {\\mbox{if}} \\quad n\\geqslant 4,\\\\{C\\over k \\ln k} \\quad {\\mbox{if}} \\quad n=3.\\end{array}\\right.$ The error in ${\\mathbb {R}}^n \\setminus \\bigcup _{j=1}^k \\left( B(\\underline{\\xi }_j , {\\delta \\over k}) \\cup B(\\bar{\\xi }_j , {\\delta \\over k}) \\right)$.",
"For $y$ in this region we have $\\begin{split}|E (y)| &\\leqslant C \\left[ \\frac{1}{(1+ |y|^2)^2} + \\left| \\sum _{j=1}^k \\left[ \\frac{\\lambda ^{\\frac{n-2}{2} } }{ |y-\\underline{\\xi }_j|^{n-2}} + \\frac{\\lambda ^{\\frac{n-2}{2} } }{ |y-\\bar{\\xi }_j|^{n-2}}\\right] \\right|^{\\frac{4}{n-2}} \\right] \\, \\\\&\\times \\left[ \\sum _{j=1}^k \\left( \\frac{ {\\lambda ^{\\frac{n-2}{2} }} }{ |y-\\underline{\\xi }_j|^{n-2}} +\\frac{\\lambda ^{\\frac{n-2}{2} } }{ |y-\\overline{\\xi }_j|^{n-2}}\\right) \\right]\\\\&\\leqslant \\ C \\frac{\\lambda ^{\\frac{n-2}{2} }}{(1+ |y|^2)^2} \\sum _{j=1}^k \\left( \\frac{1 }{ |y-\\underline{\\xi }_j|^{n-2}} +\\frac{1 }{ |y-\\overline{\\xi }_j|^{n-2}}\\right),\\end{split}$ where we have used that in this exterior region $\\sum _{j=1}^k\\frac{ {\\lambda ^{\\frac{n-2}{2} }} }{ |y-\\underline{\\xi }_j|^{n-2}}\\leqslant C,\\quad \\sum _{j=1}^k\\frac{\\lambda ^{\\frac{n-2}{2} } }{ |y-\\overline{\\xi }_j|^{n-2}}\\leqslant C.$ For $n\\geqslant 4$ , using (REF ), $\\begin{split}\\Vert \\, &(1+ |y|)^{{n+2} - \\frac{2n}{q}}\\, E\\Vert _{L^q( {\\mathbb {R}}^n \\setminus \\bigcup _{j=1}^k \\left( B(\\underline{\\xi }_j , {\\delta \\over k}) \\cup B(\\bar{\\xi }_j , {\\delta \\over k}) \\right))} \\ \\\\&\\leqslant \\ C\\lambda ^{\\frac{n-2}{2}} \\sum _{j=1}^k \\left[\\left( \\int _ { |y-\\underline{\\xi }_j| > {\\delta \\over k}} \\frac{(1+ |y|)^{(n+2)q - 2n-4q}}{|y-\\underline{\\xi }_j|^{(n-2)q }}\\, dy\\, \\right)^{\\frac{1}{q} } \\right.\\\\&\\;\\;\\;\\;\\left.+ \\left( \\int _ { |y-\\bar{\\xi }_j| > {\\delta \\over k}} \\frac{(1+ |y|)^{(n+2)q - 2n-4q}}{|y-\\overline{\\xi }_j|^{(n-2)q }}\\, dy\\, \\right)^{\\frac{1}{q} } \\right] \\\\&\\leqslant \\ C\\lambda ^{\\frac{n-2}{2}} k\\left( \\int _{ {\\delta \\over k} }^1\\frac{ t^{n-1} }{t^{(n-2)q} }\\, dt\\, \\right)^{\\frac{1}{q} }\\ \\leqslant \\, Ck^{1-\\frac{n}{q}},\\end{split} $ for some constant $C$ .",
"Similarly, if $n=3$ , we get $\\Vert (1+ |y|)^{{n+2} - \\frac{2n}{q}}\\, E \\Vert _{L^q( {\\mathbb {R}}^n \\setminus \\bigcup _{j=1}^k ( B(\\underline{\\xi }_j , {\\delta \\over k}) \\cup B(\\bar{\\xi }_j , {\\delta \\over k}) ))} \\ \\leqslant \\ \\frac{C}{\\ln k }\\ .$ Estimates (REF ), (REF ), (REF ) and (REF ) merge in the following Proposition 2.1 Assume that $\\lambda $ and $\\tau $ satisfy (REF ).",
"There exist an integer $k_0$ and a positive constant $C$ such that for all $k\\geqslant k_0$ the following estimates hold true $\\Vert E \\Vert _{**} \\leqslant C k^{1-{n\\over q}} \\quad {\\mbox{if}} \\quad n\\geqslant 4\\quad {\\mbox{and}}\\quad \\Vert E \\Vert _{**} \\leqslant C |\\ln k|^{-1} \\quad {\\mbox{if}} \\quad n = 3.$ We refer to (REF ) for the definition of the $\\Vert \\cdot \\Vert _{**}$ -norm.",
"The following result is the key point of the argument carried out for the gluing procedure and the reduction method in sections and .",
"It provides a decomposition of the interior error term in two parts: one is relatively large but symmetric, and the other is non symmetric but smaller in size.",
"Thanks to this observation we can refine the fixed point argument to fix a smaller size of the functions involved in the reduction.",
"Proposition 2.2 Assume that $\\lambda $ and $\\tau $ satisfy (REF ) and $|y|<\\frac{\\delta }{\\lambda k}$ .",
"Then, there exists a decomposition $E(\\overline{\\xi }_1+\\lambda y)=E^s(\\overline{\\xi }_1+\\lambda y)+E^*(\\overline{\\xi }_1+\\lambda y),$ such that $E^s(\\overline{\\xi }_1+\\lambda y)$ is even with respect to $y_\\alpha $ for all $\\alpha =1,\\ldots ,n$ and there exists an integer $k_0$ such that, for $k\\geqslant k_0$ , $|\\lambda ^{\\frac{n+2}{2}}E^*(\\overline{\\xi }_1+\\lambda y)|\\leqslant {\\left\\lbrace \\begin{array}{ll}C \\frac{\\lambda ^{\\frac{n-2}{2}}}{k}\\frac{1}{1+|y|^3}\\mbox{ if }n\\geqslant 4,\\\\C \\frac{\\lambda ^{1/2}}{k(\\ln k)^3}\\frac{1}{1+|y|^3}\\mbox{ if }n=3,\\end{array}\\right.",
"},\\qquad |y|<\\frac{\\delta }{\\lambda k}.$ The idea is to identify in (REF ) the main order terms using (REF ).",
"Indeed, we write, for $|y|<\\frac{\\delta }{k\\lambda }$ , $\\begin{split}\\gamma ^{-1} \\, \\lambda ^{\\frac{n+2}{2}}&E^s(\\overline{\\xi }_1+\\lambda y):=pU(y)^{p-1}\\lambda ^{\\frac{n-2}{2}}\\left[\\sum _{j\\ne 1}\\frac{2^{\\frac{n-2}{2}}\\lambda ^{\\frac{n-2}{2}}}{|\\overline{\\xi }_1-\\overline{\\xi }_j|^{n-2}}+\\sum _{j=1}^k\\frac{2^{\\frac{n-2}{2}}\\lambda ^{\\frac{n-2}{2}}}{|\\overline{\\xi }_1-\\underline{\\xi }_j|^{n-2}}-U(\\overline{\\xi }_1)\\right]\\\\&+ \\lambda ^{n+2 \\over 2} \\left[ \\sum _{j\\ne 1} \\left(\\frac{2^{\\frac{n-2}{2}}\\lambda ^{\\frac{n-2}{2}}}{|\\overline{\\xi }_1-\\overline{\\xi }_j|^{n-2}}\\right)^p+ \\sum _{j=1}^k \\left(\\frac{2^{\\frac{n-2}{2}}\\lambda ^{\\frac{n-2}{2}}}{|\\overline{\\xi }_1-\\underline{\\xi }_j|^{n-2}}\\right)^p - U^p (\\overline{\\xi }_1 )\\right],\\end{split}$ that is even with respect to every $y_\\alpha $ , $\\alpha =1,\\ldots ,n$ .",
"Let us define $E^*(\\overline{\\xi }_1+\\lambda y):=E(\\overline{\\xi }_1+\\lambda y)-E^s(\\overline{\\xi }_1+\\lambda y).$ Using (REF ) (renaming $n-1=\\tilde{n}-2$ and noticing that $\\tilde{n}\\geqslant 4$ ) we get that $\\bigg |\\lambda ^{\\frac{n-2}{2}}\\sum _{j\\ne 1}\\frac{(y,\\overline{\\xi }_1-\\overline{\\xi }_j)}{|\\overline{\\xi }_1-\\overline{\\xi }_j|^n}\\lambda \\bigg |\\leqslant \\lambda ^{\\frac{n}{2}}|y|\\sum _{j\\ne 1}\\frac{1}{|\\overline{\\xi }_1-\\overline{\\xi }_j|^{n-1}}\\leqslant C\\lambda ^{\\frac{n}{2}}k^{n-1}|y|\\leqslant {\\left\\lbrace \\begin{array}{ll}C\\frac{|y|}{k}\\mbox{ if }n\\geqslant 4,\\\\C\\frac{|y|}{k(\\ln k)^3}\\mbox{ if }n=3,\\end{array}\\right.",
"}$ and (REF ) follows for $|y|<\\frac{\\delta }{\\lambda k}$ ."
],
[
"The linear theory",
"Consider the linear problem $L_0(\\varphi )=h(y)\\quad \\hbox{in}\\quad {\\mathbb {R}}^n,\\qquad L_0(\\varphi ):=\\Delta \\varphi +p\\gamma U^{p-1}\\varphi .$ It is known that $\\ker \\lbrace L_0\\rbrace =\\mbox{span}\\lbrace Z_1,Z_2,\\ldots ,Z_{n+1}\\rbrace ,$ with $Z_\\alpha :=\\partial _{y_\\alpha }U,\\;\\; \\alpha =1,\\ldots ,n,\\quad Z_{n+1}:=y\\cdot \\nabla U+\\frac{n-2}{2}U.$ Defining the norm $\\Vert \\varphi \\Vert _*:=\\Vert (1+|y|^{n-2})\\varphi \\Vert _{L^\\infty ({\\mathbb {R}}^n)},$ in [8] the following existence result is proved.",
"Lemma 3.1 Assume $\\frac{n}{2}<q<n$ in the definition of $\\Vert \\cdot \\Vert _{**}$ .",
"Let $h$ be a function such that $\\Vert h\\Vert _{**}<+\\infty $ and $\\int _{{\\mathbb {R}}^n}Z_\\alpha h=0\\;\\;\\mbox{ for all }\\;\\alpha =1,\\ldots ,n+1.$ Then (REF ) has a unique solution $\\varphi $ with $\\Vert \\varphi \\Vert _*<+\\infty $ such that $\\int _{{\\mathbb {R}}^n}U^{p-1}Z_\\alpha \\varphi =0\\;\\;\\mbox{ for all }\\alpha =1,\\ldots ,n+1,\\quad \\mbox{ and }\\quad \\Vert \\varphi \\Vert _*\\leqslant C\\Vert h\\Vert _{**},$ for some constant $C$ depending only on $q$ and $n$ .",
"We will also need a priori estimates of the gradient of such solution.",
"Lemma 3.2 Let $\\varphi $ be the solution of (REF ) predicted by Lemma REF and assume $\\Vert (1+|y|^{n+2})h\\Vert _{L^\\infty ({\\mathbb {R}}^n)}<+\\infty $ .",
"Then, there exists $C>0$ depending only on $n$ such that $\\Vert \\nabla \\varphi \\Vert _{L^\\infty ({\\mathbb {R}}^n)}\\leqslant C(\\Vert \\varphi \\Vert _*+\\Vert (1+|y|^{n+2})h\\Vert _{L^\\infty ({\\mathbb {R}}^n)}).$ By standard elliptic estimates, $\\Vert \\nabla \\varphi \\Vert _{L^\\infty (B_1)}\\leqslant C\\left(\\Vert \\varphi \\Vert _{L^\\infty (B_2)}+\\Vert h\\Vert _{L^\\infty (B_2)}\\right)\\leqslant C\\left(\\Vert \\varphi \\Vert _*+\\Vert (1+|y|^{n+2})h\\Vert _{L^\\infty ({\\mathbb {R}}^n)}\\right).$ Defining $\\tilde{\\varphi }(y):=|y|^{2-n}\\varphi (|y|^{-2}y)$ it can be checked that $\\Delta \\tilde{\\varphi }+p\\gamma U^{p-1}\\tilde{\\varphi }=\\tilde{h}\\mbox{ in }{\\mathbb {R}}^n\\setminus \\lbrace 0\\rbrace ,$ with $\\tilde{h}(y):=|y|^{-n-2}h(y |y|^{-2})$ , and thus $\\Vert \\nabla \\tilde{\\varphi }\\Vert _{L^\\infty (B_1)}\\leqslant C\\left(\\Vert \\tilde{\\varphi }\\Vert _{L^\\infty (B_2)}+\\Vert \\tilde{h}\\Vert _{L^\\infty (B_2)}\\right).$ Noticing that $\\Vert \\tilde{\\varphi }\\Vert _{L^\\infty (B_2)}=\\Vert |y|^{n-2}\\varphi \\Vert _{L^\\infty ({\\mathbb {R}}^n\\setminus B_{1/2})}\\leqslant \\Vert \\varphi \\Vert _*,$ $\\Vert \\tilde{h}\\Vert _{L^\\infty (B_2)}=\\Vert |y|^{n+2}h\\Vert _{L^\\infty ({\\mathbb {R}}^n\\setminus B_{1/2})}\\leqslant \\Vert (1+|y|^{n+2})h\\Vert _{L^\\infty ({\\mathbb {R}}^n)},$ we obtain $\\Vert \\nabla \\tilde{\\varphi }\\Vert _{L^\\infty (B_1)}\\leqslant C\\left(\\Vert \\varphi \\Vert _*+\\Vert (1+|y|^{n+2})h\\Vert _{L^\\infty ({\\mathbb {R}}^n)}\\right).$ Writing $\\varphi (y)=|y|^{-(n-2)}\\tilde{\\varphi }(|y|^{-2}y)$ it can be seen that $|\\nabla \\varphi (y)|\\leqslant \\frac{C}{|y|^{n-1}}|\\tilde{\\varphi }(|y|^{-2}y)|+\\frac{C}{|y|^n}|\\nabla \\tilde{\\varphi }(|y|^{-2}y)|,$ and thus $\\Vert \\nabla \\varphi \\Vert _{L^\\infty ({\\mathbb {R}}^n\\setminus B_1)}\\leqslant C\\left(\\Vert \\tilde{\\varphi }\\Vert _{L^\\infty (B_1)}+\\Vert \\nabla \\tilde{\\varphi }\\Vert _{L^\\infty (B_1)}\\right)$ Putting together (REF ), (REF ) and (REF ) the estimate follows."
],
[
"The gluing scheme",
"Our goal will be to find a solution of the form $u=\\textbf {u}+\\phi ,$ with $\\phi $ a small function (in a sense that will be precised later).",
"Thus, $u$ is a solution of (REF ) if and only if $\\Delta \\phi +p\\gamma |\\textbf {u}|^{p-1}\\phi +E+\\gamma N(\\phi )=0,$ where $\\begin{split}N(\\phi )&:=|\\textbf {u}+\\phi |^{p-1}(\\textbf {u}+\\phi )-|\\textbf {u}|^{p-1}\\textbf {u}-p|\\textbf {u}|^{p-1}\\phi ,\\end{split}$ and $E$ was defined in (REF ).",
"Let $\\zeta (s)$ be a smooth function such that $\\zeta (s)=1$ for $s<1$ and $\\zeta (s)=0$ for $s>2$ , and let $\\delta >0$ be a fixed number independent of $k$ .",
"Let us define $\\overline{\\zeta }_j(y):={\\left\\lbrace \\begin{array}{ll}\\zeta (k \\delta ^{-1}|y|^{-2}|y-\\bar{\\xi }_j|y|^2|)\\quad \\hbox{ if }|y|>1,\\\\\\zeta (k \\delta ^{-1}|y-\\bar{\\xi }_j|)\\quad \\hbox{ if }|y|\\leqslant 1,\\end{array}\\right.",
"}\\quad \\underline{\\zeta }_j(y):=\\overline{\\zeta }_j(y_1,y_2,-y_3,\\ldots ,y_n)$ for $j=1,\\ldots ,k$ .",
"We notice that a function $\\phi $ of the form $\\phi =\\sum _{j=1}^k(\\overline{\\phi }_j+\\underline{\\phi }_j)+\\psi $ is a solution of (REF ) provided the functions $\\overline{\\phi }_j$ , $\\underline{\\phi }_j$ and $\\psi $ solve the following system of coupled non linear equations $\\Delta \\overline{\\phi }_j+p\\gamma |\\textbf {u}|^{p-1}\\overline{\\zeta }_j\\overline{\\phi }_j+\\overline{\\zeta }_j\\left[p\\gamma |\\textbf {u}|^{p-1}\\psi +E+\\gamma N(\\phi )\\right]=0, \\; \\;j=1,\\ldots ,k,$ $\\Delta \\underline{\\phi }_j+p\\gamma |\\textbf {u}|^{p-1}\\underline{\\zeta }_j\\underline{\\phi }_j+\\underline{\\zeta }_j\\left[p\\gamma |\\textbf {u}|^{p-1}\\psi +E+\\gamma N(\\phi )\\right]=0, \\; \\;j=1,\\ldots ,k,$ $\\begin{split}\\Delta \\psi &+p\\gamma U^{p-1}\\psi +\\left[p\\gamma (|\\textbf {u}|^{p-1}-U^{p-1})(1-\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j))\\right.\\\\&\\left.+p\\gamma U^{p-1}\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j)\\right]\\psi + p\\gamma |\\textbf {u}|^{p-1}\\sum _{j=1}^k(1-\\overline{\\zeta }_j)\\overline{\\phi }_j\\\\&+p\\gamma |\\textbf {u}|^{p-1}\\sum _{j=1}^k(1-\\underline{\\zeta }_j)\\underline{\\phi }_j+\\left(1-\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j)\\right)(E+\\gamma N(\\phi ))=0.\\end{split}$ Given our setting it is natural to ask for some symmetry properties on $\\overline{\\phi }_j$ and $\\underline{\\phi }_j$ .",
"In particular, denoting $\\hat{y}:=(y_1,y_2)$ and $y^{\\prime }:=(y_3,\\ldots ,y_n)$ , we want them to satisfy $\\overline{\\phi }_j(\\hat{y},y^{\\prime })=\\overline{\\phi }_1(e^{2\\pi \\frac{ \\leavevmode {\\color {black}(j-1)}}{k}i} \\hat{y},y^{\\prime }),\\qquad j=1,\\ldots \\,\\leavevmode {\\color {black}k},$ where $\\begin{split}&\\overline{\\phi }_1(y_1,\\ldots ,y_\\alpha ,\\ldots ,y_n)=\\overline{\\phi }_1(y_1,\\ldots ,-y_\\alpha ,\\ldots ,y_n),\\;\\; \\alpha =2,4,\\ldots ,n,\\\\&\\overline{\\phi }_1(y)=|y|^{2-n}\\overline{\\phi }_1(|y|^{-2}y),\\end{split}$ and $\\underline{\\phi }_1(y)=\\overline{\\phi }_1(y_1,y_2,-y_3,\\ldots ,y_n).$ Remark 4.1 The functions $\\overline{\\phi }_j$ and $\\underline{\\phi }_j$ are not even in the third coordinate separately but $\\overline{\\phi }_j+\\underline{\\phi }_j$ is so, that is, $(\\overline{\\phi }_j+\\underline{\\phi }_j)(y)=(\\overline{\\phi }_j+\\underline{\\phi }_j)(y_1,y_2,-y_3,\\ldots ,y_n).$ Likewise, the functions $(\\overline{\\zeta }_j+\\underline{\\zeta }_j)$ and $(\\overline{\\zeta }_j\\overline{\\phi }_j+\\underline{\\zeta }_j\\underline{\\phi }_j)$ are even in the third coordinate.",
"For $\\rho >0$ small and fixed we assume in addition $\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*\\leqslant \\rho ,$ where $\\overline{\\overline{\\phi }}_1(y):=\\lambda ^{\\frac{n-2}{2}}\\overline{\\phi }_1(\\overline{\\xi }_1+\\lambda y)$ and $\\Vert \\cdot \\Vert _*$ is defined in (REF ).",
"Proposition 4.2 There exist constants $k_0$ , $C$ and $\\rho _0$ such that, for all $k\\geqslant k_0$ and $\\rho <\\rho _0$ , if $\\overline{\\phi }_j$ and $\\underline{\\phi }_j$ satisfy conditions (REF )-(REF ) then there exists a unique solution $\\psi =\\Psi (\\overline{\\overline{\\phi }}_1)$ of (REF ) such that $\\psi (y_1,\\ldots ,y_\\alpha ,\\ldots )=\\psi (y_1,\\ldots ,-y_\\alpha ,\\ldots ),\\;\\;\\alpha =3,\\ldots ,n,$ $\\psi (\\hat{y},y^{\\prime })=\\psi (e^{\\frac{2\\pi \\leavevmode {\\color {black}(j-1)}}{k}i} \\hat{y},y^{\\prime }),\\qquad j=1,\\ldots \\,\\leavevmode {\\color {black}k},$ $\\psi (y)=|y|^{2-n}\\psi (|y|^{-2}y),$ and $\\begin{split}\\Vert \\psi \\Vert _*&\\leqslant C\\left(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+k^{1-\\frac{n}{q}}\\right)\\;\\;\\mbox{if }n\\geqslant 4,\\quad \\Vert \\psi \\Vert _*\\leqslant C\\left(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+(\\ln k)^{-1}\\right)\\;\\;\\mbox{if }n=3.\\end{split}$ Furthermore, given two functions $\\phi ^1,\\,\\phi ^2$ the operator $\\Psi $ satisfies $\\Vert \\Psi (\\phi ^1)-\\Psi (\\phi ^2)\\Vert _*\\leqslant C\\Vert \\phi ^1-\\phi ^2\\Vert _*.$ We prove the result by combining a linear theory with a fixed point argument as in [8].",
"Indeed, consider first the problem $\\Delta \\psi +p\\gamma U^{p-1}\\psi =h,$ with $h$ satisfying (REF ), (REF ), $\\Vert h\\Vert _{**}<+\\infty $ and $h(y)=|y|^{-n-2}h(|y|^{-2}y).$ Proceeding as in [8] one can apply Lemma REF to conclude the existence of a unique bounded solution $\\psi =T(h)$ of (REF ) satisfying symmetries (REF )-(REF ) and $\\Vert \\psi \\Vert _*\\leqslant C\\Vert h\\Vert _{**},$ where $C$ is a positive constant depending only on $n$ and $q$ .",
"Let us denote $\\begin{split}V(y)&:=\\underbrace{p\\gamma (|\\textbf {u}|^{p-1}-U^{p-1})\\left(1-\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j)\\right)}_{V_1(y)}+\\underbrace{p\\gamma U^{p-1}\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j)}_{V_2(y)},\\end{split}$ and $M(\\psi ):=\\left(1-\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j)\\right)(E+\\gamma N(\\phi )).$ Thus, in order to solve (REF ) by a fixed point argument, we write $\\psi =-T\\left(V\\psi + p\\gamma |\\textbf {u}|^{p-1}\\left(\\sum _{j=1}^k(1-\\overline{\\zeta }_j)\\overline{\\phi }_j+\\sum _{j=1}^k(1-\\underline{\\zeta }_j)\\underline{\\phi }_j\\right)+M(\\psi )\\right)=:{M}(\\psi ),$ where $\\psi \\in X$ , the space of continuous functions with $\\Vert \\cdot \\Vert _*<+\\infty $ and satisfying (REF )-(REF ).",
"Pointing out Remark REF and the special symmetries of $\\textbf {u}$ it can be checked that $V\\psi + p\\gamma |\\textbf {u}|^{p-1}\\left(\\sum _{j=1}^k(1-\\overline{\\zeta }_j)\\overline{\\phi }_j+\\sum _{j=1}^k(1-\\underline{\\zeta }_j)\\underline{\\phi }_j\\right)+M(\\psi )$ satisfies (REF ), (REF ) and (REF ) for every $\\psi \\in X$ and thus ${M}(\\psi )$ is well defined.",
"Let us see that ${M}$ is actually a contraction mapping in the $\\Vert \\cdot \\Vert _*$ norm in a small ball around the origin in $X$ .",
"Proceeding as in [8] we see that $\\Vert V\\psi \\Vert _{**}\\leqslant {\\left\\lbrace \\begin{array}{ll}Ck^{1-\\frac{n}{q}}\\Vert \\psi \\Vert _{*} \\quad \\mbox{ if }n\\geqslant 4,\\\\\\frac{C}{\\ln k}\\Vert \\psi \\Vert _* \\quad \\mbox{ if }n=3,\\end{array}\\right.",
"}\\Vert U^{p-1}\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j)\\Vert _{**}\\leqslant {\\left\\lbrace \\begin{array}{ll}Ck^{1-\\frac{n}{q}}\\Vert \\overline{\\overline{\\phi }}_1\\Vert _* \\quad \\mbox{ if }n\\geqslant 4,\\\\\\frac{C}{\\ln k}\\Vert \\overline{\\overline{\\phi }}_1\\Vert _* \\quad \\mbox{ if }n=3,\\end{array}\\right.",
"}$ whenever $|y-\\overline{\\xi }_j|>\\frac{\\delta }{k}$ , $|y-\\underline{\\xi }_j|>\\frac{\\delta }{k}$ .",
"Using Proposition REF we get $\\Vert M(\\psi )\\Vert _{**}\\leqslant {\\left\\lbrace \\begin{array}{ll}Ck^{1-\\frac{n}{q}}(1+\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*^2)+C\\Vert \\psi \\Vert _*^2 \\quad \\mbox{ if }n\\geqslant 4,\\\\C\\frac{1}{\\ln k}(1+\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*^2)+C\\Vert \\psi \\Vert _*^2 \\quad \\mbox{ if }n=3.\\end{array}\\right.",
"}$ Similarly, for $\\psi _1$ , $\\psi _2$ such that $\\Vert \\psi _1\\Vert _*<\\rho $ , $\\Vert \\psi _2\\Vert _*<\\rho $ it can be seen that $\\Vert M(\\psi _1)-M(\\psi _2)\\Vert _{**}\\leqslant C\\rho \\Vert \\psi _1-\\psi _2\\Vert _*.$ This estimate, together with (REF )-(REF ), allows us to conclude that for $\\rho $ small enough (independent of $k$ ) the operator ${M}$ is a contraction map in the set of functions $\\psi \\in X$ with $\\Vert \\psi \\Vert _*\\leqslant {\\left\\lbrace \\begin{array}{ll}C(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+k^{1-\\frac{n}{q}}) \\quad \\mbox{ if }n\\geqslant 4,\\\\C(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+(\\ln k)^{-1}) \\quad \\mbox{ if }n=3.\\end{array}\\right.",
"}$ Then the lemma follows by a fixed point argument.",
"We can also establish a uniform bound on the gradient of the function.",
"Proposition 4.3 Let $\\psi $ be the solution of (REF ) provided by Proposition REF .",
"Then there exists $C>0$ , depending only on $n$ , such that $\\Vert \\nabla \\psi \\Vert _{L^\\infty ({\\mathbb {R}}^n)}\\leqslant C.$ The goal is to estimate the terms in (REF ) to apply the a priori estimate in Lemma REF .",
"Consider $V_1$ and $V_2$ defined in (REF ).",
"Thus, $\\begin{split}(1+|y|^{n+2})|V_1\\psi |&\\leqslant C\\Vert \\psi \\Vert _*(1+|y|^4)U^{p-2}\\sum _{j=1}^k\\left(\\frac{\\lambda ^{\\frac{n-2}{2}}}{|y-\\overline{\\xi }_j|^{n-2}}+\\frac{\\lambda ^{\\frac{n-2}{2}}}{|y-\\underline{\\xi }_j|^{n-2}}\\right)\\leqslant C\\Vert \\psi \\Vert _*,\\\\&(1+|y|^{n+2})|V_2\\psi |\\leqslant \\Vert \\psi \\Vert _*(1+|y|^{4})U^{p-1}\\leqslant C\\Vert \\psi \\Vert _*.\\end{split}$ Analogously, noticing that $(1+|y|^{n+2})U^p\\approx 1$ , we have $(1+|y|^{n+2})|\\textbf {u}|^{p-1}\\sum _{j=1}^k(1-\\overline{\\zeta }_j)|\\overline{\\phi }_j|+ |\\textbf {u}|^{p-1}\\sum _{j=1}^k(1-\\underline{\\zeta }_j)|\\underline{\\phi }_j|\\leqslant C\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*,$ and $\\begin{split}(1+|y|^{n+2})\\left(1-\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j)\\right)|N(\\phi )|&\\leqslant (1+|y|^{n+2})U^{p-2}\\left(\\bigg |\\sum _{j=1}^k(\\overline{\\phi }_j+\\underline{\\phi }_j)\\bigg |^2+|\\psi |^2\\right)\\\\&\\leqslant C(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*^2+\\Vert \\psi \\Vert _*^2).\\end{split}$ On the other hand $(1+|y|^{n+2})(1-\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j))|E|\\leqslant C,$ as a consequence of (REF ) and (REF ).",
"Using the estimates in Proposition REF and Lemma REF we conclude (REF ).",
"Let $\\psi =\\Psi (\\overline{\\overline{\\phi }}_1)$ given by Proposition REF .",
"Notice that, thanks to the imposed conditions (REF ) and (REF ), solving the systems (REF ) and (REF ) can be reduced to solving the equation for $\\overline{\\phi }_1$ , that is, $\\Delta \\overline{\\phi }_1+p\\gamma |\\textbf {u}|^{p-1}\\overline{\\zeta }_1\\overline{\\phi }_1+\\overline{\\zeta }_1\\left[p\\gamma |\\textbf {u}|^{p-1}\\Psi (\\overline{\\overline{\\phi }}_1)+E+\\gamma N(\\phi )\\right]=0\\mbox{ in }{\\mathbb {R}}^n,$ or equivalently $\\Delta \\overline{\\phi }_1+p\\gamma |\\overline{U}_1|^{p-1}\\overline{\\phi }_1+\\overline{\\zeta }_1 E+\\gamma {N}(\\overline{\\phi }_1,\\phi )=0,$ where ${N}(\\overline{\\phi }_1,\\phi ):=p\\left(|\\textbf {u}|^{p-1}\\overline{\\zeta }_1-|\\overline{U}_1|^{p-1}\\right)\\overline{\\phi }_1+\\overline{\\zeta }_1\\left[p|\\textbf {u}|^{p-1}\\Psi (\\overline{\\overline{\\phi }}_1)+N(\\phi )\\right].$ Denote $\\overline{Z}_\\alpha (y):=\\lambda ^{-\\frac{n-2}{2}}Z_\\alpha \\left(\\frac{y-\\overline{\\xi }_1}{\\lambda }\\right),\\;\\;\\alpha =1,\\ldots ,n+1$ where $Z_\\alpha $ was defined in (REF ).",
"In order to solve (REF ) we will deal first with a projected linear version.",
"Given a general function $\\overline{h}$ we consider $\\Delta \\overline{\\phi }+p\\gamma |\\overline{U}_1|^{p-1}\\overline{\\phi }+\\overline{h}=c_3\\overline{U}_1^{p-1}\\overline{Z}_3+c_{n+1}\\overline{U}_1^{p-1}\\overline{Z}_{n+1},$ where $c_3:=\\frac{\\int _{{\\mathbb {R}}^n}\\overline{h}\\overline{Z}_3}{\\int _{{\\mathbb {R}}^n}\\overline{U}_1^{p-1}\\overline{Z}^2_3},\\qquad c_{n+1}:=\\frac{\\int _{{\\mathbb {R}}^n}\\overline{h}\\overline{Z}_{n+1}}{\\int _{{\\mathbb {R}}^n}\\overline{U}_1^{p-1}\\overline{Z}^2_{n+1}}.$ Lemma 4.4 Suppose that $\\overline{h}$ is even with respect to $y_2,y_4,\\ldots ,y_n$ and satisfies (REF ), and assume that $h(y):=\\lambda ^{\\frac{n+2}{2}}\\overline{h}(\\overline{\\xi }_1+\\lambda y)$ satisfies $\\Vert h\\Vert _{**}<+\\infty $ .",
"Then problem (REF ) has a unique solution $\\overline{\\phi }=\\overline{T}(\\overline{h})$ that is even with respect to $y_2,y_4,\\ldots ,y_n$ and satisfies $\\overline{\\phi }(y)=|y|^{2-n}\\overline{\\phi }(|y|^{-2}y),$ $\\int _{{\\mathbb {R}}^n}\\overline{\\overline{\\phi }}U^{p-1}Z_{n+1}=0,\\qquad \\int _{{\\mathbb {R}}^n}\\overline{\\overline{\\phi }}U^{p-1}Z_3=0,\\qquad \\Vert \\overline{\\overline{\\phi }}\\Vert _*\\leqslant C\\Vert h\\Vert _{**},$ with $\\overline{\\overline{\\phi }}(y):=\\lambda ^{\\frac{n-2}{2}}\\overline{\\phi }(\\overline{\\xi }_1+\\lambda y)$ .",
"Notice that, up to redefining $\\overline{h}$ as $\\overline{h}-c_3\\overline{U}_1^{p-1}\\overline{Z}_3-c_{n+1}\\overline{U}_1^{p-1}\\overline{Z}_{n+1},$ we can assume $\\int _{{\\mathbb {R}}^n}\\overline{h}\\overline{Z}_3=\\int _{{\\mathbb {R}}^n}\\overline{h}\\overline{Z}_{n+1}=0,$ i.e., $c_3=c_{n+1}=0$ and thus equation (REF ) is equivalent to $\\Delta \\overline{\\overline{\\phi }}+p\\gamma |U|^{p-1}\\overline{\\overline{\\phi }}=-h \\qquad \\hbox{in }\\mathbb {R}^n.$ We want to apply [8] to solve this problem, and therefore we need to prove that $\\int _{{\\mathbb {R}}^n} hZ_\\alpha =0\\,\\mbox{ for all }\\alpha =1,2,4,5,\\ldots ,n.$ This follows straightforward for $\\alpha =2,4,5,\\ldots ,n$ due to the evenness of $h$ .",
"The case $\\alpha =1$ holds as a consequence of (REF ) (see the proof of [8]).",
"Then the result follows by [8].",
"If instead of satisfying condition (REF ) the function $h$ is even in all its coordinates we can prove a similar result (notice that in such case $c_3=0$ ).",
"Lemma 4.5 Suppose that $h(y):=\\lambda ^{\\frac{n+2}{2}}\\overline{h}(\\overline{\\xi }_1+\\lambda y)$ is even with respect to $y_\\alpha $ for every $\\alpha =1,\\ldots , n$ and $\\Vert h\\Vert _{**}<+\\infty $ .",
"Then problem (REF ) has a unique solution $\\overline{\\phi }=\\overline{T}(\\overline{h})$ that is even with respect to $y_\\alpha $ for every $\\alpha =1,\\ldots ,n$ and satisfies $\\int _{{\\mathbb {R}}^n}\\overline{\\overline{\\phi }}U^{p-1}Z_{n+1}=0,\\qquad \\Vert \\overline{\\overline{\\phi }}\\Vert _*\\leqslant C\\Vert h\\Vert _{**},$ with $\\overline{\\overline{\\phi }}(y):=\\lambda ^{\\frac{n-2}{2}}\\overline{\\phi }(\\overline{\\xi }_1+\\lambda y)$ .",
"The result follows as in the previous Lemma just by noticing that $\\int _{{\\mathbb {R}}^n}hZ_\\alpha =0\\mbox{ for all }\\alpha =1,\\ldots ,n,$ due to the evenness of $h$ .",
"As a consequence of these lemmas we are able to solve the projected version of (REF ).",
"Indeed, consider $\\Delta \\overline{\\phi }_1+p\\gamma |\\overline{U}_1|^{p-1}\\overline{\\phi }_1+\\overline{\\zeta }_1 E+\\gamma {N}(\\overline{\\phi }_1,\\phi )=c_3\\overline{U}_1^{p-1}\\overline{Z}_3+c_{n+1}\\overline{U}_1^{p-1}\\overline{Z}_{n+1},$ with $c_3:=\\frac{\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_1 E+\\gamma {N}(\\overline{\\phi }_1,\\phi ))\\overline{Z}_3}{\\int _{{\\mathbb {R}}^n}\\overline{U}_1^{p-1}\\overline{Z}^2_3},\\qquad c_{n+1}:=\\frac{\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_1 E+\\gamma {N}(\\overline{\\phi }_1,\\phi ))\\overline{Z}_{n+1}}{\\int _{{\\mathbb {R}}^n}\\overline{U}_1^{p-1}\\overline{Z}^2_{n+1}}.$ Proposition 4.6 There exists a unique solution $\\overline{\\phi }_1=\\overline{\\phi }_1(\\ell ,t)$ of (REF ), that satisfies $\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*\\leqslant C k^{-\\frac{n}{q}}\\mbox{ if }n\\geqslant 4,\\qquad \\Vert \\overline{\\overline{\\phi }}_1\\Vert _*\\leqslant \\frac{C}{k\\ln k}\\mbox{ if }n=3,$ and $\\Vert \\overline{\\overline{{N}}}(\\overline{\\phi }_1,\\phi )\\Vert _{**}\\leqslant Ck^{-\\frac{2n}{q}}\\mbox{ if }n\\geqslant 4,\\qquad \\Vert \\overline{\\overline{{N}}}(\\overline{\\phi }_1,\\phi )\\Vert _{**}\\leqslant \\frac{C}{(k\\ln k)^2}\\mbox{ if }n=3,$ where $\\overline{\\overline{\\phi }}_1(y):=\\lambda ^\\frac{n-2}{2}\\overline{\\phi }_1(\\overline{\\xi }_1+\\lambda y)$ and $\\overline{\\overline{{N}}}(\\overline{\\phi }_1,\\phi )(y):=\\lambda ^\\frac{n+2}{2}{N}(\\overline{\\phi }_1,\\phi )(\\overline{\\xi }_1+\\lambda y)$ .",
"We will solve (REF ) by means of a fixed point argument, writting $\\overline{\\phi }_1=\\overline{T}(\\overline{\\zeta }_1 E+\\gamma {N}(\\overline{\\phi }_1,\\phi ))=:{M}(\\phi _1),$ where $\\overline{T}$ is the linear operator specified in Lemma REF .",
"To do so, we begin analyzing the nonlinear term, that can be decomposed as ${N}(\\overline{\\phi }_1,\\phi )=f_1+f_2+f_3+f_4,$ where $f_1:=p\\overline{\\zeta }_1(|{\\bf u}_2|^{p-1}-\\overline{U}_1^{p-1})\\overline{\\phi }_1,\\quad f_2:=(\\overline{\\zeta }_1-1)\\overline{U}_1^{p-1}\\overline{\\phi }_1,$ $f_3:=\\overline{\\zeta }_1 p|{\\bf u}_2|^{p-1}\\Psi (\\overline{\\phi }_1),\\quad f_4:=\\overline{\\zeta }_1N(\\phi _1).$ To estimate these terms we proceed in the same way as [8], so we just highlight the differences.",
"Given a general function $f$ , let us denote $\\tilde{f}(y):=\\lambda ^\\frac{n+2}{2}f(\\overline{\\xi }_1+\\lambda y)$ .",
"Assume first $n\\geqslant 4$ .",
"Thus, noticing that $\\sum _{j=1}^kU(y+\\lambda ^{-1}(\\underline{\\xi }_j-\\overline{\\xi }_1))\\leqslant C\\lambda ^{n-2}\\left(k^{n-2}\\sum _{j=1}^k\\frac{1}{j^{n-2}}+\\frac{1}{(2\\tau )^{n-2}}\\right)\\leqslant C\\lambda ^{n-2},$ and using Proposition REF one gets $|\\tilde{f}_1(y)|\\leqslant C\\lambda ^{\\frac{n-2}{2}}U(y)^{p-2}|\\overline{\\overline{\\phi }}_1(y)|\\;\\;\\hbox{ for }|y|<\\frac{\\delta }{\\lambda k},\\quad \\Vert \\tilde{f}_1\\Vert _{**}\\leqslant C \\lambda ^{\\frac{n}{2q}}\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*,$ $|\\tilde{f}_2(y)|\\leqslant C U(y)^{p-1}|\\overline{\\overline{\\phi }}_1(y)|\\;\\;\\hbox{ for }|y|>c\\lambda ^{-1/2},\\quad \\Vert \\tilde{f}_2\\Vert _{**}\\leqslant C\\lambda ^{\\frac{n}{2q}}\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*,$ $|\\tilde{f}_3(y)|\\leqslant C U^{p-1}(y)\\lambda ^{\\frac{n-2}{2}}|\\psi (\\overline{\\xi }_1+\\lambda y)|\\;\\;\\hbox{ for }|y|>c\\lambda ^{-1/2},\\quad \\Vert \\tilde{f}_3\\Vert _{**}\\leqslant C\\lambda ^{\\frac{n}{2q}}(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+k^{1-\\frac{n}{q}}).$ Notice that $\\tilde{N}(\\phi )=|V_*+\\hat{\\phi }|^{p-1}(V_*+\\hat{\\phi })-|V_*|^{p-1}V_* -p|V_*|^{p-1}\\hat{\\phi },$ where $\\hat{\\phi }(y):=\\lambda ^{\\frac{n-2}{2}}\\phi (\\overline{\\xi }_1+\\lambda y)$ and $V_*(y):=-U(y)-\\sum _{j\\ne 1}U(y-\\lambda ^{-1}(\\overline{\\xi }_j-\\overline{\\xi }_1))-\\sum _{j=1}^k U(y-\\lambda ^{-1}(\\underline{\\xi }_j-\\overline{\\xi }_1))+\\lambda ^\\frac{n-2}{2}U(\\overline{\\xi }_1+\\lambda y).$ Hence $|\\tilde{f}_4(y)|\\leqslant CU(y)^{p-2}(|\\overline{\\overline{\\phi }}_1(y)|^2+\\lambda ^{n-2}|\\psi (\\overline{\\xi }_1+\\lambda y)|^2)\\;\\;\\hbox{ for }|y|<\\frac{\\delta }{\\lambda k},\\quad \\Vert \\tilde{f}_4\\Vert _{**}\\leqslant C\\lambda ^{\\frac{n}{2q}}(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+k^{1-\\frac{n}{q}}).$ Finally, defining $f_5:=\\overline{\\zeta }_1E$ and using estimate (REF ) we also have $\\Vert \\tilde{f}_5\\Vert _{**}\\leqslant C\\lambda ^{\\frac{n}{2q}}.$ In the case $n=3$ one has $\\Vert \\tilde{f}_1\\Vert _{**}\\leqslant \\frac{C}{k\\ln k}\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*, \\quad \\Vert \\tilde{f}_2\\Vert _{**}\\leqslant \\frac{C}{k\\ln k}\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*,\\quad \\Vert \\tilde{f}_3\\Vert _{**}\\leqslant \\frac{C}{k\\ln k}\\left(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+\\frac{1}{k\\ln k}\\right),$ $\\Vert \\tilde{f}_4\\Vert _{**}\\leqslant \\frac{C}{k\\ln k}\\left(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+\\frac{1}{k\\ln k}\\right),\\quad \\Vert \\tilde{f}_5\\Vert _{**}\\leqslant \\frac{C}{k\\ln k}.$ Applying Proposition REF , estimates (REF )-(REF ), Proposition REF and Lemma REF , we conclude that ${M}$ is a contraction that maps functions $\\overline{\\overline{\\phi }}$ with $\\Vert \\overline{\\overline{\\phi }}\\Vert _*\\leqslant C k^{-\\frac{n}{q}}\\mbox{ if }n\\geqslant 4,\\qquad \\Vert \\overline{\\overline{\\phi }}\\Vert _*\\leqslant \\frac{C}{k\\ln k}\\mbox{ if }n=3,$ into the same class of functions whenever $\\frac{n}{2}<q<\\frac{n}{2-\\frac{2}{n-1}}$ .",
"Analogously, it can be proved the Lipschitz character of the operators and thus, applying a fixed point argument, we conclude the proof.",
"Proposition 4.7 Let $\\overline{\\phi }_1$ be the solution of (REF ) provided by Proposition REF .",
"Then there exists $C>0$ , depending only on $n$ , such that $|\\overline{\\overline{\\phi }}_1(y)|\\leqslant C\\frac{\\lambda ^{\\frac{n-2}{2}}}{(1+|y|)^\\alpha }\\quad \\mbox{ where }{\\left\\lbrace \\begin{array}{ll}\\alpha =2\\;\\;\\mbox{ if }n\\geqslant 5,\\\\\\alpha =1\\;\\;\\mbox{ if }n=4,\\\\0<\\alpha <1\\;\\;\\mbox{ if }n=3,\\end{array}\\right.",
"}$ where $\\overline{\\overline{\\phi }}_1(y):=\\lambda ^{\\frac{n-2}{2}}\\overline{\\phi }_1(\\overline{\\xi }_1+\\lambda y)$ .",
"Denote $L_0(\\phi ):=\\Delta \\phi +p\\gamma |U|^{p-1}\\phi $ .",
"Thus, (REF ) can be written in the form $L_0(\\overline{\\overline{\\phi }}_1)+a(y)\\overline{\\overline{\\phi }}_1=g(y)+c_3U^{p-1}Z_3+c_{n+1}U^{p-1}Z_{n+1},$ with $a(y):=\\lambda ^2p\\gamma (|\\textbf {u}|\\overline{\\zeta }_1-|\\overline{U}_1|^{p-1})(\\overline{\\xi }_1+\\lambda y)$ .",
"Hence $|a(y)|\\leqslant CU^{p-1}, \\qquad |g(y)|\\leqslant C\\frac{\\lambda ^\\frac{n-2}{2}}{(1+|y|)^4}.$ Applying [8] with $\\nu =4$ for $n\\geqslant 5$ , $\\nu =3$ for $n=4$ and $2<\\nu <3$ for $n=3$ we get the desired estimates on $\\overline{\\overline{\\phi }}_1$ .",
"To perform the reduction procedure in section we will need more precise estimates on the pointwise behavior of $\\overline{\\phi }_1$ , in particular on the part that will not be orthogonal to the kernel, whose size is smaller.",
"Proposition 4.8 Let $\\overline{\\phi }_1$ be the solution of (REF ) provided by Proposition REF and denote $\\overline{\\overline{\\phi }}_1(y):=\\lambda ^{\\frac{n-2}{2}}\\overline{\\phi }_1(\\overline{\\xi }_1+\\lambda y)$ .",
"Then there exists a decomposition $\\overline{\\overline{\\phi }}_1=\\overline{\\overline{\\phi ^s_1}}+\\overline{\\overline{\\phi ^*_1}}$ such that $\\overline{\\overline{\\phi ^s_1}}$ is even with respect to $y_\\alpha $ for every $\\alpha =1,\\ldots ,n$ and $|\\overline{\\overline{\\phi ^*_1}}(y)|\\leqslant {\\left\\lbrace \\begin{array}{ll}C \\frac{\\lambda ^{\\frac{n-2}{2}}}{k}\\frac{1}{1+|y|}\\mbox{ if }n\\geqslant 4,\\\\C\\frac{\\lambda ^{1/2}}{k(\\ln k)^3}\\frac{1}{1+|y|^\\alpha }\\quad 0<\\alpha <1,\\quad \\mbox{ if }n=3.\\end{array}\\right.",
"}.$ The idea of the proof is to identify in the equation (REF ) the largest terms, that happen to be symmetric, and will produce a large but symmetric solution.",
"The remaining terms will give a function $\\overline{\\overline{\\phi _1^*}}$ non symmetric but smaller.",
"Denote $V^s:=\\lambda ^{\\frac{n-2}{2}}\\left[\\sum _{j\\ne 1}\\frac{2^{\\frac{n-2}{2}}\\lambda ^{\\frac{n-2}{2}}}{|\\overline{\\xi }_1-\\overline{\\xi }_j|^{n-2}}+\\sum _{j=1}^k\\frac{2^{\\frac{n-2}{2}}\\lambda ^{\\frac{n-2}{2}}}{|\\overline{\\xi }_1-\\underline{\\xi }_j|^{n-2}}-U(\\overline{\\xi }_1)\\right].$ Consider $E^s$ given by (REF ) and let us define $\\lambda ^{\\frac{n-2}{2}}{N}^s(\\overline{\\xi }_1+\\lambda y):=\\overline{\\zeta }_1(\\overline{\\xi }_1+\\lambda y)\\left(\\tilde{f}^s_1+\\tilde{f}^s_3+\\tilde{f}^s_4\\right)+(\\overline{\\zeta }_1(\\overline{\\xi }_1+\\lambda y)-1)\\tilde{f}^s_2,$ where $\\tilde{f}^s_1:=p\\left[(U(y)+V^s)^{p-1}-U(y)^{p-1}\\right]\\overline{\\overline{\\phi _1^s}},\\quad \\tilde{f}^s_2:=U(y)^{p-1}\\overline{\\overline{\\phi _1^s}},$ $\\tilde{f}^s_3:=p\\lambda ^{\\frac{n-2}{2}}(U(y)+V^s)^{p-1}\\psi (\\overline{\\xi }_1),\\quad \\tilde{f}^s_4:=(U(y)+V^s)^{p-2}(\\hat{\\phi }^s)^2,$ being $\\hat{\\phi }^s:=\\overline{\\overline{\\phi _1^s}}+\\sum _{j\\ne 1}\\lambda ^{\\frac{n+2}{2}}\\overline{\\phi }_j^s(\\overline{\\xi }_1+\\lambda y)+\\sum _{j=1}^k\\lambda ^{\\frac{n+2}{2}}\\underline{\\phi }_j^s(\\overline{\\xi }_1+\\lambda y).$ Notice that, if $\\overline{\\overline{\\phi _1^s}}$ is even in all its coordinates, then ${N}^s(\\overline{\\xi }_1+\\lambda y)$ is also even and groups the largest terms of ${N}(\\overline{\\xi }_1+\\lambda y)$ .",
"Thus, proceeding as in the previous Proposition we can find a solution to $\\Delta \\overline{\\phi _1^s}+p\\gamma |\\overline{U}_1|^{p-1}\\overline{\\phi _1^s}+\\overline{\\zeta }_1 E^s+\\gamma {N}^s(\\overline{\\phi }_1,\\phi )=c^s_{n+1}\\overline{U}_1^{p-1}\\overline{Z}_{n+1},$ with $c^s_{n+1}:=\\frac{\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_1 E^s+\\gamma {N}^s(\\overline{\\phi }_1,\\phi ))\\overline{Z}_{n+1}}{\\int _{{\\mathbb {R}}^n}\\overline{U}_1^{p-1}\\overline{Z}^2_{n+1}},\\quad \\overline{\\overline{\\phi _1^s}}(y)=\\lambda ^{\\frac{n-2}{2}}\\overline{\\phi ^s_1}(\\overline{\\xi }_1+\\lambda y),$ by applying Lemma REF to perform a fixed point argument in the set of functions $\\overline{\\overline{\\phi ^s}}$ which are even in all their coordinates and have size $\\Vert \\overline{\\overline{\\phi ^s}}\\Vert _*\\leqslant Ck^{-\\frac{n}{q}}\\mbox{ if }n\\geqslant 4,\\quad \\Vert \\overline{\\overline{\\phi ^s}}\\Vert _*\\leqslant \\frac{C}{k\\ln k}\\mbox{ if }n= 4.$ Furthermore, proceeding as in Proposition REF we can conclude that $|\\overline{\\overline{\\phi _1^s}}(y)|\\leqslant C\\frac{\\lambda ^{\\frac{n-2}{2}}}{(1+|y|)^\\alpha }\\quad \\mbox{ where }{\\left\\lbrace \\begin{array}{ll}\\alpha =2\\;\\;\\mbox{ if }n\\geqslant 5,\\\\\\alpha =1\\;\\;\\mbox{ if }n=4,\\\\0<\\alpha <1\\;\\;\\mbox{ if }n=3.\\end{array}\\right.",
"}$ Let us define $\\overline{\\phi _1^*}:=\\overline{\\phi }_1-\\overline{\\phi _1^s}$ .",
"Hence it solves $\\Delta \\overline{\\phi _1^*}+p\\gamma |\\overline{U}_1|^{p-1}\\overline{\\phi _1^*}+\\overline{\\zeta }_1 E^*+\\gamma {N}^*(\\overline{\\phi }_1,\\phi )=c_3\\overline{U}_1^{p-1}\\overline{Z}_3+c^*_{n+1}\\overline{U}_1^{p-1}\\overline{Z}_{n+1},$ where $E^*$ is defined in Proposition REF , $ {N}^*(\\overline{\\phi }_1,\\phi ):={N}(\\overline{\\phi }_1,\\phi )-{N}^s(\\overline{\\phi }_1,\\phi ),$ and $c_3:=\\frac{\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_1 E^*+\\gamma {N}^*(\\overline{\\phi }_1,\\phi ))\\overline{Z}_3}{\\int _{{\\mathbb {R}}^n}\\overline{U}_1^{p-1}\\overline{Z}^2_3},\\quad c^*_{n+1}:=\\frac{\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_1 E^*+\\gamma {N}^*(\\overline{\\phi }_1,\\phi ))\\overline{Z}_{n+1}}{\\int _{{\\mathbb {R}}^n}\\overline{U}_1^{p-1}\\overline{Z}^2_{n+1}}=c_{n+1}-c^s_{n+1}.$ The key point is that, without the previous symmetric part, the terms left are smaller and more precise estimates con be done, Indeed, denote $V(y):=\\lambda ^{\\frac{n-2}{2}}\\left[\\sum _{j\\ne 1}\\overline{U}_j(\\overline{\\xi }_1+\\lambda y)+\\sum _{j=1}^k\\underline{U}_j(\\overline{\\xi }_1+\\lambda y)-U(\\overline{\\xi }_1+\\lambda y)\\right].$ Thus we can write $\\lambda ^{\\frac{n-2}{2}}{N}^*(\\overline{\\xi }_1+\\lambda y):=\\overline{\\zeta }_1(\\overline{\\xi }_1+\\lambda y)\\left(\\tilde{f}^*_1+\\tilde{f}^*_3+\\tilde{f}^*_4\\right)+(\\overline{\\zeta }_1(\\overline{\\xi }_1+\\lambda y)-1)\\tilde{f}^*_2,$ where $\\begin{split}\\tilde{f}_1^*(y):=&\\,p\\left[(U(y)+V^s(y))^{p-1}-U^{p-1}(y)\\right]\\overline{\\overline{\\phi _1^*}}(y)\\\\&+p\\left[(U(y)+V(y))^{p-1}-(U(y)+V^s(y))^{p-1}\\right]\\overline{\\overline{\\phi }}_1(y),\\end{split}$ $\\tilde{f}_2^*(y):=U(y)^{p-1}\\overline{\\overline{\\phi _1^*}}(y),$ $\\begin{aligned}\\tilde{f}_3^*(y):=&\\,p\\lambda ^{\\frac{n-2}{2}}\\left(U(y)+V(y)\\right)^{p-1}\\lambda \\nabla \\psi (\\eta )y\\\\&+p\\lambda ^{\\frac{n-2}{2}}\\left[\\left(U(y)+V(y)\\right)^{p-1}-\\left(U(y)+V^s(y)\\right)^{p-1}\\right]\\psi (\\overline{\\xi }_1),\\end{aligned}$ $\\tilde{f}^*_4(y):=\\tilde{f}_4(y)-\\tilde{f}^s_4(y),$ with $\\tilde{f}_4$ defined in Proposition REF .",
"Noticing that $\\bigg |\\left[(U(y)+V(y))^{p-1}-(U(y)+V^s(y))^{p-1}\\right]\\overline{\\overline{\\phi }}_1(y)\\bigg |\\leqslant CU^{p-1}(y)|V(y)-V^s|\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*$ and $|V(y)-V^s|\\leqslant C \\frac{\\lambda ^{\\frac{n-2}{2}}|y|}{k}\\mbox{ if }n\\geqslant 4,\\qquad |V(y)-V^s|\\leqslant C\\frac{\\lambda ^{1/2}|y|}{k(\\ln k)^3}\\mbox{ if }n=3,$ it can be proved that $\\bigg |\\left[(U(y)+V(y))^{p-1}-(U(y)+V^s(y))^{p-1}\\right]\\overline{\\overline{\\phi }}_1(y)\\bigg |\\leqslant {\\left\\lbrace \\begin{array}{ll}C \\frac{\\lambda ^{\\frac{n-2}{2}}}{k}\\frac{1}{1+|y|^3}\\mbox{ if }n\\geqslant 4,\\\\C\\frac{\\lambda ^{1/2}}{k(\\ln k)^3}\\frac{1}{1+|y|^3}\\mbox{ if }n=3,\\end{array}\\right.",
"}$ and the same bound can be proved for $\\tilde{f}_3^*$ and $\\tilde{f}_4^*$ by using the estimates in Proposition REF and Proposition REF .",
"Using this together with Proposition REF we can proceed as in the proof of Proposition REF to estimate the size of $|\\overline{\\overline{\\phi _1^*}}|$ .",
"That is, we can rewrite problem (REF ) as $L_0(\\overline{\\overline{\\phi _1^*}})+a^*(y)\\overline{\\overline{\\phi _1^*}}=g^*(y)+c_3U^{p-1}Z_3+c_{n+1}^*U^{p-1}Z_{n+1},$ where $L_0(\\phi ):=\\Delta \\phi +p\\gamma |U|^{p-1}\\phi ,\\qquad |a^*(y)|\\leqslant CU(y)^{p-1},$ and $|g^*(y)|\\leqslant {\\left\\lbrace \\begin{array}{ll}C \\frac{\\lambda ^{\\frac{n-2}{2}}}{k}\\frac{1}{1+|y|^3}\\mbox{ if }n\\geqslant 4,\\\\C\\frac{\\lambda ^{1/2}}{k(\\ln k)^3}\\frac{1}{1+|y|^3}\\mbox{ if }n=3.\\end{array}\\right.",
"}.$ Applying again [8] with $\\nu =3$ for $n\\geqslant 4$ and $2<\\nu <3$ for $n=3$ we obtain (REF ).",
"Likewise, we will need accurate estimates on the non symmetric part of $\\psi $ .",
"Proposition 4.9 Let $\\psi $ be the solution of (REF ) provided by Proposition REF .",
"Then $\\psi (\\overline{\\xi }_1+\\lambda y)=\\psi ^s(y)+\\psi ^*(y), \\qquad y\\in B(0,\\frac{\\delta }{\\lambda k}),$ where $\\psi ^s$ is even with respect to $y_3$ and $|\\psi ^*(y)|\\leqslant C \\left(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+\\Vert \\psi \\Vert _*+\\tau o_k(1)\\right)\\lambda |y|(1+|y|),$ where $o_k(1)$ is a function that goes to 0 when $k\\rightarrow \\infty $ .",
"Since $\\psi $ is a solution of (REF ) we can write, making the convolution with the fundamental solution of the Laplace equation, $\\begin{split}\\psi (\\overline{\\xi }_1+\\lambda y)=c_n\\int _{{\\mathbb {R}}^n}&\\frac{1}{|\\overline{\\xi }_1+\\lambda y-x|^{n-2}} W(\\psi )(x) \\,dx, \\quad y\\in B(0,\\frac{\\delta }{\\lambda k}),\\end{split}$ $W(\\psi ):=V\\psi +M(\\psi )+p\\gamma |{\\bf {u}}|^{p-1}\\sum _{j=1}^k((1-\\overline{\\zeta }_j)\\overline{\\phi }_j+(1-\\underline{\\zeta }_j)\\underline{\\phi }_j)-p\\gamma U^{p-1}\\psi ,$ where $c_n$ is a constant depending only on the dimension and $V$ , $M$ were defined in (REF ) and (REF ) respectively.",
"Furthermore, $\\begin{split}\\frac{1}{|\\overline{\\xi }_1+\\lambda y-x|^{n-2}}=:A(x,y)+B_1(x,y)+B_2(x,y),\\end{split}$ where $A(x,y):=\\frac{1}{|x-\\overline{\\xi }_1|^{n-2}}\\left[1-\\left(\\frac{n-2}{2}\\right)\\frac{\\lambda ^2 y^2+2\\lambda \\sum _{i=1, i\\ne 3}^n y_i(x-\\overline{\\xi }_1)_i}{|x-\\overline{\\xi }_1|^2}\\right],$ $B_1(x,y):=-(n-2)\\lambda y_3\\frac{x_3-\\tau }{|x-\\overline{\\xi }_1|^n},\\qquad B_2(x,y):=O\\left(\\left(\\frac{(\\lambda ^2 y^2+2\\lambda (y,x-\\overline{\\xi }_1))^2}{|x-\\overline{\\xi }_1|^{n+2}}\\right)\\right).$ Notice that $A(x,y)$ is even with respect to $y_3$ and thus we can define $\\psi ^s(y):=c_n\\int _{{\\mathbb {R}}^n}A(x,y) W(\\psi )(x)\\,dx,$ that inherits this symmetry.",
"Therefore we have to estimate $\\psi ^*(y):=c_n\\int _{{\\mathbb {R}}^n}(B_1(x,y)+B_2(x,y)) W(\\psi )(x)\\,dx.$ Since $|W(\\psi )(x)|\\leqslant \\frac{C}{(1+|x|)^4}$ and $|y|<c\\lambda ^{-1/2}$ it easily follows that $\\bigg |\\int _{{\\mathbb {R}}^n}B_2(x,y) W(\\psi )(x)\\,dx\\bigg | \\leqslant \\lambda \\tau |y|^2 o_k(1).$ To estimate the term with $B_1(x,y)$ notice first that $\\bigg |\\int _{{\\mathbb {R}}^n}B_1(x,y)U^{p-1}\\psi \\,dx\\bigg |\\leqslant C\\lambda |y|\\int _{{\\mathbb {R}}^n}\\frac{1}{|x-\\overline{\\xi }_1|^{n-1}}\\frac{1}{(1+|x|)^4}|\\psi (x)|\\,dx \\leqslant C\\lambda |y|\\Vert \\psi \\Vert _*$ and likewise $\\bigg |\\int _{{\\mathbb {R}}^n}B_1(x,y)V \\psi \\,dx \\bigg |\\leqslant C\\lambda |y|\\Vert \\psi \\Vert _*.$ Observing that $|\\overline{\\phi }_j(y)|\\leqslant C\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*\\frac{\\lambda ^{\\frac{n-2}{2}}}{|y-\\overline{\\xi }_j|^{n-2}},\\qquad |\\underline{\\phi }_j(y)|\\leqslant C\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*\\frac{\\lambda ^{\\frac{n-2}{2}}}{|y-\\underline{\\xi }_j|^{n-2}},$ we get $\\bigg |\\int _{{\\mathbb {R}}^n}B_1(x,y)|{\\bf {u}}|^{p-1}\\sum _{j=1}^k((1-\\overline{\\zeta }_j)\\overline{\\phi }_j+(1-\\underline{\\zeta }_j)\\underline{\\phi }_j)\\bigg |\\leqslant C \\lambda |y| \\Vert \\overline{\\overline{\\phi }}_1\\Vert _*,$ $\\bigg |\\int _{{\\mathbb {R}}^n}B_1(x,y)\\left(1-\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j)\\right)N(\\phi )\\bigg |\\leqslant C \\lambda |y| (\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+\\Vert \\psi \\Vert _*),$ and thus the only term left is the one concerning the error, namely $\\int _{{\\mathbb {R}}^n}B_1(x,y)\\left(1-\\sum _{j=1}^k(\\overline{\\zeta }_j+\\underline{\\zeta }_j)\\right)E\\,dx=\\underbrace{\\int _{{\\mathbb {R}}^n}B_1(x,y)\\left(1-\\sum _{j=1}^k\\overline{\\zeta }_j\\right)E\\,dx}_{B_{11}}-\\underbrace{\\sum _{j=1}^k\\int _{{\\mathbb {R}}^n}B_1(x,y)\\underline{\\zeta }_j E\\,dx}_{B_{12}}.$ Let $R>0$ large.",
"Thus it can be seen that $B_{11}=\\int _{B(\\overline{\\xi }_1,R)}B_1(x,y)(1-\\sum _{j=1}^k\\overline{\\zeta }_j)E\\,dx+O(|y| k^{-(n-1)}).$ The desired estimate will follow by noticing that the largest terms of the error are orthogonal to $B_1(x,y)$ .",
"To see this, we write the error as $\\begin{split}\\gamma ^{-1}E\\simeq &\\,pU^{p-1}\\left(\\sum _{j=1}^k\\overline{U}_j+\\sum _{j=1}^k\\underline{U}_j\\right)-\\sum _{j=1}^k\\overline{U}_j^p-\\sum _{j=1}^k\\underline{U}_j^p\\\\=&\\,pU^{p-1}(\\overline{\\xi }_1)\\sum _{j=1}^k\\overline{U}_j-\\sum _{j=1}^k\\overline{U}_j^p+pU^{p-1}(\\overline{\\xi }_1)\\sum _{j=1}^k\\underline{U}_j-\\sum _{j=1}^k\\underline{U}_j^p\\\\&\\,+p(U^{p-1}-U^{p-1}(\\overline{\\xi }_1))\\sum _{j=1}^k\\overline{U}_j+p(U^{p-1}-U^{p-1}(\\overline{\\xi }_1))\\sum _{j=1}^k\\underline{U}_j.\\end{split}$ Notice first that, since $B_1(x,y)$ is odd with respect to the hyperplane $x_3=\\tau $ , there holds $\\int _{B(\\overline{\\xi }_1,R)}B_1(x,y)(1-\\sum _{j=1}^k\\overline{\\zeta }_j)\\left(pU^{p-1}(\\overline{\\xi }_1)\\sum _{j=1}^k\\overline{U}_j-\\sum _{j=1}^k\\overline{U}_j^p\\right)\\,dx=0.$ On the other hand, writting $\\begin{split}\\underline{U}_j(x)=&\\,\\frac{\\lambda ^\\frac{n-2}{2}}{(\\lambda ^2+|\\overline{\\xi }_1-\\underline{\\xi }_j|^2+|x-\\overline{\\xi }_1|^2)^{\\frac{n-2}{2}}}\\left[1-(n-2)\\frac{(x-\\overline{\\xi }_1,\\overline{\\xi }_1-\\underline{\\xi }_j)}{\\lambda ^2+|\\overline{\\xi }_1-\\underline{\\xi }_j|^2+|x-\\overline{\\xi }_1|^2}\\right.\\\\&\\,\\left.+O\\left(\\left(\\frac{(x-\\overline{\\xi }_1,\\overline{\\xi }_1-\\underline{\\xi }_j)}{\\lambda ^2+|\\overline{\\xi }_1-\\underline{\\xi }_j|^2+|x-\\overline{\\xi }_1|^2}\\right)^2\\right)\\right],\\end{split}$ and applying again the evenness of its main terms with respect to $x_3=\\tau $ we have that $\\begin{split}\\bigg |\\int _{B(\\overline{\\xi }_1,R)}&B_1(x,y)\\left(1-\\sum _{j=1}^k\\overline{\\zeta }_j\\right)pU^{p-1}(\\overline{\\xi }_1)\\sum _{j=1}^k\\underline{U}_j\\,dx\\bigg |\\\\\\leqslant &\\,C\\lambda ^{1+\\frac{n-2}{2}}|y|\\int _{B(\\overline{\\xi }_1,R)}\\frac{|x_3-\\tau |}{|x-\\overline{\\xi }_1|^n}\\sum _{j=1}^k\\frac{\\tau |x_3-\\tau |}{(\\lambda ^2+|\\overline{\\xi }_1-\\underline{\\xi }_j|^2+|x-\\overline{\\xi }_1|^2)^{\\frac{n}{2}}}\\,dx\\\\&\\,+C\\lambda ^{1+\\frac{n-2}{2}}|y|\\int _{B(\\overline{\\xi }_1,R)}\\frac{|x_3-\\tau |}{|x-\\overline{\\xi }_1|^n}\\sum _{j=1}^k\\frac{|x-\\overline{\\xi }_1|^2|\\overline{\\xi }_1-\\underline{\\xi }_j|^2}{(\\lambda ^2+|\\overline{\\xi }_1-\\underline{\\xi }_j|^2+|x-\\overline{\\xi }_1|^2)^{\\frac{n+2}{2}}}\\,dx\\\\\\leqslant &\\, C\\lambda ^{1+\\frac{n-2}{2}}|y|\\int _{B(\\overline{\\xi }_1,R)}\\frac{1}{|x-\\overline{\\xi }_1|^{n-1}}\\sum _{j=1}^k\\left(\\frac{\\tau }{|\\overline{\\xi }_1-\\underline{\\xi }_j|^{n-1}}+\\frac{1}{|\\overline{\\xi }_1-\\underline{\\xi }_j|^{n-2}}\\right)\\,dx\\\\=&\\, \\lambda \\tau |y| o_k(1),\\end{split}$ where in the last inequality we have used (REF ).",
"Proceeding analogously with the other terms it can be concluded that $\\int _{B(\\overline{\\xi }_1,R)}B_1(x,y)\\left(1-\\sum _{j=1}^k\\overline{\\zeta }_j\\right)E\\,dx=\\lambda \\tau o_k(1)|y|,$ and therefore $B_{11}=\\lambda \\tau o_k(1)|y|.$ Likewise, $\\begin{split}|B_{12}|\\leqslant C\\lambda |y|\\sum _{j=1}^k\\int _{B(\\underline{\\xi }_j,\\frac{\\delta }{k})}\\frac{|x_3-\\tau |}{|\\overline{\\xi }_1-\\underline{\\xi }_j|^n}\\sum _{i=1}^k\\frac{\\lambda ^{\\frac{n-2}{2}}}{|\\overline{\\xi }_1-\\underline{\\xi }_j|^{n-2}}\\,dx=\\lambda \\tau o_k(1)|y|.\\end{split}$ Putting together (REF )-(REF ) with (REF ) and (REF ) the result follows."
],
[
"Proof of Theorem ",
"The goal of this section is to find positive parameters $\\ell $ and $t$ that enter in the definition of $\\lambda $ and $\\tau $ in (REF ) and are independent of $k$ , in such a way that $c_3$ and $c_{n+1}$ (defined in (REF )) vanish.",
"In fact, if such a choice is possible, then the solution $\\overline{\\phi }_1$ found in Proposition REF solves (REF ) and thus, applying Proposition REF and (REF )-(REF ), we can conclude that $\\phi $ solves (REF ) and therefore $u=\\textbf {u}+\\phi $ is a solution of (REF ).",
"Thus, we want to prove the existence of $\\ell $ and $t$ so that $\\overline{c}_3(\\ell , t):=\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_1 E+\\gamma {N}(\\overline{\\phi }_1,\\phi ))\\overline{Z}_3=0,\\;\\; \\overline{c}_{n+1}(\\ell , t):=\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_1 E+\\gamma {N}(\\overline{\\phi }_1,\\phi ))\\overline{Z}_{n+1}=0.$ It is worth pointing out that, due to some symmetry, the main order term of $\\overline{c}_3$ vanishes, what makes necessary an expansion of $\\overline{c}_3$ at lower order.",
"This is usually a delicate issue and it requires sharper estimates on the non linear term, that is, a finer control on the size of the terms $\\psi $ and $\\overline{\\phi }_1$ in the spirit of Proposition REF .",
"However in this case this type of estimates are not enough, since they do not produce a non linear term sufficiently small.",
"We need to identify a precise decomposition of $\\overline{\\phi }_1$ and $\\psi $ in one symmetric part whose contribution to the computation of $\\overline{c}_3$ is zero, and a smaller non-symmetric part (see Claim 6).",
"This decompositions were developed in Proposition REF and Proposition REF .",
"What we obtain at the end is that, for $n\\geqslant 4$ , $\\begin{split}&\\overline{c}_{3}(\\ell , t)=D_n\\frac{t\\ell ^{\\frac{n}{n-2}}}{k^{n+1-\\frac{2}{n-1}}}\\left[d_n\\frac{\\ell }{t^{n-1}}-1\\right]+\\frac{1}{k^{\\alpha }}\\Theta _{k}(\\ell ,t),\\\\&\\overline{c}_{n+1}(\\ell , t)=E_n\\frac{\\ell }{k^{n-2}}\\left[e_n\\ell ^2 -1\\right]+\\frac{1}{k^{\\beta }}\\Theta _{k}(\\ell ,t),\\end{split}$ where $\\alpha >n+1-\\frac{2}{n-1},\\qquad \\beta >n-2,$ and $\\begin{split}&\\overline{c}_3(\\ell ,t)=F\\frac{\\ell ^3 t}{\\sqrt{\\ln k}(k\\ln k)^3}\\left[f\\frac{\\ell }{t^2}-1\\right]+\\frac{1}{k^3(\\ln k)^4}\\Theta _k(\\ell ,t),\\\\&\\overline{c}_4(\\ell ,t)=G\\frac{\\ell }{k\\ln k}\\left[g\\ell ^2-1\\right]+\\frac{1}{k\\ln k}\\frac{\\ln \\left(\\frac{2\\pi }{\\sqrt{\\ln k}}\\right)}{\\ln k}\\Theta _k(\\ell ,t),\\end{split}$ for $n=3$ , where $D_n, d_n, E_n, e_n, F, f, G, g$ are fixed positive numbers (depending only on $n$ ) and $\\Theta _{k}(\\ell ,t)$ is a generic function, smooth on its variables, and uniformly bounded as $k\\rightarrow \\infty $ .",
"Hence, by a fixed point argument we can conclude the existence of $\\ell $ and $t$ such that $\\overline{c}_3(\\ell , t)=\\overline{c}_{n+1}(\\ell , t)=0.$ By simplicity we detail the argument in the case of (REF ).",
"With abuse of notation on the function $\\Theta $ , that always stands for a generic function smooth on its variables and uniformly bounded as $k\\rightarrow \\infty $ , (REF ) is equivalent to $\\begin{split}&f\\ell -t^2+o_k(1)\\Theta _k(\\ell ,t)=0,\\\\&g\\ell ^2-1+o_k(1)\\Theta _k(\\ell ,t)=0.\\end{split}$ Defining $\\rho :=t^2$ and $\\eta :=\\ell ^2$ we can rewrite the system as $\\begin{split}&\\rho =f\\eta ^{1/2}+o_k(1)\\Theta _k(\\eta ,\\rho ),\\\\&\\eta =\\frac{1}{g}+o_k(1)\\Theta _k(\\eta ,\\rho ).\\end{split}$ Suppose $0<\\rho \\leqslant C$ fixed.",
"Hence the second equation can be expressed as $\\eta =F_\\rho (\\eta ),\\quad \\mbox{ with }F_a(s):=\\frac{1}{g}+o_k(1)\\Theta _k(s,a).$ Consider the set $X:=\\lbrace \\eta \\in {\\mathbb {R}}:\\,0<\\eta \\leqslant \\frac{2}{g}\\rbrace $ .",
"Using the smoothness of $\\Theta _k$ it is easy to see that $F_\\rho $ maps $X$ into itself and that it is a contraction for $k$ large enough.",
"Thus, for any fixed $\\rho $ there exists a fixed point $\\eta _\\rho \\in X$ such that $\\eta _\\rho =F_\\rho (\\eta _\\rho )$ .",
"Replacing on the first equation this translates into $\\rho =f (F_\\rho (\\eta _\\rho ))^{1/2}+o_k(1)\\Theta _k(\\eta _\\rho ,\\rho )=\\frac{f}{g^{1/2}}+o_k(1)\\Theta _k(\\eta _\\rho ,\\rho )=:G(\\rho ).$ Considering the set $Y:=\\lbrace \\rho \\in {\\mathbb {R}}:0<\\rho \\leqslant \\frac{2f}{g^{1/2}}\\rbrace $ and using the smoothness of $\\Theta $ it can be checked that $G$ is a contraction that maps the set $Y$ into itself, and therefore we conclude the existence of a fixed point $\\rho = G(\\rho )$ , what concludes the argument.",
"The rest of the section is devoted to prove (REF ) and (REF ).",
"Let us consider first the case of $\\overline{c}_{n+1}$ , that follows analogously to [8].",
"We write $\\overline{c}_{n+1}(\\ell , t)=\\int _{{\\mathbb {R}}^n}E\\overline{Z}_{n+1}-\\int _{{\\mathbb {R}}^{n}}(1-\\overline{\\zeta }_1)E\\overline{Z}_{n+1}+\\gamma \\int _{{\\mathbb {R}}^n}{N}(\\overline{\\phi }_1,\\phi )\\overline{Z}_{n+1}=0.$ Thus, for $\\ell $ and $t$ as in (REF ) we have: Claim 1: $ \\int _{{\\mathbb {R}}^n}E\\overline{Z}_{n+1}={\\left\\lbrace \\begin{array}{ll}E_n\\frac{\\ell }{k^{n-2}}\\left[e_n\\ell ^2 -1\\right]+\\frac{1}{k^{n-2+2\\frac{n-3}{n-1}}}\\Theta _k(\\ell ,t) \\;\\;\\mbox{ if }n\\geqslant 4,\\\\G\\frac{\\ell }{k\\ln k}\\left[g\\ell ^2-1\\right]+\\frac{1}{k\\ln k}\\frac{\\ln \\left(\\frac{2\\pi }{\\sqrt{\\ln k}}\\right)}{\\ln k}\\Theta _k(\\ell ,t)\\;\\;\\mbox{ if }n=3.\\end{array}\\right.",
"}$ Claim 2: $\\int _{{\\mathbb {R}}^{n}}(1-\\overline{\\zeta }_1)E\\overline{Z}_{n+1}={\\left\\lbrace \\begin{array}{ll}\\frac{1}{k^{n-1}}\\Theta _k(\\ell ,t) \\;\\;\\mbox{ if }n\\geqslant 4,\\\\\\frac{1}{(k\\ln k)^2}\\Theta _k(\\ell ,t)\\;\\;\\mbox{ if }n=3.\\end{array}\\right.",
"}$ Claim 3: $\\int _{{\\mathbb {R}}^n}{N}(\\overline{\\phi }_1,\\phi )\\overline{Z}_{n+1}={\\left\\lbrace \\begin{array}{ll}\\frac{1}{k^{n+\\frac{n}{q}-3}}\\Theta _k(\\ell ,t) \\;\\;\\mbox{ if }n\\geqslant 4,\\\\\\frac{1}{(k\\ln k)^2}\\Theta _k(\\ell ,t)\\;\\;\\mbox{ if }n=3.\\end{array}\\right.",
"}$ Notice that these claims together give the second equation in (REF ) and (REF ).",
"Proof of Claim 1.",
"We decompose $\\int _{\\mathbb {R}^n}E\\overline{Z}_{n+1}=\\int _{B(\\overline{\\xi }_1,\\frac{\\delta }{k})}E\\overline{Z}_{n+1}+\\int _{Ext}E\\overline{Z}_{n+1}+\\sum _{j\\ne 1}\\int _{B(\\overline{\\xi }_j,\\frac{\\delta }{k})}E\\overline{Z}_{n+1}+\\sum _{j= 1}^k\\int _{B(\\underline{\\xi }_j,\\frac{\\delta }{k})}E\\overline{Z}_{n+1},$ where $\\delta $ is a positive constant independent of $k$ and $Ext:=\\lbrace \\cap _{j=1}^k\\lbrace |y-\\overline{\\xi }_j|>\\frac{\\delta }{k}\\rbrace \\rbrace \\cap \\lbrace \\cap _{j=1}^k\\lbrace |y-\\underline{\\xi }_j|>\\frac{\\delta }{k}\\rbrace \\rbrace .$ Denoting $V(y):=\\lambda ^{\\frac{n-2}{2}}\\left[\\sum _{j\\ne 1}\\overline{U}_j(\\lambda y+\\overline{\\xi }_1)-\\sum _{j=1}^k\\underline{U}_j(\\lambda y+\\overline{\\xi }_1)-U(\\lambda y+\\overline{\\xi }_1)\\right]$ we have, for some $s\\in (0,1)$ , $\\begin{split}&\\gamma ^{-1}\\int _{B(\\overline{\\xi }_1,\\frac{\\delta }{k})}E\\overline{Z}_{n+1}=\\lambda ^\\frac{n+2}{2}\\int _{B(0,\\frac{\\delta }{\\lambda k})}E(\\lambda y+\\overline{\\xi }_1)Z_{n+1}(y)\\\\&\\qquad =p\\sum _{j\\ne 1}\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}\\overline{U}_j(\\lambda y+\\overline{\\xi }_1)Z_{n+1}+p\\sum _{j= 1}^k\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}\\underline{U}_j(\\lambda y+\\overline{\\xi }_1)Z_{n+1}\\\\&\\qquad \\;\\;-p\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}U(\\lambda y+\\overline{\\xi }_1)Z_{n+1}+p\\int _{B(0,\\frac{\\delta }{\\lambda k})}\\left[(U+sV)^{p-1}-U^{p-1}\\right]VZ_{n+1}\\\\&\\qquad \\;\\;+\\sum _{j\\ne 1}\\lambda ^{\\frac{n+2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}\\overline{U}_j^p(\\lambda y+\\overline{\\xi }_1)Z_{n+1}+\\sum _{j= 1}^k\\lambda ^{\\frac{n+2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}\\underline{U}_j^p(\\lambda y+\\overline{\\xi }_1)Z_{n+1}\\\\&\\qquad \\;\\;-\\lambda ^{\\frac{n+2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^p(\\lambda y+\\overline{\\xi }_1)Z_{n+1}.\\end{split} $ Thus, defining $I_1:=\\int _{{\\mathbb {R}}^n}U^{p-1}Z_{n+1},$ from (REF ) follows that $\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}\\overline{U}_j(\\lambda y+\\overline{\\xi }_1)Z_{n+1}=\\frac{2^{\\frac{n-2}{2}}\\lambda ^{n-2}I_1}{|\\overline{\\xi }_1-\\overline{\\xi }_j|^{n-2}}\\left(1+\\frac{\\lambda ^2}{|\\overline{\\xi }_1-\\overline{\\xi }_j|^2}\\Theta _k(\\ell ,t)\\right),$ $\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}\\underline{U}_j(\\lambda y+\\overline{\\xi }_1)Z_{n+1}=\\frac{2^{\\frac{n-2}{2}}\\lambda ^{n-2}I_1}{|\\overline{\\xi }_1-\\underline{\\xi }_j|^{n-2}}\\left(1+\\frac{\\lambda ^2}{|\\overline{\\xi }_1-\\underline{\\xi }_j|^2}\\Theta _k(\\ell ,t)\\right),$ and $\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}U(\\lambda y+\\overline{\\xi }_1)Z_{n+1}=\\lambda ^{\\frac{n-2}{2}}U(\\overline{\\xi }_1)I_1\\left(1+\\frac{\\lambda ^2}{1+|\\overline{\\xi }_1|^2}\\Theta _k(\\ell ,t)\\right),$ which are the main order terms in (REF ).",
"Indeed, $\\begin{aligned}\\bigg |\\sum _{j\\ne 1}\\lambda ^{\\frac{n+2}{2}}&\\int _{B(0,\\frac{\\delta }{\\lambda k})}\\overline{U}_j^p(\\lambda y+\\overline{\\xi }_1)Z_{n+1}\\bigg |\\leqslant C\\sum _{j\\ne 1}\\frac{\\lambda ^{n+2}}{|{\\overline{\\xi }}_j-{\\overline{\\xi }}_1|^{n+2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}\\frac{1}{(1+|y|)^{n-2}}\\\\&\\leqslant C{\\left\\lbrace \\begin{array}{ll}(\\lambda k)^{-2}\\sum _{j\\ne 1}\\frac{\\lambda ^{n+2}}{|{\\overline{\\xi }}_j-{\\overline{\\xi }}_1|^{n+2}}\\;\\;\\mbox{ if }n\\geqslant 4,\\\\|\\ln (\\lambda k)|\\sum _{j\\ne 1}\\frac{\\lambda ^{n+2}}{|{\\overline{\\xi }}_j-{\\overline{\\xi }}_1|^{n+2}}\\;\\;\\mbox{ if }n=3,\\end{array}\\right.",
"}\\end{aligned}$ $\\begin{aligned}\\bigg |\\sum _{j=1}^k\\lambda ^{\\frac{n+2}{2}}&\\int _{B(0,\\frac{\\delta }{\\lambda k})}\\underline{U}_j^p(\\lambda y+\\underline{\\xi }_1)Z_{n+1}\\bigg |\\leqslant C\\sum _{j\\ne 1}\\frac{\\lambda ^{n+2}}{|{\\underline{\\xi }}_j-{\\overline{\\xi }}_1|^{n+2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}\\frac{1}{(1+|y|)^{n-2}}\\\\&\\leqslant C{\\left\\lbrace \\begin{array}{ll}(\\lambda k)^{-2}\\sum _{j\\ne 1}\\frac{\\lambda ^{n+2}}{|{\\underline{\\xi }}_j-{\\overline{\\xi }}_1|^{n+2}}\\;\\;\\mbox{ if }n\\geqslant 4,\\\\|\\ln (\\lambda k)|\\sum _{j\\ne 1}\\frac{\\lambda ^{n+2}}{|{\\underline{\\xi }}_j-{\\overline{\\xi }}_1|^{n+2}}\\;\\;\\mbox{ if }n=3,\\end{array}\\right.",
"}\\end{aligned}$ and $\\begin{aligned}\\bigg |\\lambda ^{\\frac{n+2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}&U^p(\\lambda y+\\overline{\\xi }_1)Z_{n+1}\\,dy\\bigg |\\leqslant C\\lambda ^{\\frac{n+2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}\\frac{1}{(1+|y|)^{n-2}}\\\\&\\leqslant C{\\left\\lbrace \\begin{array}{ll}\\lambda ^{\\frac{n-2}{2}}k^{-2}\\;\\;\\mbox{ if }n\\geqslant 4,\\\\\\lambda ^{\\frac{n+2}{2}}|\\ln (\\lambda k)|\\;\\;\\mbox{ if }n=3.\\end{array}\\right.",
"}\\end{aligned}$ Finally these three estimates, together with the mean value theorem, also imply $\\begin{aligned}\\bigg |p\\int _{B(0,\\frac{\\delta }{\\lambda k})}&\\left[(U+sV)^{p-1}-U^{p-1}\\right]VZ_{n+1}\\bigg |\\\\&\\leqslant C{\\left\\lbrace \\begin{array}{ll}\\lambda ^{\\frac{n-2}{2}}k^{-2}+(\\lambda k)^{-2}(\\lambda k)^{n+2}\\;\\;\\mbox{ if }n\\geqslant 4,\\\\\\lambda ^{\\frac{n+2}{2}}|\\ln (\\lambda k)|+(\\lambda k)^{n+2}|\\ln (\\lambda k)|\\;\\;\\mbox{ if }n=3.\\end{array}\\right.",
"}\\end{aligned}$ Proceeding like in [8] we obtain the estimates of the other terms in (REF ), that is, $\\begin{split}\\bigg |\\int _{Ext}E\\overline{Z}_{n+1}\\bigg |\\leqslant C{\\left\\lbrace \\begin{array}{ll}\\frac{1}{k^{n-1}}\\;\\;\\mbox{ if }n\\geqslant 4,\\\\\\frac{1}{(k\\ln k)^2}\\;\\;\\mbox{ if }n=3,\\end{array}\\right.",
"}\\end{split}$ $\\bigg |\\sum _{j\\ne 1}\\int _{B(\\overline{\\xi }_j,\\frac{\\delta }{k})}E\\overline{Z}_{n+1}\\bigg |\\leqslant C{\\left\\lbrace \\begin{array}{ll}\\frac{1}{k^{n-1}}\\;\\;\\mbox{ if }n\\geqslant 4,\\\\\\frac{1}{(k\\ln k)^2}\\;\\;\\mbox{ if }n=3,\\end{array}\\right.",
"}\\bigg |\\sum _{j=1}^k\\int _{B(\\underline{\\xi }_j,\\frac{\\delta }{k})}E\\overline{Z}_{n+1}\\bigg |\\leqslant C{\\left\\lbrace \\begin{array}{ll}\\frac{1}{k^{n-1}}\\;\\;\\mbox{ if }n\\geqslant 4,\\\\\\frac{1}{(k\\ln k)^2}\\;\\;\\mbox{ if }n=3.\\end{array}\\right.",
"}$ Claim 1 follows using estimates (REF ) and (REF ).",
"Proof of Claim 2.",
"Noticing that $\\bigg |\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_1-1)E\\overline{Z}_{n+1}\\bigg |\\leqslant C\\bigg |\\int _{\\lbrace |y-\\overline{\\xi }_1|\\geqslant \\frac{\\delta }{k}\\rbrace }E\\overline{Z}_{n+1}\\bigg |,$ the result follows using (REF ) and (REF ).",
"Proof of Claim 3.",
"Decomposing the non linear term as in the proof of Proposition REF and using Proposition REF it can be seen that $\\bigg |\\int _{{\\mathbb {R}}^n}{N}(\\overline{\\phi }_1,\\phi )\\overline{Z}_{n+1}\\bigg |\\leqslant Ck^{3-n-\\frac{n}{q}}\\int _{{\\mathbb {R}}^n}U^{p-1}|Z_{n+1}|,$ and the claim holds for $n\\geqslant 4$ .",
"If $n=3$ it follows from estimates (REF ), (REF ) and the fact that $\\bigg |\\int _{{\\mathbb {R}}^n}{N}(\\overline{\\phi }_1,\\phi )\\overline{Z}_{n+1}\\bigg |\\leqslant C\\Vert \\lambda ^{\\frac{n+2}{2}}{N}(\\overline{\\phi }_1,\\phi )(\\lambda y+\\overline{\\xi }_1)\\Vert _{**}\\left(\\int _{{\\mathbb {R}}^n}\\frac{dy}{(1+|y|)^{2n}}\\right)^{\\frac{q-1}{q}}.$ Next we proceed to compute $\\overline{c}_3$ .",
"We write $\\overline{c}_{3}(\\ell , t)=\\int _{{\\mathbb {R}}^n}E\\overline{Z}_{3}-\\int _{{\\mathbb {R}}^{n}}(1-\\overline{\\zeta }_1)E\\overline{Z}_{3}+\\gamma \\int _{{\\mathbb {R}}^n}{N}(\\overline{\\phi }_1,\\phi )\\overline{Z}_{3}=0,$ and we affirm that, for $\\ell $ and $t$ as in (REF ), Claim 4: $\\int _{{\\mathbb {R}}^n}E\\overline{Z}_3={\\left\\lbrace \\begin{array}{ll}D_n\\frac{t\\ell ^{\\frac{n}{n-2}}}{k^{n+1-\\frac{2}{n-1}}}\\left[d_n\\frac{\\ell }{t^{n-1}}-1\\right]+\\frac{1}{k^{n+1}}\\Theta _k(\\ell ,t)\\;\\;\\mbox{ if }n\\geqslant 4,\\\\F\\frac{\\ell ^3 t}{\\sqrt{\\ln k}(k\\ln k)^3}\\left[f\\frac{\\ell }{t^2}-1\\right]+\\frac{1}{k^3(\\ln k)^4}\\Theta _k(\\ell ,t)\\;\\;\\mbox{ if }n=3.\\end{array}\\right.",
"}$ Claim 5: $\\int _{{\\mathbb {R}}^{n}}(1-\\overline{\\zeta }_1)E\\overline{Z}_{3}={\\left\\lbrace \\begin{array}{ll}\\frac{1}{k^{n+1}}\\Theta _k(\\ell ,t)\\;\\;\\mbox{ if }n\\geqslant 4,\\\\\\frac{1}{k^3(\\ln k)^4}\\Theta _k(\\ell ,t)\\;\\;\\mbox{ if }n=3.\\end{array}\\right.",
"}$ Claim 6: $\\int _{{\\mathbb {R}}^n}{N}(\\overline{\\phi }_1,\\phi )\\overline{Z}_{3}={\\left\\lbrace \\begin{array}{ll}\\frac{1}{k^{\\alpha }}\\Theta _k(\\ell ,t)\\;\\;\\mbox{ if }n\\geqslant 4,\\\\\\frac{1}{k^3(\\ln k)^4}\\Theta _k(\\ell ,t)\\;\\;\\mbox{ if }n=3,\\end{array}\\right.",
"}$ where $\\alpha >n+1-\\frac{2}{n-1}$ .",
"These claims together imply the validity of the first equation in (REF ) and (REF ).",
"Proof of Claim 4.",
"We decompose again as $\\int _{\\mathbb {R}^n}E\\overline{Z}_{3}=\\int _{B(\\overline{\\xi }_1,\\frac{\\delta }{k})}E\\overline{Z}_{3}+\\int _{Ext}E\\overline{Z}_{3}+\\sum _{j\\ne 1}\\int _{B(\\overline{\\xi }_j,\\frac{\\delta }{k})}E\\overline{Z}_{3}+\\sum _{j= 1}^k\\int _{B(\\underline{\\xi }_j,\\frac{\\delta }{k})}E\\overline{Z}_{3},$ Proceeding as in (REF ) and (REF ) we get $\\bigg |\\int _{Ext}E\\overline{Z}_{3}\\bigg |\\leqslant C\\lambda ^{n-1}k^{n-1}\\lambda ^{-\\frac{n-2}{2}}k^{-n\\frac{q-1}{q}}\\Vert (1+|y|)^{n+2-\\frac{2n}{q}}E\\Vert _{L^q(Ext)},$ $\\bigg |\\sum _{j\\ne 1}\\int _{B(\\overline{\\xi }_j,\\frac{\\delta }{k})}E\\overline{Z}_{3}\\bigg |\\leqslant C\\sum _{j\\ne 1}\\frac{\\lambda ^{n-1}(\\lambda k)^{2-\\frac{n}{q}}}{|\\overline{\\xi }_j-\\overline{\\xi }_1|^{n-1}}\\Vert \\, (1+ |y|)^{{n+2} - \\frac{2n}{q}}\\, \\lambda ^{n+2 \\over 2} \\gamma ^{-1} E (\\overline{\\xi }_j + \\lambda y ) \\Vert _{L^q( |y| < {\\delta \\over \\lambda k})},$ $\\bigg |\\sum _{j= 1}^k\\int _{B(\\underline{\\xi }_j,\\frac{\\delta }{k})}E\\overline{Z}_{3}\\bigg |\\leqslant C\\sum _{j=1}^k\\frac{\\lambda ^{n-1}(\\lambda k)^{2-\\frac{n}{q}}}{|\\underline{\\xi }_j-\\overline{\\xi }_1|^{n-1}}\\Vert \\, (1+ |y|)^{{n+2} - \\frac{2n}{q}}\\, \\lambda ^{n+2 \\over 2} \\gamma ^{-1} E (\\underline{\\xi }_j + \\lambda y ) \\Vert _{L^q( |y| < {\\delta \\over \\lambda k})}.$ For the first integral in (REF ) we separate as in (REF ).",
"Noticing that $\\overline{U}_j(\\lambda y +\\overline{\\xi }_1)$ is even with respect to the third coordinate, it follows that $\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}\\overline{U}_j(\\lambda y+\\overline{\\xi }_1) Z_{3}=0.$ Furthermore, using (REF ), $\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}\\underline{U}_j(\\lambda y+\\overline{\\xi }_1) Z_{3}=c_n\\frac{\\lambda ^{n-2}\\tau \\lambda I_2}{|\\overline{\\xi }_1-\\underline{\\xi }_j|^{n}}\\left(1+\\lambda ^2\\Theta _k(\\ell ,t)\\right),$ and $\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}U(\\lambda y+\\overline{\\xi }_1) Z_3=\\tilde{c}_n\\lambda ^{\\frac{n-2}{2}}\\tau \\lambda I_2\\left(1+\\frac{\\lambda ^2}{1+|\\overline{\\xi }_1|^2}\\Theta _k(\\ell ,t)\\right),$ where $I_2:=\\int _{{\\mathbb {R}}^n}U^{p-1}y_3Z_3.$ One also can compute the lower order terms $\\bigg |\\sum _{j=1}^k\\lambda ^{\\frac{n+2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}\\underline{U}_j^p(\\lambda y+\\overline{\\xi }_1) Z_{3}\\bigg |\\leqslant C (\\lambda k)^{-1}(\\lambda k)^{n+2}$ and $\\bigg |\\lambda ^{\\frac{n+2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^p(\\lambda y+\\overline{\\xi }_1) Z_{3}\\,dy\\bigg |\\leqslant C\\lambda ^{\\frac{n}{2}}k^{-1}.$ Thus, decomposing as in (REF ) Claim 4 is obtained from estimates (REF )-(REF ) together with (REF ), (REF ) and (REF ).",
"Proof of Claim 5.",
"It follows straightforward from (REF ), (REF ) and (REF ).",
"Proof of Claim 6.",
"Assume first $n\\geqslant 5$ and let us decompose ${N}(\\overline{\\phi }_1,\\phi )=f_1+f_2+f_3+f_4$ as in the proof of Proposition REF .",
"Changing variables and using (REF ) it can be seen that $\\bigg |\\int _{{\\mathbb {R}}^n}f_1(y)\\overline{Z}_3(y)\\,dy \\bigg |\\leqslant C\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U(y)^{p-2}|\\overline{\\overline{\\phi }}_1(y)||Z_3|.$ Notice that in this region $\\lambda ^{1/2}\\leqslant \\frac{c}{|y|}$ and thus, by Proposition REF we can write $|\\overline{\\overline{\\phi }}_1(y)|\\leqslant C\\frac{\\lambda ^{\\beta }\\lambda ^{\\frac{n-2}{2}-\\beta }}{(1+|y|)^2}\\leqslant C\\frac{\\lambda ^{\\beta }}{(1+|y|)^{n-2\\beta }},$ where $\\beta :=\\frac{3}{2}-\\frac{1}{n-1}+\\varepsilon $ , $\\varepsilon >0$ small.",
"Replacing above we get $\\bigg |\\int _{{\\mathbb {R}}^n}f_1(y)\\overline{Z}_3(y)\\,dy\\bigg | \\leqslant \\frac{C}{k^{2\\beta +n-2}}\\int _{{\\mathbb {R}}^n}\\frac{1}{(1+|y|)^{-n+6}}\\frac{|y_3|}{(1+|y|)^n}\\frac{1}{(1+|y|)^{n-2\\beta }}\\,dy \\leqslant \\frac{C}{k^{2\\beta +n-2}}$ for $\\varepsilon $ small enough.",
"Notice also that $2\\beta +n-2>n+1-\\frac{2}{n-1}$ , that is the order of the main term.",
"Likewise, using the estimate on $\\overline{\\overline{\\phi }}_1$ , $\\bigg |\\int _{{\\mathbb {R}}^n}f_2(y)\\overline{Z}_3(y)\\,dy\\bigg | \\leqslant C\\lambda ^{\\frac{n-2}{2}}\\int _{\\lbrace |y|>c\\lambda ^{-1/2}\\rbrace }\\frac{U^{p-1}|Z_3|}{(1+|y|)^2}\\,dy\\leqslant C\\lambda ^{\\frac{n-2}{2}+2}\\int _{{\\mathbb {R}}^n}\\frac{|y_3|}{(1+|y|)^{n+2}}\\,dy\\leqslant \\frac{C}{k^{n+2}}.$ To estimate the projection of $f_3$ we first point out that $f_3 \\approx \\overline{\\zeta }_1 p|\\overline{U}_1|^{p-1}\\psi $ .",
"Due to the cancellation in (REF ) the main order term in Claim 4 is rather small, and this makes necessary sharp estimates on the size of the projection of the nonlinear term.",
"Indeed, to prove this claim we will have to make use of the decomposition of $\\psi $ in a large but symmetric part (that happens to be orthogonal to $\\overline{Z}_3$ ) and a non symmetric but small part specified in Proposition REF .",
"Thus, $\\begin{aligned}\\int _{{\\mathbb {R}}^n}f_3(y)\\overline{Z}_3(y)\\,dy&\\approx \\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U(y)^{p-1}\\psi (\\overline{\\xi }_1+\\lambda y)Z_3\\\\&=\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}\\psi ^s(y)Z_3\\,dy+\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}\\psi ^*(y)Z_3\\,dy.\\end{aligned}$ The first integral in the right hand side vanishes due to the oddness of $Z_3$ and the evennes of $\\psi ^s$ in the third coordinate.",
"The second can be estimated as $\\begin{split}\\bigg |\\lambda ^{\\frac{n-2}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U^{p-1}\\psi ^*(y)Z_3\\,dy\\bigg |&\\leqslant C\\lambda ^{\\frac{n-2}{2}+1}(\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*+\\Vert \\psi \\Vert _*+\\tau o_k(1))\\int _{{\\mathbb {R}}^n}U^{p-1}|y|(1+|y|)|Z_3|\\,dy\\\\&\\leqslant \\frac{C}{k^{n-1+\\frac{n}{q}}},\\end{split}$ and hence, choosing $\\frac{n}{2}<q<\\frac{n}{2-\\frac{2}{n-1}}$ we conclude that $\\bigg |\\int _{{\\mathbb {R}}^n}f_3(y)\\overline{Z}_3(y)\\,dy\\bigg |\\leqslant \\frac{C}{k^\\alpha }\\quad \\mbox{ with }\\alpha >n+1-\\frac{2}{n-1}.$ Analogously, noticing that $f_4\\approx |\\overline{U}_1|^{p-2}\\left(\\sum _{j=1}^k(\\overline{\\phi }_j+\\underline{\\phi }_j)+\\psi \\right)^2$ , estimate $\\bigg |\\int _{{\\mathbb {R}}^n}f_4(y)\\overline{Z}_3(y)\\,dy\\bigg |\\leqslant \\frac{C}{k^\\alpha }\\quad \\mbox{ with }\\alpha >n+1-\\frac{2}{n-1},$ follows by Proposition REF .",
"This completes the proof of Claim 6 for $n\\geqslant 5$ .",
"Consider now the cases $n=3,4$ .",
"Using the decomposition of $\\overline{\\overline{\\phi }}_1$ found in Proposition REF we notice that $\\int _{{\\mathbb {R}}^n}\\tilde{f}_1^s(y)Z_3(y)\\,dy =0,\\qquad \\int _{{\\mathbb {R}}^n}\\tilde{f}_2^s(y)Z_3(y)\\,dy =0,$ and we obtain, for $n=3$ , $\\begin{split}&\\bigg |\\int _{{\\mathbb {R}}^n}f_1(y)\\overline{Z}_3(y)\\,dy\\bigg |=\\bigg |\\int _{{\\mathbb {R}}^n}\\tilde{f}^*_1(y)Z_3(y)\\,dy\\bigg |\\\\&\\leqslant C\\left(\\lambda ^{\\frac{1}{2}}\\int _{B(0,\\frac{\\delta }{\\lambda k})}U(y)^{p-2}|\\overline{\\overline{\\phi _1^*}}(y)||Z_3(y)|\\,dy+\\frac{\\lambda ^{\\frac{1}{2}}}{k(\\ln k)^3}\\int _{B(0,\\frac{\\delta }{\\lambda k})}|y|U(y)^{p-2}|\\overline{\\overline{\\phi }}_1(y)||Z_3(y)|\\,dy\\right)\\\\&\\leqslant \\frac{C}{k^3(\\ln k)^5},\\end{split}$ and $\\begin{split}\\bigg |\\int _{{\\mathbb {R}}^n}f_2(y)\\overline{Z}_3(y)\\,dy\\bigg |&=\\bigg |\\int _{\\lbrace |y|>c\\lambda ^{-1/2}\\rbrace }f_2^*(y)Z_3(y)\\,dy\\bigg |=\\bigg |\\int _{\\lbrace |y|>c\\lambda ^{-1/2}\\rbrace }U(y)^{p-1}\\overline{\\overline{\\phi _1^*}}(y)Z_3(y)\\,dy\\bigg |\\\\&\\leqslant C\\frac{\\lambda ^\\frac{1}{2}\\lambda }{k(\\ln k)^3}\\int _{{\\mathbb {R}}^n}\\frac{1}{(1+|y|)^2}\\frac{|y_3|}{(1+|y|)^n}\\leqslant \\frac{C}{k^3(\\ln k)^5},\\end{split}$ where in the last inequalities we have applied Proposition REF and Proposition REF .",
"Likewise, for $n=4$ , $\\bigg |\\int _{{\\mathbb {R}}^n}f_1(y)\\overline{Z}_3(y)\\,dy\\bigg |\\leqslant \\frac{C}{k^5},\\qquad \\bigg |\\int _{{\\mathbb {R}}^n}f_2(y)\\overline{Z}_3(y)\\,dy\\bigg |\\leqslant \\frac{C}{k^5}.$ The terms involving $\\tilde{f}_3$ and $\\tilde{f}_4$ are estimated in a similar way and Claim 6 follows."
],
[
"Remark ",
"This section is devoted to the constructions described in Remark REF .",
"The first is the construction of the doubling of the equatorial $\\Gamma $ with an even number of circles, which is done in subsection REF .",
"The second is a combination of the doubling and the desingularization of the equatorial with an odd number of circles.",
"This is done in subsection REF ."
],
[
"Even number of circles.",
"Let $m $ be a fixed integer.",
"Let $\\tau _i \\in (0,1)$ , $i=1, \\ldots , m,$ and fix the points $\\overline{P}_i := (\\sqrt{1-\\tau _i^2 } , 0 , \\tau _i , 0 , \\ldots , 0) , \\quad \\underline{P}_i := (\\sqrt{1-\\tau _i^2 } , 0 , -\\tau _i , 0 , \\ldots , 0).$ Let $ \\lambda _i \\in (0,1)$ , $i=1, \\ldots , m,$ be positive numbers, and define $R_i$ as $\\lambda _i^2 + R_i^2 = 1.$ We use the notation $\\lambda = (\\lambda _1 , \\ldots , \\lambda _m) , \\quad \\tau = (\\tau _1 , \\ldots , \\tau _m).$ Let $k$ be an integer number and $\\textbf {u}_{2m} [\\lambda , \\tau ] (y) := U(y) -\\sum _{j=1}^k \\bigg [ \\sum _{i=1}^m \\underbrace{ \\lambda ^{-{n-2 \\over 2}} U\\left( {y-\\overline{\\xi }_{ij} \\over \\lambda } \\right)}_{\\overline{U}_{ij} (y)}+ \\sum _{i=1}^m \\underbrace{ \\lambda ^{-{n-2 \\over 2}} U\\left( {y- \\underline{\\xi }_{ij} \\over \\lambda } \\right)}_{ \\underline{U}_{ij} (y) } \\bigg ]$ for $y \\in {\\mathbb {R}}^n$ , where, for $i=1, \\ldots , m$ , and $j=1, \\ldots , k$ , $\\begin{aligned}\\overline{\\xi }_{ij} &:= R_i (\\sqrt{1-\\tau _i^2} \\cos \\theta _j , \\sqrt{1-\\tau _i^2} \\sin \\theta _j , \\tau _i , 0, \\ldots , 0 ), \\\\\\underline{\\xi }_{ij} &:= R_i (\\sqrt{1-\\tau _i^2} \\cos \\theta _j , \\sqrt{1-\\tau _i^2} \\sin \\theta _j , -\\tau _i , 0, \\ldots , 0 ),\\quad {\\mbox{with}} \\quad \\theta _j := 2\\pi {j-1 \\over k}.\\end{aligned}$ Observe that the function (REF ) satisfies the symmetries (REF ), (REF ) and (REF ).",
"We assume that the integer $k$ is large, and that the parameters $\\lambda $ and $\\tau $ are given by $\\begin{split} \\lambda _i &:= {\\ell _i^{2\\over n-2} \\over k^2} , \\quad \\tau _i := {t_i \\over k^{1-{2\\over n-1}} }, \\quad {\\mbox{if}} \\quad n\\geqslant 4, \\\\\\lambda _i &:= {\\ell _i^{2} \\over k^2 (\\ln k)^2 } , \\quad \\tau _i := {t_i \\over \\sqrt{\\ln k} }, \\quad {\\mbox{if}} \\quad n=3,\\end{split} \\quad \\quad {\\mbox{where}} \\quad \\eta < \\ell _i , \\, t_i < \\eta ^{-1}$ for some $\\eta $ small and fixed, independent of $k$ , for any $k$ large enough.",
"The doubling of the equatorial $\\Gamma $ with an even number of circles is the content of next Theorem 6.1 Let $n\\geqslant 3$ and let $k$ be a positive integer.",
"Then for any sufficiently large $k$ there is a finite energy solution to (REF ) of the form $u(y)=\\textbf {u}_{2m} [\\lambda , \\tau ] (y) +o_k(1)(1+|\\lambda |^{-{n-2 \\over 2}}),$ where the term $o_k(1)\\rightarrow 0$ uniformly on compact sets of ${\\mathbb {R}}^n$ as $k \\rightarrow \\infty $ .",
"The solution in Theorem REF has the form $u (y) = \\textbf {u}_{2m} [\\lambda , \\tau ] (y) + \\phi (y ), \\quad \\phi = \\psi + \\sum _{j=1}^k \\sum _{i=1}^m ( \\bar{\\phi }_{ij} + \\underline{\\phi }_{ij} )$ where $\\overline{\\phi }_{ij}$ , $\\underline{\\phi }_{ij}$ , $i=1, \\ldots , m$ , $j=1, \\ldots , k$ , and $\\psi $ solve the following system of coupled non linear equations $\\Delta \\overline{\\phi }_{ij}+p\\gamma |\\textbf {u}_{2m}|^{p-1}\\overline{\\zeta }_{ij}\\overline{\\phi }_{ij}+\\overline{\\zeta }_{ij}\\left[p\\gamma |\\textbf {u}_{2m}|^{p-1}\\psi +E_{2m}+\\gamma N(\\phi )\\right]=0, \\; \\;$ $\\Delta \\underline{\\phi }_{ij}+p\\gamma |\\textbf {u}_{2m}|^{p-1}\\underline{\\zeta }_{ij}\\underline{\\phi }_{ij}+\\underline{\\zeta }_{ij}\\left[p\\gamma |\\textbf {u}_{2m}|^{p-1}\\psi +E_{2m}+\\gamma N(\\phi )\\right]=0, \\; \\;$ $\\begin{split}\\Delta \\psi &+p\\gamma U^{p-1}\\psi +\\left[p\\gamma (|\\textbf {u}_{2m}|^{p-1}-U^{p-1})(1-\\sum _{j=1}^k \\sum _{i=1}^m(\\overline{\\zeta }_{ij}+\\underline{\\zeta }_{ij}))\\right.\\\\&\\left.+p\\gamma U^{p-1}[\\sum _{j=1}^k \\sum _{i=1}^m(\\overline{\\zeta }_{ij}+\\underline{\\zeta }_{ij})\\right]\\psi + p\\gamma |\\textbf {u}_{2m}|^{p-1}\\sum _{j=1}^k \\sum _{i=1}^m (1-\\overline{\\zeta }_{ij})\\overline{\\phi }_{ij}\\\\&+p\\gamma |\\textbf {u}_{2m}|^{p-1}\\sum _{j=1}^k \\sum _{i=1}^m (1-\\underline{\\zeta }_{ij})\\underline{\\phi }_{ij}+\\left(1-\\sum _{j=1}^k \\sum _{i=1}^m(\\overline{\\zeta }_{ij}+\\underline{\\zeta }_{ij})\\right)(E_{2m}+\\gamma N(\\phi ))=0.\\end{split}$ The functions $\\overline{\\zeta }_{ij}$ are defined as $\\overline{\\zeta }_j$ in (REF ) with $\\bar{\\xi }_j$ replaced by $\\bar{\\xi }_{ij}$ , and $\\underline{\\zeta }_{ij}(y):=\\overline{\\zeta }_{ij}(y_1,y_2,-y_3,\\ldots ,y_n),\\quad E_{2m} (y) := \\Delta \\textbf {u}_{2m} + \\gamma |\\textbf {u}_{2m}|^{p-1} \\textbf {u}_{2m}, \\quad y \\in {\\mathbb {R}}^n,$ and $N(\\phi ):=|\\textbf {u}_{2m}+\\phi |^{p-1}(\\textbf {u}_{2m}+\\phi )-|\\textbf {u}_{2m}|^{p-1}\\textbf {u}_{2m}-p|\\textbf {u}_{2m}|^{p-1}\\phi $ .",
"One can prove that $\\Vert E_{2m} \\Vert _{**} \\leqslant C k^{1-\\frac{n}{q}}\\;\\;\\mbox{if }n\\geqslant 4,\\quad \\Vert E_{2m} \\Vert _{**} \\leqslant C(\\ln k)^{-1}\\;\\;\\mbox{if }n=3.$ Denoting $\\hat{y}:=(y_1,y_2)$ and $y^{\\prime }:=(y_3,\\ldots ,y_n)$ , we assume that the functions $\\overline{\\phi }_{ij}$ and $\\underline{\\phi }_{ij}$ satisfy $\\overline{\\phi }_{ij}(\\hat{y},y^{\\prime })=\\overline{\\phi }_{i1}(e^{2\\pi \\frac{ \\leavevmode {\\color {black}(j-1)}}{k}i} \\hat{y},y^{\\prime }),\\quad \\quad \\overline{\\phi }_{i1}(y)=|y|^{2-n}\\overline{\\phi }_{i1}(|y|^{-2}y),$ $\\overline{\\phi }_{i1}(y_1,\\ldots ,y_\\alpha ,\\ldots ,y_n)=\\overline{\\phi }_{i1}(y_1,\\ldots ,-y_\\alpha ,\\ldots ,y_n),\\;\\; \\alpha =2,4,\\ldots ,n,$ and $\\underline{\\phi }_{ij}(y)=\\overline{\\phi }_{ij}(y_1,y_2,-y_3,\\ldots ,y_n).$ Moreover $(\\overline{\\phi }_{ij}+\\underline{\\phi }_{ij})(y)=(\\overline{\\phi }_{ij}+\\underline{\\phi }_{ij})(y_1,y_2,-y_3,\\ldots ,y_n),$ as well as $(\\overline{\\zeta }_{ij}+\\underline{\\zeta }_{ij})$ and $(\\overline{\\zeta }_{ij}\\overline{\\phi }_{ij}+\\underline{\\zeta }_{ij}\\underline{\\phi }_{ij})$ .",
"For $\\rho >0$ small and fixed we assume in addition $\\sum _{i=1}^m \\Vert \\overline{\\overline{\\phi }}_{i1}\\Vert _*\\leqslant \\rho ,$ where $\\overline{\\overline{\\phi }}_{i1}(y):=\\lambda ^{\\frac{n-2}{2}}\\overline{\\phi }_{i1}(\\overline{\\xi }_{i1}+\\lambda y)$ and $\\Vert \\cdot \\Vert _*$ is defined in (REF ).",
"Arguing as in Proposition REF , one proves that there exists a unique solution $\\psi =\\Psi (\\overline{\\overline{\\phi }}_{11} , \\ldots , \\overline{\\overline{\\phi }}_{m1})$ of (REF ), satisfying (REF ), (REF ) and (REF ).",
"Besides $\\begin{split}\\Vert \\psi \\Vert _*&\\leqslant C\\left(\\sum _{i=1}^m \\Vert \\overline{\\overline{\\phi }}_{i1}\\Vert _*+k^{1-\\frac{n}{q}}\\right)\\;\\;\\mbox{if }n\\geqslant 4,\\quad \\Vert \\psi \\Vert _*\\leqslant C\\left( \\sum _{i=1}^m \\Vert \\overline{\\overline{\\phi }}_{i1}\\Vert _*+(\\ln k)^{-1}\\right)\\;\\;\\mbox{if }n=3.\\end{split}$ We replace the solution $\\psi =\\Psi (\\overline{\\overline{\\phi }}_{11} , \\ldots , \\overline{\\overline{\\phi }}_{m1})$ of (REF ) in (REF ) and (REF ).",
"Using the symmetries we described before, it is enough to solve (REF ) for $j=1$ .",
"We are thus left with a system of $m$ equations in $\\overline{\\phi }_1 = (\\overline{\\phi }_{11} , \\ldots , \\overline{\\phi }_{m1} )$ unknowns $\\Delta \\overline{\\phi }_{i1}+p\\gamma |\\textbf {u}_{2m}|^{p-1}\\overline{\\zeta }_{i1}\\overline{\\phi }_{i1}+\\overline{\\zeta }_{i1}\\left[p\\gamma |\\textbf {u}_{2m}|^{p-1}\\psi +E_{2m}+\\gamma N(\\phi )\\right]=0, \\quad i=1, \\ldots , m.$ Instead of solving it directly, we first solve the auxiliary problem $\\Delta \\overline{\\phi }_{i1}+p\\gamma |\\overline{U}_{i1}|^{p-1}\\overline{\\phi }_{i1}+\\overline{\\zeta }_{i1} E_{2m} +\\gamma {N}_i(\\overline{\\phi }_1,\\phi )=c_{i3}\\overline{U}_{i1}^{p-1}\\overline{Z}_{i3}+c_{i,n+1}\\overline{U}_{i1}^{p-1}\\overline{Z}_{i,n+1},$ where ${N}_i(\\overline{\\phi }_1,\\phi ):=p(|\\textbf {u}_{2m}|^{p-1}\\overline{\\zeta }_1-|\\overline{U}_{i1}|^{p-1})\\overline{\\phi }_{i1}+\\overline{\\zeta }_1\\left[p|\\textbf {u}_{2m}|^{p-1}\\Psi (\\overline{\\overline{\\phi }}_1)+N(\\phi )\\right],$ $\\overline{Z}_{i\\alpha }(y):=\\lambda ^{-\\frac{n-2}{2}}Z_\\alpha \\left(\\frac{y-\\overline{\\xi }_{i1}}{\\lambda }\\right),\\;\\;\\alpha =3, n+1$ and $c_{i3}:=\\frac{\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_{i1} E_{2m}+\\gamma {N}_i(\\overline{\\phi }_1,\\phi ))\\overline{Z}_{i3}}{\\int _{{\\mathbb {R}}^n}\\overline{U}_{i1}^{p-1}\\overline{Z}^2_{i3}},\\qquad c_{i, n+1}:=\\frac{\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_{i1} E_{2m} +\\gamma {N}_i (\\overline{\\phi }_1,\\phi ))\\overline{Z}_{i, n+1}}{\\int _{{\\mathbb {R}}^n}\\overline{U}_{i1}^{p-1}\\overline{Z}^2_{i,n+1}}.$ Arguing as in Proposition REF , one proves that there exists a unique solution $\\overline{\\phi }_{i1}=\\overline{\\phi }_{i1}(l,t)$ of (REF ), that satisfies $\\Vert \\overline{\\overline{\\phi }}_{i1}\\Vert _*\\leqslant C k^{-\\frac{n}{q}}\\mbox{ if }n\\geqslant 4,\\qquad \\Vert \\overline{\\overline{\\phi }}_{i1}\\Vert _*\\leqslant \\frac{C}{k\\ln k}\\mbox{ if }n=3,$ and $\\Vert \\overline{\\overline{{N}}}(\\overline{\\phi }_{i1},\\phi )\\Vert _{**}\\leqslant Ck^{-\\frac{2n}{q}}\\mbox{ if }n\\geqslant 4,\\qquad \\Vert \\overline{\\overline{{N}}}(\\overline{\\phi }_{i1},\\phi )\\Vert _{**}\\leqslant \\frac{C}{(k\\ln k)^2}\\mbox{ if }n=3,$ where $\\overline{\\overline{\\phi }}_{i1}(y):=\\lambda ^\\frac{n-2}{2}\\overline{\\phi }_{i1}(\\overline{\\xi }_{i1}+\\lambda y)$ and $\\overline{\\overline{{N}}}(\\overline{\\phi }_{i1},\\phi )(y):=\\lambda ^\\frac{n+2}{2}{N}(\\overline{\\phi }_{i1},\\phi )(\\overline{\\xi }_{i1}+\\lambda y)$ .",
"Furthermore, proceeding as in Proposition REF and Proposition REF , there exist decompositions $\\overline{\\overline{\\phi }}_{i1}=\\overline{\\overline{\\phi _{i1}^s}}+\\overline{\\overline{\\phi _{i1}^*}},\\qquad \\psi (\\overline{\\xi }_{i1}+\\lambda y)=\\psi ^s(y)+\\psi ^*(y),$ where $\\overline{\\overline{\\phi _{i1}^s}}$ and $\\psi ^s$ are even with respect to $y_3$ and $|\\overline{\\overline{\\phi _{i1}^*}}(y)|\\leqslant {\\left\\lbrace \\begin{array}{ll}C\\frac{\\lambda ^{\\frac{n-2}{2}}}{k}\\frac{1}{1+|y|}\\mbox{ if }n\\geqslant 4,\\\\C\\frac{\\lambda ^{1/2}}{k(\\ln k)^3}\\frac{1}{1+|y|^\\alpha },\\quad 0<\\alpha <1,\\quad \\mbox{ if }n\\geqslant 3,\\end{array}\\right.",
"}\\qquad |\\psi ^*(y)|\\leqslant C\\lambda k^{1-\\frac{n}{q}}|y|(1+|y|).$ In order to complete the proof of Theorem REF , we need to find positive parameters $\\ell _1 , \\ldots \\ell _m$ and $t_1 , \\ldots , t_m$ entering in the definition of $\\lambda $ and $\\tau $ in (REF ) so that for all $i=1, \\ldots , m$ $c_{i3} (\\ell , t) = c_{i, n+1} (\\ell , t) = 0, \\quad \\ell = (\\ell _1 , \\ldots \\ell _m), \\quad t= ( t_1 , \\ldots , t_m ).$ In dimension $n\\geqslant 4$ , this system decouples and becomes $\\begin{aligned}&a_n \\ell _i^2-1+\\frac{1}{k^{2 {n-3 \\over n-1}}}\\Theta _{i, n+1, k}(\\ell _1 , \\ldots \\ell _m ,t_1 , \\ldots , t_m ) =0, \\quad i=1, \\ldots , m, \\\\&b_n{ \\ell _i \\over t_i^{n-1} }-1+\\frac{1}{k^{n-3 \\over n-1} }\\Theta _{i, 3, k}(\\ell _1 , \\ldots \\ell _m ,t_1 , \\ldots , t_m)=0, \\quad i=1, \\ldots , m,\\end{aligned}$ where $a_n$ , $b_n$ are positive constants that are independent of $k$ , and $\\Theta _{i, n+1, k}$ , $\\Theta _{i, 3, k}$ are smooth functions of their argument, which are uniformly bounded, together with their first derivatives, as $k \\rightarrow \\infty $ .",
"In dimension $n=3$ , the system becomes $\\begin{aligned}&a_3 \\ell _i^2-1+\\frac{\\ln \\ln k }{\\ln k }\\Theta _{i, 4, k}(\\ell _1 , \\ldots \\ell _m ,t_1 , \\ldots , t_m ) =0, \\quad i=1, \\ldots , m, \\\\&b_3{ \\ell _i \\over t_i^{2} }-1+\\frac{\\ln \\ln k }{\\ln k } \\Theta _{i, 3, k}(\\ell _1 , \\ldots \\ell _m ,t_1 , \\ldots , t_m)=0, \\quad i=1, \\ldots , m,\\end{aligned}$ where $a_3$ , $b_3$ are positive constants, and $\\Theta _{i, 4, k}$ , $\\Theta _{i, 3, k}$ are smooth functions of their argument, which are uniformly bounded, together with their first derivatives, as $k \\rightarrow \\infty $ .",
"A fixed point argument gives the existence of $\\ell $ and $t$ solutions to (REF ).",
"This concludes the proof of Theorem REF ."
],
[
"Odd number of circles.",
"Let $ \\mu \\in (0,1)$ and define $R $ so that $\\mu ^2 + R^2 = 1.$ Let $k$ , $m$ be integer numbers and $\\textbf {u}_{2m+1} [\\mu , \\lambda , \\tau ] (y) := U(y) -\\sum _{j=1}^k \\left[ U_j (y) + \\sum _{i=1}^m \\left( \\overline{U}_{ij} (y)+ \\underline{U}_{ij} (y) \\right) \\right]$ where $\\lambda $ , $\\tau $ , $\\overline{U}_{ij}$ and $\\underline{U}_{ij}$ are defined at the beginning of subsection REF and (REF ), while $U_j (y) := \\mu ^{-{n-2 \\over 2}} U({y-\\xi _j \\over \\mu } ), \\quad \\xi _j:= R ( \\cos \\theta _j , \\sin \\theta _j , 0 , 0, \\ldots , 0 ) \\quad \\theta _j := 2\\pi {j-1 \\over k}.$ The function (REF ) satisfies the symmetries (REF ), (REF ) and (REF ).",
"We assume that the integer $k$ is large, and that the parameters $\\mu $ , $\\lambda $ and $\\tau $ are given by $\\begin{split} \\mu &:= {\\ell ^{2\\over n-2} \\over k^2} \\quad {\\mbox{if }} n\\geqslant 4, \\quad \\mu := {\\ell ^{2} \\over k^2 (\\ln k)^2 } \\quad {\\mbox{if }} n=3,\\\\\\lambda _i &:= {\\ell _i^{2\\over n-2} \\over k^2} , \\quad \\tau _i := {t_i \\over k^{1-{2\\over n-1}} }, \\quad {\\mbox{if }} n\\geqslant 4, \\quad \\lambda _i := {\\ell _i^{2} \\over k^2 (\\ln k)^2 } , \\quad \\tau _i := {t_i \\over \\sqrt{\\ln k} }, \\quad {\\mbox{if }} n=3,\\end{split}$ where $\\eta < \\ell , \\ell _i \\, , \\, t_i < \\eta ^{-1}$ for some $\\eta $ small and fixed, independent of $k$ , for any $k$ large enough.",
"We have Theorem 6.2 Let $n\\geqslant 3$ and let $k$ be a positive integer.",
"Then for any sufficiently large $k$ there is a finite energy solution to (REF ) of the form $u(y)=\\textbf {u}_{2m+1} [\\mu , \\lambda , \\tau ] (y) +o_k(1)(1+(\\mu + |\\lambda |)^{-{n-2 \\over 2}}),$ where the term $o_k(1)\\rightarrow 0$ uniformly on compact sets of ${\\mathbb {R}}^n$ as $k \\rightarrow \\infty $ .",
"The solution in Theorem REF has the form $u (y) = \\textbf {u}_{2m+1} [\\mu , \\lambda , \\tau ] (y) + \\phi (y ), \\quad \\phi = \\psi + \\sum _{j=1}^k [ \\phi _j + \\sum _{i=1}^m ( \\bar{\\phi }_{ij} + \\underline{\\phi }_{ij} )],$ with $\\phi _j (\\hat{y},y^{\\prime })=\\phi _{1}(e^{2\\pi \\frac{ \\leavevmode {\\color {black}(j-1)}}{k}i} \\hat{y},y^{\\prime }), \\quad \\overline{\\phi }_{ij}(\\hat{y},y^{\\prime })=\\overline{\\phi }_{i1}(e^{2\\pi \\frac{ \\leavevmode {\\color {black}(j-1)}}{k}i} \\hat{y},y^{\\prime }),$ where $\\hat{y}:=(y_1,y_2)$ and $y^{\\prime }:=(y_3,\\ldots ,y_n)$ .",
"The functions $ \\phi _j$ , $\\overline{\\phi }_{ij}$ and $\\underline{\\phi }_{ij}$ also satisfy $\\phi _{1}(y)=|y|^{2-n}\\phi _{1}(|y|^{-2}y), \\quad \\overline{\\phi }_{i1}(y)=|y|^{2-n}\\overline{\\phi }_{i1}(|y|^{-2}y),$ $\\overline{\\phi }_{i1}(y_1,\\ldots ,y_\\alpha ,\\ldots ,y_n)=\\overline{\\phi }_{i1}(y_1,\\ldots ,-y_\\alpha ,\\ldots ,y_n),\\;\\; \\alpha =2,4,\\ldots ,n,$ and $\\underline{\\phi }_{ij}(y)=\\overline{\\phi }_{ij}(y_1,y_2,-y_3,\\ldots ,y_n).$ Moreover $\\phi _j (y) = \\phi _j (y_1,y_2,-y_3,\\ldots ,y_n), \\quad (\\overline{\\phi }_{ij}+\\underline{\\phi }_{ij})(y)=(\\overline{\\phi }_{ij}+\\underline{\\phi }_{ij})(y_1,y_2,-y_3,\\ldots ,y_n).$ Thanks to (REF ), it is enough to describe $ \\phi _1$ , $\\overline{\\phi }_{i1}$ , $i=1, \\ldots , m$ .",
"The functions $ \\phi _1$ , $\\overline{\\phi }_{i1}$ and $\\psi $ solve the following system of coupled non linear equations $\\begin{aligned}\\Delta \\psi &+p\\gamma U^{p-1}\\psi +\\left[p\\gamma (|\\textbf {u}_{2m+1}|^{p-1}-U^{p-1})\\left(1-\\sum _{j=1}^k \\left[\\zeta _j +\\sum _{i=1}^m(\\overline{\\zeta }_{ij}+\\underline{\\zeta }_{ij})\\right]\\right)\\right.\\\\&\\left.+p\\gamma U^{p-1}\\left[\\sum _{j=1}^k [ \\zeta _j + \\sum _{i=1}^m(\\overline{\\zeta }_{ij}+\\underline{\\zeta }_{ij})\\right]\\right]\\psi + p\\gamma |\\textbf {u}_{2m+1}|^{p-1}\\sum _{j=1}^k \\sum _{i=1}^m (1-\\overline{\\zeta }_{ij})\\overline{\\phi }_{ij}\\\\&+p\\gamma |\\textbf {u}_{2m+1}|^{p-1}\\sum _{j=1}^k \\sum _{i=1}^m (1-\\underline{\\zeta }_{ij})\\underline{\\phi }_{ij}+p\\gamma |\\textbf {u}_{2m+1}|^{p-1}\\sum _{j=1}^k (1-\\zeta _{j})\\phi _{j} \\\\&+\\left(1-\\sum _{j=1}^k \\left[\\zeta _j +\\sum _{i=1}^m(\\overline{\\zeta }_{ij}+\\underline{\\zeta }_{ij})\\right]\\right)(E_{2m +1}+\\gamma N(\\phi ))=0,\\end{aligned}$ $\\Delta \\phi _{1}+p\\gamma |{U}_{1}|^{p-1}\\phi _{1}+\\zeta _{1} E_{2m +1} +\\gamma {N}(\\phi )=c_{n+1}U_{1}^{p-1}Z_{n+1},$ and, for $i=1, \\ldots , m$ , $\\Delta \\overline{\\phi }_{i1}+p\\gamma |\\overline{U}_{i1}|^{p-1}\\overline{\\phi }_{i1}+\\overline{\\zeta }_{i1} E_{2m+1} +\\gamma {N}_i(\\phi )=c_{i3}\\overline{U}_{i1}^{p-1}\\overline{Z}_{i3}+c_{i,n+1}\\overline{U}_{i1}^{p-1}\\overline{Z}_{i,n+1}.$ Here $E_{2m+1} (y) := \\Delta \\textbf {u}_{2m+1} + \\gamma |\\textbf {u}_{2m+1}|^{p-1} \\textbf {u}_{2m+1}, \\quad y \\in {\\mathbb {R}}^n,$ and $N(\\phi ):=|\\textbf {u}_{2m+1}+\\phi |^{p-1}(\\textbf {u}_{2m+1}+\\phi )-|\\textbf {u}_{2m+1}|^{p-1}\\textbf {u}_{2m+1}-p|\\textbf {u}_{2m+1}|^{p-1}\\phi $ .",
"For any $j$ , $\\zeta _j$ , $\\overline{\\zeta }_{ij}$ are defined as $\\zeta _j$ in (REF ) with $\\bar{\\xi }_j$ replaced respectively by $\\xi _j$ and $\\bar{\\xi }_{ij}$ , and $\\underline{\\zeta }_{ij}(y):=\\overline{\\zeta }_{ij}(y_1,y_2,-y_3,\\ldots ,y_n)$ .",
"Moreover, ${N}(\\phi ):=p(|\\textbf {u}_{2m+1}|^{p-1}\\zeta _1-|\\overline{U}_{1}|^{p-1})\\phi _{1}+\\zeta _1\\left[p|\\textbf {u}_{2m+1}|^{p-1}\\psi +N(\\phi )\\right],$ ${N}_i(\\phi ):=p(|\\textbf {u}_{2m+1 }|^{p-1}\\overline{\\zeta }_{i1}-|\\overline{U}_{i1}|^{p-1})\\overline{\\phi }_{i1}+\\overline{\\zeta }_{i1}\\left[p|\\textbf {u}_{2m+1}|^{p-1}\\psi +N(\\phi )\\right],$ $c_{ n+1}:=\\frac{\\int _{{\\mathbb {R}}^n}(\\zeta _{1} E_{2m+1} +\\gamma {N} (\\phi ))Z_{ n+1}}{\\int _{{\\mathbb {R}}^n}U_{1}^{p-1}Z^2_{n+1}},$ and $c_{i3}:=\\frac{\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_{i1} E_{2m+1}+\\gamma {N}_i(\\phi ))\\overline{Z}_{i3}}{\\int _{{\\mathbb {R}}^n}\\overline{U}_{i1}^{p-1}\\overline{Z}^2_{i3}},\\qquad c_{i, n+1}:=\\frac{\\int _{{\\mathbb {R}}^n}(\\overline{\\zeta }_{i1} E_{2m+1} +\\gamma {N}_i (\\phi ))\\overline{Z}_{i, n+1}}{\\int _{{\\mathbb {R}}^n}\\overline{U}_{i1}^{p-1}\\overline{Z}^2_{i,n+1}}.$ where $ Z_\\alpha (y) := \\mu ^{-{n-2 \\over 2}} Z_\\alpha \\left(\\frac{y-\\xi _{1}}{\\mu }\\right), \\quad \\overline{Z}_{i\\alpha }(y):=\\lambda ^{-\\frac{n-2}{2}}Z_\\alpha \\left(\\frac{y-\\overline{\\xi }_{i1}}{\\lambda }\\right),\\;\\;\\alpha =3, n+1 .$ It can be proved that $\\begin{aligned}\\Vert \\psi \\Vert _*&\\leqslant C\\left(\\sum _{i=1}^m \\Vert \\overline{\\overline{\\phi }}_{i1}\\Vert _*+k^{1-\\frac{n}{q}}\\right)\\;\\;\\mbox{if }n\\geqslant 4,\\quad \\Vert \\psi \\Vert _*\\leqslant C\\left( \\sum _{i=1}^m \\Vert \\overline{\\overline{\\phi }}_{i1}\\Vert _*+(\\ln k)^{-1}\\right)\\;\\;\\mbox{if }n=3,\\end{aligned}$ $\\Vert \\overline{\\overline{\\phi }}_1\\Vert _*\\leqslant C k^{-\\frac{n}{q}}\\mbox{ if }n\\geqslant 4,\\qquad \\Vert \\overline{\\overline{\\phi }}_1\\Vert _*\\leqslant \\frac{C}{k\\ln k}\\mbox{ if }n=3,$ $\\Vert \\overline{\\overline{\\phi }}_{i1}\\Vert _*\\leqslant C k^{-\\frac{n}{q}}\\mbox{ if }n\\geqslant 4,\\qquad \\Vert \\overline{\\overline{\\phi }}_{i1}\\Vert _*\\leqslant \\frac{C}{k\\ln k}\\mbox{ if }n=3,$ where $\\overline{\\overline{\\phi }}_{1}(y):=\\mu ^\\frac{n-2}{2}\\phi _{1}(\\xi _{1}+\\mu y)$ and $\\overline{\\overline{\\phi }}_{i1}(y):=\\lambda ^\\frac{n-2}{2}\\overline{\\phi }_{i1}(\\overline{\\xi }_{i1}+\\lambda y)$ , and the corresponding estimates on their non symmetric part.",
"In order to complete the proof of Theorem REF , we need to find positive parameters $\\ell , \\ell _1 , \\ldots \\ell _m$ and $t_1 , \\ldots , t_m$ entering in the definition of $\\mu $ , $\\lambda $ and $\\tau $ in (REF ) so that for all $i=1, \\ldots , m$ $c_{n+1} (\\bar{\\ell }, t) = c_{i3} (\\bar{\\ell }, t) = c_{i, n+1} (\\bar{\\ell }, t) = 0, \\quad \\bar{\\ell }= (\\ell , \\ell _1 , \\ldots \\ell _m), \\quad t= ( t_1 , \\ldots , t_m ).$ In dimension $n\\geqslant 4$ , this system decouples at main order and becomes $\\begin{aligned}&a_n \\ell ^2-1+\\frac{1}{k^{2 {n-3 \\over n-1}}}\\Theta _{n+1, k}(\\ell , \\ell _1 , \\ldots \\ell _m ,t_1 , \\ldots , t_m ) =0, \\\\&a_n \\ell _i^2 -1+\\frac{1}{k^{2 {n-3 \\over n-1}}}\\Theta _{i, n+1, k}(\\ell , \\ell _1 , \\ldots \\ell _m ,t_1 , \\ldots , t_m ) =0, \\quad i=1, \\ldots , m, \\\\&b_n{ \\ell _i \\over t_i^{n-1} }-1+\\frac{1}{k^{n-3 \\over n-1} }\\Theta _{i, 3, k}(\\ell , \\ell _1 , \\ldots \\ell _m ,t_1 , \\ldots , t_m)=0, \\quad i=1, \\ldots , m,\\end{aligned}$ where $a_n$ , $b_n$ are positive constants, and $\\Theta _{n+1, k}$ , $\\Theta _{i, n+1, k}$ , $\\Theta _{i, 3, k}$ are smooth functions of their argument, which are uniformly bounded, together with their first derivatives, as $k \\rightarrow \\infty $ .",
"In dimension $n=3$ , system (REF ) decouples and becomes $\\begin{aligned}&a_3 \\ell ^2-1+\\frac{\\ln \\ln k }{\\ln k }\\Theta _{4, k}(\\ell , \\ell _1 , \\ldots \\ell _m ,t_1 , \\ldots , t_m ) =0, \\\\&a_3 \\ell _i^2-1+\\frac{\\ln \\ln k }{\\ln k }\\Theta _{i, 4, k}(\\ell , \\ell _1 , \\ldots \\ell _m ,t_1 , \\ldots , t_m ) =0, \\quad i=1, \\ldots , m, \\\\&b_3{ \\ell _i \\over t_i^{2} }-1+\\frac{\\ln \\ln k }{\\ln k } \\Theta _{i, 3, k}(\\ell , \\ell _1 , \\ldots \\ell _m ,t_1 , \\ldots , t_m)=0, \\quad i=1, \\ldots , m,\\end{aligned}$ where $a_3$ , $b_3$ are positive constants, and $\\Theta _{4, k}$ , $\\Theta _{i,4,k} $ $\\Theta _{3, k}$ are smooth functions of their argument, which are uniformly bounded, together with their first derivatives, as $k \\rightarrow \\infty $ .",
"A fixed point argument gives the existence of $\\ell $ and $t$ solutions to (REF )."
],
[
"Appendix A: Some useful computations",
"Proof of (REF ) and (REF ).",
"By definition $\\sum _{j=2}^k {1\\over |\\overline{\\xi }_1 - \\overline{\\xi }_j|^{n-2}} = {1\\over R^{n-2} (1-\\tau ^2)^{n-2 \\over 2}} \\, \\left(\\sum _{j=2}^k {1\\over [2 \\, (1-\\cos \\theta _j)]^{n-2 \\over 2} } \\right) , \\quad \\theta _j = 2\\pi {j-1 \\over k}.$ Using the symmetry of the construction we have $\\begin{split}\\sum _{j=2}^k {1\\over [2(1-\\cos \\theta _j)]^{n-2 \\over 2} }&= {\\left\\lbrace \\begin{array}{ll}\\displaystyle 2\\sum _{j=2}^{\\frac{k}{2}} {1\\over [2(1-\\cos \\theta _j)]^{n-2 \\over 2}} +\\frac{1}{4^{\\frac{n-2}{2}}}\\quad \\mbox{ if $k$ even},\\\\\\displaystyle 2\\sum _{j=2}^{\\frac{k-1}{2}} {1\\over [2(1-\\cos \\theta _j)]^{n-2 \\over 2}}\\quad \\mbox{ if $k$ odd},\\end{array}\\right.",
"}\\end{split}$ where $0<\\theta _j<\\pi $ .",
"Hence, $2(1-\\cos \\theta _j)=\\theta _j^2\\left[1-\\cos (\\eta _j)\\frac{\\theta _j^2}{12}\\right]\\quad \\mbox{for some }0\\leqslant \\eta _j\\leqslant \\theta _j,$ and $1-\\frac{\\pi ^2}{12}\\leqslant 1-\\cos (\\eta _j)\\frac{\\theta _j^2}{12}\\leqslant 1+\\frac{\\pi ^2}{12}.$ Then, for $k $ large and odd, we get $\\begin{split}\\sum _{j=2}^k {1\\over [2(1-\\cos \\theta _j)]^{n-2 \\over 2} }&= 2\\sum _{j=2}^{\\frac{k-1}{2}}\\frac{1}{\\theta _j^{n-2}}+2\\sum _{j=2}^{\\frac{k-1}{2}}\\frac{1}{\\theta _j^{n-2}}\\frac{1-\\left[1-\\cos (\\eta _j)\\frac{\\theta _j^2}{12}\\right]^{\\frac{n-2}{2}}}{\\left[1-\\cos (\\eta _j)\\frac{\\theta _j^2}{12}\\right]^{\\frac{n-2}{2}}}\\\\&={\\left\\lbrace \\begin{array}{ll}k^{n-2} A_n \\left(1+ O( \\sigma _k ) \\right) \\quad {\\mbox{if}} \\quad n \\geqslant 4,\\\\k \\ln k A_3 \\left(1+ O(\\sigma _k ) \\right) \\quad {\\mbox{if}} \\quad n =3,\\end{array}\\right.",
"}\\end{split}$ where $A_n:= {2\\over (2\\pi )^{n-2}} \\, \\sum _{j=1}^\\infty j^{2-n},\\;A_3 := \\pi ^{-1}\\;\\hbox{ and }\\;\\sigma _k := {\\left\\lbrace \\begin{array}{ll} \\begin{split}k^{-2} & \\quad {\\mbox{if}} \\quad n>5, \\\\k^{-2} \\ln k & \\quad {\\mbox{if}} \\quad n=5, \\\\k^{-1} & \\quad {\\mbox{if}} \\quad n=4,\\\\(\\ln k)^{-1} & \\quad {\\mbox{if}} \\quad n=3.\\end{split}\\end{array}\\right.",
"}$ The case of $k$ even can be analogously treated, and therefore (REF ) follows.",
"In the same spirit we also observe that, for $n\\geqslant 4$ and $k$ odd, we have $\\begin{split}\\sum _{j=1}^k {1\\over |\\overline{\\xi }_1 - \\underline{\\xi }_j|^{n-2}} &= {2\\over R^{n-2} } \\sum _{j=1}^{\\frac{k-1}{2}} {1\\over [4 \\tau ^2 + 2 (1-\\tau ^2) (1-\\cos \\theta _j ) ]^{n-2 \\over 2} }\\\\&= {2\\over (2R)^{n-2} \\tau ^{n-2 } } \\left(\\sum _{j=1}^{\\frac{k-1}{2}} {1\\over [1 + \\pi ^2 {1-\\tau ^2 \\over \\tau ^2} ({j-1 \\over k} )^2]^{n-2 \\over 2} } \\right) (1+ O(\\tau ^2) ),\\end{split}$ and for $n=3$ , $\\begin{split}\\sum _{j=1}^k {1\\over |\\overline{\\xi }_1 - \\underline{\\xi }_j|}&= {1\\over R \\tau } \\left(\\sum _{j=1}^{\\frac{k-1}{2}} {1\\over [1 + \\pi ^2 {1-\\tau ^2 \\over \\tau ^2} ({j-1 \\over k} )^2]^{n-2 \\over 2} } \\right) (1+ O(|\\ln \\tau |^{-1}) ).\\end{split}$ If $n \\geqslant 4$ , using (REF ) we get $\\begin{split}\\sum _{j=1}^{\\frac{k-1}{2}} {1\\over [1 + \\pi ^2 {1-\\tau ^2 \\over \\tau ^2} ({j-1 \\over k} )^2]^{n-2 \\over 2} } &= \\left(\\int _0^{\\frac{k-3}{2}} {dx \\over (1 + \\pi ^2 {1-\\tau ^2 \\over \\tau ^2} ({x \\over k} )^2)^{n-2 \\over 2} } \\right) \\, (1+ O ((k\\tau )^{-2})) \\\\&= {k \\tau \\over \\pi \\, \\sqrt{1-\\tau ^2} } \\left( \\int _0^{\\pi {\\sqrt{1-\\tau ^2} \\over \\tau } {{k-3} \\over {2 k}} } {ds \\over (1+ s^2)^{n-2 \\over 2} }\\right) \\, (1+ O ((k\\tau )^{-2}) )\\\\&= {k \\tau \\over \\pi \\, \\sqrt{1-\\tau ^2} } \\left( \\int _0^{\\infty } {ds \\over (1+ s^2)^{n-2 \\over 2} }\\right) \\, (1+ O ((k\\tau )^{-2})+ O(\\tau ^{n-3} ) ).\\end{split}$ If $n=3$ , $\\begin{split}\\sum _{j=1}^{\\frac{k-1}{2}} &{1\\over [1 + \\pi ^2 {1-\\tau ^2 \\over \\tau ^2} ({j-1 \\over k} )^2]^{1 \\over 2} } ={k \\tau \\over \\pi \\, \\sqrt{1-\\tau ^2} } \\left( \\int _0^{\\pi {\\sqrt{1-\\tau ^2} \\over \\tau } {k-3 \\over 2k} } {ds \\over (1+ s^2)^{1 \\over 2} }\\right) \\, (1+ O ((k\\tau )^{-2}) )\\\\&\\quad = {k \\tau \\over \\pi \\, \\sqrt{1-\\tau ^2} } \\left( \\ln ( \\pi {\\sqrt{1-\\tau ^2} \\over \\tau } {k-3 \\over 2k} + \\sqrt{1+ \\pi ^2 {1-\\tau ^2 \\over \\tau ^2 } \\left({k-3 \\over 2k}\\right)^2 } ) \\right) \\, (1+ O ((k\\tau )^{-2}) )\\\\&\\quad = {k \\tau \\over \\pi \\, \\sqrt{1-\\tau ^2} } \\ln \\left( {\\pi \\over \\tau } \\right) \\, (1+ O(\\tau ^2 |\\ln \\tau |^{-1} )).\\end{split}$ Combining (REF ) and (REF ) we obtain the validity of (REF ) for $k$ odd.",
"The even case follows in the same way.",
"Analogous computations provide $\\begin{split} \\sum _{j=1}^k {1\\over |\\overline{\\xi }_1 - \\underline{\\xi }_j|^{n}} ={\\left\\lbrace \\begin{array}{ll} C_n \\, {k \\over \\tau ^{n-1}} \\, \\left(1+ O( (\\tau k)^{-2}) \\right) \\quad {\\mbox{if}} \\quad n \\geqslant 5,\\\\C_n \\, {k \\over \\tau ^{n-1}} \\, \\left(1+ O( \\tau ^{n-1}) \\right) \\quad {\\mbox{if}} \\quad n =3,4,\\end{array}\\right.",
"}\\quad C_n := {2\\over 2^{n} \\pi } \\int _0^\\infty {ds \\over (1+ s^2)^{n \\over 2}}.\\end{split}$ We give here the proof of the linear independence of the functions $z_j(y)$ , $j = 0,\\ldots , 4n-3$ , defined in Remark REF , in the case $k$ even.",
"Indeed, we claim that, if there are $4n-2$ constants $c_j$ with $\\sum _{j=0}^{4n-3} c_j z_j (y) = 0 , \\quad \\forall y \\in {\\mathbb {R}}^n,$ then $c_j = 0 $ for all $j$ .",
"The result will follow from evaluating this expression in different points, properly chosen.",
"Notice that, by definition, (REF ) can be written as $\\begin{split}&\\left[c_0-2y_1c_{n+1}-2y_2c_{n+2}-2y_3c_{n+3}\\right]z_0(y)+\\left[c_1+|y|^2c_{n+1}-\\sum _{\\alpha =2}^ny_\\alpha c_{n+\\alpha +2}\\right]z_1(y)\\\\&\\quad +\\left[c_2+|y|^2c_{n+2}+c_{n+4}y_1-\\sum _{\\alpha =3}^nc_{2n+\\alpha }y_\\alpha \\right]z_2(y)\\\\&\\quad +\\left[c_3+|y|^2c_{n+3}+c_{n+5}y_1+c_{2n+3}y_2-\\sum _{\\alpha =4}^nc_{3n+\\alpha -3}y_\\alpha \\right]z_3(y)\\\\&\\quad +\\sum _{l=4}^n\\left[c_l+c_{n+l+2}y_1+c_{2n+l}y_2+c_{3n+l-3}y_3\\right]z_l(y)=0, \\quad \\forall y \\in {\\mathbb {R}}^n.\\end{split}$ Since our solution $u $ satisfies $u(y_1 , \\ldots , y_j , \\ldots , y_n) = u(y_1 , \\ldots ,- y_j , \\ldots , y_n), \\quad \\forall j=1, \\ldots n,$ we have that necessarily, for every $j=1, \\ldots n$ , $z_j (q) = 0 \\quad \\forall q= (q_1 , \\ldots , q_n) \\mbox{ with } q_j =0.$ Take a point of the form $q=(r, 0,...,0)$ , with $r >0$ .",
"Thus, evaluating (REF ) at $q$ one gets $c_0z_0(q)+c_1 z_1(q)+c_{n+1}(-2r z_0(q)+r^2z_1(q))=0.$ Given the decay of the functions $z_0$ and $z_1$ , there are three constants $a$ , $b$ and $c$ such that $\\begin{aligned}r^{n-2} z_0(q)&\\sim -{n-2 \\over 2} 2^{n-2 \\over 2} a , \\quad r^{n-1} z_1(q)\\sim -{n-2 \\over 2} 2^{n-2 \\over 2} b ,\\\\r^{n-3} &\\left(-2r z_0(q)+r^2z_1(q)\\right) \\sim -{n-2 \\over 2} 2^{n-2 \\over 2} c, \\quad {\\mbox{as}} \\quad r \\rightarrow \\infty .\\end{aligned}$ The constants $a$ , $b$ and $c$ can be computed explicitly $a&= (1+ o_k (1) ) , \\quad b= 2 (1+ o_k (1) ),\\\\c&= 4 k \\lambda ^{n-2 \\over 2} (1+ o_k (1) ),$ where $\\lim _{k\\rightarrow \\infty } o_k (1) =0$ .",
"Evaluating (REF ) at three points $(r_1,0,...,0)$ , $(r_2,0,...,0)$ , $(r_3,0,...,0)$ , with $r_i$ large, we arrive at a system which at main order looks like $\\left(\\begin{array}{ccc}a r_1 & b & c r_1^2\\\\a r_2 & b & c r_2^2\\\\a r_3 & b & c r_3^3\\end{array}\\right)\\left(\\begin{array}{c}c_0 \\\\ c_1 \\\\ c_{n+1}\\end{array}\\right)=\\left(\\begin{array}{c}0 \\\\ 0 \\\\ 0\\end{array}\\right).$ Choosing properly $r_1\\ne r_2\\ne r_3$ it can be seen that the determinant of the matrix is not zero and thus necessarily $c_0=c_1=c_{n+1}=0.$ Let us take now $q=(q_1, q_2,0,...,0)$ .",
"From (REF ) we obtain $c_2z_2(q)+c_{n+2}(-2q_2 z_0(q)+z_2(q)|q|^2)+c_{n+4}(-q_2z_1(q) + q_1 z_2 (q) )=0.$ If $q_2=0$ , then $-q_2z_1(q) + q_1 z_2 (q) =0.$ Since $u (y)$ is invariant under rotations of angle ${2\\pi \\over k }$ in the $(y_1,y_2)$ -plane, we have that $-q_2z_1(q) + q_1 z_2 (q) =0$ for all points of the form $q= (e^{{2\\pi \\over k} (j-1) i} \\tilde{r},0,0, 0 , ..)$ , with $\\tilde{r}=(r,0)$ , for some $j=2, \\ldots , k$ .",
"Evaluating (REF ) at these points $q$ , we get $c_2z_2(q)+c_{n+2}(-2q_2 z_0(q)+z_2(q)|q|^2)=0.$ Replacing in the above equation the points $q_1= (e^{{2\\pi \\over k} (j-1) i} \\tilde{r}_1,0,0, 0 , ..)$ , $q_2= (e^{{2\\pi \\over k} (j-1) i} \\tilde{r}_2,0,0, 0 , ..)$ , with $\\tilde{r}_1=(r_1,0)$ , $\\tilde{r}_2=(r_2,0)$ , $r_1\\ne r_2$ , we get a $2\\times 2$ system in $c_2$ and $c_{n+2}$ .",
"Arguing as before and choosing $r_1$ , $r_2$ properly, we obtain $c_2=c_{n+2}=0$ .",
"Thus so far we have proven that $c_0=c_1=c_{n+1}=c_2=c_{n+2}=0.$ Next we evaluate (REF ) at points of the form $q=(q_1,0,q_3,0,...,0)$ we deduce $c_3z_3(q)+c_{n+3}(-2q_3z_0(q)+|q|^2z_3(q))+c_{n+5}(q_1 z_3 (q) - q_3z_1(q))=0.$ Taking now $\\bar{q} = (-q_1,0,-q_3,0,...,0)$ and using that $z_0(\\bar{q})=z_0(q)$ , $z_1(\\bar{q})= -z_1 (q)$ , $z_3 (\\bar{q}) = - z_3 (q)$ , we obtain $c_3z_3(q)+c_{n+3}(-2q_3z_0(q)+|q|^2z_3(q))=0.$ Similar decay rates as in (REF ) allow us to choose two different points of the form $q=(q_1,0,q_3,0,...,0)$ that inserted in the above equation produce an invertible $2\\times 2$ system.",
"This gives $c_3=c_{n+3}=0$ .",
"Evaluate (REF ) at $q=(q_1,q_2, 0,0,0..0)$ to get $c_{n+4} [ q_2 z_1 (q) - q_2 z_2(q) ]=0.$ Since our solution $u$ is not radially symmetric in the $(y_1, y_2)$ -plane, the function $y_2 z_1(y) - y_1 z_2 (y)$ is not identically zero.",
"Thus we get $c_{n+4}=0$ .",
"Arguing similarly, we prove that $c_{n+5}=c_{2n+3}=0$ (we use (REF ) for $q=(q_1,0,q_3, 0,0,..)$ and $q=(0,q_2,q_3,0,0,...0)$ respectively, and the fact that the solution is not radially symmetric in the $(y_1,y_3)$ and $(y_2,y_3)$ planes).",
"Let now $\\ell =4, \\ldots , n$ and evaluate (REF ) at points of the form $q=(q_1, 0, .., 0 , q_\\ell , ...)$ We obtain $c_\\ell z_\\ell (q) + c_{2n+\\ell } (-q_\\ell z_1 (q) + q_1 z_\\ell (q)) =0.$ Since $u$ is not radially symmetric in the $(y_1,y_\\ell )$ -plane, the function $y \\rightarrow -y_\\ell z_1 (y) + y_1 z_\\ell (y)$ is not identically zero.",
"Choose now two different points of the form $q=(q_1,0,q_\\ell ,0,...,0)$ that inserted in the above equation produce an invertible $2\\times 2$ system.",
"This gives $c_\\ell =c_{2n+\\ell }=0$ , for $\\ell =4,\\ldots ,n$ .",
"Hence, so far we have proven that $c_0=c_1=.....=c_{3n}=0.$ It remains to see that $\\sum _{j=4}^n c_{3n+j-3} [ y_3 z_j(y) - y_j z_3 (y) ] = 0 \\quad \\forall y,$ implies $c_{3n+4}= \\ldots = c_{4n-3} = 0$ .",
"Let us evaluate this linear combination at the points $ q_{3,j} = (0,0, q_3, 0,..., q_j , 0, ...)$ , for $j =4, \\ldots , n$ .",
"We get $c_{3n+j -3 } (q_3 z_j (q_{3,j} ) - q_j z_j (q_{3,j} )) = 0.$ Since $u$ is not radially symmetric in the $(y_3, y_j)$ -plane, we can choose $q_{3,j}$ to show that $c_{3n+j - 3} = 0 $ for all $j=4,\\ldots ,n$ .",
"We then conclude that (REF ) implies that $c_j=0, \\quad {\\mbox{for all}} \\quad j=0,1, \\ldots , 4n-3.$"
]
] | 2001.03548 |
[
[
"Ricci tensor in graded geometry"
],
[
"Abstract We define the notion of the Ricci tensor for NQ symplectic manifolds of degree 2 and show that it corresponds to the standard generalized Ricci tensor on Courant algebroids.",
"We use an appropriate notion of connections compatible with the generalized metric on the graded manifold."
],
[
"Introduction",
"It has been known for some time that Courant algebroids [12] provide a natural framework for the study of several aspects of string theory.",
"First, the string sigma models can be seen [16] as particular cases of the Courant sigma models [9], [13] on manifolds with boundary.",
"Furthermore, the low-energy dynamics of string theory can be described via a suitable generalization of the Ricci tensor and scalar curvature, defined for Courant algebroids [7], [3], [5], [4], [10], [18].",
"While for some purposes it is enough to consider only a special class of the so-called exact Courant algebroids [15], for the study of dualities (such as the Poisson-Lie T-duality [11]), it is neccessary to understand the constructions in the general setup [15].",
"Such generalized Ricci tensor was introduced and studied in [4], [10], [18].",
"In particular, its properties were used to provide a proof of the compatibility of the Poisson-Lie T-duality with the RG flow and with the string background equations, extending the results of [20] and [8], as well as to find new solutions to generalized supergravity equations.",
"On the other hand, it is known [15], [14] that graded geometry offers a much more conceptual viewpoint of Courant algebroids, simplifying the formulas and providing various new insights.",
"It is therefore desirable to also find a formulation of the curvature tensors in the graded language.",
"Motivated by the recent work [1], where the graded analogues of connections, curvature and torsion were introduced and studied, we propose a simple definition of the generalized Ricci tensor in the graded setup.",
"The purpose of this is twofold – it provides a more conceptual viewpoint of the generalized Ricci tensor and at the same time opens a very concrete door towards generalizations, for example in the context of U-duality, following [2].",
"This will be explored in a future work.",
"This note is structured as follows.",
"We start by reviewing the notion of generalized metric, from the perspective of graded geometry.",
"We then recall the neccessary definitions from [1], introduce the notion of connections with invariant torsion and present the Ricci tensor.",
"Finally, we explore the relation to the more standard Courant algebroid connections [7] and discuss the exact case, where this Ricci tensor produces the usual one from Riemannian geometry."
],
[
"Acknowledgements",
"The author would like to thank Andreas Deser and Pavol Ševera for helpful discussions, suggestions, and comments on the preliminary version of the paper.",
"The author would also like to acknowledge the COST action MP 1405 Quantum structure of spacetime and the Corfu Summer Institute 2019 at EISA, where the main idea of the paper was conceived."
],
[
"Generalized metric",
"In what follows, we shall use $\\mathcal {E}$ to denote an NQ symplectic manifold of degree 2 [17].",
"This means that $\\mathcal {E}$ is an $\\mathbb {N}$ -graded manifoldi.e.",
"an ordinary manifold $\\mathcal {E}_0$ together with a sheaf of $\\mathbb {N}$ -graded commutative algebras, locally of the form $C^\\infty (\\mathcal {U})\\otimes S(V)$ , with $S(V)$ the free graded commutative algebra generated by a finite-dimensional vector space $V=\\bigoplus _{i=1}^mV_m$ , and $\\mathcal {U}$ an open subset of $\\mathcal {E}_0$, equipped with a symplectic form of degree 2, and a degree 1 symplectic vector field $Q_\\mathcal {E}$ , satisfying $Q^2_\\mathcal {E}=0$ .",
"There is an associated sequence of fibrations $\\mathcal {E}=\\mathcal {E}_2 \\rightarrow \\mathcal {E}_1 \\rightarrow \\mathcal {E}_0,$ which corresponds to the subsheafs generated by coordinates of degrees up to 2, up to 1, and up to 0, respectively.",
"In particular, the last arrow gives a vector bundle.",
"A generalized metric is, in the graded language, a symplectic involution $\\iota $ on $\\mathcal {E}$ , which preserves the basis $\\mathcal {E}_0$ , i.e.",
"it is a diffeomorphism of $\\mathcal {E}$ satisfying $\\iota ^2=id_\\mathcal {E},\\quad \\iota ^*\\omega =\\omega ,\\quad \\iota |_{\\mathcal {E}_0}=id_{\\mathcal {E}_0}.$ Given a generalized metric, it is always possible to locally choose the coordinates $x^i$ , $e^a$ , $e^{\\dot{a}}$ , $p_i$ on $\\mathcal {E}$ of degrees 0, 1, 1, 2, respectively, such that $\\omega =dp_idx^i+de_a de^a +de_{\\dot{a}}de^{\\dot{a}},$ $\\iota ^*x^i=x^i,\\quad \\iota ^*p_i=p_i,\\quad \\iota ^*e^a=e^a,\\quad \\iota ^*e^{\\dot{a}}=-e^{\\dot{a}},$ where $e_a:=g_{ab}e^b$ , $e_{\\dot{a}}:=g_{\\dot{a}\\dot{b}}e^{\\dot{b}}$ for some constants $g_{ab}$ , $g_{\\dot{a}\\dot{b}}$ .",
"We will sometimes denote the coordinates $e^a$ , $e^{\\dot{a}}$ collectively as $e^\\alpha $ .The proof goes as follows: Using the (graded) Darboux theorem we first find coordinates $x^i$ , $e^\\alpha $ , $p_i$ such that $\\omega =dp_idx^i+g_{\\alpha \\beta }de^\\alpha de^\\beta $ , for $g_{\\alpha \\beta }$ a diagonal matrix with only 1 and $-1$ on the diagonal.",
"It follows that the ($x$ -dependent) matrix $R^\\alpha _\\beta $ defined by $\\iota ^*e^\\alpha =R^\\alpha _\\beta (x) e^\\beta $ is idempotent and orthogonal w.r.t.",
"$g$ , and thus can be made into the form $\\operatorname{diag}(1,\\dots ,1,-1,\\dots ,-1)$ using $e^{\\prime \\alpha }=O^\\alpha _\\beta (x) e^\\beta $ , for $O$ orthogonal.",
"Making an appropriate shift of $p_i$ , the form of $\\omega $ is preserved in the new coordinates.",
"It is easy to see that $Q_\\mathcal {E}$ is always Hamiltonian, i.e.",
"it comes from a degree 3 function $H$ .",
"The most general such function has the form $H=\\rho _\\alpha ^i(x)p_ie^\\alpha -\\tfrac{1}{6} c_{\\alpha \\beta \\gamma }(x)e^\\alpha e^\\beta e^\\gamma ,\\vspace{-5.69046pt}$ giving $Q_\\mathcal {E}=\\rho ^i_\\alpha e^\\alpha \\partial _{x^i}+(\\rho ^i_\\alpha p_i-\\tfrac{1}{2} c_{\\alpha \\beta \\gamma }e^\\beta e^\\gamma )\\partial _{e_\\alpha }+(\\tfrac{1}{6} c_{\\alpha \\beta \\gamma ,i}e^\\alpha e^\\beta e^\\gamma -\\rho ^j_{\\alpha ,i}e^\\alpha p_j)\\partial _{p_i},$ where, in setting $e_\\alpha :=g_{\\alpha \\beta }e^\\beta $ , we extended $g_{ab}$ , $g_{\\dot{a}\\dot{b}}$ to $g_{\\alpha \\beta }$ by adding $g_{a\\dot{b}}=g_{\\dot{a} b}=0$ .",
"The condition $Q^2_\\mathcal {E}=0$ translates to the classical master equation $\\lbrace H,H\\rbrace =0$ .",
"Since $\\iota $ preserves the degree, it induces an involution on $\\mathcal {E}_1$ .",
"Because of the base-fixing condition, $\\iota $ is in turn fully determined by the corresponding fixed point set $\\mathcal {V}_1\\subset \\mathcal {E}_1$ .",
"Seeing $\\mathcal {V}_1$ as a vector bundle over $\\mathcal {E}_0$ , we can pull it back along $\\mathcal {E}\\rightarrow \\mathcal {E}_0$ to obtain a bundle $\\mathcal {V}\\rightarrow \\mathcal {E}$ .",
"The latter bundle is locally described by coordinates $x^i$ , $e^a$ , $\\xi ^a$ , $e^{\\dot{a}}$ , $p_i$ , with one copy of $\\xi ^a$ , $\\deg \\xi ^a=1$ , corresponding to each $e^a$ .",
"Finally, notice that $\\mathcal {E}_1$ , $\\mathcal {V}_1$ , and $\\mathcal {V}$ are all symplectic vector bundles.",
"The relation to Courant algebroids is as follows [14], [15].",
"The Courant algebroid is given by the ordinary vector bundle $E=\\mathcal {E}_1[-1]$ over $\\mathcal {E}_0$ .",
"(In other words, $x^i$ are coordinates on the base and $e^\\alpha $ correspond to the linear coordinates on the fibers of the algebroid.)",
"The coefficients $c$ and $\\rho $ give the structure functions of the bracket and the anchor, respectively, while $g$ encodes the fiberwise inner product.",
"Finally, the involution $\\iota $ corresponds to the usual viewpoint of generalized metric as a fiberwise reflection on the Courant algebroid, or equivalently, as a subbundle $V_+=\\mathcal {V}_1[-1]\\subset E$ ."
],
[
"Tautological section and contraction",
"Let us now denote the vector bundle morphism $\\mathcal {V}\\rightarrow \\mathcal {V}_1$ by $\\varphi $ .",
"There is a unique section $\\tau ^{\\prime }\\colon \\mathcal {E}\\rightarrow \\mathcal {V}$ such that $\\varphi \\circ \\tau ^{\\prime }$ coincides with the map $\\mathcal {E}\\rightarrow \\mathcal {E}_1 \\rightarrow \\mathcal {V}_1$ (the last arrow is the orthogonal projection).",
"Since the bundle $\\mathcal {V}$ is symplectic, we get an induced section $\\tau $ on the dual bundle $\\mathcal {V}^*\\rightarrow \\mathcal {E}$ .",
"We call $\\tau $ the tautological section [1].",
"More concretely, identifying sections of $\\mathcal {V}^*$ with functions on $\\mathcal {V}$ which are linear in the fiber coordinates, we get $\\tau =e_a\\xi ^a.$ The involution on $C^\\infty (\\mathcal {V})$ (induced by $\\iota $ ) allows us to split any vector field on $\\mathcal {V}$ into the sum of its self-dual and anti-self-dual part.",
"We will denote the anti-self-dual part of a vector field $D$ by $\\pi D$ .",
"Let us now consider a special subspace $\\text{End}_2$ of the space of vector fields, given by degree 2 bundle endomorphisms of $\\mathcal {V}^*$ .",
"Locally we have $D=D^{a\\phantom{b}i}_{\\phantom{a}b}(x)p_i\\xi ^b\\partial _{\\xi ^a}+\\tfrac{1}{2} D^a_{\\phantom{a} b \\alpha \\beta }(x)e^\\alpha e^\\beta \\xi ^b\\partial _{\\xi ^a}\\quad {\\pi }\\quad D^a_{\\phantom{a} b c\\dot{d}}(x)e^c e^{\\dot{d}}\\xi ^b\\partial _{\\xi ^a}.$ Writing $\\mathcal {V}_1^\\perp \\subset \\mathcal {E}_1$ for the subbundle (over $\\mathcal {E}_0$ ) perpendicular to $\\mathcal {V}_1\\subset \\mathcal {E}_1$ , we have an identification (we use $\\Gamma $ for the space of sections) $\\pi (\\text{End}_2)\\cong \\Gamma (\\mathcal {V}_1^*\\otimes \\mathcal {V}_1^{\\perp *}\\otimes End(\\mathcal {V}_1^*))\\cong \\Gamma (\\mathcal {V}_1^*\\otimes \\mathcal {V}_1^{\\perp *}\\otimes \\mathcal {V}_1\\otimes \\mathcal {V}_1^*).$ We define the contraction map $C\\colon \\text{End}_2\\rightarrow \\Gamma (\\mathcal {V}_1^{\\perp *}\\otimes \\mathcal {V}_1^*)$ as the projection $\\pi $ followed by the contraction of the first and third factor in the last expression.",
"Explicitly, $C\\colon D\\mapsto \\partial _{e^a} (\\pi D) \\xi ^a.$"
],
[
"Connections, torsion and curvature",
"Following [1], a connection on a $\\mathcal {V}^* \\rightarrow \\mathcal {E}$ is a degree 1 vector field $Q$ on $\\mathcal {V}$ , which projects to $Q_\\mathcal {E}$ , and which preserves the space of sections $\\Gamma (\\mathcal {V}^*)\\subset C^\\infty (\\mathcal {V})$ .",
"The torsion is then a particular section of $\\mathcal {V}^*$ , defined as $Q\\tau $ .",
"The curvature of $Q$ is the vector field $Q^2\\equiv \\tfrac{1}{2}[Q,Q]$ on $\\mathcal {V}$ .",
"One easily sees that $Q^2\\in \\text{End}_2$ .",
"We will say that a connection $Q$ on $\\mathcal {V}$ has invariant torsion if its torsion is invariant under the induced involution on $\\Gamma (\\mathcal {V}^*)$ .",
"Finally, we define the Ricci tensor $\\operatorname{Ric}$ by $\\operatorname{Ric}:=CQ^2\\in \\Gamma (\\mathcal {V}_1^{\\perp *}\\otimes \\mathcal {V}_1^*).$ Connections with invariant torsion should be seen as analogues of the usual Levi-Civita connection – the compatibility with the metric is replaced by the fact that $Q$ is a vector field on $\\mathcal {V}$ and the vanishing of the torsion is replaced by the invariance of $Q\\tau $ (as seen below, in general it is impossible to have $Q\\tau =0$ ; on the other hand the invariant torsion condition does not fix the connection uniquely, c.f.",
"[3]).",
"The contraction in (REF ) corresponds to the usual procedure for obtaining the Ricci tensor from the Riemann tensor.",
"The insertion of the projection $\\pi $ keeps only the part of the tensor which can be identified with an infinitesimal deformation of the generalized metric.It also allows us to define the contraction.",
"This is due to the identification $\\Gamma (\\mathcal {V}_1^{\\perp *}\\otimes \\mathcal {V}_1^*)\\cong \\operatorname{Hom}_{\\mathcal {E}_0}(\\mathcal {V}_1,\\mathcal {V}_1^{\\perp *})\\cong \\operatorname{Hom}_{\\mathcal {E}_0}(\\mathcal {V}_1,\\mathcal {V}_1^{\\perp }),$ with the last isomorphism provided by the symplectic form on $\\mathcal {V}_1^{\\perp }$ .",
"This deformation is the (infinitesimal) generalized Ricci flow [19], [4].",
"Passing again to a coordinate description, a general connection on $\\mathcal {V}^*$ has the form $Q=Q_\\mathcal {E}+\\psi ^a_{\\phantom{a}b\\alpha }(x)e^\\alpha \\xi ^b\\partial _{\\xi ^a}.$ Its torsion is $Q\\tau =\\rho _a^i p_i\\xi ^a-\\tfrac{1}{2} c_{a\\beta \\gamma }e^\\beta e^\\gamma \\xi ^a-\\psi _{\\phantom{a}b\\alpha }^a e^\\alpha \\xi ^b e_a.$ Since the involution preserves the coordinates $\\xi ^a$ , the invariance of torsion is equivalent to the constraint $\\psi _{\\phantom{a}b\\dot{a}}^a=c_{\\phantom{a}b\\dot{a}}^a.$ In particular, the coefficients $\\psi ^a_{\\phantom{a}bc}$ are left unrestricted.",
"However, as we will see, in the case of invariant torsion the curvature only depends on $\\psi ^a_{\\phantom{a}bc}$ through the trace $\\lambda _b:=\\psi ^a_{\\phantom{a}ba}.$ Concretely, a short calculation (see Appendix) reveals that for a connection $Q$ with invariant torsion, $\\operatorname{Ric}=(c^c_{\\phantom{c}b\\dot{a} }\\lambda _c-\\rho ^i_a c^a_{\\phantom{a}b\\dot{a},i }+\\rho ^i_{\\dot{a}} \\lambda _{b,i}+c_{\\dot{c}a\\dot{a}}c^{a\\phantom{b}\\dot{c}}_{\\phantom{a}b})\\xi ^b e^{\\dot{a}} .$ This recovers exactly the formula for the generalized Ricci tensor from [18].",
"More precisely, using the notation of [18] we have $\\operatorname{Ric}=\\operatorname{GRic}_{V_+,\\operatorname{div}}(e_c,e_{\\dot{a}})\\xi ^c e^{\\dot{a}},$ where the divergence operator is given by $\\operatorname{div}(e_a)=-\\lambda _a$ ."
],
[
"Connection with Courant algebroid connections",
"From the definition it follows that connections are in one-to-one correspondence with linear mapsWe use subscript (for $C^\\infty $ and $\\Gamma $ ) to denote the degree.",
"$\\hat{Q}\\colon \\Gamma _1(\\mathcal {V}^*) \\rightarrow \\Gamma _2(\\mathcal {V}^*)$ , satisfying $\\hat{Q}(f u)=f(\\hat{Q}u)+(Q_\\mathcal {E} f)u$ , for $f\\in C^\\infty _0(\\mathcal {E})$ , $u\\in \\Gamma _1(\\mathcal {V}^*)$ .",
"Since $C^\\infty _0(\\mathcal {E})\\cong C^\\infty (\\mathcal {E}_0),\\quad C^\\infty _1(\\mathcal {E})\\cong \\Gamma (E^*),\\quad \\Gamma _1(\\mathcal {V}^*)\\cong \\Gamma (V_+^*),\\quad \\Gamma _2(\\mathcal {V}^*)\\cong \\Gamma (E^*\\otimes V_+^*).$ we can understand $\\hat{Q}$ as a map $\\nabla \\colon \\Gamma (V_+^*)\\rightarrow \\Gamma (E^*\\otimes V_+^*),\\quad \\text{such that}\\quad \\nabla (fu)=f\\nabla u+(Q_\\mathcal {E} f)\\otimes u.$ Dually, we have a map $\\nabla \\colon \\Gamma (V_+)\\rightarrow \\Gamma (E^*\\otimes V_+)$ (satisfying the same Leibniz identity).",
"This is known in the literature as the Courant algebroid (or generalized) connection [7].",
"If $Q$ has invariant torsion, then it is uniquely determined (c.f.",
"(REF )) by the restriction of $\\nabla $ to a map $\\Gamma (V_+)\\rightarrow \\Gamma (V_+^*\\otimes V_+)$ ."
],
[
"The exact case",
"Let us now consider the following example [15], [16].Every $\\mathcal {E}$ with vanishing tangential cohomology and a generic generalized metric can be put into this form [15], [6].",
"First, $\\mathcal {E}=T^*[2]T[1]\\mathcal {E}_0, \\quad H=d+\\eta , \\quad \\text{ for }\\eta \\in \\Omega ^3_{\\text{closed}}(\\mathcal {E}_0),$ with the standard symplectic form on the cotangent bundle.",
"Here $d$ is understood as a vector field on $T[1]\\mathcal {E}_0$ (thus a function on $T^*[2]T[1]\\mathcal {E}_0$ ), and $\\eta \\in \\Omega (M)\\cong C^\\infty (T[1]M)$ is pulled back to $T^*[2]T[1]\\mathcal {E}_0$ .",
"(Such $\\mathcal {E}$ is called exact.)",
"We fix the generalized metric by the requirement that the submanifold $\\mathcal {V}_1\\subset \\mathcal {E}_1\\cong (T\\oplus T^*)[1]\\mathcal {E}_0$ corresponds to the graph of a (pseudo Riemannian) metric $\\langle \\cdot ,\\cdot \\rangle $ on $\\mathcal {E}_0$ .",
"In particular, we have a vector-bundle isomorphism $T\\mathcal {E}_0\\cong V_+$ .",
"Combining this with the result of the previous section, a connection on $\\mathcal {V}^*$ with invariant torsion is now given by an ordinaryThe term “ordinary” refers here to the usual non-graded geometry.",
"connection on $T\\mathcal {E}_0$ .",
"Let us now take $Q$ given by the ordinary Levi-Civita connection on $T\\mathcal {E}_0$ w.r.t.",
"the metric on $\\mathcal {E}_0$ .",
"Choose a local frame $E_a$ on $T\\mathcal {E}_0$ satisfying $\\langle E_a,E_b\\rangle =\\pm \\delta _{ab}$ .",
"We define the frame $F_a$ on $T^*\\mathcal {E}_0$ by $F_a:=\\langle E_a,\\cdot \\rangle $ and we denote by $E^a$ , $F^a$ the induced fiber coordinates on $\\mathcal {E}_1\\cong (T\\oplus T^*)[1]\\mathcal {E}_0$ .",
"Finally, setting $e^a:=\\tfrac{1}{2} (E^a+F^a)$ , $e^{\\dot{b}}:=\\tfrac{1}{2} (E^b-F^b)$ , we have $\\operatorname{Ric}=\\operatorname{Ric}_{\\eta }(E_a,E_b)e^a e^{\\dot{b}},$ where $\\operatorname{Ric}_\\eta $ is the ordinary Ricci tensor on $T\\mathcal {E}_0$ for the metric connection (w.r.t.",
"$\\langle \\cdot ,\\cdot \\rangle $ ) with torsion given by $\\eta $ .",
"For the proof of this fact we refer the reader to [18].",
"It can also be verified by a direct calculation."
],
[
"Expression for the Ricci tensor",
"Using the fact that $Q^2_\\mathcal {E}=0$ , we have (for a general $Q$ ) $Q^2&=(Q_\\mathcal {E} \\psi ^a_{\\phantom{a}b\\alpha }e^\\alpha \\xi ^b - \\psi ^b_{\\phantom{b}c\\beta }e^\\beta \\xi ^c\\psi ^a_{\\phantom{a}b\\alpha }e^\\alpha )\\partial _{\\xi ^a}\\\\&=(\\rho ^i_\\beta e^\\beta \\psi ^a_{\\phantom{a}b\\alpha ,i }e^\\alpha \\xi ^b+\\rho ^i_\\alpha p_i \\psi ^{a\\phantom{b}\\alpha }_{\\phantom{a}b} \\xi ^b-\\tfrac{1}{2} c_{\\alpha \\beta \\gamma }e^\\beta e^\\gamma \\psi ^{a\\phantom{b}\\alpha }_{\\phantom{a}b}\\xi ^b-\\psi ^b_{\\phantom{b}c\\beta }e^\\beta \\xi ^c\\psi ^a_{\\phantom{a}b\\alpha }e^\\alpha )\\partial _{\\xi ^a}\\\\\\pi Q^2 &= (\\rho ^i_d \\psi ^a_{\\phantom{a}b\\dot{a},i}e^de^{\\dot{a}}\\xi ^b+\\rho ^i_{\\dot{a}}\\psi ^a_{\\phantom{a}bd,i}e^{\\dot{a}}e^d\\xi ^b-c_{\\alpha d \\dot{a}}\\psi ^{a\\phantom{b}\\alpha }_{\\phantom{a}b}e^de^{\\dot{a}}\\xi ^b+\\psi ^b_{\\phantom{b}cd}\\psi ^a_{\\phantom{a}b\\dot{a}}e^de^{\\dot{a}}\\xi ^c\\\\&+\\psi ^b_{\\phantom{b}c\\dot{a}}\\psi ^a_{\\phantom{a}bd}e^{\\dot{a}}e^d\\xi ^c)\\partial _{\\xi ^a}\\\\CQ^2&=(\\rho ^i_a\\psi ^a_{\\phantom{a}b\\dot{a},i}-\\rho ^i_{\\dot{a}}\\psi ^a_{\\phantom{a}ba,i}-c_{\\alpha a\\dot{a}}\\psi ^{a\\phantom{b}\\alpha }_{\\phantom{a}b}+\\psi ^c_{\\phantom{c}ba}\\psi ^a_{\\phantom{a}c\\dot{a}}-\\psi ^c_{\\phantom{c}b\\dot{a}}\\psi ^a_{\\phantom{a}ca})e^{\\dot{a}}\\xi ^b$ Using $\\psi ^a_{\\phantom{a}b\\dot{a}}=c^a_{\\phantom{a}b\\dot{a}}$ and $\\psi ^a_{\\phantom{a}ba}=\\lambda _b$ we recover (REF )."
]
] | 2001.03382 |
[
[
"Decision tool and Sample Size Calculator for Composite Endpoints"
],
[
"Abstract Summary points: - This article considers the combination of two binary or two time-to-event endpoints to form the primary composite endpoint for leading a trial.",
"- It discusses the relative efficiency of choosing a composite endpoint over one of its components in terms of: the frequencies of observing each component; the relative treatment effect of the tested therapy; and the association between both components.",
"- We highlight the very important role of the association between components in choosing the most efficient endpoint to use as primary.",
"- For better grounded future trials, we recommend trialists to always reporting the association between components of the composite endpoint.",
"- Common fallacies to note when using composite endpoints: i) composite endpoints always imply higher power; ii) treatment effect on the composite endpoint is similar to the average effects of its components; and iii) the probability of observing the primary endpoint increases significantly."
],
[
"INTRODUCTION",
"Choosing the primary endpoint (PE) in a randomized controlled trial (RCT) is a critical decision in the design phase.",
"Composite endpoints are defined as the occurrence of any of the relevant events in trials with binary response and as the time from randomization to the first observed event among all components in time-to-event studies.",
"Some examples are the composite of death or myocardial re-infarction and major adverse cardiovascular events (MACE) generally defined as a composite of cardiovascular death, myocardial infarction, stroke and target vessel revascularization.",
"What are the reasons for using a CE?",
"On the one hand, taking into account multiple outcomes altogether could provide a broader overview of an intervention's efficacy (1).",
"On the other hand, if the event rates are too low, it could increase the study power.",
"However, CEs also hide some drawbacks.",
"For instance, the tested intervention should have a similar effect on all the components while none of them should be affected negative (2).",
"Moreover, having more outcomes does not necessarily imply an increase in power: small intervention effects on one component could mask significant effects on others (3) or strong associations may reduce the joint information.",
"We discuss the gain in efficiency of adding endpoints to form a primary CE rather than simply using a single relevant endpoint.",
"We provide a multitask tool named CompARE that can be freely access through https://cinna.upc.edu/compare/ as an aid for a more well-informed decision.",
"The code of the functions implemented in this web app is available in GitHub at https://github.com/CompARE-Composite/Functions.",
"CompARE allows to check the efficiency of different designs with more than one outcome and provides a calculator to get the required sample size based on several CEs configurations among other features.",
"This will be illustrated through two RCT published in NEJM and European Heart Journal, respectively (4,5).",
"List of abbreviations ARE: Asymptotic Relative Efficiency $\\varepsilon _1$ , $\\varepsilon _2$ , $\\varepsilon _*$ : Relevant event, $\\varepsilon _1$ ; Additional event, $\\varepsilon _2$ ; Composite event, $\\varepsilon _*$ , defined as $\\varepsilon _1$ or $\\varepsilon _2$ .",
"HR: Hazard Ratio PE: Primary Endpoint OS: Overall survival MI: Myocardial infarction"
],
[
"RELATIVE EFFICIENCY OF USING COMPOSITE ENDPOINTS",
"Should we use the most relevant endpoint (such as overall survival, OS) as the PE for a two-arm RCT?",
"Or should we base our trial on a CE that includes an additional endpoint (such as myocardial re-infarction)?",
"Both endpoints are capable of measuring the treatment effect in the same direction, are of interest to the investigator but they are not equally efficient.",
"What does efficiency mean when it refers to an endpoint that may be chosen for testing a therapy or treatment?",
"According to the Oxford dictionary, a system is efficient when it “achieves maximum productivity with minimum wasted effort or expense\".",
"A hypothesis testing procedure based on a given endpoint is efficient when it can be influenced by the treatment achieving maximum power for a given sample size.",
"The efficiency is then a measure of the quality of a given test based on an endpoint, while the relative efficiency between two endpoints is determined by their efficiency ratio.",
"Since often we can only assess the efficiency of a statistical test for large enough sample sizes, we use the asymptotic relative efficiency (ARE) as the principal comparison measure.",
"This measure has been studied in a time-to-event framework (6) as well as in a binary setup (7) and it has been used (8) to assess, in the cardiovascular research area, the characteristics that a potential endpoint should have to be part of the CE leading the trial.",
"Suppose that, two potential endpoints could satisfactorily answer the study's primary clinical question: we could use a relevant event, say $\\varepsilon _1$ , or the composite event ($\\varepsilon _*$ ), say $\\varepsilon _1$ or $\\varepsilon _2$ , where $\\varepsilon _2$ is some additional endpoint.",
"The ARE measure can be roughly interpreted as the ratio of the required sample sizes using $\\varepsilon _1$ versus $\\varepsilon _*$ to attain the same power for a given significance level (9) yielding the following criterion: whenever ARE $>$ 1, choose $\\varepsilon _*$ as PE to guide the study; otherwise, use $\\varepsilon _1$ .",
"To calculate the ARE we need to anticipate: the frequencies of observing events $\\varepsilon _1$ and $\\varepsilon _2$ in the control group; the relative treatment effect given by either the hazard ratios (HR) –survival outcome– or risk differences –binary outcome– and a measure of the association between components.",
"In very general terms, one should use the CE (ARE$>$ 1) if the relative effect of treatment on the additional endpoint is greater or about the same as that on the relevant endpoint.",
"Nevertheless, other scenarios such as those for slightly smaller relative effect of treatment on the additional endpoint together with low frequencies of observing the relevant endpoint, might yield ARE$>$ 1, thus, making the composite endpoint more efficient.",
"Furthermore, as we discuss next, the association between $\\varepsilon _1$ and $\\varepsilon _2$ plays a major role when computing the ARE.",
"The ability to make a well-informed decision can be aided by a simple and friendly tool, such as CompARE, which allows calculating the ARE in various clinical trial scenarios."
],
[
"ASSOCIATION BETWEEN THE COMPONENTS OF THE COMPOSITE ENDPOINT",
"The association between $\\varepsilon _1$ and $\\varepsilon _2$ plays a major role when computing the ARE.",
"Two endpoints are positively associated when a large value of one implies a large value of the other, and vice versa.",
"The association between two binary outcomes can be measured by means of Pearson's correlation, although the conditional probability of one, given the other, might furnish a more natural and interpretable understanding of association.",
"When we wonder about the percentage of patients ends up suffering a stroke after having a myocardial infarction (MI), the interpretation is quite clear in terms of conditional probability.",
"The positive and linear relationship between the correlation coefficient and the conditional probability allows to use the estimated conditional probability to guess the correlation.",
"For instance, if the frequency of having a MI and suffering a stroke are roughly equal and very low (say, less than 5%), the correlation between them is approximately equal to the proportion of patients having a MI among those that had a stroke.",
"ARE insights The clinical question could be satisfactorily answered by events $\\varepsilon _1$ and $\\varepsilon _*$ = $\\varepsilon _1$ or $\\varepsilon _2$ , where $\\varepsilon _2$ is some additional endpoint.",
"ARE quantifies how much more efficient the composite endpoint $\\varepsilon _*$ is over $\\varepsilon _1$ as the primary endpoint of the study: If ARE $>$ 1, choose $\\varepsilon _*$ to guide the study; If ARE $<$ 1, base the study on $\\varepsilon _1$ .",
"To compute ARE, we need to anticipate: frequencies of observing $\\varepsilon _1$ and $\\varepsilon _2$ in the control group, relative treatment effect given by the hazard ratios (in a survival trial) or risk differences (in a binary trial).",
"In the survival setting, additional assumptions should be made.",
"a measure of the association between $\\varepsilon _1$ and $\\varepsilon _2$ .",
"The association between the time-to-event outcomes for $\\varepsilon _1$ and $\\varepsilon _2$ is a little bit “tricky\" in the presence of competing risks, when one outcome could prevent the observation of the other.",
"The ARE method is based on a joint copula model built on the cause specific hazards for times to $\\varepsilon _1$ and $\\varepsilon _2$ and on Spearman's rho correlation between these times.",
"We strongly remark here that the anticipation of Spearman's rho is only done for the purpose to obtain precise sample size calculations and not assumed to be used or computed in the analysis phase when it makes no sense to get the association between times to $\\varepsilon _1$ and $\\varepsilon _2$ when one of them is death.",
"It has been shown (6,7) that the efficiency of using a CE rather than using its more relevant component decreases as the composite components become more associated; or, in other words, the more correlated the components are, the larger the sample size needed for the trial (9) and using the CE might not be worthwhile.",
"The association between the components of the composite endpoint has a significant impact both on the probability of observing the composite endpoint and on the efficiency of the composite endpoint as a primary endpoint.",
"Therefore, it is of utmost importance to consider how the components of the composite endpoint are associated, quantify this association and report them in all documents and summaries derived from the trials."
],
[
"CASE STUDIES USING COMPARE",
"We illustrate the PE decision using two RCTs and with the aid of CompARE, which is useful for different purposes as shown in Figure 1.",
"CompARE is a comprehensive and freely available web-tool intended to provide guidance on how to deal with composite endpoints in the planning stage of a randomized controlled trial.",
"CompARE can also be used to: Choose the best primary endpoint to lead the trial.",
"CompARE computes the ARE to quantify the gain in the efficiency of using –as the primary endpoint– a composite endpoint over one of its components.",
"Specify the treatment effect for the composite endpoint based on the marginal information of the composite components and to study the performance of the composite parameters according to these.",
"In a survival trial, it can also evaluate the proportional hazards assumption for the composite endpoint.",
"Determine the sample size for different situations, such as when the association between composite components is unknown or when the hazards are not proportional.",
"Calculate and interpret the different association measures among the composite components.",
"The code to reproduce the following two case studies is available in GitHub.",
"Figure: CompARE scheme with the inputs to be provided by the researcher and the outputs returned by the application."
],
[
"TUXEDO trial: binary endpoints",
"The TUXEDO study was a RCT aimed at comparing paclitaxel-eluting stents (control group) with everolimus-eluting stents (intervention group) (4).",
"The PE was target-vessel failure ($\\varepsilon _*$ ), defined as a composite of cardiac death, target vessel MI or ischemia-driven target-vessel revascularization at 1-year of follow-up.",
"The secondary endpoints were, among others, the ischemia-driven target-lesion revascularization ($\\varepsilon _1$ , revascularization for short) and the endpoint of cardiac death or target-vessel MI ($\\varepsilon _2$ , death or MI for short).",
"The TUXEDO protocol was based on the results of the SPIRIT IV trial (10), which faced similar research questions and considered the same PE and that we will use as reference values for illustrative purposes.",
"The association between the different outcomes was not reported, nor was the number of patients who had more than one event.",
"Would it have been preferable to consider revascularization ($\\varepsilon _1$ ) instead of the CE of target-vessel failure ($\\varepsilon _*$ ) as PE?",
"We discuss it considering that the frequency of $\\varepsilon _1$ and $\\varepsilon _2$ in the control group were 5.9% and 3.2%, respectively.",
"Everolimus-eluting stents as compared with paclitaxel-eluting stents resulted in a 1.96 percentage-point absolute reduction in $\\varepsilon _1$ and a 0.98 percentage-point absolute reduction in $\\varepsilon _2$ .",
"Table: Probability of target-vessel failure (ε * \\varepsilon _*) and total sample size to achieve a power of 80% at a significance level of 0.05 in a design with ε * \\varepsilon _* as PE based on the SPIRIT IV trial values and for different degrees of association between ε 1 \\varepsilon _1 and ε 2 \\varepsilon _2Table 1 presents three scenarios according to weak, moderate, strong degrees of association between revascularization and death or MI.",
"The strength of association is provided both in terms of the correlation coefficient (0.1, 0.4, 0.7) and as the conditional probability of $\\varepsilon _2$ given a $\\varepsilon _1$ .",
"For each of these scenarios and based on the values provided by the SPIRIT IV trial, the probability of observing $\\varepsilon _*$ in the control group ranges between 6% and 8%; and the expected absolute reduction for $\\varepsilon _*$ takes values between 2.3 and 2.9.",
"Observe that both the probability and the expected effect for $\\varepsilon _*$ decrease as the association increases, and this pattern is always found (11,12).",
"We have also computed the required total sample size for a trial with $\\varepsilon _*$ as PE.",
"Notice that, as the association between the composite components gets larger, the sample size is increasingly higher.",
"Figure 2 shows that the efficiency of using $\\varepsilon _*$ as PE rather than $\\varepsilon _1$ decreases as the components become more correlated.",
"Indeed, revascularization is preferred over target-vessel failure when $\\varepsilon _1$ and $\\varepsilon _2$ are strongly associated.",
"To study the extent to which adding $\\varepsilon _2$ affects the PE choice, we consider several effect values for $\\varepsilon _2$ : for a given correlation, the ARE increases as the anticipated treatment effect in $\\varepsilon _2$ gets larger.",
"Figure: ARE values depending on the correlation (horizontal axis) and the treatment effect of the additional endpoint (colored lines)."
],
[
"OASIS-6 trial: time-to-event endpoints",
"Odlgren et al.",
"(5) tested fondaparinux treatment vs. placebo or unfractionated heparin, in a pre-specified subgroup of 2,867 patients not receiving reperfusion treatment in the OASIS-6 trial.",
"The primary outcome was the composite endpoint ($\\varepsilon _*$ ) of death or myocardial re-infarction at 30 days.",
"The observed frequency in the control group was 0.125 for OS ($\\varepsilon _1$ ) and 0.037 for myocardial re-infarction ($\\varepsilon _2$ ), and the estimated HRs were 0.83 for $\\varepsilon _1$ and 0.66 for $\\varepsilon _2$ .",
"If a new similar study was planned, should we use the composite endpoint ($\\varepsilon _*$ ) to lead the trial or should we stick to OS as the PE?",
"Figure 3 shows how CompARE can help to discuss the efficiency of using death or myocardial re-infarction as PE (see also Figure 4), to compare sample sizes and to illustrate effect sizes for several scenarios with different HRs (between 0.65 and 0.90) for $\\varepsilon _2$ and several degrees of association.",
"Figure: Screenshots of CompARE using the information from the OASIS-6 Trial.Figure 4 (self-explained) shows the influence of correlation on the ARE value and, consequently, can be used as a criterion to choose between OS or the CE formed by death ($\\varepsilon _1$ ) and Myocardial re-infarction ($\\varepsilon _2$ ).",
"The ARE values (vertical axis) depend on the Spearman correlation (horizontal axis) and the treatment effect of the additional endpoint (colored lines).",
"Cause-specific hazard function for the time to Myocardial re-infarction ($\\varepsilon _2$ ) and the time to death ($\\varepsilon _1$ ) are assumed constant during the follow-up.",
"When the PE is based on the time from randomization to $\\varepsilon _*$ , the behavior of the corresponding cause-specific hazards has to be as well anticipated.",
"Table 2 presents the ARE and the required sample size under five selected scenarios, assuming constant, increasing or decreasing hazards for each outcome and strong correlation between them.",
"ARE is greater than 1 in all scenarios and hence $\\varepsilon _*$ is always recommended; however sample sizes vary a lot – in a range from 2,682 to 3,381 – exemplifying the impact of the hazard shape over time.",
"Figure: ARE values depending on the correlation (horizontal axis) and the treatment effect of the additional endpoint (colored lines).Table: ARE and the required sample size to achieve an 80% power at a significance level of 0.05 for detecting differences in the CE, depending on the cause-specific hazard behavior over time for each component: constant (exponential distribution), increasing (Weibull distribution with shape parameter equals 2) or decreasing (Weibull with shape parameter equals 0.5).",
"Other assumed parameters: i) frequencies of 0.125 and 0.05 for death and myocardial re-infarction in the control arm; ii) HRs of 0.83 and 0.66 for OS and myocardial re-infarction; and iii) Spearman's rho correlation between components equal to 0.7, considering Gumbel copula.",
"Sample size calculations are based on the Freedman formula."
],
[
"DISCUSSION",
"Composite endpoints are commonly used in clinical trials and observational studies.",
"However, contrary to popular belief, composite endpoints only barely increase total event-probability and do not necessarily yield higher power.",
"The design of a randomized controlled trial involving a composite endpoint needs a careful specification of the expected rates, the treatment effects and the correlation between the components.",
"While the frequencies and relative effects of the primary and secondary endpoints in most of the published clinical trials can often be derived from previous studies, the degree of association between them are seldom disclosed.",
"CompARE can be used to explore different scenarios and for each one calculate the treatment effect for the CE and the needed sample size.",
"Because the association between the composite components plays an important role in the overall expected frequency of the CE, but also in the statistical power of the study, this information is critical in designing a trial, it has a large impact on the design efficiency, notably on the PE choice and, hence, on the trial's sample size.",
"For example, the SAMPL guidelines (13) enumerate the items to be reported for each type of analysis, but currently do not make any reference to this point.",
"We urge trialists to report, as accurately as possible, all the primary and secondary parameter information as well as a measure of the association between the endpoints considered in the trial."
],
[
"Acknowledgements",
"This work is partially supported through grant MTM2015-64465-C2-1-R (MINECO/FEDER, UE) from the SecretarÃa de Estado de Investigación, Desarrollo e Innovación del Ministerio de EconomÃa y Competitividad (Spain) and through grant 2017 SGR 622 (GRBIO) from the Departament d'Economia i Coneixement de la Generalitat de Catalunya (Spain).",
"Marta Bofill Roig acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the MarÃa de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445).",
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]
] | 2001.03396 |
[
[
"Parameter learning and fractional differential operators: application in\n image regularization and decomposition"
],
[
"Abstract In this paper, we focus on learning optimal parameters for PDE-based image regularization and decomposition.",
"First we learn the regularization parameter and the differential operator for gray-scale image denoising using the fractional Laplacian in combination with a bilevel optimization problem.",
"In our setting the fractional Laplacian allows the use of Fourier transform, which enables the optimization of the denoising operator.",
"We prove stable and explainable results as an advantage in comparison to other machine learning approaches.",
"The numerical experiments correlate with our theoretical model setting and show a reduction of computing time in contrast to the ROF model.",
"Second we introduce a new image decomposition model with the fractional Laplacian and the Riesz potential.",
"We provide an explicit formula for the unique solution and the numerical experiments illustrate the efficiency."
],
[
"Introduction",
"Your text comes here.",
"Separate text sections with"
],
[
"Section title",
"Text with citations [2] and [1]."
],
[
"Subsection title",
"as required.",
"Don't forget to give each section and subsection a unique label (see Sect. ).",
"Use paragraph headings as needed.",
"$a^2+b^2=c^2$ Figure: Please write your figure caption hereFigure: Please write your figure caption hereTable: Please write your table caption here"
]
] | 2001.03394 |
[
[
"A rasterized ray-tracer pipeline for real-time, multi-device sonar\n simulation"
],
[
"Abstract Simulating sonar devices requires modeling complex underwater acoustics, simultaneously rendering time-efficient data.",
"Existing methods focus on basic implementation of one sonar type, where most of sound properties are disregarded.",
"In this context, this work presents a multi-device sonar simulator capable of processing an underwater scene by a hybrid pipeline on GPU: Rasterization computes the primary intersections, while only the reflective areas are ray-traced.",
"Our proposed system launches few rays when compared to a full ray-tracing based method, achieving a significant performance gain without quality loss in the final rendering.",
"Resulting reflections are then characterized as two sonar parameters: Echo intensity and pulse distance.",
"Underwater acoustic features, such as speckle noise, transmission loss, reverberation and material properties of observable objects are also computed in the final generated acoustic image.",
"Visual and numerical performance assessments demonstrated the effectiveness of the proposed simulator to render underwater scenes in comparison to real-world sonar devices."
],
[
"Introduction",
"The number of underwater structures in the offshore industry has significantly increased over the last decades, and so the need of monitoring, inspection and intervention of these structures .",
"Since autonomy is necessary to reduce mission expenses, the offshore industry has leading the development of autonomous underwater vehicles (AUVs) to accomplish the main field tasks.",
"With a pre-programmed mission and onboard sensors, AUVs are able to perform completely autonomous decisions, returning to surface only for servicing.",
"The accomplishment of AUV tasks demands to deal with challenges inherent to undersea environment.",
"For instance, beneath the water, optical cameras are affected by turbidity and lightning conditions, thus restricting the image quality to short visible ranges.",
"On the other hand, imaging sonars take advantage of the low attenuation of sound waves in order to cover larger areas than those ones covered by optical cameras, although producing noisy data with low resolution.",
"AUV real-world experimentation is challenging, mainly due to human resources, time consumption and hazards involved on deployment and testing the underwater vehicles in the target domain.",
"While initial experiments can be performed in water tanks (e.g., low-level control and basic prototyping), high-level tests require trials in deep open waters (e.g., way-point navigation, mapping and autonomous control).",
"An unexpected behavior of an AUV may result in an unrecoverable equipment, causing a considerable financial loss.",
"This way, simulation of underwater sensors and reproducible environments is essential to cope with insufficient data, as well as to develop effective algorithms before tests in the wild.",
"To contribute with the development of underwater acoustic-based systems, this paper introduces a novel simulator able to reproduce the operation of different sonar devices.",
"Rendering of a virtual scene is accelerated by a selective rasterization and ray-tracing scheme on GPU, where the computational resources are allocated only for reflective regions.",
"Subsequently, the resulting reflections are converted to the acoustic scene representation on CPU, including several phenomena present on the sonar images.",
"By considering the complexity in the process of transmitting sound through the water, several mathematical and computational models have been proposed to approximate the calculation of acoustic propagation .",
"Ray-based methods are the most common solutions to simulate underwater sonar systems , , , , , , , , although other approaches can also be considered , , .",
"All simulation methods try to mimic one or more types of sonar devices.",
"Side scan sonar (SSS) simulation: presented a simulator for SSS imagery based on optical ray-tracing, where a group of rays is projected to insonify the scene and produce the acoustic data; fractal models are used to represent the roughness surface of the seafloor; stochastic influences as noise and reverberation are neglected in that work.",
"Instead of propagating many individual rays, developed a volume-based approach with a tube tracing technique; the tubes are composed of four rays, which intersect a certain area to allow computing the backscattered energy; the few launched rays optimized the sonar rendering, while the surface details and transmitting signal characteristics are suppressed.",
"By using a frequency domain-based method, produced frames from a virtual SSS by using Fourier transform; the returned intensity relies on the angle of incidence applied to a basic Lambert illumination model; physical effects, such as noise and multi-path returns, are considered, although the method was not designed to operate online.",
"With a simplified Lambert diffusion model, generated SSS data integrated with UWSim simulator and ROS framework; acoustic frames are degraded with speckle (low frequency) and Rayleigh (high frequency) noises; although the performance of feature matching methods decays in images containing multiplicative noise, due to the variance for intensity and affine changes, the authors applied SIFT, SURF, ORB and AKAZE algorithms to evaluate the similarity between two consecutive frames, obtaining a very low number of inliers for all feature extractors.",
"Forward-looking sonar (FLS) simulation: modeled an FLS system, where the rays are comprised of basic lines, equivalent to the number of pixels of sonar image to be emulated; the reflection representation is severally reduced to three colors only (black, gray and white).",
"improved Gu et al.",
"'s method by introducing sound attenuation effect in order to produce gray-scale sonar images; by assuming a mirror-like reflection model, the sonar system only considers specular reflections, so that the method is only successful for smooth surfaces.",
"introduced an acoustic model by combining ray-tracing in frequency domain; the intensity and the range of sonar data are calculated by Lambert diffusion model and Euclidean distance respectively; the high average time to compute a single FLS frame prevents the use of the method in real-time operations.",
"detailed an FLS simulator integrated with Gazebo simulator and ROS for diving assistance; the ray path mimics the sound wave to generate a point cloud; the simulated images are compared with real ones, although the reflectivity of the objects and the noise models are analytically defined.",
"conceived a simulator based on ray propagation to produce acoustic data; by assuming only the freshwater component, the sound attenuation is partially considered, while other physical properties of sound are ignored; time consumption to calculate one single frame has not been well established by .",
"Mechanical scanning imaging sonar (MSIS) simulation: fused the ray-tracing and additive noise models, proposed in and , respectively, to produce single beam data; in that work, no image distortion induced by robot movement was considered; the simulated frames were later used to feed an underwater localization system based on Hilbert maps.",
"introduced a GPU-based simulator to reproduce the operation of two sonar devices; by deferred shading, the rasterization rendering was exploited to compute the acoustic parameters (i.e., echo intensity, pulse distance and azimuth angle); sound phenomena such as multiplicative noise and material properties were addressed, while multipath returns, attenuation and additive noise did not; experiments comparing real-world acoustic images certified the use of the simulator by real-time applications.",
"Table: References"
]
] | 2001.03539 |
[
[
"One-body entanglement as a quantum resource in fermionic systems"
],
[
"Abstract We show that one-body entanglement, which is a measure of the deviation of a pure fermionic state from a Slater determinant (SD) and is determined by the mixedness of the single-particle density matrix (SPDM), can be considered as a quantum resource.",
"The associated theory has SDs and their convex hull as free states, and number conserving fermion linear optics operations (FLO), which include one-body unitary transformations and measurements of the occupancy of single-particle modes, as the basic free operations.",
"We first provide a bipartitelike formulation of one-body entanglement, based on a Schmidt-like decomposition of a pure $N$-fermion state, from which the SPDM [together with the $(N-1)$-body density matrix] can be derived.",
"It is then proved that under FLO operations, the initial and postmeasurement SPDMs always satisfy a majorization relation, which ensures that these operations cannot increase, on average, the one-body entanglement.",
"It is finally shown that this resource is consistent with a model of fermionic quantum computation which requires correlations beyond antisymmetrization.",
"More general free measurements and the relation with mode entanglement are also discussed."
],
[
"Introduction",
"Quantum entanglement and identical particles are two fundamental concepts in quantum mechanics.",
"Entanglement in systems of distinguishable components is particularly valuable in the field of quantum information theory [1] because it can be considered as a resource within the Local Operations and Classical Communication (LOCC) paradigm [1], [2].",
"Extending the notion of entanglement to the realm of indistinguishable particles is, however, not straightforward because the constituents of the system cannot be individually accessed.",
"Different approaches have been considered, like mode entanglement [3], [4], [5], where subsystems correspond to a set of single-particle (SP) states in a given basis, extensions based on correlations between observables [6], [7], [8], [9], [10] and entanglement beyond symmetrization [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], which is independent of the choice of SP basis.",
"Several studies on the relation between these types of entanglement [5], [16], [20], [22], [23], [24], [25], [26], [27], [28] and on whether exchange correlations can be associated with entanglement [29], [30], [31], [32], [33] have been recently made.",
"There is also a growing interest in quantum chemistry simulations based on optical lattices [34], [35], which would benefit from a detailed characterization of fermionic correlations.",
"In this paper we will focus on entanglement beyond antisymmetrization in fermionic systems and analyze its consideration as a quantum resource.",
"Quantum resource theories [36], [37] have recently become a topic of great interest since they essentially describe quantum information processing under a restricted set of operations.",
"Standard entanglement theory in systems of distinguishable components is just one of these theories, amongst which we may include others like quantum thermodynamics [38], [39], coherence [40], [41], nonlocality [42] and non-Gaussianity [43].",
"In the usual entanglement theory a multipartite quantum system shared by distant parties is considered.",
"These parties can operate each on their own subsystem and are allowed to communicate via classical channels [2].",
"From these restrictions the LOCC set arises naturally as the set of free operations of the resource theory, and the set of free (separable) states is then derived.",
"In our case ignoring antisymmetrization correlations defines Slater determinants (SDs) and their convex hull as the set of “free” states $\\cal S$ and we are looking for a set of free operations $\\cal O$ consistent with this set.",
"With this aim, we first define a partial order relation on the Fock space $\\cal F$ of the system, based on the mixedness of the corresponding single-particle density matrix (SPDM) $\\rho ^{(1)}$ [also denoted as the one-particle or one-body density matrix (DM)], which determines whether a given pure fermionic state can be considered more entangled than another state.",
"A bipartite like formulation for this one-body entanglement, involving $\\rho ^{(1)}$ and the $(N-1)$ -body density matrix (isospectral for pure states of $N$ fermions) is also provided.",
"Next we define a class of operations consistent with $\\cal S$ and the previous partial order, through a majorization relation to be fulfilled by the initial and final SPDMs, which ensures that one-body entanglement will not be increased by such operations.",
"We then show that number conserving Fermion linear optics (FLO) operations [44], [45], [46], which include one-body unitary transformations and measurement of the occupancy of a SP state, are indeed within this class.",
"One-body entanglement then plays the role of a resource in a theory where $\\cal S$ is the convex hull of SDs and $\\cal O$ is that of FLO operations.",
"Possible extensions of the set of free operations and connection of this resource with a quantum computation model and with mode entanglement are also discussed.",
"We consider a SP space ${\\cal H}$ of finite dimension $n$ and a set of fermion creation and annihilation operators $c^\\dag _k$ and $c_k$ associated with an orthogonal basis of ${\\cal H}$ , satisfying the anticommutation relations $\\lbrace c_k,c_{k^{\\prime }}^\\dag \\rbrace =\\delta _{kk^{\\prime }}$ , $\\lbrace c_k,c_{k^{\\prime }}\\rbrace =\\lbrace c^\\dag _k,c^\\dag _{k^{\\prime }}\\rbrace =0$ .",
"The elements of the SPDM $\\rho ^{(1)}$ in a general fermionic state $\\rho $ are given by $\\rho ^{(1)}_{kk^{\\prime }}=\\langle c^\\dag _{k^{\\prime }} c_k\\rangle ={\\rm Tr}\\,\\rho \\, c^\\dag _{k^{\\prime }} c_k.$ They form a Hermitian matrix with eigenvalues $\\lambda _\\nu =\\langle c^\\dag _\\nu c_\\nu \\rangle \\in [0,1]$ , where $c^\\dag _\\nu =\\sum _k U_{k\\nu }c^\\dag _k$ creates a fermion in one of the “natural” SP orbitals diagonalizing $\\rho ^{(1)}$ ($\\langle c^\\dag _{\\nu ^{\\prime }}c_{\\nu }\\rangle =\\lambda _\\nu \\delta _{\\nu \\nu ^{\\prime }}$ ).",
"For pure states $\\rho =|\\Psi \\rangle \\langle \\Psi |$ , the “mixedness” of $\\rho ^{(1)}$ then reflects the deviation of $|\\Psi \\rangle $ from a SD $[\\prod _\\nu (c^\\dag _\\nu )^{n_\\nu }]|0\\rangle $ , since for the latter $\\lambda _\\nu =n_\\nu =0$ or 1 $\\forall \\, \\nu $ and hence $(\\rho ^{(1)})^2=\\rho ^{(1)}$ .",
"Such mixedness can be rigorously characterized through majorization [47], [48], [49], [50].",
"For states $|\\Psi \\rangle $ and $|\\Phi \\rangle $ with the same fermion number $N={\\rm Tr}\\,\\rho ^{(1)}_{\\Psi }={\\rm Tr}\\,\\rho ^{(1)}_{\\Phi }$ , we will say that $|\\Psi \\rangle $ is not less one-body entangled than $|\\Phi \\rangle $ if $\\rho ^{(1)}_{\\Psi }$ is more (or equally) mixed than $\\rho ^{(1)}_{\\Phi }$ , i.e.",
"if their eigenvalues $\\mathbf {\\lambda }=(\\lambda _1,\\ldots ,\\lambda _n)$ , sorted in decreasing order, satisfy the majorization relation $\\mathbf {\\lambda }(\\rho ^{(1)}_{\\Psi })\\prec \\mathbf {\\lambda }(\\rho ^{(1)}_{\\Phi })\\,,$ which means $\\sum _{\\nu =1}^m\\lambda _\\nu (\\rho _\\Psi ^{(1)})\\le \\sum _{\\nu =1}^m\\lambda _\\nu (\\rho ^{(1)}_\\Phi )$ for $m=1,\\ldots ,n-1$ , with identity for $m=n$ .",
"Thus SDs are the least one-body entangled states, as their SPDM majorizes any other $\\rho ^{(1)}$ with the same trace.",
"Relation (REF ) is analogous to that imposed by LOCC operations on reduced states of systems of distinguishable components, which in the bipartite case lead to the celebrated Nielsen's theorem: $|\\Psi _{AB}\\rangle $ can be converted by LOCC to $|\\Phi _{AB}\\rangle $ (and hence is not less entangled than $|\\Phi _{AB}\\rangle $ ) if and only if their reduced states satisfy $\\mathbf {\\lambda }(\\rho ^{A(B)}_{\\Psi })\\prec \\mathbf {\\lambda }(\\rho ^{A(B)}_{\\Phi })$ [51], [48].",
"Local measurements reduce the ignorance about the state of the measured subsystem, decreasing the mixedness of reduced states and hence bipartite entanglement.",
"Similarly, we will show that one-body entanglement will decrease under operations which reduce the ignorance about the SPDM.",
"We first remark that one-body entanglement also admits a bipartite like formulation: A pure state $|\\Psi \\rangle $ of $N$ fermions ($\\sum _k c^\\dag _k c_k|\\Psi \\rangle =N|\\Psi \\rangle $ ) can be expanded as $|\\Psi \\rangle &=&\\frac{1}{N}\\sum _{k,l}\\Lambda _{kl}c^\\dag _kC^\\dag _l|0\\rangle $ where $C^\\dag _l=c^\\dag _{l_1}\\ldots c^\\dag _{l_{N-1}}$ , $l=1,\\ldots ,(^{\\;\\;\\;n}_{N-1})$ , are operators creating $N-1$ fermions in specific SP states labeled by $l$ , satisfying $\\langle 0|C_l C^\\dag _{l^{\\prime }}|0\\rangle =\\delta _{ll^{\\prime }}$ , while the coefficients $\\Lambda _{kl}$ form an $n\\times (^{\\;\\;\\;n}_{N-1})$ matrix $\\Lambda $ satisfying ${\\rm Tr}\\,\\Lambda \\Lambda ^\\dag =N$ .",
"Thus, each term in the sum (REF ) is a SD which is repeated $N$ times, such that $c_k|\\Psi \\rangle = \\sum _{l}\\Lambda _{kl}C^\\dag _l|0\\rangle $ is the (unnormalized) state of remaining fermions when SP state $k$ is occupied, while $C_l|\\Psi \\rangle =(-1)^{N-1}\\sum _k \\Lambda _{kl}c^\\dag _k|0\\rangle $ is that of remaining fermion when the $N-1$ SP states $l$ are occupied.",
"In this way, $\\langle \\Psi |\\Psi \\rangle =\\frac{1}{N}{\\rm Tr}\\,\\Lambda \\Lambda ^\\dag =1$ .",
"Moreover, Eqs.",
"(REF )–(REF ) allow us to express the elements of both the SPDM $\\rho ^{(1)}$ and the $(N-1)$ -body DM $\\rho ^{(N-1)}$ in terms of $\\Lambda $ as $\\rho ^{(1)}_{kk^{\\prime }}&=&\\langle \\Psi |c^\\dag _{k^{\\prime }} c_k|\\Psi \\rangle =(\\Lambda \\Lambda ^\\dag )_{kk^{\\prime }}\\,,\\\\\\rho ^{(N-1)}_{ll^{\\prime }}&=&\\langle \\Psi | C^\\dag _{l^{\\prime }} C_l|\\Psi \\rangle =(\\Lambda ^T\\Lambda ^*)_{ll^{\\prime }}\\,.$ Eqs.",
"(REF )–() are analogous to those for the reduced states $\\rho ^{A(B)}$ of distinguishable subsystems in a standard pure bipartite state $|\\Psi _{AB}\\rangle =\\sum _{i,j}C_{ij}|i_A,j_B\\rangle $ , where $\\rho ^{A}_{ii^{\\prime }}=\\langle |i^{\\prime }_A\\rangle \\langle i_A|\\rangle =(CC^\\dag )_{ii^{\\prime }}$ , $\\rho ^{B}_{jj^{\\prime }}=\\langle |j^{\\prime }_B\\rangle \\langle j_B|\\rangle =(C^{_T}C^*)_{jj^{\\prime }}$ [1].",
"The only difference is that ${\\rm Tr}\\,\\rho _{A(B)}={\\rm Tr}\\,CC^\\dag =1$ whereas ${\\rm Tr}\\,\\rho ^{(1)}={\\rm Tr}\\,\\rho ^{(N-1)}=N$ .",
"Eqs.",
"(REF )–() imply that $\\rho ^{(1)}$ and $\\rho ^{(N-1)}$ have the same nonzero eigenvalues $\\lambda _\\nu $, which are just the square of the singular values of $\\Lambda $ .",
"Moreover, by means of the singular value decomposition $\\Lambda =UDV^\\dag $ , with $D_{\\nu \\nu ^{\\prime }}=\\sqrt{\\lambda _\\nu }\\delta _{\\nu \\nu ^{\\prime }}$ and $U$ and $V$ unitary matrices (of $n\\times n$ and $(^{\\;\\;\\;n}_{N-1})\\times (^{\\;\\;\\;n}_{N-1})$ respectively), we may now obtain from (REF ) the 1–$(N-1)$ Schmidt-like decomposition of the $N$ -fermion state: $|\\Psi \\rangle &=&\\frac{1}{N}\\sum _\\nu \\sqrt{\\lambda _\\nu } c^\\dag _\\nu C^\\dag _\\nu |0\\rangle \\,, $ where $c^\\dag _\\nu =\\sum _{k}U_{k\\nu }c^\\dag _k\\;\\;\\;\\;, \\;\\;\\;\\;\\;C^\\dag _\\nu =\\sum _l V_{l\\nu }^*C^\\dag _l$ are the “natural” one- and $N-1$ - fermion creation operators satisfying $\\langle 0|c_\\nu c^\\dag _{\\nu ^{\\prime }}|0\\rangle &=&\\delta _{\\nu \\nu ^{\\prime }}=\\langle 0|C_\\nu C^\\dag _{\\nu ^{\\prime }}|0\\rangle \\,.\\\\\\langle \\Psi |c^\\dag _\\nu c_{\\nu ^{\\prime }}|\\Psi \\rangle &=&\\lambda _\\nu \\delta _{\\nu \\nu ^{\\prime }}= \\langle \\Psi |C^\\dag _\\nu C_{\\nu ^{\\prime }}|\\Psi \\rangle \\,.$ Thus, $c_\\nu |\\Psi \\rangle =\\sqrt{\\lambda _\\nu }C^\\dag _\\nu |0\\rangle \\,, \\;\\;\\;C_\\nu |\\Psi \\rangle =(-1)^{N-1}\\sqrt{\\lambda _\\nu }c^\\dag _\\nu |0\\rangle \\,,$ i.e.",
"the orthogonal natural $N-1$ -fermion states $C^\\dag _\\nu |0\\rangle $ are those of remaining fermions when the natural SP orbital $\\nu $ is occupied, while $c^\\dag _\\nu |0\\rangle $ are the orthogonal states of the remaining fermion when the natural $N-1$ -fermion state $C^\\dag _{\\nu }|0\\rangle $ (which in general is no longer a SD) is occupied.",
"Therefore, in an $N$ -fermion state one-body entanglement is actually the 1–$(N-1)$ -body entanglement, associated with the correlations between one- and $N-1$ -body observables.",
"In the case of a SD $|\\Psi \\rangle =(\\prod _{\\nu =1}^N c^\\dag _\\nu )|0\\rangle $ , $\\lambda _\\nu =1$ (0) for $\\nu \\le N$ $(>N)$ , with $C^\\dag _\\nu \\propto \\prod _{\\nu ^{\\prime }\\ne \\nu }^N c^\\dag _{\\nu ^{\\prime }}$ such that $c^\\dag _\\nu C^\\dag _\\nu |0\\rangle =|\\Psi \\rangle $ for $\\nu \\le N$ .",
"On the other hand, for $N=2$ Eq.",
"(REF ) becomes the Slater decomposition of a two-fermion state [11], [12], [13], $|\\Psi \\rangle =\\sum _{\\nu }\\sqrt{\\lambda _\\nu }c^\\dag _\\nu c^\\dag _{\\bar{\\nu }}|0\\rangle =\\frac{1}{2}\\sum _\\nu \\sqrt{\\lambda _\\nu }(c_\\nu C^\\dag _\\nu +c_{\\bar{\\nu }}C^\\dag _{\\bar{\\nu }})|0\\rangle $ where $C^\\dag _\\nu =c^\\dag _{\\bar{\\nu }}$ , $C^\\dag _{\\bar{\\nu }}=-c^\\dag _\\nu $ .",
"In this case one-body entanglement is directly related to that between the set of normal $\\nu $ and $\\bar{\\nu }$ modes, which contain each just one-fermion (see sec.",
"REF )."
],
[
"One-body entanglement entropies",
"We may now define a general one-body entanglement entropy $E(|\\Psi \\rangle )\\equiv E^{(1)}(|\\Psi \\rangle )$ as $E(|\\Psi \\rangle )=S(\\rho ^{(1)}_{\\Psi })=S(\\rho ^{(N-1)}_{\\Psi })\\,,$ where $S(\\rho ^{(1)})$ is a Schur-concave function [49], [50] of $\\rho ^{(1)}$ .",
"These entropies will all satisfy $E(|\\Psi \\rangle )\\ge E(|\\Phi \\rangle )\\,,$ whenever the majorization relation of Eq.",
"(REF ) is fulfilled.",
"For instance, trace-form entropies $S(\\rho ^{(1)})={\\rm Tr}f(\\rho ^{(1)})=\\sum \\limits _\\nu f(\\lambda _\\nu )$ where $f:[0,1]\\rightarrow \\mathbb {R}$ is concave and satisfies $f(0)=f(1)=0$ [52], will fulfill (REF ), with $E(|\\Psi \\rangle )\\ge 0$ $\\forall $ $|\\Psi \\rangle $ and $E(|\\Psi \\rangle )=0$ if and only if $|\\Psi \\rangle $ is a SD.",
"Such $E(|\\Psi \\rangle )$ will then be one-body entanglement monotones.",
"Examples are the von Neumann entropy of $\\rho ^{(1)}$ , $S(\\rho ^{(1)})=-\\sum _\\nu \\lambda _\\nu \\log _2\\lambda _\\nu $ , a quantity of interest in various fields [53], [54], [55], [56], [57], and the one-body entropy [20], [25] $S_1(\\rho ^{(1)})=-\\sum _\\nu \\lambda _\\nu \\log _2\\lambda _\\nu +(1-\\lambda _\\nu )\\log _2(1-\\lambda _\\nu ),$ which represents, for $|\\Psi \\rangle $ of definite fermion number $N$ , the minimum relative entropy (in the grand canonical ensemble) between $\\rho =|\\Psi \\rangle \\langle \\Psi |$ and any fermionic Gaussian state $\\rho _g$ : $\\!S_1(\\rho ^{(1)}_{\\Psi })=\\mathop {\\rm Min}_{\\rho _g}S(\\rho ||\\rho _g)$ [25], for $S(\\rho ||\\rho ^{\\prime })=-{\\rm Tr}\\rho (\\log _2\\rho ^{\\prime }-\\log _2\\rho )$ and $\\rho _g\\propto \\exp [-\\sum _{k,k^{\\prime }}\\alpha _{kk^{\\prime }}c^\\dag _k c_{k^{\\prime }}]$ (pair creation and annihilation terms in $\\rho _g$ are not required for such $|\\Psi \\rangle $ [25]).",
"It is also the minimum over all SP bases of the sum of all single mode entropies [20] $-p_k\\log _2 p_k-(1-p_k)\\log _2 (1-p_k)$ , where $p_k=\\langle c^\\dag _k c_k\\rangle $ .",
"We remark that the SPDM and hence any measure (REF ) are in principle experimentally accessible.",
"Measurement of the fermionic SPDM in optical lattices has been recently reported [58].",
"All measures (REF ) can be extended to mixed states $\\rho =\\sum _\\alpha p_\\alpha |\\Psi _\\alpha \\rangle \\langle \\Psi _\\alpha |$ of definite $N$ through their convex roof extension $E(\\rho )={\\rm Min}\\sum _\\alpha p_\\alpha E(|\\Psi _\\alpha \\rangle )$ , where the minimum is over all representations $\\lbrace p_\\alpha \\ge 0,\\,|\\Psi _\\alpha \\rangle \\rbrace $ of $\\rho $ [20].",
"Such $E(\\rho )$ represents a one-body entanglement of formation, vanishing if and only if $\\rho $ is a convex mixture of SDs."
],
[
"Definition and basic properties",
"We now define a class of operations which do not generate one-body entanglement, i.e., which do not increase, on average, the mixedness of the SPDM.",
"Definition 1.",
"Let $\\varepsilon (\\rho )=\\sum _j {\\cal K}_j\\rho {\\cal K}_j^\\dagger $ be a quantum operation on a fermion state $\\rho $ , with $\\lbrace {\\cal K}_j,\\,\\sum _j {\\cal K}_j^\\dag {\\cal K}_j=\\mathbb {1}\\rbrace $ a set of Kraus operators, assumed number conserving.",
"Let $\\rho ^{(1)}$ and $\\rho ^{(1)}_j$ be the SPDMs determined by $\\rho $ and $\\rho _j={\\cal K}_j\\rho {\\cal K}^\\dagger _j/p_j$ , with $p_j={\\rm Tr}[\\rho {\\cal K}^\\dagger _j {\\cal K}_j]$ .",
"We say that $\\varepsilon $ is one-body entanglement nongenerating (ONG) if it admits a set of Kraus operators $\\lbrace {\\cal K}_j\\rbrace $ satisfying $\\forall \\,\\rho $ the relation $\\mathbf {\\lambda }(\\rho ^{(1)})\\prec \\sum _j p_j\\, \\mathbf {\\lambda }(\\rho _j^{(1)})\\,,$ where eigenvalues $\\mathbf {\\lambda }(\\rho ^{(1)}_j)$ are sorted in decreasing order.",
"This majorization relation is analogous to that satisfied by reduced local states under local operations in the standard entanglement theory [48] and implies $S(\\rho ^{(1)})\\ge S\\left[\\sum _j p_j\\mathbf {\\lambda }(\\rho ^{(1)}_j)\\right]\\ge \\sum _j p_j S(\\rho ^{(1)}_{j})\\,,$ for any concave entropy $S(\\rho ^{(1)})$ , such as those of Eq.",
"(REF ).",
"For pure states $\\rho =|\\Psi \\rangle \\langle \\Psi |$ , $\\rho _j=|\\Phi _j\\rangle \\langle \\Phi _j|$ is also pure $\\forall j$ , with $|\\Phi _j\\rangle \\propto {\\cal K}_j|\\Psi \\rangle $ , and Eqs.",
"(REF ), (REF ) imply $E(|\\Psi \\rangle )\\ge \\sum _j p_j E(|\\Phi _j\\rangle )\\ge E(\\varepsilon (|\\Psi \\rangle \\langle \\Psi |)) \\,,$ showing that any one-body entanglement monotone (REF ) will not increase, on average, after ONG operations.",
"In particular, if $|\\Psi \\rangle $ is a SD, $E(|\\Psi \\rangle )=0$ and Eq.",
"(REF ) implies that all states $|\\Phi _j\\rangle \\propto {\\cal K}_j|\\Psi \\rangle $ must be SDs or zero, i.e.",
"all Kraus operators fulfilling (REF ) should map free states onto free states.",
"And for the one-body entanglement of formation of general mixed states $\\rho $ , Eq.",
"(REF ) implies $E(\\rho )\\ge E(\\varepsilon (\\rho ))$ since by using the minimizing representation, $E(\\rho )=\\sum _\\alpha p_\\alpha E(|\\Psi _\\alpha \\rangle )\\ge \\sum _{\\alpha ,j}p_\\alpha p_{\\alpha j}E(|\\Phi _{\\alpha j}\\rangle )\\ge E(\\varepsilon (\\rho ))$ .",
"It also follows from (REF ) that the set of ONG operations is convex and closed under composition, i.e., $\\mathbf {\\lambda }(\\rho ^{(1)})\\prec \\sum _{i,j} p_{ij}\\mathbf {\\lambda }(\\rho ^{(1)}_{ij})$ for ${\\cal K}_{ij}={\\cal K}^b_i{\\cal K}^a_j$ and $\\varepsilon (\\rho )=\\varepsilon ^b[\\varepsilon ^a(\\rho )]$ .",
"This property ensures that ONG operations can be applied any number of times in any order.",
"Proposition 1 The conversion of a pure state $|\\Psi \\rangle \\in {\\cal F}$ into another pure state $|\\Phi \\rangle \\in {\\cal F}$ by means of ONG operations is possible only if the majorization relation (REF ) is satisfied by the corresponding SPDMs.",
"The state conversion will consist in some sequence of ONG operations, which can be resumed in just one ONG operation due to the closedness under composition.",
"Let $\\lbrace {\\cal K}_j\\rbrace $ be a set of associated Kraus operators satisfying (REF ).",
"After this operation is performed, we should have ${\\cal K}_j|\\Psi \\rangle =\\sqrt{p_j}|\\Phi \\rangle \\,\\forall j$ , with $p_j=\\langle \\Psi |{\\cal K}^\\dagger _j {\\cal K}_j|\\Psi \\rangle $ , implying $\\rho ^{(1)}_j =\\rho ^{(1)}_\\Phi \\,\\forall j$ and hence Eq.",
"(REF ) when (REF ) is fulfilled.",
"Then, maximally one-body entangled states are those pure states whose SPDM is majorized by that of any other state.",
"Due to Eq.",
"(REF ) they will also maximize $E(|\\Psi \\rangle )$ for any choice of $S$ .",
"At fixed fermion number $N\\ge 2$ and $n=mN$ they are states leading to $\\rho ^{(1)}=\\mathbb {1}_n/m\\,,$ for which any SP basis is natural.",
"For $m$ integer such $\\rho ^{(1)}$ emerges, for instance, from Greenberger-Horne-Zeilinger (GHZ)-like states involving superpositions of SDs in orthogonal subspaces: $|\\Psi \\rangle =\\frac{1}{\\sqrt{m}}\\sum _{l=0}^{m-1} c^\\dag _{Nl+1}\\ldots c^\\dag _{Nl+N}|0\\rangle =\\frac{1}{N\\sqrt{m}}\\sum _{\\nu =1}^nc^\\dag _\\nu C^\\dag _\\nu |0\\rangle \\,,$ which lead to $\\langle c^\\dag _\\nu c_{\\nu ^{\\prime }}\\rangle =\\delta _{\\nu \\nu ^{\\prime }}/m$ ."
],
[
"Fermion linear optics operations as ONG",
"We now show that number conserving FLO operations [44], [45], [46], which include one-body unitary transformations and measurement of the occupancy of a SP mode, are included in the ONG set.",
"First, any number conserving one-body unitary transformation ${\\cal U}=\\exp [-i\\sum _{k,k^{\\prime }}H_{k^{\\prime }k}c^\\dag _{k^{\\prime }}c_{k}]\\,,$ with ${\\cal U}^\\dag {\\cal U}=\\mathbb {1}$ ($H^\\dag =H$ ) is obviously ONG: Since ${\\cal U}c^\\dag _k{\\cal U}^\\dag =\\sum _{k^{\\prime }} U_{k^{\\prime }k}c^\\dag _{k^{\\prime }}$ , with $U=e^{-iH}$ , it will map the SPDM as $\\rho ^{(1)}\\rightarrow U^\\dag \\rho ^{(1)}U$ , leaving its eigenvalues unchanged (and hence transforming SDs into SDs).",
"It can be implemented through composition of phaseshifting and beamsplitters unitaries [44], [45], [46] ${\\cal U}_{p}(\\phi )=e^{-i\\varphi \\, c^\\dagger _kc_k}$ , ${\\cal U}_{b}(\\theta )=e^{-i\\theta \\,(c^\\dagger _k c_{k^{\\prime }} + c^\\dagger _{k^{\\prime }} c_k)}$ , which are the basic unitary elements of the FLO set.",
"FLO operations also include measurements of the occupancy of single-particle modes, described by projectors ${\\cal P}_k=c^\\dagger _k c_k\\,,\\; {\\cal P}_{\\bar{k}}=c_k c^\\dagger _k\\,, $ which satisfy ${\\cal P}_k+{\\cal P}_{\\bar{k}}=\\mathbb {1}$ .",
"We now show explicitly the following fundamental result.",
"Theorem 1 The measurement of the occupancy of a single-particle state $|k\\rangle =c^\\dagger _k|0\\rangle \\in {\\cal H}$ , described by the operators (REF ), is a ONG operation.",
"Consider a general pure fermionic state $|\\Psi \\rangle $ with SPDM $\\rho ^{(1)}$ .",
"Let $\\rho _k^{(1)}$ and $\\rho _{\\bar{k}}^{(1)}$ be the SPDMs after SP mode $|k\\rangle $ is found to be occupied or empty, respectively, determined by the states $|\\Psi _k\\rangle ={\\cal P}_k |\\Psi \\rangle /\\sqrt{p_k}\\,, \\;\\;|\\Psi _{\\bar{k}}\\rangle ={\\cal P}_{\\bar{k}}|\\Psi \\rangle /\\sqrt{p_{\\bar{k}}}\\,,$ with $p_k=\\langle \\Psi |{\\cal P}_k|\\Psi \\rangle =1-p_{\\bar{k}}$ .",
"Then $|\\Psi \\rangle =\\sqrt{p_k}|\\Psi _k\\rangle +\\sqrt{p_{\\bar{k}}}|\\Psi _{\\bar{k}}\\rangle \\,.$ We will prove relation (REF ), i.e.",
"(Fig REF ), $\\mathbf {\\lambda }(\\rho ^{(1)})\\prec p_k\\mathbf {\\lambda }(\\rho _k^{(1)}) + p_{\\bar{k}}\\mathbf {\\lambda }(\\rho ^{(1)}_{\\bar{k}})\\,.$ If the measured state $|k\\rangle $ is a natural orbital, such that $\\langle c^\\dag _k c_{k^{\\prime }}\\rangle =p_k\\delta _{kk^{\\prime }}$ with $p_k=\\lambda _k$ an eigenvalue of $\\rho ^{(1)}$ , Eq.",
"(REF ) is straightforward: In this case (REF ) leads to $\\rho ^{(1)}=p_k\\rho ^{(1)}_k+p_{\\bar{k}}\\rho ^{(1)}_{\\bar{k}}\\,,$ Figure: Measurement of the occupancy of a single fermion mode kk.",
"It reduces (or does not increase), on average, the mixedness of the SP density matrix ρ (1) \\rho ^{(1)} (λ\\mathbf {\\lambda } denotes its spectrum) and hence the one-body entanglement.as $\\langle \\Psi _k|c^\\dag _{k^{\\prime \\prime }}c_{k^{\\prime }} |\\Psi _{\\bar{k}}\\rangle =\\delta _{k^{\\prime \\prime }k}(1-\\delta _{k^{\\prime }k})\\langle c^\\dag _k c_{k^{\\prime }}\\rangle =0$ $\\forall $ $k^{\\prime },k^{\\prime \\prime }$ .",
"Eq.",
"(REF ) implies (REF ) since $\\mathbf {\\lambda }(A+B)\\prec \\mathbf {\\lambda }(A)+\\mathbf {\\lambda }(B)$ for any two Hermitian $n\\times n$ matrices $A$ and $B$ [48] (this case includes the trivial situation $p_k=1$ or 0, where $|\\Psi \\rangle =|\\Psi _k\\rangle $ or $|\\Psi _{\\bar{k}}\\rangle $ ; in the following we consider $p_k\\in (0,1)$ ).",
"Otherwise Eq.",
"(REF ) no longer holds.",
"Nevertheless, since $\\langle \\Psi _k|c^\\dag _{k^{\\prime \\prime }}c_{k^{\\prime }} |\\Psi _{\\bar{k}}\\rangle =0$ for any two SP states $|k^{\\prime }\\rangle $ and $|k^{\\prime \\prime }\\rangle $ orthogonal to $|k\\rangle $ , Eq.",
"(REF ) implies, for any SP subspace ${\\cal S}_{\\perp }\\subset {\\cal H}$ orthogonal to the measured state $|k\\rangle $ , $\\rho ^{(1)}_{{\\cal S}_{\\perp }}=p_k\\rho ^{(1)}_{k{\\cal S}_{\\perp }}+p_{\\bar{k}}\\rho ^{(1)}_{\\bar{k}{\\cal S}_{\\perp }}\\,,$ where $\\rho ^{(1)}_{{\\cal S}_\\perp }=P_{{\\cal S}_\\perp }\\rho ^{(1)}P_{{\\cal S}_\\perp }$ and $\\rho ^{(1)}_{k(\\bar{k}){\\cal S}_\\perp }$ are the restrictions of $\\rho ^{(1)}$ and $\\rho ^{(1)}_{k(\\bar{k})}$ to ${\\cal S}_\\perp $ , and $P_{{\\cal S}_\\perp }$ is the associated projector.",
"This result is expected since the measurement is “external” to ${\\cal S}_\\perp $ (if $[{\\cal K}_j,O]=0$ $\\forall \\, j$ $\\Rightarrow $ ${\\rm Tr}\\,[\\rho O]= {\\rm Tr}\\sum _j {\\cal K}_j\\rho {\\cal K}_j^\\dag O=\\sum _j p_j{\\rm Tr}[\\rho _j O]$ ; for ${\\cal K}_j={\\cal P}_{k(\\bar{k})}$ and $O=c^\\dag _{k^{\\prime \\prime }}c_{k^{\\prime }}$ with $k^{\\prime }$ and $k^{\\prime \\prime }$ orthogonal to $k$ , this result implies Eq.",
"(REF )).",
"And for any ${\\cal S}\\subset {\\cal H}$ containing the state $|k\\rangle $ , we have, as $\\langle \\Psi _k|c^\\dag _{k^{\\prime }}c_{k^{\\prime }}|\\Psi _{\\bar{k}}\\rangle =0$ for $k^{\\prime }=k$ or $k^{\\prime }$ orthogonal to $k$ , ${\\rm Tr}\\,\\rho ^{(1)}_{\\cal S}={\\rm Tr}\\,[p_k \\rho ^{(1)}_{k{\\cal S}}+p_{\\bar{k}}\\rho ^{(1)}_{\\bar{k}{\\cal S}}]\\,.$ We can now prove the $m^{\\rm th}$ inequality in (REF ), $\\sum _{\\nu =1}^m \\lambda _\\nu (\\rho ^{(1)})\\le \\sum _{\\nu =1}^m \\;\\left(p_k\\lambda _\\nu (\\rho ^{(1)}_k)+p_{\\bar{k}}\\lambda _\\nu (\\rho ^{(1)}_{\\bar{k}})\\right)\\,.$ Let ${\\cal S}_m\\subset {\\cal H}$ be the subspace spanned by the first $m$ eigenstates of $\\rho ^{(1)}$ , such that $\\lambda _\\nu (\\rho ^{(1)}_{{\\cal S}_m}) =\\lambda _\\nu (\\rho ^{(1)})$ for $\\nu \\le m$ and hence ${\\rm Tr}\\,\\rho ^{(1)}_{{\\cal S}_m}=\\sum _{\\nu =1}^m \\lambda _\\nu (\\rho ^{(1)})$ .",
"If ${\\cal S}_m$ is either orthogonal to $|k\\rangle $ or fully contains $|k\\rangle $ , Eq.",
"(REF ) or Eq.",
"(REF ) holds for ${\\cal S}={\\cal S}_m$ , implying (REF ) since ${\\rm Tr}\\,\\rho ^{(1)}_{k(\\bar{k}){\\cal S}_m}\\le \\sum _{\\nu =1}^m\\lambda _\\nu (\\rho ^{(1)}_{k(\\bar{k})})$ by the Ky Fan maximum principle [48] (the $m$ largest eigenvalues $\\lambda _\\nu $ of a Hermitian matrix $O$ satisfy $\\sum _{\\nu =1}^m\\lambda _\\nu \\ge {\\rm Tr}P^{\\prime }_m O=\\sum _{\\nu =1}^m\\lambda ^{\\prime }_\\nu $ for any rank $m$ orthogonal projector $P^{\\prime }_m$ , with $\\lambda ^{\\prime }_\\nu $ the sorted eigenvalues of $P^{\\prime }_m OP^{\\prime }_m$ ).",
"Otherwise we add to ${\\cal S}_m$ the component $|k_{\\perp }\\rangle $ of $|k\\rangle $ orthogonal to ${\\cal S}_m$ , obtaining an $m+1$ dimensional SP subspace ${\\cal S}^{\\prime }_{m}$ where Eq.",
"(REF ) holds and still $\\lambda _\\nu (\\rho ^{(1)}_{{\\cal S}^{\\prime }_m})=\\lambda _\\nu (\\rho ^{(1)})$ for $\\nu \\le m$ .",
"It is proved in the Appendix that the remaining smallest eigenvalue satisfies $\\lambda _{m+1}(\\rho ^{(1)}_{{\\cal S}^{\\prime }_{m}})\\ge p_k\\lambda _{m+1}(\\rho ^{(1)}_{k{\\cal S}^{\\prime }_{m}})+p_{\\bar{k}}\\lambda _{m+1}(\\rho ^{(1)}_{\\bar{k}{\\cal S}^{\\prime }_{m}})\\,.",
"$ Hence $\\sum _{\\nu =1}^m \\lambda _\\nu (\\rho ^{(1)})$ $ \\le \\sum _{\\nu =1}^m\\, p_k\\lambda _\\nu (\\rho ^{(1)}_{k{\\cal S}^{\\prime }_m})+p_{\\bar{k}}\\lambda _\\nu (\\rho ^{(1)}_{\\bar{k}{\\cal S}^{\\prime }_m})$ due to the trace conservation (REF ) for ${\\cal S}={\\cal S}^{\\prime }_m$ , which implies Eq.",
"(REF ) due to previous Ky Fan inequality.",
"This completes the proof for pure states.",
"It is easily verified (see Appendix ) that these measurements map SDs onto SDs, as implied by Eq.",
"(REF ).",
"The previous proof actually shows the general relation $\\mathbf {\\lambda }(\\rho ^{(1)}_{\\cal S})\\prec p_k\\mathbf {\\lambda }(\\rho _{k{\\cal S}}^{(1)}) + p_{\\bar{k}}\\mathbf {\\lambda }(\\rho ^{(1)}_{{\\bar{k}}{\\cal S}})\\,, $ valid for the restriction of $\\rho ^{(1)}$ to any subspace ${\\cal S}\\subset {\\cal H}$ either containing or orthogonal to the measured state $|k\\rangle $ ($[P_{\\cal S},|k\\rangle \\langle k|]=0$ ).",
"Moreover $\\rho _{\\cal S}^{(1)}$ will be determined by a mixed reduced state $\\rho _{\\cal S}={\\rm Tr}_{{\\cal S}_{\\perp }}|\\Psi \\rangle \\langle \\Psi |$ satisfying $\\langle \\Psi |O_{\\cal S}|\\Psi \\rangle ={\\rm Tr}\\,\\rho _{\\cal S}\\,O_{\\cal S}$ for any operator $O_{\\cal S}$ involving just creation and annihilation of SP states $\\in {\\cal S}$ (see Appendix ).",
"Eq.",
"(REF ) then shows that (REF ) holds for general mixed fermionic states $\\rho $ (assumed to commute with the fermion number $\\hat{N}=\\sum _k c^\\dag _k c_k$ or the number parity $e^{i\\pi \\hat{N}}$ ) since they can be purified and seen as a reduced state $\\rho _{\\cal S}$ of a pure fermionic state $|\\Psi \\rangle $ in an enlarged SP space (Eq.",
"(REF ) in Appendix )."
],
[
"More general ONG measurements and operations",
"By composing the basic measurements (REF ), more complex operations satisfying (REF ) are obtained.",
"In particular, a measurement in a basis of SDs, which is obviously ONG, results from the composition of all measurements $\\lbrace P_k,P_{\\bar{k}}\\rbrace $ in a given SP basis.",
"Extension of the set of free operations beyond the standard FLO set can also be considered.",
"The proof of theorem REF can be extended to more general single-mode measurements: Corollary 1 A general measurement on single-particle mode $k$ described by the operators ${\\cal M}_k=\\alpha \\, {\\cal P}_k+\\beta \\,{\\cal P}_{\\bar{k}}\\,,\\;\\; {\\cal M}_{\\bar{k}}=\\gamma \\,{\\cal P}_{k}+\\delta \\, {\\cal P}_{\\bar{k}}\\,, $ where ${\\cal M}_k^\\dag {\\cal M}_k+{\\cal M}_{\\bar{k}}^\\dag {\\cal M}_{\\bar{k}}=\\mathbb {1}$ ($|\\alpha |^2+|\\gamma |^2= |\\beta |^2+|\\delta |^2=1$ ) is also a ONG operation.",
"The proof is given in Appendix A and implies that Eq.",
"(REF ) will be satisfied for $\\rho _{k(\\bar{k})}^{(1)}$ the SPDMs obtained from $|\\Psi ^{\\prime }_{k(\\bar{k})}\\rangle \\propto {\\cal M}_{k(\\bar{k})}|\\Psi \\rangle $ and $p_k\\rightarrow p^{\\prime }_k=\\langle \\Psi |{\\cal M}_k^\\dag {\\cal M}_k|\\Psi \\rangle =1-p^{\\prime }_{\\bar{k}}$ .",
"This result entails that any pair of Kraus operators ${\\cal M}_k$ and ${\\cal M}_{\\bar{k}}$ for the occupation measurement operation $\\varepsilon (\\rho )={\\cal P}_k\\rho {\\cal P}_{k}+{\\cal P}_{\\bar{k}}\\rho {\\cal P}_{\\bar{k}}={\\cal M}_k\\rho {\\cal M}_k^\\dag +{\\cal M}_{\\bar{k}}\\rho {\\cal M}_{\\bar{k}}^\\dag $ will also be ONG operations, since they are a special case of (REF ) [$\\alpha \\beta ^*+\\gamma \\delta ^*=0$ , i.e.",
"$(^{\\alpha \\;\\beta }_{\\gamma \\;\\delta })$ unitary].",
"It is also possible to consider in the present context operations which do not conserve the fermion number $N$ but still generate states with definite particle number when applied to such states.",
"In this case it becomes necessary to extend the partial order (REF ) to states with different particle number.",
"We then notice that Eq.",
"(REF ) implies a similar majorization relation (see Appendix ) $\\mathbf {\\lambda }(D^{(1)}_\\Psi )\\prec \\mathbf {\\lambda }(D^{(1)}_{\\Phi })$ for the sorted eigenvalues of the extended $2n\\times 2n$ SPDM $D^{(1)}=\\rho ^{(1)}\\oplus (\\mathbb {1}-\\rho ^{(1)\\,T})=\\begin{pmatrix}\\rho ^{(1)}&0\\\\0&\\mathbb {1}-\\rho ^{(1)\\,T}\\end{pmatrix}\\,, $ with spectrum $(\\mathbf {\\lambda }(\\rho ^{(1)}),1-\\mathbf {\\lambda }(\\rho ^{(1)}))$ and elements $\\langle c^\\dag _{k^{\\prime }}c_k\\rangle $ and $\\langle c_{k^{\\prime }}c^\\dag _{k}\\rangle $ , the trace ${\\rm Tr}\\,D^{(1)}=n$ of which is fixed by the SP space dimension and is $N$ -independent.",
"For general states we then say that $|\\Psi \\rangle $ is not less one-body entangled than $|\\Phi \\rangle $ if Eq.",
"(REF ) holds.",
"Note that all SDs lead to the same sorted spectrum $\\mathbf {\\lambda }(D^{(1)})$ regardless of $N$ , all being then least entangled states, with $(D^{(1)})^2=D^{(1)}$ if and only if $|\\Psi \\rangle $ is a SD.",
"Similarly, Eq.",
"(REF ) for number conserving ONG operations implies $\\mathbf {\\lambda }(D^{(1)})\\prec \\sum _j p_j\\mathbf {\\lambda }(D^{(1)}_j)$ for the extended densities.",
"We then say that an operation not conserving fermion number is ONG if it admits a set of Kraus operators ${\\cal K}_j$ such that (REF ) is satisfied.",
"Prop.",
"1 remains then valid for general ONG operations replacing (REF ) by (REF ).",
"All previous properties satisfied by the entropies (REF ) extend to entropies $E(|\\Psi \\rangle )=S(D^{(1)})\\,,$ with ${\\rm Tr}f(D^{(1)})={\\rm Tr}f(\\rho ^{(1)})+{\\rm Tr}f(\\mathbb {1}-\\rho ^{(1)})$ .",
"In particular, Eq.",
"(REF ) becomes just the von Neumann entropy of $D^{(1)}$ .",
"Maximally one-body entangled states are now those leading to $D^{(1)}=\\mathbb {1}_{2n}/2\\,,$ which will maximize all entropies (REF ).",
"Examples are previous GHZ-like states (REF ) in half-filled SP spaces ($m=2$ , $N=n/2\\ge 1$ ).",
"Theorem 1 then implies the following result for the basic measurement having $c_k$ and $c^\\dag _k$ as operators: Corollary 2 A measurement on single-particle mode $k$ described by the operators $c_k$ and $c^\\dag _k$ , which satisfy $c^\\dag _kc_k+c_k c^\\dag _k=\\mathbb {1}$ , is a ONG operation: $\\mathbf {\\lambda }(D^{(1)})\\prec p_k\\mathbf {\\lambda }(D^{(1)}_k)+p_{\\bar{k}}\\mathbf {\\lambda }(D^{(1)}_{\\bar{k}})\\,.$ Here $p_k=\\langle c^\\dag _k c_k\\rangle =1-p_{\\bar{k}}$ , and $D^{(1)}$ and $D^{(1)}_{k(\\bar{k})}$ are the extended SPDMs determined by $\\rho $ , $\\rho _k=c_k\\rho c_k^\\dag /p_k$ and $\\rho _{\\bar{k}}=c^\\dag _k\\rho c_k/p_{\\bar{k}}$ .",
"Since these extended densities $D^{(1)}_{k(\\bar{k})}$ have clearly the same spectrum as those obtained from ${\\cal P}_k\\rho {\\cal P}_k/p_k$ and ${\\cal P}_{\\bar{k}}\\rho {\\cal P}_{\\bar{k}}/p_{\\bar{k}}$ , Eq.",
"(REF ) directly follows from Theorem 1 and Eq.",
"(REF ).",
"On the other hand, previous basic occupation measurement (REF ) is just the composition of this measurement with itself (see Appendix ).",
"This extension then enables us to consider the addition of free ancillas (SDs of arbitrary $N$ ) as a free operation, as it will not alter the spectrum of the extended SPDM $D^{(1)}$ in the full SP space.",
"We remark, however, that general “active” FLO operations which do not conserve the fermion number (for instance, a Bogoliubov transformation) may increase the one-body entanglement determined by $\\rho ^{(1)}$ .",
"While we will not discuss these operations here, we mention that if they are also regarded as free one should consider instead the generalized one-body entanglement, determined by the mixedness of the full quasiparticle DM [20], as the associated resource (see also Appendix ).",
"The identification of number conserving FLO operations as ONG implies that they map SDs onto SDs, as verified in App.",
".",
"It has been noted [59], [60] that this fact ultimately explains why the pure state FLO computation model can be efficiently simulated classically, as matrix elements of free unitaries and outcome probabilities of free measurements can be reduced to overlaps $\\langle \\Psi |\\Phi \\rangle $ between SDs, which can be computed in polynomial time through a determinant [44].",
"In contrast, the simultaneous measurement of the occupancy of two SP modes $k$ and $k^{\\prime }$ , described by operators $\\lbrace {\\cal M}_0={\\cal P}_{\\bar{k}}{\\cal P}_{\\bar{k}^{\\prime }},{\\cal M}_1={\\cal P}_{\\bar{k}}{\\cal P}_{k^{\\prime }}+{\\cal P}_{k}{\\cal P}_{\\bar{k}^{\\prime }}, {\\cal M}_2={\\cal P}_{k}{\\cal P}_{k^{\\prime }}\\rbrace $ , is not free since ${\\cal M}_1$ can map a SD onto a state with Slater number 2 [59], i.e.",
"a one-body entangled state.",
"A similar measurement with $m$ such outcomes may return a state with an exponentially large ($2^m$ ) Slater number [59], the expectation values of which would be hard to evaluate classically.",
"In [61] this operation is identified with a charge detection measurement in a system of free electrons, showing that it is possible to build a controlled-NOT gate with just beamsplitters, spin rotations and charge detectors.",
"The extended set of FLO plus charge detection operations then enables quantum computation.",
"If the computational power of this model is to be linked to the presence of a quantum resource, the ensuing free states and operations would be $\\cal S$ and $\\cal O$ , respectively, and the results derived here entail that one-body entanglement would be an associated resource.",
"One-body entanglement can also be considered as a resource for mode entanglement.",
"In particular, mode entanglement with definite particle number $N$ or definite number parity $e^{i\\pi N}$ at each component requires one-body entanglement.",
"A first example was seen with the normal form (REF ) for a general two-fermion state [11], [12], [13], where the entanglement between the modes $k$ ($A$ ) and $\\bar{k}$ ($B$ ), containing each one fermion, is directly linked to one-body entanglement: The entanglement entropy $E(A,B)=S(\\rho _A)=S(\\rho _B)$ of this partition is just $E(A,B)=\\sum _\\nu f(\\lambda _\\nu )=\\tfrac{1}{2}E(|\\Psi \\rangle )$ for any entropy $S(\\rho )={\\rm Tr}\\,f(\\rho )$ , where $E(|\\Psi \\rangle )=S(\\rho ^{(1)})$ is the corresponding one-body entanglement entropy (REF ) (as $\\langle c^\\dag _\\nu c_{\\nu ^{\\prime }}\\rangle =\\langle c^\\dag _{\\bar{\\nu }} c_{\\bar{\\nu }^{\\prime }}\\rangle =\\lambda _\\nu \\delta _{\\nu \\nu ^{\\prime }}$ , $\\langle c^\\dag _\\nu c_{\\bar{\\nu }^{\\prime }}\\rangle =0$ ).",
"In particular any one-body entangled state of two fermions in a SP space of dimension 4 can be seen as an entangled state of two distinguishable qubits, allowing then the realization of tasks like quantum teleportation [23].",
"In this case the one-body entanglement entropy also provides a lower bound to any bipartite mode entanglement entropy [23].",
"For a general pure fermionic state $|\\Psi \\rangle $ (with definite particle number or number parity) we now show that one-body entanglement is always required in order to have bipartite mode entanglement entropy $E(A,B)>0$ with definite particle number, or in general definite number parity, at each side $A$ and $B$ : In such a case, and assuming sides $A$ and $B$ correspond to orthogonal subspaces ${\\cal H}_A$ and ${\\cal H}_B$ of the SP space ${\\cal H}={\\cal H}_A\\oplus {\\cal H}_B$ , the ensuing SPDM takes the block-diagonal form $\\rho ^{(1)}=\\rho _A^{(1)}\\oplus \\rho _B^{(1)}$ , i.e., $\\rho ^{(1)}=\\begin{pmatrix} \\rho ^{(1)}_A&0\\\\0&\\rho ^{(1)}_B\\end{pmatrix}$ since for any $k_A\\in {\\cal H}_A$ , $k_B\\in {\\cal H}_B$ , $c^\\dag _{k_A}c_{k_B}$ connects states with different number parity at each side and hence $\\langle c^\\dag _{k_A}c_{k_B}\\rangle =0$ in such a state.",
"Then, if $|\\Psi \\rangle $ is a SD, $(\\rho ^{(1)})^2=\\rho ^{(1)}$ , implying $(\\rho ^{(1)}_{A(B)})^2=\\rho ^{(1)}_{A(B)}$ , i.e.",
"the state at each side must be a SD (a pure state) and no $A$ –$B$ entanglement is directly present.",
"For instance, a single fermion state $\\frac{1}{\\sqrt{2}}(c^\\dag _{k_A}+c^\\dag _{k_B})|0\\rangle $ implies entanglement between $A$ and $B$ but at the expense of involving zero and one fermion at each side, i.e., no definite local number parity.",
"In contrast, $|\\Psi \\rangle =\\frac{1}{\\sqrt{2}}(c^\\dag _{k_A}c^\\dag _{k_B}+c^\\dag _{k^{\\prime }_A}c^\\dag _{k^{\\prime }_B})|0\\rangle $ leads to entanglement between $A$ and $B$ with definite fermion number (and hence number parity) at each side, but it is not a SD, i.e., it has nonzero one-body entanglement.",
"Expanding the state in a SD basis as $|\\Psi \\rangle =\\sum _{\\mu ,\\nu } C_{\\mu \\nu }A^\\dag _\\mu B^\\dag _{\\nu }|0\\rangle $ , where $A^\\dag _\\mu =\\prod _k(c^\\dag _{k_A})^{n_{k_\\mu }}$ and $B^\\dag _\\nu =\\prod _k(c^\\dag _{k_B})^{n_{k_\\nu }}$ involve creation operators just on ${\\cal H}_A$ and ${\\cal H}_B$ respectively (with $n_{k_{\\mu (\\nu )}}=0,1$ and $\\mu $ and $\\nu $ labeling all possible sets of occupation numbers, such that $\\langle 0|A_\\mu A^\\dag _{\\mu ^{\\prime }}|0\\rangle =\\delta _{\\mu \\mu ^{\\prime }}$ , $\\langle 0|B_\\nu B^\\dag _{\\nu ^{\\prime }}|0\\rangle =\\delta _{\\nu \\nu ^{\\prime }}$ ), states with definite number parity at each side correspond to $(-1)^{\\sum _k n_{k_\\mu }}$ and $(-1)^{\\sum _k n_{k_\\nu }}$ fixed for all $\\mu $ and $\\nu $ with $C_{\\mu \\nu }\\ne 0$ .",
"The reduced DM of side $A$ is $\\rho _A=\\sum _{\\mu ,\\mu ^{\\prime }}(CC^\\dag )_{\\mu \\mu ^{\\prime }}A^\\dag _{\\mu }|0\\rangle \\langle 0|A_{\\mu ^{\\prime }}$ (and similarly for $\\rho _B$ ; see Appendix ), and there is entanglement between $A$ and $B$ whenever $\\rho _A$ has rank $\\ge 2$ , i.e., rank$(C)\\ge 2$ .",
"In such a case, previous argument shows that such $|\\Psi \\rangle $ cannot be a SD if the fermion number or number parity is fixed at each side, i.e.",
"$N_A$ or $e^{i\\pi N_A}$ fixed.",
"Due to the fermionic number parity superselection rule [5], [62], fixed number parity at each side is required in order to be able to form arbitrary superpositions, i.e., arbitrary unitary transformations of the local states, and hence to have entanglement fully equivalent to the distinguishable case.",
"Fixed particle number at each side may be also required if the particle number or charge superselection rule applies for the fermions considered.",
"The extension to multipartite mode entanglement is straightforward: For a decomposition ${\\cal H}=\\bigoplus _i {\\cal H}_i$ of the SP space into orthogonal subspaces ${\\cal H}_i$ , and for component $i$ associated to subspace ${\\cal H}_i$ , all elements of $\\rho ^{(1)}$ connecting different components $i$ and $j$ will vanish if each component is to have definite fermion number or number parity in a state $|\\Psi \\rangle $ : $\\langle c^\\dag _{k_i}c_{k_j}\\rangle =0$ $\\forall $ $k_i,k_j$ if $i\\ne j$ .",
"Thus, $\\rho ^{(1)}=\\bigoplus _i\\rho _i^{(1)}\\,.$ If $|\\Psi \\rangle $ were a SD, each $\\rho ^{(1)}_i$ should then satisfy $(\\rho ^{(1)}_i)^2=\\rho ^{(1)}_i$ and hence each subsystem would be in a pure SD state, implying no entanglement between them.",
"A one-body entangled state is then required.",
"And when all previous subsystems contain just one fermion, one-body entanglement is directly linked to standard multipartite entanglement.",
"Consider an $N$ -partite system with Hilbert space ${\\cal L}=\\bigotimes _{i=1}^{N}{\\cal L}_i$ , where ${\\cal L}_i$ is the Hilbert space of the $i$ -th (distinguishable) constituent.",
"Consider also an $N$ -fermion system with SP space ${\\cal H}=\\bigoplus _{i=1}^N {\\cal H}_i$ , such that ${\\rm dim}\\,{\\cal H}_i={\\rm dim}\\,{\\cal L}_i$ .",
"This enables the definition of an isomorphism $\\Theta _i:{\\cal L}_i\\rightarrow {\\cal H}_i$ between these two spaces: Any pure separable state in $\\cal L$ , $|S\\rangle _{\\cal L}=\\bigotimes _{i=1}^{N}|\\phi _i\\rangle $ , with $|\\phi _i\\rangle \\in {\\cal L}_i$ , can be mapped to a SD $|\\Psi \\rangle =[\\prod _{i=1}^N c^\\dagger _{i,\\phi }]|0\\rangle ,=\\Theta (|S\\rangle _{\\cal L}),$ where $c^\\dagger _{i, \\phi }$ creates a fermion in the state $\\Theta _i(|\\phi _i\\rangle )\\in {\\cal H}_i$ .",
"The map $\\Theta :{\\cal L}\\rightarrow {\\cal F}$ is an isomorphism between $\\cal L$ and the subspace of $\\cal F$ determined by the fermion states having occupation number $N_i=1$ in ${\\cal H}_i\\subset {\\cal H}$ .",
"Hence any pure state $|\\Phi \\rangle _{\\cal L}$ of the multipartite system is mapped to a state $|\\Psi \\rangle =\\Theta (|\\Phi \\rangle _{\\cal L})$ in $\\cal F$ .",
"Since there is a fixed fermion number (1) in each constituent, the SPDM $\\rho ^{(1)}$ determined by $|\\Psi \\rangle $ will satisfy Eq.",
"(REF ), implying here $\\rho ^{(1)}=\\bigoplus _i \\rho _i, $ where the elements of the matrix $\\rho _i$ are those of the reduced state of the $i$ -th subsystem associated to $|\\Phi \\rangle _{\\cal L}$ .",
"Thus, one-body entanglement monotones become $E(|\\Psi \\rangle )={\\rm Tr}\\,f(\\rho ^{(1)})=\\sum _i {\\rm Tr}\\,f(\\rho _i)$ , being then equivalent to the multipartite version of the linear entropy of entanglement [6], [64], [63], [65] and constituting monotones for the multipartite entanglement of the tensor product representation.",
"This link between one-body entanglement and multipartite entanglement is not a surprise if we recall that performing local operations on the multipartite system cannot increase, on average, the mixedness of the local eigenvalues.",
"Relation (REF ) then implies that these operations cannot increase the mixedness of the SPDM associated to the fermionic representation, in agreement with Eq.",
"(2).",
"And any local unitary in the tensor product representation can be implemented as a one-body unitary in the fermion system, while local projective measurements can be performed as occupation measurements, both FLO operations which are one-body entanglement nongenerating.",
"The overlap between one-body entanglement and multipartite entanglement described here reinforces the idea that the first could be the resource behind the computational power of the quantum computation model described in [61], since it consists of a mapping from qubits to fermions just like the map $\\Theta $ defined above.",
"We have shown that one-body entanglement, a measure of the deviation of a pure fermionic state from a SD determined by the mixedness of the SPDM $\\rho ^{(1)}$ , can be considered as a quantum resource.",
"We have first provided a basis-independent bipartite like formulation of one-body entanglement in general $N$ -fermion states, which relates it with the correlation between one and ($N-1$ )-body observables and is analogous to that of systems of distinguishable components.",
"Such formulation leads to a Schmidt-like decomposition of the state, which contains the common eigenvalues of the one and $(N-1)$ -body DMs.",
"We have then identified the class of one-body entanglement nongenerating operations through the rigorous majorization relation (REF ).",
"And we have shown in Theorem 1 that single mode occupation measurements satisfy indeed this relation, implying they will not increase, on average, the one-body entanglement here defined.",
"The ensuing theory then has SDs as free states and number conserving FLO operations as free operations.",
"We have also considered in corollary 1 and 2 more general occupation measurements, showing they are also ONG.",
"Connections with mode entanglement and multipartite entanglement have also been discussed, showing in particular that one-body entanglement is required for entanglement with fixed fermion number or number parity at each subsystem.",
"Present results provide the basis for a consistent resource theory associated to quantum correlations beyond antisymmetrization in fermionic systems, which should play a fundamental role in fermionic protocols beyond the FLO model.",
"The authors acknowledge support from CONICET (N.G.",
"and M.DT.)",
"and CIC (R.R.)",
"of Argentina.",
"This work was supported by CONICET PIP Grant No.",
"11220150100732."
],
[
"Proof of inequality (",
"We first prove Eq.",
"(REF ) for the lowest eigenvalue of $\\rho ^{(1)}_{{\\cal S}^{\\prime }_m}$ , where ${\\cal S}^{\\prime }_m$ is the subspace containing the measured SP state $|k\\rangle $ and the first $m$ eigenstates of $\\rho ^{(1)}$ .",
"We write the lowest eigenstate of $\\rho ^{(1)}_{{\\cal S}^{\\prime }_m}$ as $\\alpha |k\\rangle +\\beta |k^{\\prime }\\rangle $ , with $|k^{\\prime }\\rangle \\in {\\cal S}^{\\prime }_m$ orthogonal to $|k\\rangle $ , such that $\\lambda _{m+1}(\\rho ^{(1)}_{{\\cal S}^{\\prime }_m})$ is the smallest eigenvalue $\\lambda _-$ of the $2\\times 2$ matrix $\\rho ^{(1)}_{kk^{\\prime }}=\\begin{pmatrix}\\langle c^\\dagger _k c_k\\rangle &&\\langle c^\\dagger _{k^{\\prime }} c_k\\rangle \\\\\\langle c^\\dagger _k c_{k^{\\prime }}\\rangle &&\\langle c^\\dagger _{k^{\\prime }} c_{k^{\\prime }}\\rangle \\end{pmatrix}\\,.$ Setting $\\langle c^\\dagger _{k} c_{k}\\rangle =p_k$ , its eigenvalues are $\\lambda _\\pm ={\\textstyle \\frac{p_k+p_{k^{\\prime }}}{2}\\pm \\sqrt{\\frac{(p_k-p_{k^{\\prime }})^2}{4}+|\\langle c^\\dag _{k^{\\prime }} c_{k}\\rangle |^2}}\\,.$ Writing again ${\\cal P}_k=c^\\dag _k c_k$ , ${\\cal P}_{\\bar{k}}=c_kc^\\dagger _k=\\mathbb {1}-{\\cal P}_k$ , a general state $|\\Psi \\rangle $ can be expanded as ${\\textstyle |\\Psi \\rangle =\\sum _{\\mu ,\\mu ^{\\prime }}{\\cal P}_\\mu {\\cal P}_{\\mu ^{\\prime }}|\\Psi \\rangle =\\sum _{\\mu ,\\mu ^{\\prime }}\\sqrt{p_{\\mu \\mu ^{\\prime }}}|\\Psi _{\\mu \\mu ^{\\prime }}\\rangle \\,,}$ where $\\mu =k,\\bar{k}$ ; $\\mu ^{\\prime }=k^{\\prime },\\bar{k}^{\\prime }$ (so that $\\sum _{\\mu ,\\mu ^{\\prime }}{\\cal P}_\\mu {\\cal P}_{\\mu ^{\\prime }}=\\mathbb {1}$ ) and $|\\Psi _{\\mu \\mu ^{\\prime }}\\rangle ={\\cal P}_{\\mu }{\\cal P}_{\\mu ^{\\prime }}|\\Psi \\rangle /\\sqrt{p_{\\mu \\mu ^{\\prime }}}$ , with $p_{\\mu \\mu ^{\\prime }}=\\langle \\Psi |{\\cal P}_\\mu {\\cal P}_{\\mu ^{\\prime }}|\\Psi \\rangle $ , are states with definite occupation of SP states $|k\\rangle $ and $|k^{\\prime }\\rangle $ .",
"We then have $p_k=p_{kk^{\\prime }}+p_{k\\bar{k^{\\prime }}}$ , $p_{k^{\\prime }}= p_{kk^{\\prime }}+p_{\\bar{k}k^{\\prime }}$ and $\\langle c^\\dag _{k^{\\prime }}c_{k}\\rangle = r\\sqrt{p_{\\bar{k}k^{\\prime }}p_{k\\bar{k^{\\prime }}}}$ , with $|r|\\le 1$ .",
"Thus, $|\\langle c^\\dag _{k^{\\prime }}c_{k}\\rangle |^2\\le p_{\\bar{k}k^{\\prime }}p_{k\\bar{k}^{\\prime }}$ , and (REF ) implies $\\lambda _-\\ge \\frac{p_k+p_{k^{\\prime }}}{2}-\\frac{p_{k\\bar{k}^{\\prime }}+p_{\\bar{k}k^{\\prime }}}{2}$ , i.e., $\\lambda _-\\ge p_{kk^{\\prime }}\\ge p_k \\lambda _{m+1}(\\rho ^{(1)}_{k{\\cal S}^{\\prime }_m})+p_{\\bar{k}}\\lambda _{m+1}(\\rho ^{(1)}_{\\bar{k}{\\cal S}^{\\prime }_m})\\,,$ since $p_k \\lambda _{m+1}(\\rho ^{(1)}_{k{\\cal S}^{\\prime }_m})\\le p_{kk^{\\prime }}$ and $\\lambda _{m+1}(\\rho ^{(1)}_{\\bar{k}{\\cal S}^{\\prime }_m})=0$ (as the state $k$ is empty in $\\rho ^{(1)}_{\\bar{k}{\\cal S}^{\\prime }_m}$ ).",
"This implies Eq.",
"(REF ) since $\\lambda _{m+1}(\\rho ^{(1)}_{{\\cal S}^{\\prime }_m})=\\lambda _-$ .",
"We also mention that by considering the largest eigenvalue $\\lambda _+$ in (REF ) of a similar $2\\times 2$ block, such that the first eigenstate (largest eigenvalue) of $\\rho ^{(1)}$ is spanned by $|k\\rangle $ and $|k^{\\prime }\\rangle $ , it is verified, using again $|\\langle c^\\dag _{k^{\\prime }}c_{k}\\rangle |^2\\le p_{\\bar{k}k^{\\prime }}p_{k\\bar{k}^{\\prime }}$ , that $\\lambda _1(\\rho ^{(1)})=\\lambda _+\\le p_k+p_{\\bar{k}k^{\\prime }}\\le p_k\\lambda _1(\\rho ^{(1)}_k)+p_{\\bar{k}}\\lambda _1(\\rho ^{(1)}_{\\bar{k}})$ , since $\\lambda _1(\\rho ^{(1)}_k)=1$ and $p_{\\bar{k}k^{\\prime }}\\le p_{\\bar{k}}\\lambda _1(\\rho ^{(1)}_{\\bar{k}})$ , which is the first ($m=1$ ) inequality in Eq.",
"(REF ).",
"The present proof of Theorem 1 also holds for general mixed fermionic states $\\rho $ (assumed to commute with the fermion number $\\hat{N}=\\sum _k c^\\dag _k c_k$ or in general the number parity $e^{i\\pi \\hat{N}}$ ) since they can always be purified, i.e.",
"considered as reduced states $\\rho _{\\cal S}$ of a pure fermionic state $|\\Psi \\rangle =\\sum _{\\mu ,\\nu }C_{\\mu \\nu }A^\\dag _\\mu B^\\dag _\\nu |0\\rangle \\,, $ of definite number parity.",
"Here $A^\\dag _{\\mu }$ and $B^\\dag _\\nu $ contain creation operators in ${\\cal S}$ and in an orthogonal SP space ${\\cal S}_\\perp $ respectively, satisfying $\\langle 0|A_{\\mu ^{\\prime }}A^\\dag _{\\mu }|0\\rangle =\\delta _{\\mu \\mu ^{\\prime }}$ , $\\langle 0|B_{\\nu ^{\\prime }}B^\\dag _{\\nu }|0\\rangle =\\delta _{\\nu \\nu ^{\\prime }}$ , $\\langle 0|B_{\\nu ^{\\prime }}A^\\dag _{\\nu }|0\\rangle =0$ .",
"We may assume, for instance, that $\\lbrace A^\\dag _\\mu B^\\dag _\\nu |0\\rangle \\rbrace $ is a complete set of orthogonal SDs.",
"Then $\\rho _{\\cal S}={\\rm Tr}_{{\\cal S}_{\\perp }}|\\Psi \\rangle \\langle \\Psi |=\\sum _{\\mu ,\\mu ^{\\prime }}(C C^\\dag )_{\\mu \\mu ^{\\prime }}|\\mu \\rangle \\langle \\mu ^{\\prime }|$ , with $|\\mu \\rangle =A^\\dag _\\mu |0\\rangle $ a SD in ${\\cal S}$ , satisfies $\\langle \\Psi |O_S|\\Psi \\rangle ={\\rm Tr}\\rho _{\\cal S}O_{\\cal S}$ $\\forall $ operators $O_S$ containing creation and annihilation operators of SP states $\\in {\\cal S}$ .",
"Given then an arbitrary mixed fermionic state $\\rho _{\\cal S}$ Eq.",
"(REF ) is a purification of $\\rho _{\\cal S}$ for any matrix $C$ satisfying $(CC^\\dag )_{\\mu \\mu ^{\\prime }}= \\langle \\mu |\\rho _{\\cal S}|\\mu ^{\\prime }\\rangle $ (requires dim$\\,{\\cal S}_\\perp \\ge \\,$ dim$\\,{\\cal S}$ ).",
"Let us now prove Corollary 1, i.e.",
"the extension of Theorem 1 to the more general occupancy measurement operators of Eq.",
"(REF ).",
"In terms of the states (REF ), the ensuing postmeasurement states $|\\Psi ^{\\prime }_{k(\\bar{k})}\\rangle \\propto M_{k(\\bar{k})}|\\Psi \\rangle $ are $|\\Psi ^{\\prime }_k\\rangle &=&\\left(\\alpha \\sqrt{p_k}\\, |\\Psi _k\\rangle +\\beta \\sqrt{p_{\\bar{k}}}\\, |\\Psi _{\\bar{k}}\\rangle \\right)/\\sqrt{p^{\\prime }_k},\\\\|\\Psi ^{\\prime }_{\\bar{k}}\\rangle &=&\\left(\\gamma \\sqrt{p_k}\\, |\\Psi _k\\rangle +\\delta \\sqrt{p_{\\bar{k}}}\\, |\\Psi _{\\bar{k}}\\rangle \\right)/\\sqrt{p^{\\prime }_{\\bar{k}}},$ where $|\\alpha |^2+|\\gamma |^2=1$ , $|\\beta |^2+|\\delta |^2=1$ and $p^{\\prime }_k&=&p_k|\\alpha |^2+p_{\\bar{k}}|\\beta |^2\\,,\\;\\;\\;p^{\\prime }_{\\bar{k}}=p_k|\\gamma |^2+p_{\\bar{k}}|\\delta |^2\\,,$ with $p^{\\prime }_k+p^{\\prime }_{\\bar{k}}=1$ .",
"We have to prove $\\mathbf {\\lambda }(\\rho ^{(1)})\\prec p^{\\prime }_k\\mathbf {\\lambda }({\\rho ^{\\prime }}_k^{(1)}) + p^{\\prime }_{\\bar{k}}\\mathbf {\\lambda }({\\rho ^{\\prime }}^{(1)}_{\\bar{k}})\\,,$ where ${\\rho ^{\\prime }}^{(1)}_{k(\\bar{k})}$ are now the SPDM's determined by the states (REF )–().",
"The generalization of Eq.",
"(REF ), ${\\rm Tr}\\,\\rho _{\\cal S}^{(1)}={\\rm Tr}\\,[p^{\\prime }_k{\\rho ^{\\prime }}^{(1)}_{k{\\cal S}}+p^{\\prime }_{\\bar{k}}{\\rho ^{\\prime }}^{(1)}_{\\bar{k}{\\cal S}}]\\,,$ still holds for any subspace ${\\cal S}$ either orthogonal to or containing the SP state $|k\\rangle $ , as $[M_{k(\\bar{k})},c^\\dag _{k^{\\prime }} c_{k^{\\prime }}]=0$ for both $k^{\\prime }=k$ or $k^{\\prime }$ orthogonal to $k$ [see comment below Eq.",
"(REF )].",
"Proceeding in the same way and using previous notation, we see that $p^{\\prime }_k\\lambda _{m+1}({\\rho ^{\\prime }}^{(1)}_{k {\\cal S}^{\\prime }_m})$ is less than or equal to the smallest eigenvalue $\\lambda _{k-}$ of the $2\\times 2$ matrix $\\begin{pmatrix}|\\alpha |^2p_k&&\\alpha \\beta ^*\\langle c^\\dag _{k^{\\prime }}c_k\\rangle \\\\\\alpha ^*\\beta \\langle c^\\dag _{k}c_{k^{\\prime }}\\rangle && |\\alpha |^2 p_{kk^{\\prime }}+|\\beta |^2p_{\\bar{k} k^{\\prime }}\\end{pmatrix},$ while $p^{\\prime }_{\\bar{k}}\\lambda _{m+1}({\\rho ^{\\prime }}^{(1)}_{\\bar{k} {\\cal S}^{\\prime }_m})$ is less than or equal to the smallest eigenvalue $\\lambda _{\\bar{k}-}$ of a similar matrix with $\\alpha \\rightarrow \\gamma $ , $\\beta \\rightarrow \\delta $ .",
"It is then straightforward to prove, using Eq.",
"(REF ) for $\\lambda _-$ , that $\\lambda _{m+1}(\\rho ^{(1)}_{{\\cal S}^{\\prime }_m})=\\lambda _-\\ge \\lambda _{k-}+\\lambda _{\\bar{k}-}$ , since $\\lambda _--\\lambda _{k-}-\\lambda _{\\bar{k}-}&=& {\\textstyle \\sqrt{\\frac{(|\\alpha |^2 p_{k\\bar{k^{\\prime }}}-|\\beta |^2 p_{\\bar{k}k^{\\prime }})^2}{4}+|\\alpha \\beta \\langle c^\\dag _{k^{\\prime }}c_k\\rangle |^2}}\\nonumber \\\\&&+{\\textstyle \\sqrt{\\frac{(|\\gamma |^2 p_{k\\bar{k^{\\prime }}}-|\\delta |^2 p_{\\bar{k}k^{\\prime }})^2}{4}+|\\gamma \\delta \\langle c^\\dag _{k^{\\prime }}c_k\\rangle |^2}}\\nonumber \\\\&&{\\textstyle -\\sqrt{\\frac{(p_{k\\bar{k^{\\prime }}}-p_{\\bar{k}k^{\\prime }})^2}{4}+|\\langle c^\\dag _{k^{\\prime }}c_k\\rangle |^2}}\\nonumber \\\\&\\ge &0\\,,$ with equality for $|r|=1$ ($|\\langle c^\\dag _{k^{\\prime }}c_k\\rangle |=\\sqrt{p_{\\bar{k}k^{\\prime }}p_{k\\bar{k^{\\prime }}}}$ ) or $|\\alpha |=|\\beta |$ .",
"Then the $m^{\\rm th}$ inequality in (REF ), $\\sum _{\\nu =1}^m\\lambda _\\nu (\\rho ^{(1)})\\le \\sum _{\\nu =1}^m p^{\\prime }_k\\lambda _\\nu ({\\rho ^{\\prime }}^{(1)}_k)+ p^{\\prime }_{\\bar{k}}\\lambda _\\nu ({\\rho ^{\\prime }}^{(1)}_{\\bar{k}})\\,,$ follows due to (REF ) and the previously used Ky Fan inequality.",
"Eq.",
"(REF ) also holds within any subspace ${\\cal S}$ containing (or orthogonal to) the SP state $|k\\rangle $ .$\\!\\!$ $\\Box $"
],
[
"Occupation measurements on free states",
"For a one-body entanglement nongenerating operation, Eq.",
"(REF ) implies that the Kraus operators ${\\cal K}_j$ satisfying it should convert free states onto free states, i.e.",
"SDs onto SDs.",
"For the occupation measurements of Eq.",
"(REF ) (Theorem 1), Eq.",
"(REF ) (Corollary 1), and Corollary 2, this property can be easily verified.",
"Let $|\\Psi \\rangle =(\\prod _{\\nu =1}^N c^\\dag _\\nu )|0\\rangle \\,,$ be a general SD for $N$ fermions, with $\\lbrace c_\\nu ,c^\\dag _{\\nu ^{\\prime }}\\rbrace =\\delta _{\\nu \\nu ^{\\prime }}$ .",
"A general fermion creation operator $c^\\dag _k=\\sum _{\\nu =1}^n \\alpha _\\nu c^\\dag _\\nu $ , with $\\alpha _\\nu =\\lbrace c_\\nu ,c^\\dag _k\\rbrace $ and $\\lbrace c_k,c^\\dag _k\\rbrace =\\sum _{\\nu }|\\alpha _\\nu |^2=1$ , can be written as $c^\\dag _k=\\sqrt{p_k} c^\\dag _{k_\\parallel }+\\sqrt{p_{\\bar{k}}}c^\\dag _{k_\\perp }\\,,$ where $\\sqrt{p}_k c^\\dag _{k_{\\parallel }}=\\sum _{\\nu \\le N} \\alpha _\\nu c^\\dag _\\nu $ is the component in the subspace occupied in $|\\Psi \\rangle $ , with $p_k=\\sum _{\\nu \\le N} |\\alpha _\\nu |^2=\\langle \\Psi |c^\\dag _k c_k|\\Psi \\rangle $ the occupation probability of SP state $k$ and $c^\\dag _{k_\\parallel }|\\Psi \\rangle =0$ , while $\\sqrt{p}_{\\bar{k}} c^\\dag _{k_{\\perp }}=\\sum _{\\nu > N} \\alpha _\\nu c^\\dag _\\nu $ is the orthogonal complement, with $p_{\\bar{k}}=\\sum _{\\nu >N}|\\alpha _\\nu |^2=1-p_k$ and $c_{k_\\perp }|\\Psi \\rangle =0$ .",
"If $p_k>0$ , through a unitary transformation of the $c^\\dag _\\nu $ for $\\nu \\le N$ , they can be chosen such that $c^\\dag _{k\\parallel }=c^\\dag _{\\nu =N}$ .",
"Hence, for the measurement operators of corollary 2, we see that $c_k|\\Psi \\rangle &=&\\sqrt{p_k} c_{k_\\parallel }|\\Psi \\rangle , \\;\\; c^\\dag _k|\\Psi \\rangle =\\sqrt{p_{\\bar{k}}}c^\\dag _{k_\\perp }|\\Psi \\rangle \\,,$ are clearly orthogonal SDs.",
"For the measurement operators of Theorem 1, Eq.",
"(REF ) implies $c^\\dag _k c_k|\\Psi \\rangle &=&\\sqrt{p_k} c^\\dag _k c_{k_\\parallel }|\\Psi \\rangle \\,,\\\\ c_kc^\\dag _k|\\Psi \\rangle &=&\\sqrt{p_{\\bar{k}}} c_k c^\\dag _{k_\\perp }|\\Psi \\rangle = \\sqrt{p_{\\bar{k}}} c^\\dag _{k^{\\prime }}c_{k_\\parallel }|\\Psi \\rangle \\,,$ where $c^\\dag _{k^{\\prime }}=\\sqrt{p_{\\bar{k}}}c^\\dag _{k_{\\parallel }}-\\sqrt{p_k}c^\\dag _{k_\\perp }$ , which are also orthogonal SD's ($\\lbrace c_k,c^\\dag _{k^{\\prime }}\\rbrace =0$ ).",
"And in the case of the generalized measurement based on the operators (REF ), we see from (REF )–() that $M_k|\\Psi \\rangle &=& (\\alpha \\sqrt{p_k}c^\\dag _k+\\beta \\sqrt{p_{\\bar{k}}}c^\\dag _{k^{\\prime }}) c_{k_{\\parallel }}|\\Psi \\rangle \\\\M_{\\bar{k}}|\\Psi \\rangle &=& (\\gamma \\sqrt{p_k}c^\\dag _k+\\delta \\sqrt{p_{\\bar{k}}}c^\\dag _{k^{\\prime }}) c_{k_{\\parallel }}|\\Psi \\rangle $ are as well SDs, not necessarily orthogonal."
],
[
"Comparing one-body entanglement of states with different particle number",
"Given two pure fermionic states $|\\Psi \\rangle $ and $|\\Phi \\rangle $ with the same fermion number $N$ , such that their associated SPDMs have the same trace ${\\rm Tr}\\,\\rho ^{(1)}_\\Psi ={\\rm Tr}\\,\\rho ^{(1)}_\\Phi =N$ , $|\\Psi \\rangle $ is considered not less entangled than $|\\Phi \\rangle $ if Eq.",
"(REF ) ($\\mathbf {\\lambda }(\\rho ^{(1)}_{\\Psi })\\prec \\mathbf {\\lambda }(\\rho ^{(1)}_{\\Phi })$ ) is satisfied.",
"Here $\\mathbf {\\lambda }(\\rho ^{(1)})$ denotes the spectrum of $\\rho ^{(1)}$ sorted in decreasing order.",
"It can be shown that $\\mathbf {\\lambda }(\\rho ^{(1)}_{\\Psi })\\prec \\mathbf {\\lambda }(\\rho ^{(1)}_{\\Phi })\\;\\Longrightarrow \\; \\mathbf {\\lambda }(D^{(1)}_{\\Psi })\\prec \\mathbf {\\lambda }(D^{(1)}_{\\Phi })\\,,$ for the extended DM defined in Eq.",
"(REF ), the spectrum of which is $(\\mathbf {\\lambda },1-\\mathbf {\\lambda })$ .",
"Eq.",
"(REF ) follows from the straightforward properties (see for instance [49], [50]) i) $\\mathbf {\\lambda }\\prec \\mathbf {\\lambda }^{\\prime }$ $\\Longrightarrow $ $1-\\mathbf {\\lambda }\\prec 1-\\mathbf {\\lambda }^{\\prime }$ , ii) $\\mathbf {\\lambda }\\prec \\mathbf {\\lambda }^{\\prime }$ and $\\mathbf {\\mu }\\prec \\mathbf {\\mu }^{\\prime }$ $\\Longrightarrow $ $(\\mathbf {\\lambda },\\mathbf {\\mu })\\prec (\\mathbf {\\lambda }^{\\prime },\\mathbf {\\mu }^{\\prime }),$ where $\\mathbf {\\lambda },\\mathbf {\\lambda }^{\\prime }, \\mathbf {\\mu },\\mathbf {\\mu }^{\\prime }\\in \\mathbb {R}^n$ are sorted in decreasing order and $(\\mathbf {\\lambda },\\mathbf {\\mu })\\in \\mathbb {R}^{2n}$ denotes the sorted vector resulting from the union of $\\mathbf {\\lambda }$ and $\\mathbf {\\mu }$ .",
"The converse relation in (REF ) does not hold.",
"These properties also entail that the majorization relation (REF ) implies $\\mathbf {\\lambda }(D^{(1)})\\prec \\sum _j p_j\\mathbf {\\lambda }(D^{(1)}_j),$ for the corresponding extended densities.",
"The advantage of Eqs.",
"(REF )–(REF ) is that within a fixed SP space the extended vectors can always be compared through majorization, regardless of the particle number $N$ , since ${\\rm Tr}\\,D^{(1)}=n={\\rm dim}\\,{\\cal H}$ is fixed by the SP space dimension.",
"For two states $|\\Psi \\rangle $ and $|\\Phi \\rangle $ with definite but not necessarily coincident fermion number, we then say that $|\\Psi \\rangle $ is not less entangled than $|\\Phi \\rangle $ if Eq.",
"(REF ) holds.",
"In particular, it is clear that —up to a permutation— the same eigenvalue vector $\\mathbf {\\lambda }(D^{(1)})$ is assigned to all SD states in the Fock space of the system, irrespective of $N$ , so that they are all least entangled states.",
"The extension of Definition 1 to general ONG operations, not necessarily conserving the particle number, is now straightforward: a quantum operation is ONG if it admits a set of Kraus operators $\\lbrace {\\cal K}_j\\rbrace $ satisfying Eq.",
"(REF ) $\\forall $ $\\rho $ , with $D^{(1)}$ and $D^{(1)}_j$ the extended SPDMs determined by $\\rho $ and $\\rho _j$ respectively.",
"This extension allows us to consider operations such as that of Corollary 2, with Kraus operators $c_k$ and $c^\\dag _k$ .",
"The extended SPDMs $D^{(1)}$ of the postmeasurement states $\\frac{1}{\\sqrt{p_k}} c_k|\\Psi \\rangle $ and $\\frac{1}{\\sqrt{p_{\\bar{k}}}} c^\\dag _k|\\Psi \\rangle $ have clearly the same spectrum as those obtained from $\\frac{1}{\\sqrt{p_k}} c^\\dag _kc_k|\\Psi \\rangle $ and $\\frac{1}{\\sqrt{p_{\\bar{k}}}} c_kc^\\dag _k|\\Psi \\rangle $ —up to a permutation—, with the same probabilities, such that theorem 1 directly implies corollary 2.",
"In fact, the number conserving occupation measurement is just a composition of the former with itself, as $(c_k,c^\\dag _k)\\circ (c_k,c^\\dag _k)=(c_kc^\\dag _k,c^\\dag _k c_k)$ .",
"We can therefore embed the fermion number preserving resource theory within a more general theory in which the set of free states is the convex hull of SD states —of all possible particle number— and where the free operations are one-body unitaries and operations based on the $\\lbrace c_k,c^\\dag _k\\rbrace $ measurement mapping SDs onto SDs.",
"Any SD and hence any free state can be prepared from an arbitrary state $\\rho $ by means of free operations only, i.e., by applying one-body unitaries and successive $\\lbrace c_k,c^\\dag _k\\rbrace $ measurements with postselection.",
"Since the starting state is arbitrary, any free state in this theory can be converted into any other free state by free operations.",
"Allowing the particle number to vary implies that appending free states is also a free operation, since this will not alter the spectrum of the associated $D^{(1)}$ in the full SP space.",
"We have here considered pure states $|\\Psi \\rangle $ with definite fermion number and operators ${\\cal K}_j$ which produce states ${\\cal K}_j|\\Psi \\rangle $ with definite fermion number when applied to such states, suitable for systems where a particle number superselection rule applies.",
"The extension to the case where general fermionic Gaussian states (with no fixed particle number but definite number parity) and active FLO operations are also considered free is straightforward.",
"It requires the consideration of the full extended $2n\\times 2n$ quasiparticle density matrix containing in addition the pair creation and annihilation contractions $\\langle c^\\dag _k c^\\dag _{k^{\\prime }}\\rangle $ and $\\langle c_{k^{\\prime }} c_k\\rangle $ , the eigenvalues of which remain invariant under general Bogoliubov transformations.",
"Its mixedness determines a generalized one-body entanglement [20] which vanishes if and only if the state is a SD or a quasiparticle vacuum, i.e., a general pure fermionic Gaussian state."
]
] | 2001.03570 |
[
[
"Trend Filtering -- II. Denoising Astronomical Signals with Varying\n Degrees of Smoothness"
],
[
"Abstract Trend filtering---first introduced into the astronomical literature in Paper I of this series---is a state-of-the-art statistical tool for denoising one-dimensional signals that possess varying degrees of smoothness.",
"In this work, we demonstrate the broad utility of trend filtering to observational astronomy by discussing how it can contribute to a variety of spectroscopic and time-domain studies.",
"The observations we discuss are (1) the Lyman-$\\alpha$ forest of quasar spectra; (2) more general spectroscopy of quasars, galaxies, and stars; (3) stellar light curves with planetary transits; (4) eclipsing binary light curves; and (5) supernova light curves.",
"We study the Lyman-$\\alpha$ forest in the greatest detail---using trend filtering to map the large-scale structure of the intergalactic medium along quasar-observer lines of sight.",
"The remaining studies share broad themes of: (1) estimating observable parameters of light curves and spectra; and (2) constructing observational spectral/light-curve templates.",
"We also briefly discuss the utility of trend filtering as a tool for one-dimensional data reduction and compression."
],
[
"Introduction",
"Many astronomical analyses can be described by the following problem setup.",
"Suppose we collect noisy measurements of an observable quantity (e.g., flux, magnitude, photon counts) according to the data generating process $f(t_i) = f_0(t_i) + \\epsilon _i, \\hfill i=1,\\dots ,n,$ where $t_1,\\dots ,t_n$ is an arbitrarily-spaced grid of one-dimensional inputs (e.g., times or wavelengths), $\\epsilon _i$ is mean zero noise, and $f_0$ is a signal that may exhibit varying degrees of smoothness across the input domain (e.g., a smooth signal with abrupt dips/spikes).",
"Given the observed data sample, we then attempt to estimate (or denoise) the underlying signal $f_0$ from the observations by applying appropriate statistical methods.",
"In [50]—hereafter referred to as Paper I—we introduced trend filtering [62], [61] into the astronomical literature.",
"When the underlying signal is spatially heterogeneous, i.e.",
"possesses varying degrees of smoothness, trend filtering is superior to popular statistical methods such as Gaussian process regression, smoothing splines, kernels, LOESS, and many others [42], [41], [61].",
"Furthermore, the trend filtering estimate can be computed via a highly efficient and scalable convex optimization algorithm [54] and only requires data-driven selection of a single scalar hyperparameter.",
"In this paper, we directly demonstrate the broad utility of trend filtering to observational astronomy by using it to carry out a diverse set of spectroscopic and time-domain analyses.",
"The outline of this paper is as follows.",
"In Section , we use trend filtering to study the Lyman-$\\alpha $ forest of quasar spectra—a series of absorption features that can be used as a tracer of the matter density distribution along quasar-observer lines of sight.",
"We choose to study this application in depth and then illustrate the breadth of trend filtering's utility through our discussions in Section .",
"The applications we discuss in Section can be grouped into two broad (and often intertwined) categories: (1) deriving estimates of observable parameters from trend-filtered observations; and (2) using trend filtering to construct spectral/light-curve templates of astronomical objects/events.",
"In Section REF , we discuss constructing spectral template libraries for astronomical objects by trend filtering coadded spectroscopic observations.",
"We illustrate our approach on quasar, galaxy, and stellar spectra from the Sloan Digital Sky Survey.",
"Emission-line parameters can also be robustly estimated by fitting radial basis functions (e.g., Gaussians) to trend-filtered estimates near emission lines.",
"In Section REF , we use relaxed trend filtering to model the detrended, phase-folded light curve of a Kepler stellar system with a transiting exoplanet.",
"We derive estimates and full uncertainty distributions for the transit depth and total transit duration.",
"In Section REF , we use trend filtering to denoise a detrended, phase-folded Kepler light curve of an eclipsing binary (EB) system.",
"We illustrate that trend filtering provides significant improvements upon the popular polyfit method of [52] that is used to model Kepler EB light curves and derive observable parameters.",
"In Section REF , we discuss using trend filtering to construct light-curve templates of supernova (SN) explosions.",
"We illustrate this approach on a SN light curve obtained from the Open Supernova Catalog ([28]).",
"Furthermore, we derive estimates and full uncertainty distributions for a set of observable parameters—namely, the maximum apparent magnitude, the time of maximum, and the decline rate.",
"Finally, in Section REF , we briefly discuss a different, non-data-analysis application of trend filtering.",
"Specifically, we discuss the use of trend filtering as a tool for fast and flexible one-dimensional data reduction and compression.",
"The flexibility of the trend filtering estimator, paired with its efficient speed and storage capabilities, make it a potentially powerful tool to include in large-scale (one-dimensional) astronomical data reduction and storage pipelines.",
"We utilize the [4] glmgen R implementation of the [54] trend filtering optimization algorithm in this work.",
"See Paper I for implementations in other programming languages."
],
[
"MAIN APPLICATION: QUASAR LYMAN-$\\alpha $ FOREST",
"The Lyman-$\\alpha $ (Ly$\\alpha $ ) forest is the name given to the absorption features seen in quasar spectra which are caused by neutral hydrogen (H I) in the intergalactic medium between a quasar and an observer.",
"When emitted from an accretion disk close to the central black hole, the light from the quasar has a relatively smooth spectrum—a continuum—caused by the summed black-body emission of gas with different temperatures at different disk radii [43].",
"Emission lines are also seen, and their intensities and line ratios supply information on the physical conditions in the line emitting gas.",
"At least twenty broad emission lines, broadened by high velocities and temperatures can be measured in a single AGN, along with a similar number of narrow lines from colder gas [37].",
"The emitted spectrum therefore already consists of a superposition of components with varying degrees of smoothness.",
"The Ly$\\alpha $ forest arises when this spectrum is further processed with the addition of absorption lines.",
"Light moving towards the observer is redshifted into resonance with the Ly$\\alpha $ transition of H I, and the strength of absorption features is dictated by the densities of intergalactic material along the line of sight [55].",
"The smoothness of the absorption lines varies depending on the gas pressure, and thermal doppler broadening [27], [47].",
"Sharper absorption features, metal lines, are also caused by other intergalactic species, such as C IV, O VI and Mg II [29], [49].",
"The usefulness of the Ly$\\alpha $ forest as a cosmological probe [45] stems from its relationship to the matter density field in the Universe, effectively mapping out structure along each quasar-observer line of sight [18], [34].",
"In order to extract this information from noisy spectra and separate it from other components, it is useful to have a method that can deal with the complexities outlined above, i.e.",
"one that can naturally adapt to varying degrees of smoothness without extensive tuning.",
"The relative fluctuations in the Ly$\\alpha $ forest transmitted flux fraction are of primary interest since they possess a monotonic relationship with the relative distribution of the absorbing H I.",
"We utilize trend filtering to first denoise the spatially heterogeneous flux signal in an observed Ly$\\alpha $ forest.",
"Estimates for the fluctuations in transmitted flux due to absorbing H I are then typically produced by coupling the denoised Ly$\\alpha $ forest with estimates for the quasar continuum and the cosmic mean transmitted flux in the Ly$\\alpha $ forest.",
"We take an alternative approach: directly estimating the mean flux level—defined as the product of the continuum and cosmic mean transmitted flux—as in [19].",
"The mean flux level is a very smooth, spatially homogeneous function within the truncated Ly$\\alpha $ forest restframe.",
"It is therefore appropriate to use a linear smoother (see Paper I, Section 2) for this stage of estimation.",
"Specifically, we use local polynomial regression (LOESS; [16]; [24]; [35]).",
"In this section, we illustrate these methods on a mock quasar Ly$\\alpha $ forest from [6] and a real quasar Ly$\\alpha $ forest from the Baryon Oscillation Spectroscopic Survey Data Release 12 (BOSS DR12; [3]) of the Sloan Digital Sky Survey III , .",
"Historically, Ly$\\alpha $ forest analyses have typically utilized kernel smoothers [19], [32], wavelets [60], or Gaussian processes [44]."
],
[
"Notation",
"Suppose we observe a quasar located at redshift $z = z_0$ .",
"Ignoring systematic effects such as sky contamination and interstellar extinction for the moment, the observational DGP of the Ly$\\alpha $ forest can be assumed to follow the model $f(\\lambda ) &= f_0(\\lambda ) + \\epsilon (\\lambda ), &\\;\\;\\;\\;\\;\\; \\lambda \\in \\Lambda (z_0), \\\\&= \\overline{F}(\\lambda )\\cdot C(\\lambda )\\cdot \\big ( 1+ \\delta _{F}(\\lambda )\\big ) + \\epsilon (\\lambda ), $ where $f(\\lambda )$ is the observed flux at wavelength $\\lambda $ , $f_0(\\lambda )$ is the flux signal, $\\epsilon (\\lambda )$ is zero mean white Gaussian noise, $\\Lambda (z_0) = (\\lambda _{\\text{Ly}\\beta }, \\;\\lambda _{\\text{Ly}\\alpha })\\cdot (1+z_0)$ is the redshifted Ly$\\alpha $ forest, $C(\\lambda )$ is the flux of the unabsorbed quasar continuum, $F(\\lambda ) = f_0(\\lambda )/C(\\lambda )$ is the transmitted flux fraction, $\\overline{F}(\\lambda ) = \\mathbb {E}[F(\\lambda )]$ is the mean transmitted flux fraction (over the sky) in the Ly$\\alpha $ forest at redshift $z = \\lambda /\\lambda _{\\text{Ly}\\alpha } - 1$ , and $\\delta _{F}(\\lambda ) = F(\\lambda )/\\overline{F}(\\lambda ) -1 $ is the fluctuation about the mean Ly$\\alpha $ transmitted flux at redshift $z = \\lambda /\\lambda _{\\text{Ly}\\alpha } - 1$ .",
"Here, $\\delta _{F}$ is the quantity we are primarily interested in estimating since $\\delta _F \\propto \\delta _{\\text{H\\hspace{2.125pt}{\\scriptsize I}}}^{-1}$ at each fixed redshift, where $\\delta _{\\text{H\\hspace{2.125pt}{\\scriptsize I}}}$ is the density of H I.",
"The estimation of the flux signal $f_0$ is viewed as an ancillary step.",
"Although, in principle, it is preferable to study the full spectral range $\\Lambda (z_0)$ we have found that, in the nonparametric setting, estimating the quasar continuum near the localized Ly$\\alpha $ and Ly$\\beta $ emission peaks at the boundaries of the Ly$\\alpha $ forest reduces the estimation accuracy in the interior of $\\Lambda (z_0)$ .",
"Therefore, in this work we limit our analysis to the truncated Ly$\\alpha $ forest range $\\overline{\\Lambda }(z_0) = (1045 \\;\\text{Å}\\;, 1195 \\;\\text{Å})\\cdot (1+z_0).",
"$ We simplify notation in this work by changing the input space of the functions introduced above by merely altering the input variable.",
"For example, with respect to $\\delta _F$ , we maintain the notation $\\delta _F(\\cdot )$ for all inputs $\\lambda $ , $\\nu $ , $z$ , $\\zeta $ , while it is understood that a proper change of input spaces has taken place.",
"The various input spaces are defined in Table REF .",
"Table: Various input spaces utilized for the Lyα\\alpha forest analysis.",
"Notation of functions is held constant, e.g.",
"δ F (·)\\delta _F(\\cdot ), and an alteration of the input variable implicitly indicates a change of input spaces.",
"Logarithmic wavelengths are scaled for numerical stability of the trend filtering optimization algorithm."
],
[
"Trend filtering the observed flux",
"We use quadratic trend filtering [62], [61] to estimate the flux signal $f_0$ of the observational model (REF ).",
"In both BOSS DR12 and the [6] mock catalog, the quasar spectra are sampled on equally-spaced grids in logarithmic wavelength space with $\\Delta \\log _{10}(\\lambda _i) = 10^{-4}$ dex (in logarithmic angstroms).",
"Furthermore, flux measurement variances are provided by the BOSS pipeline ([10]), accounting for the statistical uncertainty introduced by photon noise, CCD read noise, and sky-subtraction error.",
"We correct the BOSS spectrum for interstellar extinction with the [15] extinction law and the [56] dust map.",
"We fit the trend filtering estimator on the equally-spaced logarithmic grid and tune the complexity by minimizing Stein's unbiased risk estimate (SURE) of the fixed-input mean-squared error (see Paper I).",
"More precisely, we fit the trend filtering estimator in the input space $Z(z_0)=10^4\\cdot \\log _{10}(\\overline{\\Lambda }(z_0))$ , as defined in Table REF , where we add the scaling to unit spacing for numerical stability of the trend filtering convex optimization."
],
[
"Nonparametric continuum estimation",
"We utilize a modified [19] approach to propagate the trend filtering estimate for the flux signal $f_0$ from Section REF into an estimate for the fluctuation field $\\delta _F$ along a line of sight to an observed quasar.",
"Namely, given the trend filtering estimate $\\widehat{f}_0$ , we directly estimate the smooth mean flux level defined as the product $m = \\overline{F}\\cdot C$ and then define the $\\delta _F$ estimates via the transformation $\\widehat{\\delta }_F := \\widehat{f}_0/\\widehat{m} - 1$ .",
"We carry out the estimation of $m$ via a wide-kernel LOESS smooth of the trend filtering estimate, with the specific bandwidth of the kernel selected by optimizing over a large sample of mock spectra (detailed in Section REF ).",
"We find that regressing on the fitted values of the trend filtering estimate—instead of the observational DGP (REF )—significantly improves the accuracy and robustness of the $\\delta _F$ estimates.",
"We carry out the LOESS estimation in the Ly$\\alpha $ restframe $\\overline{\\Lambda }_{\\text{rest}}(z_0)$ (see Table REF ) in order to remove the effect of redshifting on the smoothness of $m$ .",
"The LOESS estimation of $m$ is a fully nonparametric procedure and provides a reduction in bias over popular parametric approaches such as low-order power laws and principal components analyses (PCA).",
"The sole assumption of our LOESS approach is that, in the restframe, the mean flux level $m$ always has a fixed degree of smoothness, defined by an optimal fixed kernel bandwidth.",
"There is of course a bias-variance tradeoff here; namely, the decreased bias comes with a modest increase in variance compared to a parametric power law or low-dimensional PCA model.",
"Our decision in favor of the LOESS approach directly reflects our stance that low bias is preferable to low variance in this context since statistical uncertainty due to estimator variability is tracked by our uncertainty quantification (Section REF ), while uncertainty due to modeling bias is not easily quantifiable.",
"Therefore, we can be more confident that significant fluctuations in the estimated $\\delta _F$ field are in fact real, and not due to statistical bias in the quasar continuum estimate.",
"To be explicit, the LOESS estimator for $m$ is a regression on the data set $\\lbrace (\\nu _i,\\;\\widehat{f}_0(\\nu _i; \\widehat{\\gamma })\\rbrace _{i=1}^{n},\\hfill \\nu _i\\in \\overline{\\Lambda }_{\\text{rest}}(z_0), $ which can be viewed as arising from the DGP $\\widehat{f}_0(\\nu _i; \\widehat{\\gamma }) = m(\\nu _i) + \\rho _i,$ where $\\widehat{f}_0$ is the trend filtering estimate fixed at the minimum SURE hyperparameter $\\widehat{\\gamma }$ , $e_i = \\widehat{f}_0(\\nu _i; \\widehat{\\gamma }) - f_0(\\nu _i)$ are the errors of the trend filtering estimate, and $\\rho _i = m(\\nu _i)\\cdot \\delta _{F}(\\nu _i) + e_i$ are autocorrelated fluctuations about zero.",
"The LOESS estimator is the natural extension of kernel regression ([40]; [63]) to higher-order local polynomials.",
"Given a kernel function $K(\\cdot )$ with bandwidth $h>0$ , for each $i=1,\\dots ,n$ , the LOESS estimator is obtained by minimizing $\\sum _{j=1}^{n}{\\Big (\\widehat{f}_0(\\nu _j; \\widehat{\\gamma }) - \\phi _{\\nu _i}(\\nu _{j};\\beta _0,\\dots ,\\beta _d)\\Big )^2}K\\Bigg (\\frac{|\\nu _j - \\nu _i|}{h}\\Bigg ),$ and letting $\\widehat{m}(\\nu _i) = \\widehat{\\beta }_0$ , where $\\phi _{\\nu _i}(\\cdot \\;;\\beta _0,\\dots ,\\beta _d)$ is a $d$ th order polynomial centered at $\\nu _i$ .",
"Specifically, we utilize the local linear regression estimator (LLR; $d=1$ ) and the Epanechnikov kernel ([23]) $K(t) = \\frac{3}{4}(1-t^2)\\mathbb {1}\\lbrace |t|<1\\rbrace .$ The LLR estimator is described in full detail by Algorithm REF .",
"Given the trend filtering estimate $\\widehat{f}_0$ and the LLR estimate $\\widehat{m}$ , the $\\delta _F$ estimates are then defined as $\\widehat{\\delta }_{F}(z_i; \\widehat{\\gamma }) = \\frac{\\widehat{f}_0(z_i; \\widehat{\\gamma })}{\\widehat{m}(z_{i})} - 1, \\hfill z_1,\\dots ,z_n\\in \\Pi (z_0), $ where we deliberately express $\\widehat{m}$ as “hyperparameter-less” since $\\gamma $ has already been fixed at the minimum SURE value $\\widehat{\\gamma }$ and we provide the optimal LOESS bandwidth value $h_0 = 74$ Å—optimized over the large mock sample.",
"Here, we have also done a change of variables to redshift space—our desired input domain for studying the H I density fluctuations in the IGM.",
"LOESS (local linear) estimator for mean flux level [1] Training Data $\\lbrace (\\nu _i,\\widehat{f}_0(\\nu _i; \\widehat{\\gamma }))\\rbrace _{i=1}^{n}$ , $\\;$ Bandwidth $h_0 = 74$ Å $i$ Let $\\widehat{\\beta }_0$ , $\\widehat{\\beta }_1$ minimize the locally weighted sum of squares $\\sum _{j=1}^{n}{\\Big (\\widehat{f}_0(\\nu _j; \\widehat{\\gamma }) - \\beta _0 - \\beta _1(\\nu _j-\\nu _i)\\Big )^2}\\cdot K\\Bigg (\\frac{|\\nu _j - \\nu _i|}{h_0}\\Bigg ).$ Let $\\widehat{m}(\\nu _i) = \\widehat{\\beta }_0(\\nu _i)$ .",
"Output: $\\lbrace \\widehat{m}(\\nu _i)\\rbrace _{i=1}^{n}$ Although $h_0$ is chosen to directly optimize $\\widehat{\\delta }_F$ accuracy, an estimate for the quasar continuum $C(\\cdot )$ arises intrinsically: $\\widehat{C}(\\nu _i) = \\overline{F}(\\nu _i)^{-1}\\cdot \\widehat{m}(\\nu _i),\\hfill \\nu _i\\in \\overline{\\Lambda }_{\\text{rest}}(z_0),$ where precise estimates of $\\overline{F}$ follow from a rich literature (e.g.",
"[9]; [39]; [25]; [20]; [46]; [7]).",
"The $\\delta _F$ estimates could then be equivalently restated as $\\widehat{\\delta }_F(z_i; \\widehat{\\gamma }) = \\frac{\\widehat{F}(z_i; \\widehat{\\gamma })}{\\overline{F}(z_i)} - 1, \\hfill z_1,\\dots ,z_n\\in \\Pi (z_0),$ where $\\widehat{F}(z_i; \\widehat{\\gamma }) = \\widehat{f}_0(z_i; \\widehat{\\gamma })/\\widehat{C}(z_i).",
"$"
],
[
"Calibrating continuum smoothness",
"We utilize a sample of 124,709 mock quasar spectra from the [6] catalog to optimize the bandwidth of the LOESS estimator for the mean flux level that intrinsically removes the effect of the quasar continuum.",
"Our mock data reduction is detailed in the appendix and the redshift distribution of the quasars is shown in the top panel of Figure REF .",
"Figure: Top: Distribution of mock quasar redshifts (data reduction detailed in the appendix).",
"We utilize this sample of 124,709 quasars to calibrate the optimal nonparametric continuum smoothness.",
"Bottom: Mean absolute deviation error curve for selecting the optimal kernel bandwidth for the LOESS (local linear) estimator of the mean flux level, averaged over the 124,709 spectra in the mock sample.",
"The optimal choice of bandwidth is h 0 =74h_0 = 74 Å.For each mock quasar Ly$\\alpha $ forest with DGP $Q_j$ , $j=1$ , $\\dots $ , 124,709, we first compute the trend filtering hyperparameter value that minimizes the true fixed-input mean-squared prediction error $\\gamma _0^j = \\operatornamewithlimits{arg\\,min}_{\\gamma >0} \\frac{1}{n}\\sum _{i=1}^{n}\\mathbb {E}_{Q_j}\\Big [\\big (f(\\zeta _i) - \\widehat{f}_0(\\zeta _i;\\gamma )\\big )^2\\;\\Big |\\;\\zeta _1,\\dots ,\\zeta _n\\Big ].",
"$ Then, given the trend filtering restframe fitted values $\\lbrace (\\nu _i,\\;\\widehat{f}_0(\\nu _i; \\gamma _0^j)\\rbrace _{i=1}^{n}$ , we fit a LOESS estimator with bandwidth $h$ and define the error (as a function of $h$ ) of the resulting $\\delta _F$ estimator as the fixed-input mean absolute deviation (MAD) error $R_j(h) = \\frac{1}{n}\\sum _{i}\\mathbb {E}_{Q_j}\\Big [\\big |\\delta _{F}(\\nu _i) - \\widehat{\\delta }_F(\\nu _i; \\gamma _0^j,h)\\big |\\;\\Big |\\;\\nu _1,\\dots ,\\nu _n\\Big ], $ where $\\mathbb {E}_{Q_j}$ denotes the mathematical expectation over the randomness arising from the observational DGP $Q_j$ .",
"Because we can repeatedly sample from each mock quasar DGP, the expectations in (REF ) and (REF ) can be computed to an arbitrary precision.",
"Here, we utilize 300 realizations of each DGP to approximate the mathematical expectations.",
"We then define the optimal choice of $h$ as the minimizer of the summed error over the full sample of $m = $ 124,709 mock quasar spectra: $h_0 = \\operatornamewithlimits{arg\\,min}_{h>0} \\sum _{j=1}^{m} R_j(h).$ The aggregate error curve is shown in the bottom panel of Figure REF , yielding an optimal value of $h_0 = 74$ Å.",
"We find that defining $R(h)$ as the conditional MAD error—instead of the conditional MSE—provides an essential boost in robustness that keeps the error from being dominated by a very small proportion of worst-case estimates.",
"More complex choices of bandwidth, e.g.",
"variable bandwidths that depend on the S/N ratio of the trend filtering estimator, the restframe pixel spacing, and/or the redshift of the quasar, do not significantly improve upon the $h_0 = 74$ Å$\\;$ restframe fixed bandwidth."
],
[
"Uncertainty quantification",
"Given the assumed noise model $\\epsilon _i \\sim N(0, \\sigma _i^2)$ , $i=1,\\dots ,n$ provided by the BOSS pipeline ([10]), we can construct a pointwise variability band for $\\widehat{\\delta }_F$ via an augmentation of the parametric bootstrap outlined in Algorithm 2 of Paper I.",
"Specifically, given the parametric bootstrap ensemble of trend filtering estimates provided by the parametric bootstrap algorithm, we fit the mean flux level of each with the LOESS estimator detailed in Algorithm REF , and then define the $\\widehat{\\delta }_F$ bootstrap ensemble $\\widehat{\\delta }_{F,b}^{*}(z_i) = \\frac{\\widehat{f}_b^{*}(z_i)}{\\widehat{m}_b^{*}(z_i)} - 1, \\hfill i=1,\\dots ,n,\\;\\;b=1,\\dots ,B,$ where, for each $b=1,\\dots ,B$ , $\\widehat{m}_b^{*}$ is the LOESS estimate fit to the data set $\\big \\lbrace (\\nu _i, \\widehat{f}_b^{*}(\\nu _i))\\big \\rbrace _{i=1}^{n}$ .",
"Note that refitting the LOESS estimator on each bootstrap realization allows us to track the extra variability introduced into the $\\delta _F$ estimates by the uncertainty in the continuum estimate.",
"A $(1-\\alpha )\\cdot 100\\%$ quantile-based pointwise variability band for $\\widehat{\\delta }_F$ is then given by $V_{1-\\alpha }(z_i) = \\Big (\\widehat{\\delta }_{F,\\alpha /2}^{*}(z_i), \\;\\widehat{\\delta }_{F,1-\\alpha /2}^{*}(z_i)\\Big ), \\hfill i = 1,\\dots ,n. $ where $\\widehat{\\delta }_{F,\\beta }^{*}(z_i) = \\inf _{g}\\Bigg \\lbrace g : \\frac{1}{B} \\sum _{b=1}^{B} \\mathbb {1}\\big \\lbrace \\widehat{\\delta }_{F,b}^{*}(z_i) \\le g\\big \\rbrace \\ge \\beta \\Bigg \\rbrace ,$ for any $\\beta \\in (0,1)$ ."
],
[
"Results",
"A mock quasar spectrum from [6] and a real quasar spectrum from the Baryon Oscillation Spectroscopic Survey (BOSS; [3]) are displayed in Figure REF , with the results of our Ly$\\alpha $ forest analysis overlaid.",
"Starting with the top panel, the quadratic trend filtering estimate is fit on the equally-spaced observations in logarithmic wavelength space and then transformed to the restframe wavelength space, where the LOESS estimate for the mean flux level—the product of the continuum and cosmic mean Ly$\\alpha $ absorption—is then fit to the trend filtering fitted values.",
"The $\\delta _F$ estimates are then computed according to (REF ) and displayed in redshift space in the second panel, where they closely track the true $\\delta _F$ defined by (REF ).",
"Furthermore, the 95% bootstrap variability band defined in (REF ) can be seen to almost fully cover the true $\\delta _F$ signal.",
"The estimated $\\delta _F$ can be interpreted as an inversely proportional proxy for the deviations from the mean H I density in the intervening intergalactic medium at each redshift—with negative values of $\\delta _F$ corresponding to (epoch-relative) over-densities of H I and positive values corresponding to under-densities.",
"Figure: Results of Lyα\\alpha forest analysis.",
"Top panel: Lyα\\alpha forest of a mock quasar spectrum () in the restframe, with the quadratic trend filtering estimate shown in orange and the LOESS (local linear) estimate for the mean flux level shown in blue.",
"Second panel: The redshift-space fluctuations in the Lyα\\alpha transmitted flux fraction, with our estimate superposed.",
"The fluctuations inversely trace the relative under- and over-densities of H I in the intervening intergalactic medium between Earth and the quasar.",
"Third and Fourth panels: Analogous plots for a real quasar Lyα\\alpha forest from the twelfth data release of the Baryon Oscillation Spectroscopic Survey (; Plate = 6487, MJD = 56362, Fiber = 647).",
"The quasar is located at (RA, Dec, zz) ≈\\approx (196.680 ∘ ^{\\circ }, 31.078 ∘ ^{\\circ }, 2.560).The third and fourth panels are the analogous plots for a BOSS quasar spectrum Ly$\\alpha $ forest (Data Release 12, Plate = 6487, MJD = 56362, Fiber = 647).",
"The quasar is located in the northern galactic cap at (RA, Dec, $z$ ) $\\approx $ (196.680$^{\\circ }$ , 31.078$^{\\circ }$ , 2.560)."
],
[
"Further Applications",
"The analysis of signals possessing varying degrees of smoothness permeates many areas of astronomy.",
"In this section, we discuss a variety of further applications for which trend filtering may find use.",
"For the sake of brevity, we discuss these applications in less detail than the Ly$\\alpha $ forest analysis in Section .",
"Naturally, we expect trend filtering may find many uses in astronomy beyond those we explicitly discuss."
],
[
"Spectral template generation and estimation of emission-line parameters",
"Automated spectral classification and redshift estimation pipelines require rich template libraries that span the full space of physical objects in the targeted set in order to achieve high statistical accuracy.",
"In this section, we discuss using trend filtering to construct spectral templates from observational spectra.",
"We describe the procedure here for generating a single spectral template from a well-sampled observed spectrum.",
"Suppose we observe coadded flux measurements of a targeted source at wavelengths $\\lambda _1,\\dots ,\\lambda _n\\in \\Lambda $ according to the data generating process $f(\\lambda _i) = f_0(\\lambda _i) + \\epsilon _i, \\hfill i=1,\\dots ,n,$ where the observations have been corrected for systematic effects (e.g., sky subtraction, interstellar extinction, etc.)",
"and the $\\epsilon _i$ are mean-zero errors that arise from instrumental uncertainty and systematic miscalibrations.",
"After removal of potentially problematic pixels (e.g., near bright sky lines), let $\\widehat{f}_0(\\lambda ),\\;\\lambda \\in \\Lambda $ denote the continuous trend filtering estimate fit to the observations.",
"Given a confident object classification and redshift estimate $z_0$ (e.g., determined by visual inspection), we then define the restframe spectral template $b(\\lambda _{\\text{rest}}) = \\widehat{f}_0(\\lambda / (z_0 + 1)), \\hfill \\lambda _{\\text{rest}}\\in \\Lambda / (z_0 + 1),$ and store it in the respective object template library.",
"Figure: Optical coadded spectra collected by the Baryon Oscillation Spectroscopic Survey () of the Sloan Digital Sky Survey III (; ).",
"From top to bottom, a quasar (DR12, Plate = 7140, MJD = 56569, Fiber = 58) located at (RA, Dec, zz) ≈\\approx (349.737 ∘ ^\\circ , 33.414 ∘ ^\\circ , 2.3992.399), a galaxy (DR12, Plate = 7140, MJD = 56569, Fiber = 68) located at (RA, Dec, zz) ≈\\approx (349.374 ∘ ^\\circ , 33.617 ∘ ^\\circ , 0.1380.138), and a star (DR12, Plate = 4055, MJD = 55359, Fiber = 84) located at (RA, Dec, zz) ≈\\approx (236.834 ∘ ^\\circ , 0.680 ∘ ^\\circ , 0.0000.000).",
"We fit a quadratic trend filtering estimate to each spectrum in the logarithmic wavelength space in which the observations are equally spaced, and optimize the hyperparameter by minimizing Stein's unbiased risk estimate.",
"Given confidently determined redshifts (e.g., determined by visual inspection), the trend filtering estimate for each object can be scaled to the restframe and stored as a spectral template.",
"Furthermore, emission-line parameter estimates for a spectrum can be obtained by fitting Gaussian radial basis functions to the emission lines of the trend filtering estimate.In Figure REF , we show three optical coadded spectra from the twelfth data release of the Baryon Oscillation Spectroscopic Survey (BOSS DR12; [3]) of the Sloan Digital Sky Survey III (SDSS-III; [21]; [1]).",
"The figure is zoomed to a subinterval of the optical range for visual clarity and the spectra are easily identifiable as a quasar, a galaxy, and a star, respectively.",
"We fit an error-weighted quadratic trend filtering estimate to each spectrum in the logarithmic-angstrom wavelength space in which the BOSS spectra are gridded, and tune the hyperparameter to minimize Stein's unbiased risk estimate (see Section 3.5 of Paper I).",
"The trend filtering estimates give good results, adequately adapting to even the strongest emission and absorption features in each spectrum.",
"As in the BOSS spectroscopic pipeline [10], after a spectral classification and redshift measurement have been precisely determined, emission-line parameter estimates can then be obtained by fitting Gaussian radial basis functions to the emission lines of the spectrum “best fit”—nonlinearly optimizing the amplitudes, centroids, and widths.",
"We propose that the trend filtering estimate of a spectrum be used as the “best fit” for this type of procedure, e.g.",
"instead of the low-dimensional principal components models currently used by the BOSS pipeline.",
"The relative magnitudes of the emission-line parameter estimates can then be used to determine object subclassifications."
],
[
"Exoplanet transit modeling",
"In this section we discuss the use of trend filtering for modeling the photometric time series of an extrasolar planet transiting its host star.",
"Given a phase-folded stellar light curve, corrected for stellar variability and spacecraft motion systematics, trend filtering can be used to automatically produce fully nonparametric estimates and uncertainties for the transit depth and total transit duration.",
"Figure: Kepler-10c transit light curve analysis.",
"Top: Long-cadence (30-min.",
"), quarter-stitched, median detrended, relative flux light curve (LC_DETREND) of the confirmed exoplanet host Kepler-10 (KOI-072, KIC 11904151; ), processed by the Kepler pipeline () and obtained from the NASA Exoplanet Archive ().",
"Vertical lines indicate the observed transit events of the system's second confirmed planet Kepler-10c (KOI-072 c, KIC 11904151 c; ).",
"Middle: Phase-folded transit light curve for Kepler-10c (∼\\sim 45.29 day orbital period) with 1σ1\\sigma error bars.",
"The error-weighted relaxed trend filtering estimate, optimized by sequential KK-fold cross validation, is superposed with 95% variability bands.",
"The estimated inception and termination of the transit event are indicated by the vertical dashed lines.",
"The estimated transit depth and total transit duration are δ ^=488.292\\widehat{\\delta } = 488.292 ppm and T ^=6.927\\widehat{T} = 6.927 hours, respectively.",
"Bottom: Bootstrap sampling distributions of the transit depth and transit duration estimates.We demonstrate our approach on long-cadence photometric observations of Kepler-10 (KOI-72, KIC 11904151; [11]).",
"Kepler-10 is confirmed to host at least two exoplanets: Kepler-10b (KOI-72 b, KIC 11904151 b; [5]) and Kepler-10c (KOI-72 c, KIC 11904151 c; [26]).",
"Each planet was first detected via the transit method—a measurable periodic dimming in the photometry caused by the planet crossing in front of the host star.",
"Here, we use trend filtering to estimate a nonparametric transit model for Kepler-10c and derive depth and duration measurements.",
"The results of our analysis are displayed in Figure REF .",
"The top panel displays a sample of the Kepler-10 long-cadence (30-min.",
"increment), quarter-stitched, median detrended, relative flux light curve processed by the Kepler pipeline [31], which we accessed from the NASA Exoplanet Archive [2].",
"The observed transit events of Kepler-10c are indicated by the vertical dashed lines.",
"The middle panel displays the light curve (with $1\\sigma $ error bars) after phase-folding with respect to the $\\sim $ 45.29 day orbital period of Kepler-10c and zooming in on the transit event.",
"We fit a relaxed quadratic trend filtering estimate (see Section 3.7 of Paper I) to the phased data, weighted by the measurement variances provided by the Kepler pipeline and tuned by sequential $K$ -fold cross validation.",
"The optimal relaxation hyperparameter is $\\widehat{\\phi } = 0.14$ , indicating that relaxation significantly improves the traditional trend filtering estimate in this setting.",
"In particular, we find that relaxation allows the estimate to faithfully capture the sharp transitions corresponding to the beginning of the ingress phase and the end of the egress phase.",
"The relaxed trend filtering estimate is overlaid on the phase-folded light curve, along with 95% variability bands.",
"Estimates for the transit depth and total transit duration follow immediately from the relaxed trend filtering estimate, as detailed below.",
"The estimated inception and termination of the transit event are indicated by the vertical dashed lines in the middle panel.",
"The nonparametric bootstrap sampling distributions (see Algorithm 1 of Paper I) of the transit depth and total transit duration measurements are displayed in the bottom panel.",
"Our point estimates for the transit depth and total transit duration for Kepler-10c are $\\widehat{\\delta } = 488.292$ ppm and $\\widehat{T} = 6.927$ hours, respectively.",
"The knot-selection property of trend filtering is particularly appealing in this setting because it provides interpretation as to where the transit event begins and ends.",
"Specifically, we define our estimate of the inception of the transit event $\\widehat{T}_0$ as the leftmost knot selected by the trend filtering estimator and we define our estimate of the termination of the transit event $\\widehat{T}_1$ as the rightmost knotNote that the boundary points of the input space are not considered knots..",
"The total transit duration estimate then follows as $\\widehat{T} = \\widehat{T}_1 - \\widehat{T}_0$ .",
"Naturally, we define our transit depth estimate $\\widehat{\\delta }$ to be the minimum of the relaxed trend filtering estimate.",
"It is thus far unclear to us whether trend filtering can also reliably detect the end of the ingress and beginning of the egress phases—e.g.",
"via knot selection or examining changes in the estimated derivates—and therefore also provide nonparametric ingress/egress duration measurements.",
"We recommend pairing trend filtering with the traditional analytical transit model search [36] for these particular measurements.",
"That is, given the transit depth and total transit duration measurements provided by trend filtering, an analytical planet model can be fit over a parameter space that is constrained by the trend filtering parameter measurements.",
"Coupling the two methods in this way therefore also provides significant computational speedups over a traditional single-stage analytical model search since it greatly reduces the dimensionality of the parameter space to be searched over.",
"Furthermore, the relaxed trend filtering estimate may provide a benchmark $\\chi ^2$ statistic for the constrained analytical model comparison.",
"A dedicated paper on this application of trend filtering is forthcoming."
],
[
"Eclipsing binary modeling",
"In this section we discuss how trend filtering can further improve the work of the Kepler Eclipsing Binary working group, specifically in regard to the Eclipsing Binaries via Artificial Intelligence (EBAI) pipeline used to characterize Kepler eclipsing binary stars via their phase-folded light curves [52], [53], [58], [38].",
"The EBAI pipeline utilizes an artificial neural network (ANN) to estimate a set of physical parameters for each binary pair (the temperature ratio, sum of fractional radii, photometric mass ratio, radial and tangential components of the eccentricity, fillout factor, and inclination) from the observables of the phase-folded light curve (e.g., the eclipse widths, depths, and separations).",
"[52] outline the EBAI light curve pre-processing algorithm, which they call polyfit, that provides the crucial step of taking a noisy, irregularly-spaced, phase-folded light curve (detrended for spacecraft motion and normalized by the median flux) and outputting a denoised and gridded phase-folded light curve, which is then fed to the ANN.",
"We propose that trend filtering be used for this pre-processing step instead of the polyfit algorithm for the reasons detailed below.",
"An eclipsing binary (EB) light curve is characterized by periodic dips in the observed brightness that correspond to eclipse events along the line of sight to an observer.",
"In particular, there are two eclipses per orbital period—a primary and a secondary eclipse.",
"The primary eclipse occurs when the hotter star is eclipsed by the cooler star and produces a comparatively deep dip in observed brightness.",
"Conversely, the secondary eclipse occurs when the cooler star is eclipsed by the hotter star and produces a comparatively shallow dip in the observed brightness.",
"Depending on the effective temperature ratio and orbital period of the EB, the dips may range from very narrow and abrupt to very wide and smooth.",
"In Figure REF , we display a detrended, median-normalized, long-cadence (30 min.",
"increment) light curve of a Kepler EB (KIC 6048106; [11]; [53]), with the primary eclipses and secondary eclipses designated by dashed red and blue lines, respectively.",
"The orbital period of KIC 6048106 is $\\sim $ 1.559 days.",
"After phase-folding with respect to the estimated EB orbital period and centering the primary eclipse at Phase = 0, the purpose of the EBAI light-curve pre-processing step is to faithfully extract the signal of the phase-folded light curve and evaluate it on a regular grid so that it can then be input into the EBAI ANN.",
"We show a comparison of the polyfit algorithm of [52] and our trend filtering approach in Figure REF on the phase-folded KIC 6048106 light curve.",
"We choose a relatively high S/N light curve here in order to elucidate the significant statistical bias that underlies polyfit.",
"The polyfit algorithm fits a piecewise quadratic polynomial by weighted least-squares with four knots selected by a randomized computational search over the phase space.",
"The piecewise quadratic polynomial is forced to be continuous, but no smoothness constraints are placed on the derivatives of the estimate at the knots.",
"Recalling our discussion in Section 2.1 of Paper I, this overly-stringent modeling assumption leads to significant statistical bias in the light-curve estimate.",
"The bias is particularly apparent by examining the residuals of the polyfit estimate, which we display below the light curve.",
"Moreover, recalling our discussion of variable-knot regression splines in Section 3.1.1 of Paper I, a randomized partial search over the space of feasible knots inherently provides no guarantee of finding a global solution—leaving the algorithm susceptible to extreme failure scenarios.",
"We display a quadratic trend filtering estimate of the KIC 6048106 phase-folded light curve in the bottom panel of Figure REF , with the hyperparameter chosen by $K$ -fold cross validation.",
"The trend filtering estimate accurately recovers the signal of the light curve (clearly apparent here by examining a high S/N ratio light curve) and produces a desirable random residual scatter about zero.",
"Since the statistical bias introduced by the polyfit pre-processing stage propagates through to the EBAI ANN as systematic bias in the input data, we are confident that the use of trend filtering will in turn improve the error rate of the EBAI ANN output-parameter estimates.",
"Figure: Comparison of the polyfit algorithm of and our trend filtering approach for denoising phase-folded eclipsing binary light curves.",
"The light curve shown in this example comes from the Kepler eclipsing binary system KIC 6048106 (; ).",
"Top: The polyfit algorithm fits a piecewise quadratic polynomial by weighted least-squares with four knots selected by a randomized search over the phase space.",
"The estimate is constrained to be continuous but no constraints are enforced on the derivatives at the knots.",
"The overly-stringent assumed model leads to significant statistical bias, which is readily apparent by examining the autocorrelation in the residuals.",
"Bottom: Trend filtering is sufficiently flexible to accurately denoise the diverse set of signals observed in phase-folded eclipsing binary light curves.",
"Here, the goodness-of-fit is clear by the random, mean-zero residual scatter."
],
[
"Supernova light-curve template generation and estimation of observable parameters",
"In this section we demonstrate the use of trend filtering for generating light-curve templates of supernova (SN) events and estimating observable parameters.",
"We illustrate our approach on SN 2016coi (ASASSN-16fp; [30]) by constructing a B-band light-curve template from the well-sampled observations of [12] and [51] and deriving nonparametric estimates and full uncertainty distributions for the maximum apparent brightness, the time of maximum, and the decline rate parameter $\\Delta m_{15}(B)$ introduced by [48].",
"The improvement yielded by trend filtering as a tool for SN light-curve template generation, compared to, for example, the SNooPy cubic smoothing spline approach of the Carnegie Supernova Project [17], [13], [14] primarily corresponds to light curves with an especially bright peak magnitude and fast decline rate.",
"In such cases, trend filtering is better able to recover the abruptness of the peak, the initial sharp decline, and the subsequent slow decay.",
"This behavior is particularly characteristic of Type Ia SNe [64].",
"In cases where the peak is not particularly prominent, trend filtering and cubic smoothing splines produce nearly identical estimates.",
"Our procedure for generating the SN light-curve templates requires reasonably well-sampled observations (in particular, with the initial observation occurring before maximum light).",
"The resulting template libraries can then be used to classify SNe with partially-sampled light curves and derive parameter estimates [8].",
"The SN light-curve template generation procedure is analogous to the spectral template generation procedure discussed in Section REF , so we discuss it in less formal detail here.",
"Naturally, the same procedure can also be implemented for generating fixed-time spectral templates of SN events.",
"Given a well-sampled light curve, corrected for systematic effects (e.g., $K$ -corrections and interstellar reddening), we propose the use of quadratic trend filtering to generate a “best fit” to the observations.",
"Given a confident type classification, the trend filtering estimate can then be stored as a light-curve template in the respective library.",
"Figure: SN light-curve analysis (SN 2016coi).",
"Top: B-band photometry of the supernova SN 2016coi (ASASSN-16 fp; ) discovered on May 27, 2016 by the All Sky Automated Survey for SuperNovae (ASAS-SN; ) in the galaxy UGC 11868 at redshift z≈0.0036z \\approx 0.0036.",
"We fit a quadratic trend filtering estimate, tuned by KK-fold cross validation, and overlay 95% nonparametric bootstrap variability bands.",
"Bottom: Univariate/bivariate nonparametric bootstrap sampling distributions of the observable parameter estimates derived from the trend filtering light-curve estimate.",
"The bimodality in the bootstrap parameter distributions arises from systematic discrepancies between the observations of the two separate observers , .We show a B-band light curve for SN 2016coi (ASASSN-16fp; [30]) in the top panel of Figure REF .",
"The light curve is an aggregation of observations collected by [12] and [51], which we accessed from the Open Supernova Catalog ([28]).",
"SN 2016coi was discovered on May 27, 2016 (UT 2016-05-27.55) by the All Sky Automated Survey for SuperNovae (ASAS-SN; [57]) in the galaxy UGC 11868 at redshift $z \\approx 0.0036$ .",
"The classification of SN 2016coi currently remains intermediate between Type Ib and Type Ic [22], [33], [51], [59].",
"We fit a quadratic trend filtering estimate to the observations, weighted by measurement uncertainties and with the hyperparameter selected by $K$ -fold cross validation.",
"The trend filtering estimate, along with 95% nonparametric bootstrap variability bands, is overlaid in the top panel of the figure.",
"In the bottom panels we show the univariate nonparametric bootstrap sampling distributions for the estimates of the maximum apparent magnitude, the time of maximum, and the decline rate—defined as the relative change in the B-magnitude light curve from maximum light to the magnitude 15 days after the maximum [48].",
"We also show a bivariate bootstrap sampling distribution for maximum apparent magnitude versus decline rate.",
"The bimodality in the bootstrap sampling distributions arises from systematic discrepancies between the two separate sets of B-band observations that form the light curve [12], [51]."
],
[
"Data reduction and compression",
"Although the primary focus of this paper is the use of trend filtering as a tool for astronomical data analysis, it also possesses potential utility for large-scale reduction and compression of one-dimensional data sets.",
"This owes to several factors: its speed, scalability, flexibility, and representation as a sum of basis functions with a sparse coefficient vector.",
"Given a one-dimensional set of $n$ observations $(t_1,f(t_1)),\\dots ,(t_n,f(t_n))\\in (a,b)\\times \\mathbb {R}$ , trend filtering can quickly provide a flexible, lower-dimensional approximation of the data where the dimensionality is controlled by the choice of the hyperparameter $\\gamma $ .",
"In this context $\\gamma $ is a subjective choice that specifies the amount of (lossy) compression desired—unrelated to the discussion in Section 3.5 of Paper I.",
"For any given choice of $\\gamma $ , let $p$ be the number of knots selected by the trend filtering estimator.",
"The corresponding continuous-time representation of this lower-dimensional approximation is fully encoded by the knot locations and the sparse basis vector with $p+k+1$ nonzero entries, which can be stored efficiently.",
"The falling factorial basis then serves as the “dictionary” from which the continuous-time representation can be losslessly recovered.",
"Gridded uncertainty measurements for the reduced observations can also be computed and stored, though not in a sparse format, via the methods discussed in Section 3.6 of Paper I."
],
[
"Concluding remarks",
"In order to illustrate the broad utility of trend filtering to astronomy, we demonstrated its promising performance on a wide variety of problems across time-domain astronomy and astronomical spectroscopy.",
"We studied the Lyman-$\\alpha $ forest of quasar spectra with the most depth—using trend filtering to map the relative distribution of neutral hydrogen in the high redshift intergalactic medium along quasar-observer lines of sight.",
"Furthermore, we discussed how trend filtering can be used to (1) generate galaxy, quasar, and stellar spectral templates and estimate emission-line parameters; (2) produce nonparametric models for exoplanet transit events in phase-folded stellar light curves, providing estimates for the transit depth and total duration; (3) improve upon the polyfit algorithm utilized by the Kepler Eclipsing Binary via Artificial Intelligence (EBAI) pipeline for denoising phase-folded eclipsing binary light curves (as a preliminary step to estimating the physical parameters); (4) generate supernova light-curve templates and produce nonparametric estimates of the maximum apparent magnitude, the time of maximum, and the decline rate; (5) quickly and efficiently compress large collections of one-dimensional data sets.",
"Naturally, we expect trend filtering will find uses in astronomy beyond those that we explicitly discussed."
],
[
"ACKNOWLEDGEMENTS",
"We gratefully thank Ryan Tibshirani for his inspiration and generous feedback on this topic.",
"This work was partially supported by NASA ATP grant NNX17AK56G, NASA ATP grant 80NSSC18K1015, and NSF grant AST1615940.",
"This research contains data collected by the Sloan Digital Sky Survey III (SDSS-III).",
"Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science.",
"The SDSS-III web site is http://www.sdss3.org/.",
"SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.",
"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.",
"This paper includes data collected by the Kepler mission.",
"Funding for the Kepler mission is provided by the NASA Science Mission Directorate.",
"This research contains data that was accessed from the Open Supernova Catalog, a centralized, open repository for supernova metadata, light curves, and spectra.",
"The Open Supernova Catalog web site is https://sne.space.",
"The original sources of the supernova light curve studied in this paper are credited in the text.",
"tocsectionBibliography authoryear,open=[,close=] Appendix Mock quasar Lyman-$\\alpha $ forest reduction The [6] mock quasar catalog is designed to mimic the observational data generating processes of the quasar spectra released in Data Release 11 of the Baryon Oscillation Spectroscopic Survey ([3]).",
"We pool the first three realizations of the mock catalog, i.e.",
"M3_0_3/000, M3_0_3/001, and M3_0_3/002 and remove all damped Ly$\\alpha $ systems (DLAs), Lyman-limit systems (LLS), and broad absorption line quasars (BALs).",
"We assume no metal absorption in the Ly$\\alpha $ forest and correct estimation and subtraction of the sky.",
"We mask all pixels with and_mask $\\ne $ 0 or or_mask $\\ne $ 0.",
"Finally, we retain only the spectra with $\\ge 500$ pixels in the truncated Ly$\\alpha $ forest and those with a fixed-input-optimal trend filtering hyperparameter satisfying $\\gamma _0 < 5.25$ .",
"Spectra with $\\gamma _0 \\ge 5.25$ correspond to the very lowest S/N ratio observations, where the trend filtering estimate typically reduces to a global power law fit (zero knots).",
"The final mock sample contains 124,709 quasar spectra.",
"The [6] mock quasar catalog is designed to mimic the observational data generating processes of the quasar spectra released in Data Release 11 of the Baryon Oscillation Spectroscopic Survey ([3]).",
"We pool the first three realizations of the mock catalog, i.e.",
"M3_0_3/000, M3_0_3/001, and M3_0_3/002 and remove all damped Ly$\\alpha $ systems (DLAs), Lyman-limit systems (LLS), and broad absorption line quasars (BALs).",
"We assume no metal absorption in the Ly$\\alpha $ forest and correct estimation and subtraction of the sky.",
"We mask all pixels with and_mask $\\ne $ 0 or or_mask $\\ne $ 0.",
"Finally, we retain only the spectra with $\\ge 500$ pixels in the truncated Ly$\\alpha $ forest and those with a fixed-input-optimal trend filtering hyperparameter satisfying $\\gamma _0 < 5.25$ .",
"Spectra with $\\gamma _0 \\ge 5.25$ correspond to the very lowest S/N ratio observations, where the trend filtering estimate typically reduces to a global power law fit (zero knots).",
"The final mock sample contains 124,709 quasar spectra."
]
] | 2001.03552 |
[
[
"An Improvement to Chv\\'atal and Thomassen's Upper Bound for Oriented\n Diameter"
],
[
"Abstract An orientation of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$.",
"The oriented diameter of a graph $G$ is the smallest diameter among all the orientations of $G$.",
"The maximum oriented diameter of a family of graphs $\\mathscr{F}$ is the maximum oriented diameter among all the graphs in $\\mathscr{F}$.",
"Chv\\'atal and Thomassen [JCTB, 1978] gave a lower bound of $\\frac{1}{2}d^2+d$ and an upper bound of $2d^2+2d$ for the maximum oriented diameter of the family of $2$-edge connected graphs of diameter $d$.",
"We improve this upper bound to $ 1.373 d^2 + 6.971d-1 $, which outperforms the former upper bound for all values of $d$ greater than or equal to $8$.",
"For the family of $2$-edge connected graphs of diameter $3$, Kwok, Liu and West [JCTB, 2010] obtained improved lower and upper bounds of $9$ and $11$ respectively.",
"For the family of $2$-edge connected graphs of diameter $4$, the bounds provided by Chv\\'atal and Thomassen are $12$ and $40$ and no better bounds were known.",
"By extending the method we used for diameter $d$ graphs, along with an asymmetric extension of a technique used by Chv\\'atal and Thomassen, we have improved this upper bound to $21$."
],
[
"Introduction",
"An orientation of an undirected graph $G$ is an assignment of exactly one direction to each of the edges of $G$ .",
"A given undirected graph can be oriented in many different ways ($2^m$ , to be precise, where m is the number of edges).",
"The studies on graph orientations often concern with finding orientations which achieve a predefined objective.",
"Some of the objectives while orienting graphs include minimization of certain distances, ensuring acyclicity, minimizing the maximum in-degree, maximizing connectivity, etc.",
"One of the earliest studies regarding graph orientations were carried out by H.E.",
"Robbins in 1939.",
"He was trying to answer a question posed by Stanislaw Ulam.",
"“When may the arcs of a graph be so oriented that one may pass from any vertex to any other, traversing arcs in the positive sense only?\".",
"This led to a seminal work [1] of Robbins in which he proved the following theorem, “A graph is orientable if and only if it remains connected after the removal of any arc\"'.",
"A directed graph $\\vec{G}$ is called strongly connected if it is possible to reach any vertex starting from any other vertex using a directed path.",
"An undirected graph $G$ is called strongly orientable if it has a strongly connected orientation.",
"A bridge in a connected graph is an edge whose removal will disconnect the graph.",
"A 2-edge connected graph is a connected graph which does not contain any bridges.",
"The theorem of Robbins stated earlier says that it is possible for a graph $G$ to be strongly oriented if and only if $G$ is 2-edge connected.",
"Though Robbins stated the necessary and sufficient conditions for a graph to have a strong orientation, no comparison between the diameter of a graph and the diameter of an orientation of this graph was given in this study.",
"This was taken up by Chvátal and Thomassen in 1978 [2].",
"In order to discuss these quantitative results, we introduce some notation.",
"Let $G$ be an undirected graph.",
"The distance between two vertices $u$ and $v$ of $G$ , $d_G(u,v)$ is the number of edges in a shortest path between $u$ and $v$ .",
"For any two subsets $A$ , $B$ of $V(G)$ , let $d_G(A,B) = \\min \\lbrace d_G(u,v) : u \\in A, v \\in B \\rbrace $ .",
"The eccentricity of a vertex $v$ of $G$ is the maximum distance between $v$ and any other vertex $u$ of $G$ .",
"The diameter of $G$ is the maximum of the eccentricities of its vertices.",
"The radius of $G$ is the minimum of the eccentricities of its vertices.",
"Let $\\vec{G}$ be a directed graph and $u,v \\in V(\\vec{G})$ .",
"Then the distance from a vertex $u$ to $v$ , $d_{\\vec{G}}(u,v)$ , is defined as the length of a shortest directed path from $u$ to $v$ .",
"For any two subsets $A$ , $B$ of $V(\\vec{G})$ , let $d_{\\vec{G}}(A,B)=\\min \\lbrace d_{\\vec{G}}(u,v) : u \\in A, v \\in B \\rbrace $ .",
"The out-eccentricity of a vertex $v$ of $\\vec{G}$ is the greatest distance from $v$ to a vertex $u \\in V(\\vec{G})$ .",
"The in-eccentricity of a vertex $v$ of $\\vec{G}$ is the greatest distance from a vertex $u \\in V(\\vec{G})$ to $v$ .",
"The eccentricity of a vertex $v$ of $\\vec{G}$ is the maximum of its out-eccentricity and in-eccentricity.",
"The diameter of $\\vec{G}$ , denoted by $d(\\vec{G})$ , is the maximum of the eccentricities of its vertices.",
"The radius of $\\vec{G}$ is the minimum of the eccentricities of its vertices.",
"The oriented diameter of an undirected graph $G$ , denoted by $\\vec{d}(G)$ , is the smallest diameter among all strong orientations of $G$ .",
"That is, $ \\vec{d}(G) := \\text{min}\\lbrace d(\\vec{G}): \\vec{G} \\text{ is an orientation of } $ G$\\rbrace $ .",
"The oriented radius of an undirected graph $G$ is the smallest radius among all strong orientations of $G$ .",
"The maximum oriented diameter of the family ${F}$ of graphs is the maximum oriented diameter among all the graphs in ${F}$ .",
"Let $f(d)$ denote the maximum oriented diameter of the family of 2-edge connected diameter $d$ graphs.",
"That is, $ f(d) := \\text{max}\\lbrace \\vec{d}(G): G \\in {F}\\rbrace $ , where ${F}$ is the family of 2-edge connected graphs with diameter $d$ .",
"The following theorem by Chvátal and Thomassen [2] gives an upper bound for the oriented radius of a graph.",
"Theorem 1 [2] Every 2-edge connected graph of radius $r$ admits a strong orientation of radius at most $r^2+r$ .",
"The above bound was also shown to be tight.",
"In the same paper, they also proved that the problem of deciding whether an undirected graph admits an orientation of diameter 2 is NP-hard.",
"Motivated by the work of Chvátal and Thomassen [2], Chung, Garey and Tarjan [3] proposed a linear-time algorithm to check whether a mixed multigraph has a strong orientation or not.",
"They have also proposed a polynomial time algorithm which provides a strong orientation (if it exists) for a mixed multigraph with oriented radius at most $4r^2 + 4r$ .",
"Studies have also been carried out regarding the oriented diameter of specific subclasses of graphs like AT-free graphs, interval graphs, chordal graphs and planar graphs [4], [5], [6].",
"Bounds on oriented diameter in terms of other graph parameters like minimum degree and maximum degree are also available in literature [7], [8], [9], [10].",
"The following bounds for $f(d)$ were given by Chvátal and Thomassen [2].",
"Theorem 2 [2] $ \\frac{1}{2} d^{2} + d \\le f(d) \\le 2d^{2} + 2d $ .",
"Chvátal and Thomassen [2] has also proved that $f(2) = 6$ .",
"By Theorem REF , $8 \\le f(3) \\le 24$ .",
"In 2010, Kwok, Liu and West [11] improved these bounds to $9 \\le f(3) \\le 11$ .",
"To prove the upper bound of 11, Kwok, Liu and West partitioned the vertices of $G$ into a number of sets based on the distances from the endpoints of an edge which is not part of any 3-cycle.",
"Our study on the oriented diameter of 2-edge connected graphs with diameter $d$ uses this idea of partitioning the vertex set into a number of sets based on their distances from a specific edge."
],
[
"Our Results",
"In this paper we establish two improved upper bounds.",
"Firstly in Section 2, we show that $f(d) \\le 1.373d^2 + 6.971d - 1$ (Theorem REF ).",
"This is the first general improvement to Chvátal and Thomassen's upper bound $f(d) \\le 2d^2 + 2d$ from 1978.",
"For all $d \\ge 8$ , our upper bound outperforms that of Chvátal and Thomassen.",
"Their lower bound $f(d) \\ge \\frac{1}{2} d^{2} + d$ still remains unimproved.",
"We do not believe that our upper bound is tight.",
"Secondly in Section 3, for the case of $d = 4$ , we further sharpen our analysis and show that $f(4) \\le 21$ (Theorem REF ).",
"This is a considerable improvement from 40, which follows from Chvátal and Thomassen's general upper bound.",
"Here too, our upper bound is not yet close to the lower bound of 12 given by Chvátal and Thomassen and we believe that there is room for improvement in the upper bound."
],
[
"Oriented Diameter of Diameter $d$ Graphs",
"A subset $D$ of the vertex set of $G$ is called a $k$ -step dominating set of $G$ if every vertex not in $D$ is at a distance of at most $k$ from at least one vertex of $D$ .",
"An oriented subgraph $\\vec{H}$ of $G$ is called a $k$ -step dominating oriented subgraph if $V(\\vec{H})$ is a $k$ -step dominating set of $V(G)$ .",
"To obtain upper bounds for the oriented diameter of a graph $G$ with $n$ vertices and minimum degree $\\delta \\ge 2$ , Bau and Dankelmann [7] and Surmacs [8] first constructed a 2-step dominating oriented subgraph $\\vec{H}$ of $G$ .",
"They used this together with the idea in the proof of Theorem REF on $\\vec{H}$ to obtain the upper bounds of $\\frac{11n}{\\delta +1}+9$ and $\\frac{7n}{\\delta +1}$ , respectively, for the oriented diameter of graphs with minimum degree $\\delta \\ge 2$ .",
"We are using the algorithm OrientedCore described below to produce a 2-edge connected oriented subgraph $\\vec{H}$ of $G$ with some distance guarantees between the vertices in $\\vec{H}$ (Lemma REF ) and some domination properties (Lemma REF )."
],
[
"Algorithm ",
"Input: A 2-edge connected graph $G$ and a specified edge $pq$ in $G$ .",
"Output: A 2-edge connected oriented subgraph $\\vec{H}$ of $G$ .",
"Terminology: Let $d$ be the diameter of $G$ , let $k$ be the length of a smallest cycle containing $pq$ in $G$ and let $h = \\left\\lfloor k/2 \\right\\rfloor $ .",
"Notice that $k \\le 2d+1$ and $h \\le d$ .",
"Define $S_{i,j} = \\lbrace v \\in V(G) : d_G(v,p) = i, d_G(v,q) = j\\rbrace $ .",
"Since $S_{i,j}$ is non-empty only if $0\\le i, j \\le d$ and $|i-j| \\le 1$ , we implicitly assume these restrictions on the subscripts of $S_{i,j}$ wherever we use it.",
"For a vertex $v \\in S_{i,j}$ , its level $L(v)$ is $(j-i)$ and its width $W(v)$ is $\\max (i,j)$ .",
"We will always refer to an edge $\\lbrace u,v\\rbrace $ between two different $S_{i,j}$ 's as $uv$ when either $L(u) > L(v)$ or $L(u) =L(v)$ and $W(u) < W(v)$ (downward or rightward in Fig.",
"REF ).",
"Moreover the edge $uv$ is called vertical in the first case and horizontal in the second.",
"Observations based on the first edge of shortest paths from a vertex $v$ to $p$ or $v$ to $q$ : Every vertex $v \\in S_{i,i+1}$ , $1 \\le i \\le d-1$ , is incident to a horizontal edge $uv$ with $u \\in S_{i-1, i}$ .",
"Every vertex $v \\in S_{i+1,i}$ , $1 \\le i \\le d-1$ , is incident to a horizontal edge $uv$ with $u \\in S_{i,i-1}$ .",
"Every vertex $v \\in S_{i,i}$ , $1 \\le i \\le d$ , is incident either to a horizontal edge $uv$ with $u \\in S_{i-1,i-1}$ or two vertical edges $uv$ and $vx$ with $u \\in S_{i-1,i}$ and $x \\in S_{i, i-1}$ .",
"Consequently for any $v$ in Level 1, all the shortest $p\\mbox{--}v$ path consists of Level 1 horizontal edges only and for any vertex in $v$ in Level 1, all the shortest $v\\mbox{--}q$ path consists of Level $-1$ horizontal edges alone.",
"For any vertex $v$ in Level 0, all the shortest $p\\mbox{--}v$ path consists of horizontal edges in levels 1 and 0 and exactly one vertical edge; while all the shortest $v\\mbox{--}q$ path consists of horizontal edges in levels 0 and $-1$ and exactly one vertical edge.",
"Stage 1.",
"Initialise $\\vec{H}$ to be empty.",
"For each vertical edge $uv$ with $L(u) = 1$ and $L(v) \\in \\lbrace 0,-1\\rbrace $ , and for each shortest $p\\mbox{--}u$ path $P_u$ and shortest $v\\mbox{--}q$ path $P_v$ , do the following: Let $P$ be the $p\\mbox{--}q$ path formed by joining $P_u$ , the edge $uv$ and $P_v$ .",
"Orient the path $P$ as a directed path $\\vec{P}$ from $p$ to $q$ and add it to $\\vec{H}$ .",
"Notice that even though two such paths can share edges, there is no conflict in the above orientation since, in Stage 1, every vertical edge is oriented downward, every horizontal edge in Level 1 is oriented rightward and every horizontal edge in levels 0 and $-1$ is oriented leftward.",
"Stage 2.",
"For each vertical edge $uv$ with $L(u) = 0$ and $L(v) = -1$ not already oriented in Stage 1, and for each shortest $p\\mbox{--}u$ path $P_u$ and shortest $v\\mbox{--}q$ path $P_v$ do the following: Let $x$ be the last vertex in $P_u$ (nearest to $u$ ) that is already in $V(\\vec{H})$ and let $P_u^{\\prime }$ be the subpath of $P_u$ from $x$ to $u$ .",
"Similarly let $y$ be the first vertex in $P_v$ (nearest to $v$ ) that is already in $V(\\vec{H})$ and let $P_v^{\\prime }$ be the subpath of $P_v$ from $v$ to $y$ .",
"Let $P$ be the $x\\mbox{--}y$ path formed by joining $P_u^{\\prime }$ , the edge $uv$ and $P_v^{\\prime }$ .",
"Orient the path $P$ as a directed path $\\vec{P}$ from $x$ to $y$ and add it to $\\vec{H}$ .",
"Notice that $P$ does not share any edge with a path added to $\\vec{H}$ in Stage 1, but it can share edges with paths added in earlier steps of Stage 2.",
"However there is no conflict in the orientation since, in Stage 2, every vertical edge is oriented downward, every horizontal edge in Level 0 is oriented rightward, every horizontal edge in Level $-1$ is oriented leftward, and no horizontal edges in Level 1 is added.",
"Stage 3.",
"Finally orient the edge $pq$ from $q$ to $p$ and add it to $\\vec{H}$ .",
"This completes the construction of $\\vec{H}$ , the output of the algorithm."
],
[
"Distances in $\\vec{H}$",
"First we analyse the (directed) distance from $p$ and to $q$ of vertices added to $\\vec{H}$ in Stage 1.",
"The following bounds on distances in $\\vec{H}$ follow from the construction of each path $P$ in Stage 1.",
"Let $w$ be any vertex that is added to $\\vec{H}$ in Stage 1.",
"Then $d_{\\vec{H}}(p, w) \\le {\\left\\lbrace \\begin{array}{ll}\\begin{array}{ll}i, & w \\in S_{i,i+1}, \\\\h, & w \\in S_{h,h}, \\\\2d - i, & w \\in S_{i,i},~ i > h, \\text{ and} \\\\2d - i, & w \\in S_{i+1,i}.",
"\\\\\\end{array}\\end{array}\\right.",
"}$ $d_{\\vec{H}}(w,q) \\le {\\left\\lbrace \\begin{array}{ll}\\begin{array}{ll}2d - i, & w \\in S_{i,i+1}, \\\\h, & w \\in S_{h,h}, \\\\2d - i, & w \\in S_{i,i},~ i > h, \\text{ and} \\\\i, & w \\in S_{i+1,i}.",
"\\\\\\end{array}\\end{array}\\right.",
"}$ It is easy to verify the above equations using the facts that $w$ is part of a directed $p\\mbox{--}q$ path of length at most $2d$ (at most $2h$ if $w \\in S_{h,h}$ ) in $\\vec{H}$ .",
"No new vertices from Level 1 or $S_{h,h}$ are added to $\\vec{H}$ in Stage 2.",
"Still the distance bounds for vertices added in Stage 2 are slightly more complicated since a path $P$ added in this stage will start from a vertex $x$ in Level 0 and end in a vertex $y$ in Level $-1$ , which are added to $\\vec{H}$ in Stage 1.",
"But we can complete the analysis since we already know that $d_{\\vec{H}}(p,x) \\le 2d - h -1$ and $d_{\\vec{H}}(y, q) \\le i$ where $i$ is such that $y \\in S_{i+1,i}$ from the analysis of Stage 1.",
"Let $w$ be any vertex that is added to $\\vec{H}$ in Stage 2.",
"Then $d_{\\vec{H}}(p, w) \\le {\\left\\lbrace \\begin{array}{ll}(2d-h- 1) + (i-h-1) \\\\\\quad = 2d - 2h - 2 + i, \\quad w \\in S_{i,i},~ i > h,\\text{ and} \\\\(2d-h- 1) + (d-h-1) + (d-i) \\\\\\quad = 4d - 2h - 2 - i, \\quad w \\in S_{i+1,i}.",
"\\\\\\end{array}\\right.",
"}$ The distance from $w$ to $q$ in $\\vec{H}$ is not affected even though we trim the path $P_v$ at $y$ since $y$ already has a directed shortest path to $q$ from Stage 1.",
"Hence $d_{\\vec{H}}(w,q) \\le {\\left\\lbrace \\begin{array}{ll}\\begin{array}{ll}2d - i, & w \\in S_{i,i},~ i > h, \\text{ and} \\\\i, & w \\in S_{i+1,i}.",
"\\\\\\end{array}\\end{array}\\right.",
"}$ The first part of the next lemma follows from taking the worst case among (REF ) and (REF ).",
"Notice that $\\forall i >h,~ (2h+2-i \\le i)$ and $(4d - 2h -2 \\ge 2d)$ when $h < d$ .",
"New vertices are added to $\\vec{H}$ in Stage 2 only if $h < d$ .",
"The second part follows from (REF ) and (REF ).",
"The subsequent two claims are easy observations.",
"Lemma 3 Let $G$ be a 2-edge connected graph, $pq$ be any edge of $G$ and let $\\vec{H}$ be the oriented subgraph of $G$ returned by the algorithm OrientedCore.",
"Then for every vertex $w \\in V(\\vec{H})$ we have $d_{\\vec{H}}(p, w) \\le {\\left\\lbrace \\begin{array}{ll}\\begin{array}{ll}i, & w \\in S_{i,i+1}, \\\\h, & w \\in S_{h,h}, \\\\2d - 2h - 2 + i, & w \\in S_{i,i},~ i > h, \\text{ and} \\\\4d - 2h - 2 - i, & w \\in S_{i+1,i}.",
"\\\\\\end{array}\\end{array}\\right.",
"}$ $d_{\\vec{H}}(w,q) \\le {\\left\\lbrace \\begin{array}{ll}\\begin{array}{ll}2d - i, & w \\in S_{i,i+1}, \\\\h, & w \\in S_{h,h}, \\\\2d - i, & w \\in S_{i,i},~ i > h, \\text{ and} \\\\i, & w \\in S_{i+1,i}.",
"\\\\\\end{array}\\end{array}\\right.",
"}$ Moreover, $d_{\\vec{H}}(q,p) = 1$ and $d_{\\vec{H}}(p,q) \\le k-1$ .",
"We can see that if $S_{h,h}$ is non-empty, then all the vertices in $S_{h,h}$ are captured into $\\vec{H}$ .",
"Notice that when $k \\ge 4$ , $S_{1,2}$ and $S_{2,1}$ are non empty.",
"Thus the bound on the diameter of $\\vec{H}$ follows by the triangle inequality $d_{\\vec{H}}(x,y) \\le d_{\\vec{H}}(x, q) + d_{\\vec{H}}(q,p) + d_{\\vec{H}}(p,y)$ and the fact that $\\forall k \\ge 4$ the worst bounds for $d_{\\vec{H}}(x, q)$ and $d_{\\vec{H}}(p, y)$ from Lemma REF are when $x \\in S_{1,2}$ and $y \\in S_{2,1}$ .",
"Hence the following corollary.",
"Corollary 4 Let $G$ be a 2-edge connected graph, $pq$ be any edge of $G$ and let $\\vec{H}$ be the oriented subgraph of $G$ returned by the algorithm OrientedCore.",
"If the length of the smallest cycle containing $pq$ is greater than or equal to 4, then the diameter of $\\vec{H}$ is at most $6d-2h-3$ .",
"Let us call the vertices in $V(\\vec{H})$ as captured and those in $V(G) \\setminus V(\\vec{H})$ as uncaptured.",
"For each $i \\in \\lbrace 1,0, -1\\rbrace $ let $L_i^c$ and $L_i^u$ denote the captured and uncaptured vertices in level i, respectively.",
"Since $L_i^c$ contains every level $i$ vertex incident with a vertical edge, $L_i^c$ separates $L_i^u$ from rest of $G$ .",
"Let $d_i$ denote the maximum distance between a vertex in $L_i^u$ and the set $L_i^c$ .",
"If $u_i \\in L_i^u$ and $u_j \\in L_j^u$ such that $d_G(u_i, L_i^c) = d_i$ and $d_G(u_j, L_j^c) = d_j$ for distinct $i, j \\in \\lbrace 1,0,-1\\rbrace $ , the distance $d_G(u_i, u_j)$ is bounded above by $d$ , the diameter of $G$ , and bounded below by $d_i+ 1+d_j$ .",
"Hence $d_i + d_j \\le d-1$ for every distinct $i, j \\in \\lbrace 1,0,-1\\rbrace $ .",
"For any vertex $u \\in L_0^u$ , the last Level 0 vertex in a shortest (undirected) $u\\mbox{--}q$ path is in $L_0^c$ .",
"Hence if Level 0 is non-empty then $ d_0 \\le (d - h)$ .",
"In order to bound $d_1$ and $d_{-1}$ , we take a close look at a shortest cycle $C$ containing the edge $pq$ .",
"Let $C = (v_1, \\ldots , v_{k}, v_1)$ with $v_1= q$ and $v_{k} = p$ .",
"Each $v_i$ is in $S_{i,i-1}$ when $2i < k+1$ , $S_{i-1,i-1}$ if $2i = k+1$ and $S_{k-i, k-i+1}$ when $2i > k+1$ .",
"Let $t =\\left\\lceil k/4 \\right\\rceil $ .",
"The Level $-1$ vertex $v_t$ is special since it is at a distance $t$ from Level 1 and thus $L_1^c$ .",
"If $u_1$ is a vertex in $L_1^u$ such that $d_G(u_1, L_1^c) = d_1$ , the distance $d_G(u_1, v_t)$ is bounded above by $d$ and below by $d_1+t$ .",
"Hence $d_1 \\le d - t$ .",
"Similarly we can see that $d_{-1} \\le (d-t)$ .",
"Putting all these distance bounds on domination together, we get the next lemma.",
"Lemma 5 Let $G$ be a 2-edge connected graph, $pq$ be any edge of $G$ and let $\\vec{H}$ be the oriented subgraph of $G$ returned by the algorithm OrientedCore.",
"For each $i \\in \\lbrace 1,0,-1\\rbrace $ , let $d_i$ denote the maximum distance of a level $i$ vertex not in $V(\\vec{H})$ to the set of level $i$ vertices in $V(\\vec{H})$ .",
"Then $d_0 \\le d - \\left\\lfloor k/2 \\right\\rfloor $ , $d_1,d_{-1} \\le d - \\left\\lceil k/4 \\right\\rceil $ and for any distinct $i, j \\in \\lbrace 1, 0, -1\\rbrace $ , $d_i + d_j \\le d-1$ ."
],
[
"The Upper Bound",
"Consider a 2-edge connected graph $G$ with diameter $d$ .",
"Let $\\eta (G)$ denote the smallest integer such that every edge of a graph $G$ belongs to a cycle of length at most $\\eta (G)$ .",
"Sun, Li, Li and Huang [12] proved the following theorem.",
"Theorem 6 [12] $\\vec{d}(G) \\le 2r(\\eta -1)$ where $r$ is the radius of $G$ and $\\eta = \\eta (G)$ .",
"We know that $r \\le d$ and hence we have $\\vec{d}(G) \\le 2d(\\eta -1)$ as our first bound.",
"Let $pq$ be an edge in $G$ such that the length of a smallest cycle containing $pq$ is $\\eta $ .",
"If $\\eta \\le 3$ , then $\\vec{d}(G) \\le 4d$ which is smaller than the bound claimed in Theorem REF .",
"So we assume $\\eta \\ge 4$ .",
"By Corollary REF , $G$ has an oriented subgraph $\\vec{H}$ with diameter at most $6d-2\\left\\lfloor \\frac{\\eta }{2} \\right\\rfloor -3$ .",
"Moreover by Lemma REF , $\\vec{H}$ is a $(d - \\left\\lceil \\frac{\\eta }{4} \\right\\rceil )$ -step dominating subgraph of $G$ .",
"Let $G_0$ be a graph obtained by contracting the vertices in $V(\\vec{H})$ into a single vertex $v_H$ .",
"We can see that $G_0$ has radius at most $(d - \\left\\lceil \\frac{\\eta }{4} \\right\\rceil )$ .",
"Thus by Theorem REF , $G_0$ has a strong orientation $\\vec{G_0}$ with radius at most $(d-\\left\\lceil \\frac{\\eta }{4} \\right\\rceil )^2 + (d-\\left\\lceil \\frac{\\eta }{4} \\right\\rceil )$ .",
"Since $d \\le 2r$ , we have $d(\\vec{G_0}) \\le 2(d-\\left\\lceil \\frac{\\eta }{4} \\right\\rceil )^2 + 2(d-\\left\\lceil \\frac{\\eta }{4} \\right\\rceil )$ .",
"Notice that $\\vec{G_0}$ and $\\vec{H}$ do not have any common edges.",
"Hence $G$ has an orientation with diameter at most $2(d-\\left\\lceil \\frac{\\eta }{4} \\right\\rceil )^2 + 2(d-\\left\\lceil \\frac{\\eta }{4} \\right\\rceil ) + (6d- 2\\left\\lfloor \\frac{\\eta }{2} \\right\\rfloor -3)$ by combining the orientations in $\\vec{H}$ and $\\vec{G_0}$ .",
"Let $\\eta = 4 \\alpha d$ .",
"Hence we get $\\vec{d}(G) \\le \\min \\lbrace 8\\alpha d^2 - 2d, 2(1-\\alpha )^2d^2 + 8d-6\\alpha d-1\\rbrace $ .",
"We can see that the dominant term in the first bound is $8 \\alpha d^2$ while the dominant term in the second bound is at most $2(1-\\alpha )^2 d^2$ .",
"Notice that $0 < \\frac{3}{4d} \\le \\alpha \\le \\frac{2d+1}{4d} < 1$ .",
"Thus by optimizing for $\\alpha $ in the range $(0,1)$ , we obtain the following theorem.",
"Theorem 7 $ f(d) \\le 1.373 d^2 + 6.971 d - 1 $ .",
"For any $d \\ge 8$ , the above upper bound is an improvement over the upper bound of $2 d ^2 + 2d$ provided by Chvátal and Thomassen."
],
[
"Oriented Diameter of Diameter 4 Graphs",
"Throughout this section, we consider $G$ to be an arbitrary 2-edge connected diameter 4 graph.",
"We will show that the oriented diameter of $G$ is at most 21 and hence $f(4) \\le 21$ .",
"The following lemma by Chvátal and Thomassen [2] is used when $\\eta (G) \\le 4$ .",
"Lemma 8 [2] Let $\\Gamma $ be a 2-edge connected graph.",
"If every edge of $\\Gamma $ lies in a cycle of length at most $k$ , then it has an orientation $\\vec{\\Gamma }$ such that $ d_{\\vec{\\Gamma }}(u,v) \\le ((k-2) 2^{\\left\\lfloor (k-1)/2 \\right\\rfloor } + 1) d_{\\Gamma }(u,v) \\quad \\forall u,v \\in V(\\vec{\\Gamma }) $ Hence if all edges of the graph $G$ lie in a 3-cycle or a 4-cycle, the oriented diameter of $G$ will be at most 20.",
"Hence we can assume the existence of an edge $pq$ which is not part of any 3-cycle or 4-cycle as long as we are trying to prove an upper bound of 20 or more for $f(4)$ .",
"We apply algorithm OrientdCore on $G$ with the edge $pq$ to obtain an oriented subgraph $\\vec{H_1}$ of $G$ .",
"Fig.",
"REF shows a coarse representation of $\\vec{H_1}$ .",
"Figure: A coarse representation of H 1 →\\vec{H_1} which shows the orientation of edges between various subsets of V(G)V(G).",
"A single arrow from one part to another indicates that all the edges between these parts are oriented from the former to latter.",
"A double arrow between two parts indicates that the edges between the two parts are oriented in either direction or unoriented.",
"An unoriented edge between two parts indicate that no edge between these two parts are oriented."
],
[
"Oriented Diameter and 2-Step Domination Property of $\\vec{H_1}$",
"Let $\\vec{H_1}$ be the oriented subgraph of $G$ returned by the algorithm OrientedCore.",
"Since the smallest cycle containing $pq$ is of length greater than or equal to 5, by Corollary REF , we can see that the diameter of $\\vec{H_1}$ is at most 17.",
"Moreover, from equations REF and REF of Lemma REF , we get the upper bounds on the distances of $\\vec{H_1}$ in Table REF .",
"Hence, the following corollary.",
"Corollary 9 $d(\\vec{H_1}) \\le 17$ .",
"Moreover $\\forall w \\in V(\\vec{H_1})$ , $d_{\\vec{H_1}}(p,w)$ and $d_{\\vec{H_1}}(w,q)$ obey the bounds in Table REF .",
"Remark 1 If $k > 5$ $(h > 2)$ , then $S_{2,2}$ is empty.",
"Moreover if $S_{2,2}$ is non-empty, then all the vertices in $S_{2,2}$ are captured into $\\vec{H_1}$ .",
"Table: Upper bounds on the distances of H 1 →\\vec{H_1}Furthermore, applying Lemma REF on $\\vec{H_1}$ shows that $\\vec{H_1}$ is a 2-step dominating subgraph of $G$ .",
"Let $G_0$ be a graph obtained by contracting the vertices in $V(\\vec{H_1})$ into a single vertex $v_H$ .",
"We can see that $G_0$ has radius at most 2.",
"Thus by Theorem REF , $G_0$ has a strong orientation $\\vec{G_0}$ with radius at most 6.",
"Since $d \\le 2r$ , we have $d(\\vec{G_0}) \\le 12$ .",
"Since $\\vec{G_0}$ and $\\vec{H_1}$ do not have any common edges we can see that $G$ has an orientation with diameter at most 29 by combining the orientations in $\\vec{H_1}$ and $\\vec{G_0}$ .",
"But we further improve this bound to 21 by constructing a 1-step dominating oriented subgraph $\\vec{H_2} $ of $G$ .",
"We propose the following asymmetric variant of a technique by Chvátal and Thomassen [2] for the construction and analysis of $\\vec{H_2}$ ."
],
[
"Asymmetric Chvátal-Thomassen Lemma",
"For any subset $A$ of $V(G)$ , let $N(A)$ denote the set of all vertices with an edge incident on some vertex in $A$ .",
"Let $H$ be a subgraph of $G$ .",
"An ear of $H$ in $G$ is a sequence of edges $uv_1, v_1v_2, \\ldots , v_{k-1}v_k, v_kv$ such that $u,v \\in V(H)$ , $k \\ge 1$ and none of the vertices $v_1, \\ldots , v_k$ and none of the edges in this sequence are in $H$ .",
"In particular we allow $u = v$ .",
"Lemma 10 (Asymmetric Chvátal-Thomassen Lemma) Let $G$ be an undirected graph and let $A \\subseteq B \\subseteq V(G)$ such that $B$ is a $k$ -step dominating set in $G$ , $G/B$ is 2-edge connected, and $N(A) \\cup B$ is a $(k-1)$ -step dominating set of $G$ .",
"Then there exists an oriented subgraph $\\vec{H}$ of $G \\setminus G[B]$ such that $N(A) \\setminus B \\subseteq V(\\vec{H})$ and hence $V(\\vec{H}) \\cup B$ is a $(k-1)$ -step dominating set of $G$ , and $\\forall v \\in V(\\vec{H})$ , we have $d_{\\vec{H}}(A, v)\\le 2k$ and either $d_{\\vec{H}}(v, A) \\le 2k$ or $d_{\\vec{H}}(v, B \\setminus A) \\le 2k-1$ .",
"We construct a sequence $\\vec{H}_0, \\vec{H}_1, \\ldots $ of oriented subgraphs of $G \\setminus G[B]$ as follows.",
"We start with $\\vec{H}_0 =\\emptyset $ and add an oriented $A\\mbox{--}B$ ear $\\vec{Q}_i$ in each step.",
"Let $i \\ge 0$ .",
"If $N(A) \\setminus B \\subseteq V(\\vec{H_i})$ , then we stop the construction and set $\\vec{H} = \\vec{H}_i$ .",
"Since $N(A) \\cup B$ is a $(k-1)$ -step dominating set of $G$ , the first conclusion of the lemma is satisfied when the construction ends with $N(A) \\setminus B \\subseteq V(\\vec{H})$ .",
"If $N(A) \\setminus B \\lnot \\subseteq V(\\vec{H_i})$ , then let $v \\in (N(A) \\setminus B) \\setminus V(\\vec{H}_i)$ and let $u$ be a neighbour of $v$ in $A$ .",
"Since $G/B$ is 2-edge connected, there exists a path in $G^{\\prime } = (G/B) \\setminus \\lbrace uv\\rbrace $ from $v$ to $B$ .",
"Let $P_i$ be a shortest $v\\mbox{--}B$ path in $G^{\\prime }$ with the additional property that once $P_i$ hits a vertex in an oriented ear $\\vec{Q}_j$ that was added in a previous step, $P_i$ continues further to $B$ along the shorter arm of $Q_j$ .",
"It can be verified that $P_i$ is still a shortest $v\\mbox{--}B$ path in $G^{\\prime }$ .",
"The ear $Q_i$ is the union of the edge $uv$ and the path $P_i$ .",
"If $P_i$ hits $B$ without hitting any previous ear, then we orient $Q_i$ as a directed path $\\vec{Q_i}$ from $u$ to $B$ .",
"If $Q_i \\cap Q_j\\ne \\emptyset $ , then we orient $Q_i$ as a directed path $\\vec{Q}_i$ by extending the orientation of $Q_i \\cap Q_j$ .",
"Notice that, in both these cases, the source vertex of $\\vec{Q}_i$ is in $A$ .",
"We add $\\vec{Q}_i$ to $\\vec{H}_i$ to obtain $\\vec{H}_{i+1}$ .",
"Let $Q_i = (v_0, \\ldots , v_q)$ with $v_0 \\in A$ and $v_q \\in B$ be the ear added in the $i$ -th stage above.",
"Since $(v_1, \\ldots , v_q)$ is a shortest $v_1\\mbox{--}B$ path in $G^{\\prime } = (G/B) \\setminus \\lbrace v_0v_1\\rbrace $ and since $B$ is a $k$ -step dominating set, $q \\le 2k + 1$ .",
"Moreover, if $v_q \\in B\\setminus A$ , then $q \\le 2k$ since $N(A) \\cup B$ is a $(k-1)$ -step dominating set.",
"These bounds on the length of $Q_i$ along with the observation that the source vertex of $\\vec{Q}_i$ is in $A$ , verifies the second conclusion of the lemma.",
"Remark 2 If we flip the orientation of $\\vec{H}$ we get the bounds $d_{\\vec{H}}(v, A)\\le 2k$ and either $d_{\\vec{H}}(A, v) \\le 2k$ or $d_{\\vec{H}}(B \\setminus A, v) \\le 2k-1$ , $\\forall v \\in V(\\vec{H})$ in place of Conclusion (ii) of Lemma REF .",
"Setting $A = B$ in Lemma REF gives the key idea which is recursively employed by Chvátal and Thomassen to prove Theorem REF [2].",
"Notice from the above proof that, in this case $B \\subseteq V(\\vec{H})$ .",
"We can summarize their idea as follows.",
"Lemma 11 (Chvátal-Thomassen Lemma) Let $G$ be an undirected graph and let $B \\subseteq V(G)$ such that $B$ is a $k$ -step dominating set in $G$ , and $G/B$ is 2-edge connected.",
"Then there exists an oriented subgraph $\\vec{H}$ of $G \\setminus G[B]$ such that $V(\\vec{H})$ is a $(k-1)$ -step dominating set of $G$ , and $\\forall v \\in V(\\vec{H})$ , we have $d_{\\vec{H}}(B, v)\\le 2k$ and $d_{\\vec{H}}(v, B) \\le 2k$ .",
"Let $G$ be any 2-edge connected graph with radius $r$ .",
"Chvátal and Thomassen showed that $\\vec{d}(G) \\le 2r + 2(r-1) + \\cdots + 2 = r(r+1)$ by $r$ applications of Lemma REF ; starting with $B = \\lbrace v\\rbrace $ , where $v$ is any central vertex of $G$ and $B$ in each subsequent application being the vertex-set of the oriented subgraph $\\vec{H}$ returned by the current application."
],
[
"A 1-Step Dominating Oriented Subgraph $\\vec{H_2}$ of {{formula:8a9b1438-75c5-46cf-9a18-5bde9c82b9c4}}",
"Let $\\vec{H_1}$ be the oriented subgraph of $G$ returned by the algorithm OrientedCore.",
"We will add further oriented ears to $\\vec{H_1}$ to obtain a 1-step dominating oriented subgraph $\\vec{H_2}$ of $G$ .",
"We have already seen that $\\vec{H_1}$ is a 2-step dominating oriented subgraph of $G$ .",
"By Lemma REF , we also have $d_i + d_j \\le 3$ for any distinct $i,j \\in \\lbrace 1,0,-1\\rbrace $ .",
"Now consider the first case where we have vertices in Level 0 which are at a distance 2 from $S_{2,2}$ .",
"Notice that in this case, $d_0 = 2$ and hence $d_1, d_{-1} \\le 1$ .",
"Let $B = L_{0}^c$ , $A = S_{2,2}$ and $G_0 = G[L_0]$ .",
"By Remark REF , $A \\subseteq B$ .",
"Notice that $B = L_{0}^c$ is a cut-set separating $L_0^u$ from the rest of $G$ and hence the graph $G_0 / B $ is 2-edge connected.",
"Since $S_{3,3}^u \\subseteq N(S_{2,2})$ , we can see that $N(A) \\cup B = N(S_{2,2}) \\cup L_0^c $ is a 1-step dominating subgraph of $G_0$ .",
"Therefore we can apply Lemma REF on $G_0$ .",
"Every edge of the oriented subgraph of $G_{0} \\backslash G_{0}[B]$ obtained by applying Lemma REF is reversed to obtain the subgraph $\\vec{H_2^0}$ .",
"Now consider the vertices captured into $\\vec{H_2^0}$ .",
"From Lemma REF and Remark REF , we get the following bounds $d_{\\vec{H_2^0}}(v, A)\\le 4$ and either $d_{\\vec{H_2^0}}(A, v) \\le 4$ or $d_{\\vec{H_2^0}}(B \\setminus A, v) \\le 3$ , $\\forall v \\in V(\\vec{H_2^0})$ .",
"Here $B \\setminus A = S_{3,3}^c \\cup S_{4,4}^c$ and from Table REF , we have the bounds $d_{\\vec{H_1}}(p, x) \\le 5$ , $\\forall x \\in S_{3,3}^c$ , $d_{\\vec{H_1}}(p, y) \\le 6$ , $\\forall y \\in S_{4,4}^c$ and $d_{\\vec{H_1}}(p, z) = 2$ , $\\forall z \\in S_{2,2}$ .",
"Hence $d_{\\vec{H_1} \\cup \\vec{H_2^0}}(p, v) \\le 9$ , $\\forall v \\in V(\\vec{H_2^0})$ .",
"Since $d_{\\vec{H_2^0}}(v, A) \\le 4$ and $d_{\\vec{H_1}}(x, q) = 2$ , $\\forall x \\in A$ , we also have $d_{\\vec{H_1} \\cup \\vec{H_2^0}}(v,q) \\le 6$ , $\\forall v \\in V(\\vec{H_2^0})$ .",
"Let $\\vec{H_2} = \\vec{H_1} \\cup \\vec{H_2^0}$ .",
"By the above discussion, in combination with the distances in Table REF and Corollary REF , we get the bounds in Table REF for $d_{\\vec{H_2}}(p,w)$ and $d_{\\vec{H_2}}(w,q)$ when $V(\\vec{H_2^0}) \\ne \\phi $ .",
"Moreover, $d(\\vec{H_2}) \\le 17$ .",
"Table: Upper bounds on the distances of H 2 →\\vec{H_2} when V(H 2 0 →)≠φV(\\vec{H_2^0}) \\ne \\phi Now consider the second case where $d_1 = 2$ or $d_{-1} = 2$ .",
"Since $d_1 + d_{-1} \\le 3$ , uncaptured vertices at a distance 2 from $H_1$ can exist either in Level 1 or in Level $-1$ but not both.",
"By flipping the role of the vertices $p$ and $q$ in Algorithm OrientedCore if necessary, without loss of generality, we can assume vertices which are at a distance 2 from $\\vec{H_1}$ exists only in Level $-1$ and not in Level 1.",
"Let $G_{-1}=G[L_{-1}]$ and $B = L_{-1}^c$ .",
"Further let $r$ be any vertex in $S_{1,2}^c$ and $A = \\lbrace v \\in B: d_{G}(r,v)=2\\rbrace $ .",
"Since $q\\in A$ , $A$ is never empty.",
"Note that $A \\subseteq B \\subseteq V(G_{-1})$ .",
"Also $G_{-1} / B$ is 2-edge connected since $B = L_{-1}^c$ is a cut-set which separates $L_{-1}^u$ from the rest of $G$ .",
"Now consider a vertex $z$ in Level $-1$ which is exactly at a distance 2 from $B$ .",
"Since the graph $G$ is of diameter 4, there exists a 4-length path $P$ from $z$ to $r$ .",
"Since $B$ separates $L_{-1}^u$ from $r$ , $P$ intersects $B$ , say at a vertex $b$ .",
"Further, we have $d_G(b,r) = 2$ and thus $b \\in A$ .",
"Hence $z$ has a 2-length path to a vertex $b \\in A$ .",
"Thus $N(A) \\cup B$ is a 1-step dominating subgraph of $G_{-1}$ .",
"Hence we can apply Lemma REF on $G_{-1}$ to obtain $\\vec{H_2^{-1}}$ , an oriented subgraph of $G_{-1} \\backslash G_{-1}[B]$ .",
"Now consider the vertices captured into $\\vec{H_2^{-1}}$ .",
"From Lemma REF , we get the following bounds $\\forall v \\in V(\\vec{H_2^{-1}})$ , $d_{\\vec{H_2^{-1}}}(A, v)\\le 4$ and $d_{\\vec{H_2^{-1}}}(v, B) \\le 4$ .",
"Since $d_{\\vec{H_1}}(x,q) \\le 3$ , $\\forall x \\in B$ , we have $d_{\\vec{H_1} \\cup \\vec{H_2^{-1}}}(v,q) \\le 7$ , $\\forall v \\in V(\\vec{H_2^{-1}})$ .",
"Vertices in $A$ can be from $S_{2,1}^c$ , $S_{3,2}^c$ or $\\lbrace q\\rbrace $ .",
"By the definition of $A$ there is an undirected path in $G$ of length 3 from $p$ to any vertex $v_a$ in $(A \\setminus \\lbrace q\\rbrace )$ , going through $r$ .",
"It can be verified that this undirected path is oriented from $p$ to $v_a$ by Algorithm OrientedCore.",
"Hence $d_{\\vec{H_1}}(p,v_a) \\le 3$ , $\\forall v_a \\in (A \\setminus \\lbrace q\\rbrace )$ and hence $\\forall v \\in V(\\vec{H_2^{-1}})$ with $d_{\\vec{H_2^{-1}}}(A \\setminus \\lbrace q\\rbrace , v) \\le 4$ , $d_{\\vec{H_1} \\cup \\vec{H_2^{-1}}}(p,v) \\le 7$ .",
"But if a vertex $v \\in V(\\vec{H_2^{-1}})$ has $d_{\\vec{H_2^{-1}}}(A \\setminus \\lbrace q\\rbrace , v) > 4$ , then $d_{\\vec{H_2^{-1}}}(q,v) \\le 4$ .",
"In this case, since $d_{\\vec{H_1}}(p,q) \\le 8$ , we get $d_{\\vec{H_1} \\cup \\vec{H_2^{-1}}}(p,v) \\le 12$ .",
"Notice that this is the only situation where $d_{\\vec{H_1} \\cup \\vec{H_2^{-1}}}(p,v) > 9$ and in this particular case $d_{\\vec{H_2^{-1}}}(q,v) \\le 4$ .",
"Now consider two vertices $x, y \\in V(\\vec{H_1} \\cup \\vec{H_2^{-1}})$ .",
"We can see that $d_{\\vec{H_1} \\cup \\vec{H_2^{-1}}}(x,y)$ $ \\le d_{\\vec{H_1} \\cup \\vec{H_2^{-1}}}(x,q) + d_{\\vec{H_1} \\cup \\vec{H_2^{-1}}}(q,y)$ .",
"We have already proved that $d_{\\vec{H_1} \\cup \\vec{H_2^{-1}}}(x,q) \\le 7$ .",
"Now let us consider the $q-y$ path.",
"If $y \\in V(\\vec{H_1})$ , from Table REF , we can see that $d_{\\vec{H_1}}(p,y) \\le 9$ and therefore $d_{\\vec{H_1} \\cup \\vec{H_2^{-1}}}(x,y) \\le 17$ .",
"Now suppose if $y \\in (V(\\vec{H_2^{-1}}) \\setminus V(\\vec{H_1}))$ .",
"In this case we have already shown that $d_{\\vec{H_2^{-1}}}(p,y) \\le 9$ or $d_{\\vec{H_2^{-1}}}(q,y) \\le 4$ .",
"So, we either have a directed path of length 10 from $q$ to $y$ through $p$ or a directed path of length 4 to $y$ directly from $q$ .",
"Hence, $d_{\\vec{H_1} \\cup \\vec{H_2^{-1}}}(x,y) \\le 17$ .",
"Let $\\vec{H_2} = \\vec{H_1} \\cup \\vec{H_2^{-1}}$ .",
"By the above discussion, we get the bounds in Table REF for $d_{\\vec{H_2}}(p,w)$ and $d_{\\vec{H_2}}(w,q)$ when $V(\\vec{H_2^{-1}}) \\ne \\phi $ .",
"Moreover, $d(\\vec{H_2}) \\le 17$ .",
"Table: Upper bounds on the distances of H 2 →\\vec{H_2} when V(H 2 -1 →)≠φV(\\vec{H_2^{-1}}) \\ne \\phi In both the cases we get an oriented subgraph $\\vec{H_2}$ of $G$ with $d(\\vec{H_2}) \\le 17$ .",
"Moreover, it is clear from Conclusion (i) of Lemma REF that $\\vec{H_2}$ is a 1-step dominating subgraph of $G$ .",
"Hence the following Lemma.",
"Lemma 12 For every 2-edge connected graph $G$ with diameter 4 and $\\eta (G) \\ge 5$ , there exists a 1-step dominating oriented subgraph $\\vec{H_2}$ of $G$ with $d(\\vec{H_2}) \\le 17$ ."
],
[
"The Upper Bound",
"Now the main theorem of the section follows.",
"Theorem 13 $f(4) \\le 21$ .",
"By Lemma REF , we get a 1-step dominating oriented subgraph $\\vec{H_2}$ of $G$ with $d(\\vec{H_2}) \\le 17$ .",
"Let $G_0$ be a graph obtained by contracting the vertices in $V(\\vec{H_2})$ into a single vertex $v_H$ .",
"We can see that $G_0$ has radius at most 1.",
"Thus by Theorem REF , $G_0$ has a strong orientation $\\vec{G_0}$ with radius at most 2.",
"Since $d \\le 2r$ , we have $d(\\vec{G_0}) \\le 4$ .",
"Notice that $\\vec{G_0}$ and $\\vec{H_2}$ do not have any common edges.",
"Now we can see that $G$ has an orientation with diameter at most 21 by combining the orientations in $\\vec{H_2}$ and $\\vec{G_0}$ ."
]
] | 2001.03448 |
[
[
"Measuring Similarity between Brands using Followers' Post in Social\n Media"
],
[
"Abstract In this paper, we propose a new measure to estimate the similarity between brands via posts of brands' followers on social network services (SNS).",
"Our method was developed with the intention of exploring the brands that customers are likely to jointly purchase.",
"Nowadays, brands use social media for targeted advertising because influencing users' preferences can greatly affect the trends in sales.",
"We assume that data on SNS allows us to make quantitative comparisons between brands.",
"Our proposed algorithm analyzes the daily photos and hashtags posted by each brand's followers.",
"By clustering them and converting them to histograms, we can calculate the similarity between brands.",
"We evaluated our proposed algorithm with purchase logs, credit card information, and answers to the questionnaires.",
"The experimental results show that the purchase data maintained by a mall or a credit card company can predict the co-purchase very well, but not the customer's willingness to buy products of new brands.",
"On the other hand, our method can predict the users' interest on brands with a correlation value over 0.53, which is pretty high considering that such interest to brands are high subjective and individual dependent."
],
[
"Introduction",
"Billions of photographs are uploaded to the Internet every day through various photo & video sharing services.",
"Social media is already a part of our everyday life.",
"The rapidly increasing popularity of social media has given rise to new marketing approaches.",
"Social media marketing is the use of social media platforms and websites to promote a product or service [11].",
"Companies make use of platforms such as Twitterhttps://twitter.com and Instagramhttps://www.instagram.com to reach a much wider spectrum of audience than that of traditional advertisements such as newspaper, television, and radio.",
"By 2018, 91% of the Fortune 500 companies were actively using Twitter, and 63% of them had corporate Instagram accountshttps://www.umassd.edu/cmr/social-media-research/2018-fortune-500.",
"With these newfound trends, we can use data from social media to analyze customers' preferences and the implicit relationship between brands.",
"Understanding the relationship between brands, products, and people can have a profound effect on marketing.",
"For example, when a brand wants to find celebrities for endorsements, it is important to consider whether his/her followers are potential customers for its products.",
"Similarly, when a shopping mall is in the process of choosing the brands it wishes to carry, considering the specific combinations of brands that have the potential to attract more customers is a lucrative proposition.",
"However, such design or decision making has been done by only experts with sense of trend so far.",
"Co-purchasing history from credit cards or point cards in a shopping mall can be analyzed to measure the similarity between brands.",
"Unfortunately, there are some demerits of this method.",
"Some customers, especially teenagers, may not own a credit card.",
"Secondly, people prefer using cash for small purchase amounts.",
"Consequently, brands that offer products in the low price range may seldom appear in the purchasing history.",
"Many shopping malls have their own point card systems, which allow customers to earn points every time they make a purchase from one of the franchises.",
"But these analyses are limited by the number of brands existing at the location of the purchase.",
"Namely, similarity between brands that are not co-located in the same region can not be calculated.",
"Because of these reasons, we chose social network data to measure the similarity between brands.",
"Unlike the prerequisites for credit cards or point cards, social media has no age or income requirement, also there are no limitations to the number of brands being advertised by marketers or users.",
"One might think that SNS penetration is lower among elder generation and therefore, it is hard to get information from them.",
"But according to the investigationhttps://www.pewinternet.org/fact-sheet/social-media, 64% of Americans in the age group 50-64 use SNS actively as well as 37% in the age group of 65 years and above.",
"We briefly summarize the main contributions and findings of this study: We have presented an approach to predict the similarity between brands using tags and images posted by the brands' followers.",
"We show that such off-the-shelf multimedia technologies, though not technically novel, can be a new marketing tool.",
"We have created an Instagram brand datasets including 233,000 unique users and their most recent 100 posts for brand similarity measurement.",
"We release Instagram dataset to the publichttps://github.com/yiwei51/brand_dataset.",
"We have also created two evaluation datasets based on the purchasing histories of two real-world customers for results comparison.",
"We have conducted user studies by questionnaires to obtain the users' co-purchasing, interest, and knowledge tendencies.",
"It provides an additional angle of evaluation besides point card and credit card purchasing histories.",
"There are many related studies on measuring latent similarity with multimedia data.",
"Item2Vec [2] measured similarity between item for recommendation.",
"They considered items purchased by one user as a positive example then used a skip-gram model to produce embeddings.",
"Eisner et al.",
"used a similar method to convert an emoji to a vector then measured similarity between emoji [10].",
"[16] measured latent similarity between social images by metric learning.",
"Wang et al.",
"used graph embedding techniques to find similar items [21].",
"Hsiao et al.",
"used the topic models to learn style-coherent representations for fashion images retrieval [15]."
],
[
"Brand Analysis using Social Media",
"Researches on brands are increasingly focused on the content on social media.",
"Goh et al.",
"showed that engagement in social media brand communities leads to a positive increase in sales [13].",
"In a study of understanding brands' perceptions, Culotta et al.",
"used the data on Twitter to predict how consumers perceive brands' attributes including eco-friendliness, nutrition, and luxury [7].",
"Several researchers analyzed the popularity of brands and their social media posts.",
"De Vries et al.",
"reported that brand posts might be popular due to several cues such as vividness, interactivity, and valence of comments [8].",
"Mazloom et al.",
"investigated similar indicators with text, image and social features on Instagram [17]."
],
[
"Content-based Recommendation",
"Recommendation is another topic of interest in the study of social media, both commercially and academically.",
"Besides widely used collaborative filtering methods [6], another common approach is content-based recommendation [18].",
"Apart from users-item interactions, content-based methods analyze information such as user profiles and item relations [22].",
"Gelli et al.",
"presented a study on image recommendations for brands.",
"They recommended images which were consistent with a brand's style, by learning features from images uploaded by these brands [12]."
],
[
"Brand Similarity",
"The pairwise similarity between brands was measured previously by Bijmolt et al.",
"[3].",
"They used questionnaires to ask people regarding the likeness of two products to identify the similarities between the products of various brands.",
"The limitation of this method was that it required time and monetary investment to collect sufficient data.",
"In [23], they aimed to find similar brands in the same category using customers' reviews, while their approach is not suitable for finding the relationship between brands in diverse product categories.",
"In [20], the researchers predicted user's personality based on his or her profile picture on Facebook.",
"We also believe that images and hashtags posted by users are indicative of their personalities and users' choice of brands represents their preferences.",
"Different from [3] and [23], our proposed method could find the relationship between two diverse product categories without tremendous time and monetary investment."
],
[
"Dataset Construction",
"Instagram is a social photo sharing service.",
"We chose to collect posts from Instagram because many brands have created official accounts for promoting the brand image.",
"We show posts from the followers of three brands separately in Figure REF .",
"Though these posts are not from the brands' official accounts, we can see that followers of different brands have different styles.",
"Their posts indicate the everyday activities of these followers, which, we believe, can be utilized to represent the similarity between the brands.",
"We chose to use followers' posts instead of brands' posts because brands in the same category may have greater similarity based on their posts.",
"For example, a fashion brand will be more similar to another fashion brand than to a cosmetic brand.",
"However, finding the relationship between two diverse product categories is of greater interest to us.",
"The problem of measuring the pairwise similarity between brands transforms to the problem of measuring the pairwise similarity between their followers.",
"We randomly selected 1,000 followers per brand and used their posts to represent this brand.",
"We chose to associate a unique user to just one brand.",
"We collected data on these followers by web crawling their Instagram posts.",
"We created three datasets in our study.",
"The difference between the datasets is shown in Table REF .",
"Pointcard100 and Creditcard81 datasets are created for the evaluation.",
"The Pointcard100 dataset included 100 brands that was obtained from customers' purchasing histories at a shopping mall in Japan.",
"The Creditcard81 dataset included 81 brands and comprised customers' purchasing histories maintained by a credit card company in Japan.",
"They were both anonymized before processing.",
"Both datasets included information on products of specific brands purchased by the users.",
"The Popular108 dataset included the official accounts of 108 of the most popular companies in Japan.",
"The number of followers in Instagram was directly used to express the popularity."
],
[
"Proposed Methods",
"We present a new scale for measuring the similarity between brands by analyzing posts from brand followers.",
"A flowchart for the proposed methods consisting of the feature extraction, brand representation and similarity measurement processes is depicted in Figure REF ."
],
[
"Image Feature",
"For each brand, we used the 10 most recent photographs posted by 1,000 users to represent the brand.",
"We used a 50-layer ResNet [14] pretrained on ImageNet [9] to extract the image features.",
"In this step, we transform each image into a feature vector of column size 2048."
],
[
"Tag Feature",
"To extract efficient tag features, we took two steps: (1) tag embedding to covert each tag into a vector representation; (2) tag ranking to select the top ranked tags as the typical tag feature for brand representation.",
"Tag embedding process is inspired by the work Item2Vec [2].",
"It produces embedding for items in a latent space using Word2Vec.",
"We considered tags belonging to one image as words in one sentence.",
"Then, we used fasttext™ to convert each tag to a 100-dimension vector [5].",
"In the tag ranking, we select the top 3,000 tags to represent the brand.",
"The first method is to rank by frequency and the second method is to rank by tag score.",
"Rank by Frequency: In this method, we rank tags by the number of users who have used it.",
"In [4], the researchers observed that a large number of users use the same tags to every photo.",
"Therefore, we decided to rank the tags by the number of users who have used the tag at least once.",
"In other words, regardless of the number of times the user has used the tag, we only counted the tag once for one user.",
"Rank by Tag Score: Instead of utilizing the number of users who have used the tag at least once, we apply term frequency–inverse document frequency (TF–IDF) [19] for selecting the tags.",
"TF–IDF is a numerical statistic, which indicates the importance of a word to a document in a collection or corpus [1].",
"In our tag ranking algorithm, we considered a brand as a document, and tags belong to a brand as words inside a document.",
"We utilized the number of users who have used the tag at least once to represent the TF.",
"The TF score is higher if more people have used the tag.",
"In contrast, we consider that the IDF score would be lower if the tag has been used for other brands frequently.",
"The score for the tag is calculated using the following equation: tag_score(ti, bj) = tf(i, j) idf(i) , tf(i, j) = k ni, jnk, j , idf(i) = Bj ci, j+ 1 , ci, j = {ll 1 $r_{i, j}$ $\\le $ $l$ 0 $r_{i, j}$ $>$ $l$ , .",
"where $n_{i,j}$ is the number of users that have used the tag $t_i$ in brand $b_j$ , and $B$ is the number of brands.",
"$r_{i,j}$ is the frequency ranking of tag $t_i$ in the brand $b_j$ , $c_{i,j}$ is 1 if the tag $t_i$ appears in the top $l$ tags of brand $b_j$ , else $c_{i,j}$ is 0.",
"We set $l = 1,000$ in this experiment.",
"After converting the images and tags into vectors, we addressed the problem of representing a brand using these vectors.",
"In the following section, we introduce several ways to handle image and tag features, including histogram-based methods and average vector based methods, and compare their performances.",
"Histogram-based Method: We used mini-batch K-means to cluster features after feature extraction to form a bag of features.",
"We have tried using $K$ from 10 to 10,000 and employed the best one.",
"After clustering, we use the number of images/tags in each cluster as the brand vector.",
"Average Vector based Method: In the case of images, we average the image vectors directly and use it as the brand vector.",
"In the case of tags, after tag ranking, we select top 3,000 tags and average their tag vectors, then we use this averaged vector as the brand vector."
],
[
"Similarity Measurement",
"We considered two ways for histogram-based methods in similarity measurement.",
"The first is the Pearson correlation and the second is histogram intersection similarity.",
"For the average vector based methods, we only considered the Pearson correlation.",
"We calculated similarity between each pair of brand vectors.",
"Figure REF shows the visualization of the brand relationships based on the histogram method of ranking by the tag score.",
"We can easily find relationship between brands from this figure.",
"For example, for those who like “Starbucks,” they might also be interested in fashion brands “titivate.” Figure: Visualization of the brand relationships (full version of the visualization)."
],
[
"Stability",
"Because our proposed methods are based on 1,000 followers randomly selected for each brand, we conducted an experiment to evaluate the stability of our proposed methods.",
"For each brand, we randomly split the 1,000 followers into two groups for 5 times and then computed the similarity between the brands for each group.",
"The average Spearman's ranking correlation coefficient based on the two groups was calculated as 0.98.",
"It indicates that our proposed methods are stable.",
"We also conducted an experiment to show that less number of followers might be enough for our purpose.",
"We randomly selected a small group of followers from each brand for 5 times and then computed the similarity between using a small group of followers and using 1,000 followers.",
"We found that the average Spearman's ranking correlation coefficient was calculated as 0.93 when using 100 followers per brand, and 0.88 when using 50 followers per brand.",
"We think 100 followers might be enough for our purpose.",
"To prove that our results have implications for real-world business, we compared our results with the purchasing histories of customers in the Pointcard100 and Creditcard81 datasets.",
"We were able to create a brand-user matrix $M$ based on the co-purchasing data.",
"${M}_{b_{i}u_{k}} = t$ indicates that user $u_k$ purchased brand $b_i$ $t$ times.",
"Then, we calculated the Pearson correlation coefficient between each pair of brands as similarity score.",
"For example, the similarity score between brand $b_i$ and $b_j$ was calculated using vector ${M}_{b_i}$ and ${M}_{b_j}$ .",
"Then, we compared the results based on our proposed methods with the results calculated from the co-purchasing data using the Spearman's ranking correlation."
],
[
"Results of Evaluation using Co-Purchasing Data",
"Table REF shows the Spearman's ranking correlation coefficient for brand similarity based on our proposed methods and based on co-purchasing results from the point card and credit card usage.",
"In all tables, hist stands for the histogram-based methods; avg stands for the average vector methods; p stands for using the Pearson correlation; hi stands for using histogram intersection similarity; freq stands for ranking by frequency; score stands for ranking by tag score.",
"For histogram-based methods, results changed a little when choosing different $K$ and we empirically found that using $K=500$ has better result in our proposed methods.",
"We also compared our method with brand_img by [12].",
"Their method used images posted by the brand as the input and produced a brand vector that aimed to discovered images that match the brand style.",
"We used their implementation to train the brand vector then calculate the Pearson correlation between each pair of brands as similarity score.",
"In the case of images, the average vector based method yields better results than the histogram-based methods.",
"Also, proposed methods show better performance than brand_img [12], proving that using followers' posts is better.",
"We could see that the tag feature works better than the image feature when compared with the co-purchasing results.",
"When using the histogram-based methods, histogram intersection similarity yields better results than the Pearson correlation.",
"With regard to the average vector based methods, ranking by tag score gives better results than ranking by frequency.",
"The highest Spearman's ranking correlation coefficient achieves 0.50, which is pretty high considering that this result was calculated among different groups of users (point/credit card users and brand followers are totally different persons).",
"The correlation coefficients of the Creditcard81 is lower than those of Pointcard100 because the users of the credit card are distributed all over the country and it is not guaranteed that all the brands have their shops within their shopping territory.",
"Table: Spearman's ranking correlation coefficient betweenproposed methods and co-purchasing results."
],
[
"Method",
"To further evaluate our results, we used questionnaires to ask users regarding their awareness of certain brands and their past purchases of the products of those brands through a crowdsourcing service.",
"Two separate questionnaires were asked for the Pointcard100 and Creditcard81 datasets.",
"Nine hundred people answered the questionnaire for Pointcard100 dataset, and 890 people answered the questionnaire for Creditcard81 dataset.",
"For both questionnaires, we selected the top 50 brands from the dataset according to their number of followers on Instagram.",
"The two questionnaires were similar.",
"We asked people three questions related to each brand.",
"Have you purchased this brand by yourself before?",
"Are you interested in this brand?",
"Do you know this brand?",
"https://yiwei51.github.io/brand_visualization We obtained the users' co-purchasing, interest and knowledge tendencies from the responses to the questionnaires.",
"Similar to the purchasing data in the previous section, we created three brand-user matrices, which represented the co-purchasing, interest and knowledge tendencies separately.",
"For example, for the first question, the response ${M}_{b_{i}u_{k}} = 1$ indicated that user $u_{k}$ had bought brand $b_{i}$ before, whereas the response ${M}_{b_{i}u_{k}} = 0$ indicated that user $u_{k}$ had never bought brand $b_{i}$ before.",
"Then, we calculated the Pearson correlation coefficient between each pair of brands."
],
[
"Results of Evaluation using Questionnaires",
"Table REF shows the results for the two datasets.",
"For Pointcard100 dataset, we can see that the Spearman's ranking correlation coefficient between the co-purchasing results obtained from the point card data and from the questionnaires is 0.52, which is high considering that co-purchasing activity is strongly affected by locality, personality, etc.",
"The co-purchasing results obtained from the point card data have higher correlation than the results calculated from our proposed methods.",
"However, comparison of the customer interest obtained from the questionnaires and that from the knowledge results shows that the proposed method based on histogram has the highest Spearman's ranking correlation coefficient.",
"The highest correlation with the customer interest results obtained from the questionnaires is 0.53, whereas the highest correlation with the knowledge results obtained from the questionnaires is 0.50.",
"For Creditcard81 dataset, when compared with the customer co-purchasing, interest and knowledge results obtained from the questionnaires, the proposed methods based on the tags show stronger correlation than the co-purchasing results obtained from the credit card data.",
"From these results, it can be said that we have grasped the tendency of customer preferences.",
"Our proposed methods can predict customers' interests more accurately than the predictions made from the customer's purchasing history.",
"We considered the reason for this result might be related to age or finances.",
"For example, women in their early twenties might be unable to purchase some high-end fashion brands because of age-related and insufficient finances, but they might be interested in these brands to prepare for their late twenties or thirties.",
"We believe that our proposed methods could reflect a customer's future purchasing plan.",
"Table: Spearman's ranking correlation coefficient between the proposed methods and questionnaires."
],
[
"Applications",
"We have introduced that different brands' followers might have different interest tendencies as shown in Figure REF .",
"Based on followers' interest tendencies, we could recommend suitable content to the brand for popularity enhancement.",
"We cooperated with a fast food company from the end of 2018.",
"We provided results of analysis to the brand including tags ranking by tag score, similar and dissimilar brands to it.",
"And the fast food company changed their posts based on our recommendations.",
"We recorded the number of favorites of every post from that fast food company's official account.",
"The average number of favorites is 477 from 2015 to 2018, and increased to 3,788 in 2019.",
"The number of favorites suddenly increased after the consulting with us proves that our proposed methods could be a novel marketing tool for popularity boosting."
],
[
"Conclusions",
"In this paper, we proposed a new scale for measuring the similarity between brands using image features and tag features of social media posts made by their followers.",
"Then, we evaluated our results by comparing with the results of the co-purchasing history of real-world customers and the results from the questionnaires.",
"We found that our proposed methods have moderate correlations with the customers' co-purchasing histories.",
"In addition, we found that results based on our proposed methods show stronger correlation with users' interest tendencies obtained from questionnaires than results obtained from the customers' co-purchasing histories.",
"We believe that this multimedia-based follower analysis in social networks can become a novel marketing tool.",
"This work was partially supported by the Grants-in-Aid for Scientific Research Number JP19J22939, JP19K20289, and JP18H03339.",
"A part of research is a result of collaboration between CyberBuzz, Inc. and The University of Tokyo."
]
] | 2001.03353 |
[
[
"Trace Clustering on Very Large Event Data in Healthcare Using Frequent\n Sequence Patterns"
],
[
"Abstract Trace clustering has increasingly been applied to find homogenous process executions.",
"However, current techniques have difficulties in finding a meaningful and insightful clustering of patients on the basis of healthcare data.",
"The resulting clusters are often not in line with those of medical experts, nor do the clusters guarantee to help return meaningful process maps of patients' clinical pathways.",
"After all, a single hospital may conduct thousands of distinct activities and generate millions of events per year.",
"In this paper, we propose a novel trace clustering approach by using sample sets of patients provided by medical experts.",
"More specifically, we learn frequent sequence patterns on a sample set, rank each patient based on the patterns, and use an automated approach to determine the corresponding cluster.",
"We find each cluster separately, while the frequent sequence patterns are used to discover a process map.",
"The approach is implemented in ProM and evaluated using a large data set obtained from a university medical center.",
"The evaluation shows F1-scores of 0.7 for grouping kidney injury, 0.9 for diabetes, and 0.64 for head/neck tumor, while the process maps show meaningful behavioral patterns of the clinical pathways of these groups, according to the domain experts."
],
[
"Introduction",
"Clinical pathways are known to be enormously complex and flexible.",
"Process mining techniques are often applied to analyze event data related to clinical pathways, in order to obtain valuable insights [1].",
"The resulting findings can help to improve process quality, patient outcomes and satisfaction, and optimizing resource planning, usages, and reallocation [2].",
"Finding coherent, relatively homogenous patient groups helps process mining techniques to obtain accurate insights [3], [7], [4].",
"Many existing systems have tried to classify patients and provide such a well-defined group, known as Patient classification systems (PCSs).",
"PCSs provide a categorization of patients based on clinical data (i.e.",
"diagnoses, procedures), demographic data (i.e.",
"age, gender), and resource consumption data (i.e.",
"costs, length of stay) [5].",
"While useful, such systems often do not align well with the patient groups as clinicians would define them.",
"Patients who have received the same diagnosis (codes) may be treated for different purposes.",
"For example, patients who get reconstructive breast surgery caused by breast cancer or by gender change can be assigned to the same group, while they have different characteristics and should be assigned to different groups, according to medical experts [6].",
"Consequently, the process models derived for such a patient group is often also inaccurate and not aligned with the pathways which clinicians would have in their mind.",
"As a result, much manual work involving medical experts is needed to obtain meaningful patient groups.",
"Emerged from the process mining discipline, trace clustering techniques aim to help finding such homogenous groups of process instances (in our case the patients) [3], [7], [8], [4].",
"These techniques cluster the process instances based on the similarity between the sequences of executed activities.",
"However, when applied on hospital data, these approaches face several challenges.",
"Firstly, they have difficulties in scaling-up to handle such large data sets, which may contain hundreds of thousands of patients and millions of events per year.",
"Secondly, they assume that the cases within a group show more homogenous behavior than the cases of different groups, whereas in healthcare, patients treated for the same purpose could have disjoint paths and vice versa.",
"Thirdly, feature vectors (or other intermediate models) are often used to represent the cases and to compute similarity measures; the resulting clusters are often based on an average of the measures and, therefore, may not have clear criteria and may be difficult to explain.",
"Finally, a resulting cluster of patients could still have thousands of distinct activities, which prevents any process discovery algorithm to find a reasonable process model.",
"In this paper, we propose a novel perspective to the trace clustering problem.",
"We use sample sets to find one patient cluster at a time by exploiting frequent sequential pattern mining techniques, exemplified in fig:Untitled2.",
"Figure: An example of the partial trace clustering problem and an overview of our approach using sample sets and frequent sequence patterns.More specifically, we assume for each group a small sample set of patients (i.e., patient ids) that belongs to the group is made available by medical experts.",
"Using the available event data of patient pathways of the sample set, we compute frequent sequence patterns (FSPs) to learn the behavioral criteria of the group.",
"All patients are ranked based on the behavioral criteria, and we use thresholds to automatically determine whether they belong to the group (see fig:example, discussed in sec:approach).",
"Each group is clustered independently.",
"The obtained sequence patterns are used to discover simple process maps.",
"The approach is implemented in the Process Mining toolkit ProMhttp://www.promtools.org/, in the TraceClusteringFSM package (see source code) and evaluated using three real-life cases obtained from a large academic hospital in the Netherlands.",
"The results are validated with a semi-medical expert and a data analyst of the hospital, both of them work closely with medical experts; the semi-medical expert is a manager who has acquired relevant medical knowledge regarding the patient pathways.",
"The contribution of this work is that it gives a concrete method to identify patient clusters from a wealth of data and high variety of pathways with relatively little input from experts.",
"Moreover, this clustering will lead to simple process maps of frequent behavioral patterns in the clinical pathways that can be used in the communication with medical experts.",
"Such a method may be useful to reason about clinical pathways within hospitals for the sake of process improvement or quality control.",
"In the remainder, we first discuss related work in sec:relatedwork.",
"We recall the concepts and define the research problem in sec:research The proposed approach is described in sec:approach.",
"The evaluation results are presented sec:evaluation, and sec:conclusion concludes the paper."
],
[
"Related Work",
"In this section, we discuss three streams of trace clustering techniques, categorizing them by their similarity measures.",
"Feature-vector-based similarity Early work in trace clustering has followed the ideas in traditional data clustering.",
"Each trace is transformed into a vector of features based on, for example, the frequency of activities, the frequency of directly-followed relations, the resources involved, etc.",
"Between these feature vectors, various distance metrics in data mining are reused to estimate the similarity between the traces.",
"Subsequently, distance-based clustering algorithms are deployed, such as k-means or agglomerative hierarchical clustering algorithms [3], [7], [8].",
"In line with feature-vector based trace clustering techniques, the work of Greco et al.",
"[3] was one of the first approaches that incorporated trace clustering into process discovery algorithms.",
"Their work uses frequent (sub)sequences of activities to constitute feature vectors that represent traces.",
"Hierarchical clusters are then built using a top-down approach which iteratively refines the most imprecise process model (represented as disjunctive workflow schemas).",
"Song et al.",
"[7] present a technique that generalizes the feature space by considering data attributes in other dimensions than solely focusing on the control-flow.",
"Features of traces in one dimension are grouped into a so-called trace-profile, e.g., resource, performance, case attribute profiles, etc.",
"Furthermore, a multitude of vector-based distance metrics and clustering techniques (both partitioning and hierarchical) are deployed.",
"In [8], Bose and van der Aalst compute reoccurring sequences of activities, known as tandem arrays, and used these patterns as features in the feature space model in order to improve the way the control-flow information is taken into account in trace clustering.",
"Trace-sequence based clustering The second category proposes that the similarity can be measured by the syntax similarity between two trace sequences.",
"A trace can be edited into another trace by adding and removing events.",
"The similarity between two traces is measured by the number of the edit operations needed.",
"An example of this category of measure is the Levenshtein Edit Distance (LED).",
"Bose and van der Aalst [9] propose a trace-sequence distance by generalizing the LED and use the agglomerative clustering technique.",
"Chatain et al.",
"[10] assume that a normative process model is available and align the traces with the runs of the model.",
"In essence, the traces that are close to the same run (in terms of sequence distance) are clustered into the same group.",
"Model-based trace clustering Recently, the aim of trace clustering to discover better models has become more prominent.",
"Consequently, the definition of the similarity between traces has shifted from the traces themselves to the quality of the models discovered from those traces.",
"In essence, it is proposed that a trace is more similar to a cluster of traces, if a more fitting, precise, and simple model can be discovered from the cluster [11], [12], [4].",
"Early work in sequence clustering used first-order Markov models as the intermediate models to represent the clusters.",
"In 2003, Cadez et al.",
"[11] proposed to learn a mixture of first-order Markov models from user behavior by applying the Expectation Maximization problem.",
"The approach is evaluated on a web navigation data set.",
"Later, Ferreira et al.",
"[12] followed the same idea and qualitatively evaluated the clustering algorithm in a process mining setting using two additional data sets.",
"De Weerdt et al.",
"[4] use Petri nets as intermediate models and optimize a F-measure of the models discovered from the clusters.",
"The algorithm, called ActiTraC, first samples distinct traces, based on frequency or distance, as initial clusters.",
"The traces that “fit” into the intermediate-model of a cluster are assigned to the cluster.",
"The remaining noisy traces either are distributed over the clusters or returned as a garbage cluster.",
"More recently, De Koninck et al.",
"[13] proposed to incorporate domain knowledge by assuming that a complete clustering solution is provided by experts.",
"The proposed technique then aims to improve the quality of such a complete expert-driven clustering in terms of the model qualities.",
"Discussion Regarding the feature-based techniques, the number of possible features can be immense, especially in process mining [7].",
"For example, for $n$ activities, we could have $n^2$ number of directly-followed relations and $n^3$ if we consider three activities.",
"With thousands of distinct activities, it can be computationally expensive if we consider the full feature space.",
"Moreover, as the clusters are calculated based on the average distances between feature vectors, it is often difficult to explain the reason of a particular clustering.",
"In many cases, the feature-based techniques have difficulties in finding clusters that are in line with those of domain experts [13].",
"Sequence-based trace clustering faces similar limitations as the feature-based.",
"Furthermore, patients who have disjoint sets of activities and diagnosis (codes) may belong to the same group.",
"Both feature-based and sequence-based would have difficulties finding those.",
"For model-based trace clustering techniques, it would be difficult to handle the complexity of the intermediate models.",
"The clinical pathway of a well-defined patient group could still be extremely complex with thousands distinct activities being executed and each patient following a unique path tailored towards their conditions (see sec:data).",
"Assuming that a complete expert-driven clustering is available would put too much effort on medical experts and is not feasible for this reason.",
"Our approach, therefore, needs to be scalable and able to deal with this complexity.",
"The approach should also put more emphasis on the abundant domain knowledge available and find clear behavioral criteria of the clusters such that the behavioral criteria are meaningful for domain or medical experts."
],
[
"Research Problem",
"In this section, we first recall the preliminary concepts such as event logs, traces, and activities.",
"Using these concepts, we define our research problem."
],
[
"Preliminaries",
"A process describes a set of activities executed in a certain order.",
"For example, each patient in a hospital follows a certain process to treat a certain diagnosis of a disease, also known as clinical pathway.",
"An event log is a set of traces, each describing a sequence of events through the process.",
"Each event records additional information regarding the executed activity.",
"For example, Table REF shows a snippet of an event log of a healthcare process.",
"Each row records an executed event, which contains information such as the event id, the patient id, the activity, the timestamps, the diagnosis code (also known as diagnosis-related group (DRG) [5], or DBC in Dutch), and potentially some additional attributes regarding the event.",
"Table: An example of an event log of a healthcare process[Universes] We write the following notations for universes: $\\mathcal {E}$ denotes the universe of unique events, i.e., the set of all possible event identifiers.",
"$U$ denotes the set of all possible attribute names.",
"$\\mathit {Val}$ denotes the set of all possible attribute values.",
"$\\mathit {Act} \\subset \\mathit {Val}$ denotes the set of all possible activity names.",
"$\\mathit {PI} \\subset \\mathit {Val}$ denotes the set of all possible process instance identifiers.",
"[Event, Attribute, Label] For each event $e \\in \\mathcal {E}$ , for each attribute name $d \\in U$ , the attribute function $\\pi _{d}(e)$ returns the value of attribute $d$ of event $e$ .",
"A labeling function $\\pi _{l}: \\mathcal {E} \\rightarrow {Val}$ is a function that assigns the label to each event $e \\in \\mathcal {E}$ .",
"If the value is undefined, $\\pi _{d}(e) = \\bot $ .",
"Examples of attribute names used in this paper are listed as follows: $\\pi _{\\mathit {pi}}(e) \\in {PI}$ denotes the process instance identifier of $e$ ; $\\pi _{\\mathit {act}}(e) \\in {Act}$ is the activity associated with $e$ ; $\\pi _{\\mathit {time}}(e)$ denotes the timestamp of $e$ ; $\\pi _{\\mathit {}}(e)$ denotes the diagnosis code of $e$ .",
"For example, given the log listed in tab:eventloglistexample, $\\pi _{\\mathit {pi}}(e_{1}) = 1001$ , $\\pi _{\\mathit {act}}(e_{1}) =$ Registeration, $\\pi _{{}}(e_{1}) =$ 1.",
"The labeling function $\\pi _{l}(e)$ returns the activity label of event $e$ in the process (also known as an event classifier).",
"In this paper, we combine both the activities and the diagnosis codes and use them as labels, i.e., $\\pi _{l}(e) := \\pi _{{act}}(e) + \\pi _{{}}(e)$ , because the data analyst from the hospital indicated that both are important for the clinical pathway.",
"For example, given the log listed in tab:eventloglistexample, the label of event 1 is $\\pi _{l}(e_1) := \\pi _{{act}}(e_1) + \\pi _{{}}(e_1) =$ Reg-1; the label of event 23 is $\\pi _{l}(e_{23}) =$ Srg-4.",
"A trace $\\sigma = \\langle e_1, e_2, \\cdots , e_n\\rangle \\in \\mathcal {E}^*$ is a sequence of events, where for $1 \\le i < n$ , $\\pi _{time}(e_i) \\le \\pi _{time}(e_{i+1})$ and $\\pi _{pi}(e_i)$ = $\\pi _{pi}(e_{i+1})$ .",
"An event log $L = \\lbrace \\sigma _1, \\cdots , \\sigma _{{L}} \\rbrace \\subseteq \\mathcal {E}^*$ is a set of traces.",
"[Simplified Trace, Simplified Log] Let $\\pi _{l}$ be the labeling function.",
"Let $L = \\lbrace \\sigma _1, \\cdots , \\sigma _{{L}}\\rbrace $ be a log and $\\sigma = \\langle e_1, e_2, \\cdots , e_{{\\sigma }}\\rangle $ a trace.",
"We overload the labeling function such that, given $\\sigma $ , the labeling function returns the sequence of labels of the events in $\\sigma $ , i.e., $\\pi _{l}(\\sigma ) = \\langle \\pi _{l}(e_1), \\pi _{l}(e_2), \\cdots , \\pi _{l}(e_{{\\sigma }})\\rangle \\in {Val}^*$ .",
"Furthermore, given the log $L$ , the labeling function returns the multi-set of the sequences of the labels of the traces in $L$ , i.e., $\\pi _{l}(L) = [\\pi _{l}(\\sigma _1), \\cdots , \\pi _{l}(\\sigma _{{L}})]$ .",
"Let $\\sigma = \\langle e_1, \\cdots , e_n \\rangle \\in L$ be a trace.",
"For $1 \\le i < n$ , we say event $e_i$ is directly-followed by $e_{i+1}$ .",
"For $1 \\le i < j \\le n$ , we say event $e_i$ is eventually-followed by $e_j$ .",
"For the sake of brevity, we write $L^{l} = \\pi _{l}(L)$ and $\\sigma ^{l} = \\pi _{l}(\\sigma )$ .",
"For instance, the simplified trace of patient 1001 listed in tab:eventloglistexample is $\\pi _{l}(\\sigma _{1001}) = \\sigma _{1001}^{act,} = \\langle $ Reg-1, Doc-1, Lab-1, Srg-2, Doc-2$\\rangle $ .",
"Note that a patient (an activity) could be associated with multiple diagnosis codes [5], e.g., patient 1001 (activity Doc)."
],
[
"Research Problem - Grouping Patients",
"Traditional trace clustering aims to divide the traces of a log into clusters, such that the traces of the same cluster show more homogenous behavior than the traces of different clusters.",
"In the healthcare domain, we are facing a very large, complex data set and abundant domain knowledge.",
"As discussed at the end of sec:relatedwork, we would like to (1) handle such a large data set, to (2) incorporate, leverage, and put more emphasis on the domain knowledge, in order to obtain clusters that are more in line with those of medical experts, while requiring little effort from such experts, and to (3) be able to find the clusters accurately and validate clusters quality, we propose the following.",
"We assume that medical experts can provide a small sample set $P$ of the patients that belong to a patient group $\\hat{C}$ of interest.",
"Giving a sample requires little effort from their side.",
"We assume that $\\hat{C}$ is unknown (because when $\\hat{C}$ gets large, it would require too much effort for medical experts to exhaustively list all patients that belong to $\\hat{C}$ and to repeat this process).",
"We use the available traces of all patients in the sample $P$ , and the objective is to find a cluster $C$ in such a way that $C$ is as close to the group $\\hat{C}$ as possible (i.e., the highest recall and precision possible).",
"We do this separately for each group $\\hat{C}_i$ where the sample set $P_i$ is available.",
"To generalize, we define the partial trace clustering formally as follows.",
"[Partial Trace Clustering] Let $L = \\lbrace \\sigma _1, \\cdots , \\sigma _n\\rbrace $ be the event log, and ${PI} = \\lbrace \\pi _{pi}(\\sigma _1), \\cdots , \\pi _{pi}(\\sigma _n)\\rbrace $ the set of case ids of $L$ .",
"Let $P_1, P_2, \\cdots , P_x \\subset {PI}$ be the sets of samples that respectively belong to clusters $\\hat{C}_1, \\hat{C}_2, \\cdots , \\hat{C}_x$ , provided by experts (e.g., a doctor), with $x \\in \\mathbb {N}$ .",
"We would like to find the clusters $C_1, C_2, \\cdots , C_x \\subset {PI}$ , such that the set difference between $C_i$ and $\\hat{C}_i$ is minimized.",
"Note that clusters $C_1, \\cdots , C_x$ can be non-overlapping or form an incomplete clustering of ${PI}$ (i.e., $ C_1 \\cup \\cdots \\cup C_x \\subseteq {PI}$ ), and $x$ could be 1.",
"Based on these properties, we do not have to find all clusters or to compute a complete clustering of all traces.",
"It allows us to mine, cluster, and validate each cluster independently."
],
[
"Approach",
"As explained above, we assume that for each cluster $C$ to be found we have a small sample $P$ of the cases that belongs to the true-but-unknown cluster $\\hat{C}$ .",
"For all other cases it is unknown whether they belong to the cluster or not.",
"By exploiting the available sample set $P$ and the event log $L$ of all cases, the objective is to find behavioral criteria for determining the cluster.",
"To find the behavioral criteria and to handle the large number of features, we compute frequent behavioral patterns.",
"In sec:findfsp, we first explain the use of sequence pattern mining to learn the frequent sequence patterns (FSPs) of the sample set.",
"In sec:ranktraces, we then match the FSPs to the other cases in the sample to train our parameters.",
"Finally, we match all cases to the clustering criteria to return the computed cluster in sec:determinecriteria.",
"fig:example shows an overview of the approach."
],
[
"Finding Frequent Sequence Patterns",
"The first step of the approach is to find frequent sequence patterns repeated among the samples.",
"A frequent sequence pattern is a sequence that occurs in the traces with a frequency no less than a specified threshold.",
"We adapt the definition of sequence patterns in our context as follows.",
"[Sequence Pattern] A sequence pattern $^= \\langle a_1, \\cdots , a_m \\rangle \\in Val^*$ is a sequence of labels in which $a_i$ is said to be eventually followed by $a_{i+1}$ for $1 \\le i < m$ .",
"When a trace matches a sequence pattern, it means that the trace contains a sub sequence where the labels occur in the same order.",
"[Support of Sequence Pattern] Let $L$ be an event log and $\\pi _{l}$ the labeling function.",
"Let $\\sigma \\in L$ be a trace, with $\\pi _{l}(\\sigma ) = \\langle a_1, a_2, \\cdots , a_n \\rangle $ .",
"Let $\\vec{s} = \\langle s_1, \\cdots , s_m \\rangle \\in {Val}^*$ be a sequence pattern.",
"We say $\\sigma $ matches $^$ if and only if there exist integers $i_1, i_2, \\cdots , i_m$ such that $1 \\le i_1 < i_2 < \\cdots < i_m \\le n$ and $s_1 = a_{i_1} , s_2 = a_{i_2} ,..., s_m = a_{i_m}$ .",
"We write $^\\sqsubseteq \\pi _{l}(\\sigma )$ .",
"The support of sequence $^$ in $L$ is the number of traces in $L$ that matches $^$ , i.e., ${supp}(^, L) = \\frac{{\\lbrace ^\\sqsubseteq \\pi _{l}(\\sigma ) \\mid \\sigma \\in L\\rbrace }}{{L}}$ Let $\\phi _s$ denote the minimum support threshold.",
"A sequence pattern $^$ is said to be frequent if and only if ${supp}(^, L) \\ge \\phi _s$ .",
"We write ${SP}(L, \\phi _s)$ to denote the set of all sequence patterns in $L$ with a support of at least $\\phi _s$ , i.e., ${SP}(L, \\phi _s) = \\lbrace ^\\in Val^* \\mid {supp}(^, L) \\ge \\phi _s\\rbrace $ Step 1 in fig:example exemplifies mining frequent sequence patterns, with the minimum support $\\phi _s = 0.8$ .",
"Let $L^{\\prime } = \\lbrace \\sigma _1, \\sigma _2, \\sigma _3\\rbrace $ , as shown in fig:example.",
"We have ${SP}(L^{\\prime }, 0.8) = \\lbrace {A}, {C}, {D}, {E}, {F}, {A,C}, {A,D}, {A,E}, \\cdots , {A, C, D, F} \\rbrace $ .",
"There are several well-known algorithms to compute frequent sequence patterns.",
"In this paper, we use the CloFAST algorithm [14] and its SPMF implementation [15] due to its fast run-time, which is also used in [16] for next activity prediction.",
"Figure: An example of our approach applied on the event log (on the left), with the FSPs mined (in the middle), and the scores of the traces (on the right)."
],
[
"Trace Ranking By Sequence Pattern Matching",
"To automatically find behavioral criteria for determining the cluster $C$ , we divide the sample set $P$ into a training set $P_{tr}$ and use the entire $P$ as our test set.",
"On the training set $P_{tr}$ , we compute the set of frequent sequence patterns (FSPs).",
"The FSPs mined on the training set $P_{tr}$ could still be very large.",
"Therefore, we select a subset of the FSPs.",
"We use the FSPs of length 1, 2, and the closed sequence patterns as our behavioral criteria.",
"Note that this step can be generalized with ease and any other subset of the FSPs can be selected as behavioral criteria.",
"We select these three subsets because of the following.",
"The FSPs of length 1 represent the frequent activity labels occurred in the training set; the FSPs of length 2 represent the frequent eventually-followed relations occurred.",
"The closed sequence patterns $^\\in {SP}(L, \\phi _s)$ are the sequence patterns $^$ such that for all other patterns which $^$ satisfies have a lower support.",
"Thus, these three provide a good coverage of all FSPs with less redundancy.",
"[Closed Sequence Pattern] Let ${SP}(L, \\phi _s)$ denote all frequent sequence patterns in $L$ with a support of at least $\\phi _s$ .",
"A sequence pattern $^\\in {SP}(L, \\phi _s)$ is a closed sequence pattern if and only if for all $^^{\\prime }\\in {SP}(L, \\phi _s)$ , $^\\sqsubseteq ^^{\\prime } \\Leftrightarrow (^= ^^{\\prime } \\vee supp(^, L) > supp(^^{\\prime }, L))$ .",
"Next, using these subsets of these patterns, we rank each trace in $P$ based on the number of patterns it satisfies.",
"Let ${SP}(P, \\phi _s) = \\lbrace ^_1, ^_2, \\cdots ^_n\\rbrace $ be the set of frequent sequence patterns of $P$ above support threshold $\\phi _s$ .",
"Let ${SP}_1, {SP}_2, {SP}_{{clo}}\\subseteq {SP}(P, \\phi _s)$ be the set of patterns of size 1, size 2, and closed sequence patterns, respectively.",
"We give each case a score based on the number of patterns in ${SP}_1, {SP}_2$ , and ${SP}_{{clo}}$ the trace satisfies and rank the cases based on their score.",
"Thus, ${score}_k(\\sigma ) = {\\lbrace ^\\in {SP}_k | ^\\sqsubseteq \\pi _{l}(\\sigma ) \\rbrace }$ For example, see fig:example, step 1 shows ${SP}_1$ of five sequence patterns, ${SP}_2$ of eight, and ${SP}_{{clo}}$ of two, which are mined on the $P_{{tr}} = \\lbrace \\sigma _1, \\sigma _2, \\sigma _3\\rbrace $ using a minimum support of 0.8.",
"Given trace $\\sigma _4 \\notin P_{{tr}}$ , it matches to $\\langle A \\rangle $ , $\\langle C \\rangle $ , $\\langle D \\rangle $ , and $\\langle E \\rangle $ in ${SP}_1$ , to $\\langle A, C \\rangle $ , $\\langle A, D \\rangle $ , $\\langle A, F \\rangle $ , $\\langle C, D\\rangle $ , $\\langle C, F\\rangle $ , and $\\langle D, F\\rangle $ in ${SP}_2$ , and to $\\langle A, C, D, F\\rangle $ in ${SP}_{clo}$ .",
"Thus, ${score}_1(\\sigma _4) = 4$ , ${score}_2(\\sigma _4) = 6$ , ${score}_{clo}(\\sigma _4) = 1$ ."
],
[
"Computing Criteria Threshold",
"For each case, we have now computed ${score}_1$ , ${score}_2$ , and ${score}_{clo}$ , as explained above.",
"For the three scores, we respectively introduce three thresholds, $\\phi _1 \\in \\mathbb {N}$ , $\\phi _2\\in \\mathbb {N}_0$ , and $\\phi _{clo} \\in \\mathbb {N}_0$ .",
"We decide on whether a case belongs to cluster $C$ based on whether the scores of the trace are above the corresponding thresholds, i.e., $C_{\\phi _{1}, \\phi _{2}, \\phi _{clo}} = \\lbrace \\pi _{pi}(\\sigma )| \\sigma \\in L \\wedge {score}_1(\\sigma ) \\ge \\phi _1 \\wedge {score}_2(\\sigma ) \\ge \\phi _2 \\wedge {score}_{clo}(\\sigma ) \\ge \\phi _{clo} \\rbrace $ To estimate the quality of $C_{\\phi _{1}, \\phi _{2}, \\phi _{clo}}$ , we then compute the estimated recall with respect to $P$ , i.e., $\\overline{{recall}}_{\\phi _{1}, \\phi _{2}, \\phi _{clo}}= \\frac{{C_{\\phi _{1}, \\phi _{2}, \\phi _{clo}} \\cap P}}{{P}}$ .",
"When the sample set $P$ gets closer to the ideal cluster $\\hat{C}$ , the estimated $\\overline{{recall}}$ gets closer to the true recall.",
"When we decrease $\\phi _{1}, \\phi _{2},$ and $\\phi _{clo}$ , more cases are included in $C$ .",
"After a certain point, the increase of $\\overline{{recall}}$ starts to flatten, which suggests that further lowering the thresholds does not help to retrieve a large number of true positive cases, which is likely to result in a low precision.",
"To approximate such a point, we use ${max}_{\\phi _{1}, \\phi _{2}, \\phi _{clo}}\\frac{\\overline{{recall}}^2_{\\phi _{1}, \\phi _{2}, \\phi _{clo}}}{ {{C}_{\\phi _{1}, \\phi _{2}, \\phi _{clo}}}}$ [17], but only consider the thresholds when $\\overline{{recall}} \\ge 0.8$ .",
"The number of iterations to find such a maximum depends on the maximal values of ${score}_1$ , ${score}_2$ , and ${score}_{clo}$ ."
],
[
"Evaluation",
"We implemented the described approach in the process mining toolkit ProM.",
"We used a real-life data set to evaluate our approach with respect to the following three objectives: How accurate (in terms of F1-scores) are the clusters returned by our automated approach, compared to the optimal scores?",
"How accurate can we find the clusters using our approach, compared to a related approach that uses frequent item sets (FIS) [6]?",
"Can we discover a simple and insightful behavioral criteria for each patient group such that the criteria can be used to communicate with medical experts?",
"In the following, we first discuss the data set in sec:data and then report our results sec:results with respect to these three objectives.",
"All experiments are run on an Intel Core i7- 8550U 1.80GHZ with a processing unit of 16GB running Windows 10 Enterprise.",
"The maximal queue size of CloFAST algorithm [14], [15] is set to $10^5$ .",
"The obtained results were discussed with the semi-medical expert and the data expert in the hospital who cooperate closely with medical experts in their daily work."
],
[
"Experimental Setup",
"For the evaluation, we used anonymized patient records provided by the VU University Medical Center Amsterdam, a large academic hospital in the Netherlands.",
"All patients that have a diagnosis code registered between 2013 and 2017 are selected.",
"The administrative and dummy activities are filtered out.",
"As a result, we have in total 328,256 patients over the five years.",
"There are 7,426 unique activities and 2,251 unique diagnosis codes registered.",
"In total more than 15.5 million events are recorded in the logs.",
"Table: General information of the real-life data set and the ground truth clusters.In addition, lists of patients of three groups divided over the five years are provided by the analyst, patients with kidney failure, with diabetes, or with head/neck-tumor.",
"We use $\\hat{C}_{KidneyYY}$ , $\\hat{C}_{DiabetesYY}$ , and $\\hat{C}_{HNTumorYY}$ to refer to them, respectively, where $YY$ denotes the particular year.",
"tab:allpatients lists the number of cases (c), distinct process instances (dpi), events (e), activities (acts), and other statistical information related to the event logs of 2013 and 2017 as examples.",
"For instance, in tab:allpatients row 2, 3, and 4 show an overview of $\\hat{C}_{Kidney17}$ , $\\hat{C}_{Diabetes17}$ , and $\\hat{C}_{HNTumor17}$ , respectively.",
"These 15 clusters are used as the ground truth.",
"For each cluster, 30 patient ids are provided by medical experts as the sample set (i.e., $\\mid \\!\\!",
"P \\!\\!\\mid \\, = 30$ ), the same as a previous study [6].",
"For finding the clusters, we use all the patient records of the same year and the provided $P$ to compute our cluster $C$ .",
"The quality of $C$ is evaluated against the corresponding ground truth cluster $\\hat{C}$ by calculating the recall, precision, and F1-score, i.e., $recall(C, \\hat{C}) = \\frac{{C \\cap \\hat{C}}}{{\\hat{C}}}$ , $precision(C, \\hat{C}) = \\frac{{C \\cap \\hat{C}}}{{C}}$ , and $F1\\_measure(C, \\hat{C}) = 2 \\cdot \\frac{precision(C, \\hat{C}) \\cdot recall(C, \\hat{C})}{precision(C, \\hat{C}) + recall(C, \\hat{C})}$ .",
"It is worthwhile to mention that the data analyst stated that it took a lot of time and effort to obtain each of these ground truth clusters.",
"Multiple intensive discussion sessions were scheduled with different groups of medical experts to come to the definitions and criteria for each of these clusters.",
"This suggests that if our algorithmic approach can identify the behavioral criteria and the clusters with a reasonable accuracy using only a small sample set, it would help reducing the workload of both the analysts and the medial experts and making this process feasible to be repeated for other patient clusters or in other hospitals.",
"Another important remark is that the ground-truth clusters which we are trying to find are very small and unbalanced compared to the full event logs, making this trace clustering problem a very challenging task.",
"For instance, the $\\hat{C}$ _Kidney17 ($\\hat{C}$ _Diabetes17) contains only 140 (1521) patients, about 0.1% (1.2%) of the 128,505 patients in the log of 2017.",
"To determine the support threshold $\\phi _s$ , we started with 1.0 and decreased the value by 0.1 until a reasonable large amount of patterns are found and the F1 scores stopped increasing.",
"For the $C\\_{Kidney}$ groups, $\\phi _s$ ranges from 1.0 down-to 0.6, for $C\\_{Diabetes}$ , 0.4 down-to 0.2, and for $C\\_{HNTumor}$ , 0.5 down-to 0.2.",
"We used either 10 or 15 (of the 30 in the sample) as the training set to learn the frequent sequential patterns, i.e., $k = \\, \\mid \\!\\!",
"{P_{{tr}}} \\!\\!\\mid \\, \\in \\lbrace 10, 15\\rbrace $ .",
"We write for our approach with the automatically determined $\\phi _1$ , $\\phi _2$ , and $\\phi _{clo}$ ; for the maximum F1 score using the same $\\phi _s$ and $k$ but based on the optimal $\\phi _1, \\phi _2,$ and $\\phi _{clo}$ .",
"fig:res2 shows the difference in F1-scores between (dotted lines) and (filled lines).",
"We observe that in most cases the F1-scores of the automated (dotted line) are very close to the ones of the optimal (filled line).",
"For some clusters, for example Diabetes16&17 and HNTumor15&17, returns the exact same F1-scores as the maximum for all $\\phi _s$ and $k$ .",
"Only in a few cases, for example, for Diabetes13 when $k$ is 15 and the support $\\phi _s$ is 0.2, scores considerably lower than with a difference of 0.26.",
"Nevertheless, for the same $\\phi _s$ when $k$ is set to 10, this difference is immediately decreased to 0.01.",
"Taking into account that we only have a sample set of 30 patients and the number of activities ranges in the thousands, the is able to approximate the optimal F1-scores very well.",
"Figure: The differences in the F1-scores of the automated approach () shown in dotted lines and the maximum scores achieved () shown in filled lines, using various support threshold (on the x-axis) and training sample size kk.For the support threshold $\\phi _s$ , we observe overall a slight increase in the F1-scores for the Kidney and HNTumor groups when we decrease $\\phi _s$ .",
"For the Diabetes groups, there is a considerable increase in F1-scores during the beginning (when $\\phi _s$ is decreased from 0.4 to 0.2), but this improvement also fades out.",
"One reason for this is because when the support threshold is low, more patterns are found and used as criteria; thus, more patients are included in the cluster including false positives.",
"While the recall increases, the precision becomes lower, which led to a small increase in the F1-scores.",
"For the diabetes group, when the support $\\phi _s$ is 0.4, the number of sequence patterns is extremely small (1 or 3).",
"When the support decreases, it allowed the algorithm to find a consider number of defining patterns that is significant to retrieve the patients of the ground truth clusters.",
"This increases the recall dramatically without a significant decrease in precision.",
"Furthermore, fig:res2 also shows that using fewer training samples ($k = 10$ , denoted using light blue), our approach can achieve the same scores as when using a larger training sample set ($k = 15$ , denoted using darker blue).",
"In many cases, the former (i.e., $k=10$ ) even achieved a better result.",
"This may be due to that the training test set $P\\backslash P_{tr}$ is larger."
],
[
"EO2) Comparing the F1-scores achieved.",
"We write to refer to the maximum score of our approach using the above settings.",
"We write to denote our approach with a single setting (0.8 for kidney, 0.2 for diabetes, and 0.2 for NH-tumor, with sample size 30 and $k_{tr} = 10$ ), to compare our results to the previous work [6].",
"The parameters are selected on the results of EO1.",
"We write for referring to the previous approach that uses frequent item sets [6].",
"fig:Book1 shows the maximum F1-scores of , , and on the 15 clusters over 5 years and three groups.",
"For the diabetes group, we achieved a considerable improvement of 0.2-0.3 in the F1-scores, compared to the approach [6].",
"Overall, our approach achieved a better result.",
"One reason for this improvement is with the use of sequential patterns (instead of frequent item sets), our approach is able to decrease the number of false positives and find the clusters with a higher precision.",
"Figure: F1-measures achieved by our approaches TC-FSM1 and TC-FCM * ^* for the three groups, compared to the ones achieved by the previous approach FIS ."
],
[
"EO3) Frequent Sequence Patterns to Simple Process Maps.",
"We used the closed sequence patterns mined on the samples as the traces that represent the frequent behavior shared by the group.",
"Using these sequence patterns, the discovered process maps overall seem to be simple and insightful, representing only the crucial behavioral criteria of the patient groups.",
"We show three process maps in fig:patternlogNierschade2017k10s08, fig:patternlogHNTumor2016k10s03 and fig:patternlogDiabetes2016k10s02, for $C_{kidney17}$ , $C_{HNTumor16}$ , and $C_{Diabetes16}$ to illustrate our results.",
"All the process maps contain all activities and paths (thus no filters applied).",
"As can be seen in fig:patternlogNierschade2017k10s08, the number of activity labels in the process is reduced from about 4,700 to 11.",
"The number of distinct variants is reduced from 140 to 8.",
"The process maps are shown to the semi-medical expert and the data analyst.",
"The semi-medical expert observes and confirms that the activities shown (e.g., kalium (potassium), kreatinine (creatinine), calcium (calcium), fosfaat (phosphate), albumine (albumin), natrium (sodium), ureum bloed (ureum blood), etc.)",
"are important activities (e.g., lab activities) in the clinical pathway of the kidney groups (patients with renal insufficiency).",
"The data analyst confirms that the diagnosis code Chronic renal failure eGFR $<30$ ml/min associated with these activities is a crucial criteria for defining the kidney groups.",
"In fig:patternlogHNTumor2016k10s03 and fig:patternlogDiabetes2016k10s02, we also observe that multiple distinct diagnosis codes are used for the HNTumor and diabetes group, respectively.",
"In the process map for $C_{Diabetes16}$ , we found the process map being divided into three sub processes based on the diagnosis codes: [SG1] diabetes mellitus without secondary complications, [SG2] diabetes mellitus with secondary complications, and [SG3] diabetes mellitus chronic pump therapy (see fig:patternlogDiabetes2016k10s02, highlighted in red).",
"The semi-medical expert also observes and confirms that some of these activities are important indicators for different groups.",
"For example, creatinine is important for both the kidney and diabetes groups.",
"Nevertheless, because our approach is able to combine and handle the activities with their diagnosis codes as activity labels (in terms of the large variety of distinct labels and process variants), it enabled us to accurately distinguish the creatinine for the kidney group (i.e., creatinine$||$ Chronic renal failure eGFR $ <30 $ ml/min) versus the same creatinine but for the diabetes group (i.e., creatinine$||$ SG1 and creatinine$||$ SG3, see fig:patternlogDiabetes2016k10s02, highlighted in blue).",
"Figure: NO_CAPTIONFigure: NO_CAPTIONThe results have shown that our approach using the discovered and selected frequent sequence patterns can help to cluster the patient groups with a reasonably high accuracy (e.g., a maximum of 0.75 for the Kidney group, 0.94 for Diabetes, and 0.67 for HNTumor), despite the very large data sets (on average, about 130,000 of patients and $3.9$ millions of events per year) and the relatively very small and unbalanced clusters.",
"Moreover, the proposed approach that automatically determines the parameters achieved F1-scores that are very close to the optimal scores.",
"This means after setting the support $\\phi _s$ and with no further input, we can find the clusters with a reasonable quality as well.",
"Using the process maps, we show the meaningful and insightful behavioral patterns and criteria in the clinical pathways of the patient groups.",
"According to the semi-medical expert, the maps can be a useful tool in the communication with the domain experts regarding the pathways.",
"Note that we do not have any prior knowledge of the specific activities or diagnosis codes of the patient groups.",
"A remark is that these process maps of FSPs have a different semantics than the formal process models discovered using the traces.",
"For example, an edge from A to B in the map, in essence, means that such an eventually-followed relation is frequent.",
"To obtain formal models, we may project the patterns on the traces and use the instances of the patterns in the traces to discover models [18]."
],
[
"Conclusion and Future Work",
"In this paper, we investigated the trace clustering problem in healthcare and proposed an approach that can handle the characteristics of healthcare data.",
"Using a small sample set of patients, the proposed approach finds frequent sequence patterns and uses these as behavioral criteria for determining a cluster.",
"The results of the evaluations show that the approach is able to identify patient clusters with a very reasonable quality on the basis of very limited input from medical experts, despite the very large data sets and the small, unbalanced clusters (ground-truth).",
"The obtained behavioral criteria also led to the generation of simple process maps, where we have some first insights that these could be actually used by medical experts.",
"The semi-medical expert who works closely with medical experts was able to recognize the important activities in the clinical pathways of the patient groups.",
"Such a method may be useful to reason about clinical pathways within hospitals for the sake of process improvement or quality control.",
"For future work, we plan to investigate other strategies for selecting sequence patterns as behavioral criteria to further improve the F1-scores.",
"Also, the effect of sample size on the F1-scores is worth investigating.",
"Another interesting direction is to exploit the frequent sequence patterns to discover formal process models for the clinical pathways of each cluster.",
"Finally, we would like to validate the maps with medical experts and apply our approach to other patient clusters and other hospitals.",
"Acknowledgments.",
"This research was supported by the NWO TACTICS project (628.011.004) and Lunet Zorg in the Netherlands.",
"We would also like to thank the experts from the VUMC for their extremely valuable assistance and feedback in the evaluation."
]
] | 2001.03411 |
[
[
"Two Applications of Deep Learning in the Physical Layer of Communication\n Systems"
],
[
"Abstract Deep learning has proved itself to be a powerful tool to develop data-driven signal processing algorithms for challenging engineering problems.",
"By learning the key features and characteristics of the input signals, instead of requiring a human to first identify and model them, learned algorithms can beat many man-made algorithms.",
"In particular, deep neural networks are capable of learning the complicated features in nature-made signals, such as photos and audio recordings, and use them for classification and decision making.",
"The situation is rather different in communication systems, where the information signals are man-made, the propagation channels are relatively easy to model, and we know how to operate close to the Shannon capacity limits.",
"Does this mean that there is no role for deep learning in the development of future communication systems?"
],
[
"Deep learning has proved itself to be a powerful tool to develop data-driven signal processing algorithms for challenging engineering problems.",
"By learning the key features and characteristics of the input signals, instead of requiring a human to first identify and model them, learned algorithms can beat many man-made algorithms.",
"In particular, deep neural networks are capable of learning the complicated features in nature-made signals, such as photos and audio recordings, and use them for classification and decision making.",
"The situation is rather different in communication systems, where the information signals are man-made, the propagation channels are relatively easy to model, and we know how to operate close to the Shannon capacity limits.",
"Does this mean that there is no role for deep learning in the development of future communication systems?"
],
[
"Relevance",
"The answer to the question above is “no” but for the aforementioned reasons, we need to be careful not to reinvent the wheel.",
"We must identify the right problems to tackle with deep learning and, even then, not start from a blank sheet of paper.",
"There are many signal processing problems in the physical layer of communication systems that we already know how to solve optimally, for example, using well-established estimation, detection, and optimization theory.",
"Nonetheless, there are also important practical problems where we lack acceptable solutions, for example, due to a lack of appropriate models or algorithms.",
"In this lecture note, we first introduce the key properties of artificial neural networks and deep learning.",
"The focus is not on technicalities around the training process or choice of network structure, but on what we can practically achieve, assuming the training is carried out successfully.",
"We will then describe three application categories in communication engineering, whereof one exposes some fundamental weaknesses of deep learning and two illustrate important advances that can be made by utilizing deep learning."
],
[
"Prerequisites",
"This lecture note requires basic knowledge of linear algebra, digital communications, and probability.",
"Figure: The gray-box input-output model in (a) is characterized by f ^\\hat{f} and a parameter vector θ\\theta .",
"It is called an artificial neural network if f ^\\hat{f} has a particular structure, such as the one illustrated in (b)."
],
[
"Problem Statement and Solution",
"We begin by briefly describing what artificial neural networks are and formulating the problem of using them as function approximators.",
"Consider a system that takes an $n_0$ -length input vector $x_0 \\in \\mathbb {R}^{n_0}$ and produces a $k$ -length output vector $y \\in \\mathbb {R}^{k}$ , as illustrated in Fig.",
"REF (a).",
"The output is determined by the input via a deterministic function $\\hat{f}$ : $y = \\hat{f}(x_0;\\theta ).$ The function is fixed but is characterized by an $m$ -dimensional parameter vector $\\theta \\in \\mathbb {R}^{m}$ .",
"Many different input-output relations can be modeled in this way by changing the parameter vector $\\theta $ , but they all share an underlying structure determined by the initial choice of $\\hat{f}$ .",
"This is called a gray-box model.",
"When the function $\\hat{f}$ is selected to resemble the biological neural networks in human brains, the gray box is called an artificial neural network.",
"The input vector $x_0$ is then viewed as the values in $n_0$ neurons from which the function $\\hat{f}$ produces the values of $y$ in $k$ other neurons.",
"There are many different examples of this.",
"The classical one is a fully-connected feed-forward network, which is illustrated in Fig.",
"REF (b).",
"In this case, $\\hat{f}$ is a composition of $L$ functions, $\\hat{f}_1,\\ldots ,\\hat{f}_L$ , which describes transitions between neurons in an input layer to neurons in an output layer via $L-1$ intermediate “hidden” layers.",
"$L$ characterizes how deep the network is.",
"The function $\\hat{f}_l$ is determined by the parameters $\\theta _l = \\lbrace W_l, b_l \\rbrace $ and modeled as $\\hat{f}_l ( x_{l-1} ; \\theta _l ) = \\sigma _l ( W_l x_{l-1} + b_l ),$ where $ W_l \\in \\mathbb {R}^{n_l \\times n_{l-1}}$ is called a weight matrix, $b_l \\in \\mathbb {R}^{n_l}$ is called a bias vector, and $\\sigma _l : \\mathbb {R}^{n_l} \\rightarrow \\mathbb {R}^{n_l}$ is an element-wise non-linear function that is called an activation function.",
"With inspiration from the structure of the human brain, the function $\\hat{f}_l$ can be interpreted as taking the values $x_{l-1}$ in the $n_{l-1}$ neurons of layer $l-1$ , mixing the values together according to the affine transition relation $W_l x_{l-1} + b_l $ , and finally applying the activation function $\\sigma _l$ to the determine values of the $n_l$ neurons of layer $l$ .",
"If there are four layers as in Fig.",
"REF (b), then $L=3$ and the complete input-output relation is $y = \\hat{f}_3 \\left( \\hat{f}_2 \\left( \\hat{f}_1 \\left( x_0 ; \\theta _1 \\right); \\theta _2 \\right) ; \\theta _3 \\right).$ Hence, the composite function $\\hat{f}$ is determined by the parameter vector $\\theta $ containing the $\\sum _{l=1}^{L} n_{l} (n_{l-1} +1)$ parameter values from $\\theta _1,\\theta _2,\\theta _3$ (i.e., the weights and biases from all layers)."
],
[
"Problem Statement",
"Artificial neural networks are generally used to approximate other functions, by selecting the parameter vector $\\theta $ to somehow minimize the approximation error.",
"In particular, the category of fully-connected feed-forward networks is capable of approximating any continuous function arbitrarily well by utilizing a (possibly) large but finite number of parameters (and neurons) [1].",
"This important result can be viewed as a generalization of Taylor polynomial approximations to functions with vector inputs and vector outputs.",
"Two other categories are convolutional neural networks and recurrent neural networks [2].",
"Each category is believed to be better at approximating certain types of functions, in the sense of requiring fewer parameters to achieve a certain approximation error and/or it being easier to find appropriate parameter values in practice.",
"Selecting the right category of neural network is important but beyond the scope of this lecture note.",
"Instead, our problem statement is: what are the important use cases where the function approximation capability can be utilized in the physical layer of communication systems, to achieve large improvements compared to conventional techniques?"
],
[
"Solution",
"To identify practically important use cases, we first need to understand how the function approximation is carried out.",
"The parameter vector of an artificial neural network can be tuned/trained to approximate a (possibly unknown) function that we call $f$ ; that is, $\\hat{f}$ should be trained to become a good estimate of $f$ .",
"This is preferably done by supervised learning using a set of $T$ training examples consisting of input vectors $x_t^{\\mathrm {train}}$ and the corresponding output vectors $y_t^{\\mathrm {train}} = f(x_t^{\\mathrm {train}})$ that we want the neural network to reproduce, for $t=1,\\ldots ,T$ .",
"Let us represent these training examples as the columns of two matrices: $X^{\\mathrm {train}} = \\begin{bmatrix} x_1^{\\mathrm {train}} & \\ldots & x_T^{\\mathrm {train}} \\end{bmatrix}, \\quad \\quad Y^{\\mathrm {train}} = \\begin{bmatrix} y_1^{\\mathrm {train}} & \\ldots & y_T^{\\mathrm {train}}\\end{bmatrix}.$ The inputs should ideally be selected independently at random from the distribution of inputs that appears when using $f$ in reality.",
"The training basically consists of finding the parameter $\\theta ^*$ that minimizes a loss function $\\ell $ that measures the approximation mismatch: $ \\theta ^* = {\\underset{{\\theta }}{\\mathrm {arg\\,min}}} \\,\\, \\ell \\left( \\theta , X^{\\mathrm {train}}, Y^{\\mathrm {train}} \\right).$ For example, the loss can be measured in the mean-squared sense as $\\ell \\left( \\theta , X^{\\mathrm {train}}, Y^{\\mathrm {train}} \\right) = \\frac{1}{T} \\sum _{t=1}^{T} \\left\\Vert y_t^{\\mathrm {train}} - \\hat{f}(x_t^{\\mathrm {train}};\\theta ) \\right\\Vert ^2.$ The goal is that the trained neural network $\\hat{f}(x_0;\\theta ^*)$ will provide approximately the right outputs not only for the training examples, but for any input signal $x_0$ generated in the same way.",
"This desired property is called generalization.",
"Intuitively, if the unknown function $f$ is continuous and has limited variability, we should be able to approximate it well from a large training set.",
"We can once again make a parallel to polynomial approximations; any scalar polynomial of order $T-1$ is uniquely determined by $T$ samples (training examples) of the inputs and outputs.",
"If the polynomial order is unknown, or if the function is only approximately polynomial, we need a larger number of samples to ensure a good approximation.",
"Since the training in (REF ) is a complicated non-convex optimization problem, huge efforts have been dedicated to finding computationally and performance-wise acceptable suboptimal solutions.",
"Moreover, the generalization to unseen inputs can be improved by various regularizations, hyper-parameter choices, and network designs [2].",
"These choices affect the model complexity.",
"A simple model cannot capture complex dependencies.",
"A too complex model explains the training data only (this is called overfitting).",
"A correct complexity trade-off gives good generalization and is typically found using cross-validation.",
"However, such empirical craftsmanship is not the focus of this lecture note, but we conclude: Artificial neural networks can approximate any continuous function.",
"The supervised training requires a large training set with inputs/outputs to achieve a low approximation error.",
"There are many functionalities in communication systems that can be described by a mathematical function $f$ and, thus, can be approximated by a neural network.",
"To identify the promising use cases, we will first explain the basic methodology and its weaknesses by giving a concrete example."
],
[
"A Deep-Learning Solution to Signal Detection",
"The physical layer of a communication system determines how an information-bearing signal is sent from the transmitter to the receiver over a physical channel.",
"A critical task is the signal detection, where the receiver tries to identify what information was sent.",
"To describe some key properties of deep learning, we will exemplify how it can be used for signal detection.",
"We consider a classical additive white Gaussian noise (AWGN) channel, where a two-dimensional signal vector $s \\in \\mathbb {R}^2$ is sent.",
"The received signal $r \\in \\mathbb {R}^2$ is given by $r = s + n,$ where $n \\sim \\mathcal {N}(0,\\sigma ^2I)$ is an independent Gaussian noise vector where the entries have variance $\\sigma ^2$ .",
"We assume two bits of information are encoded into $s$ using a quadrature phase-shift keying (QPSK) constellation.",
"Hence, there are four possible signal points that are equally spaced on the unit circle: $ s \\in \\left\\lbrace \\begin{bmatrix} 1/\\sqrt{2} \\\\ 1/\\sqrt{2} \\end{bmatrix}, \\begin{bmatrix} 1/\\sqrt{2} \\\\ -1/\\sqrt{2} \\end{bmatrix} , \\begin{bmatrix} -1/\\sqrt{2} \\\\ 1/\\sqrt{2} \\end{bmatrix} , \\begin{bmatrix} -1/\\sqrt{2} \\\\ -1/\\sqrt{2} \\end{bmatrix} \\right\\rbrace .$ The mapping between information bits and signals is illustrated in Fig.",
"REF (a).",
"Due to the additive noise, the received signal $r$ can take any value, but the Gaussian distribution makes values close to one of the signal points in (REF ) more likely than values far away.",
"This can be seen from the red dots in Fig.",
"REF (a), which represent $r$ for 10,000 noise realizations with $\\sigma ^2 = 0.2$ that are added to each signal point.",
"Figure: We send QPSK signals over an AWGN channel, as shown in (a), and try to detect the signals at the receiver.",
"The detection regions produced by a trained neural network is shown in (b) and the optimal regions obtained from detection theory are shown in (c).Based on the received signal $r$ , the receiver needs to guess (detect) what signal $s$ was sent.",
"We have trained a neural network for this task, by taking the received signal $x_0=r$ as input and letting the output $y$ be a four-dimensional vector that is one for the detected signal and has zeroes elsewhere.",
"We used the 40,000 red dots in Fig.",
"REF (a), and the signals $s$ that generated these $r$ , to train a fully-connected neural network using standard training methods.",
"We then applied the neural network to a wide range of possible received signals to illustrate how it is making its detection.",
"The colored areas in Fig.",
"REF (b) show in which regions the received signals are mapped to the respective information signals.",
"The regions are separated by lines, which is expected since each layer performs linear algebra operations; in particular, each activation function determines if the input is below/above a line that has been selected by training.",
"Note that we have “zoomed out” and the range of values that was shown in Fig.",
"REF (a) is indicated by the black square.",
"The colored detection regions produced by the neural network have peculiar asymmetric shapes, which are not optimal.",
"In fact, the optimal detection regions for AWGN channels are well known [3]: the received signal should be mapped to the closest signal point in terms of Euclidean distance.",
"The optimal detection regions are shown in Fig.",
"REF (c).",
"The regions are quite similar within the black square, but greatly deviates further away.",
"Several important observations can be made from this example: If there is a known optimal algorithm, a trained neural network cannot outperform it.",
"The detection error probability is, however, almost the same in this particular example since most received signals appear within the black square where the neural network has a decent behavior.",
"There are two reasons why the detection regions in Fig.",
"REF (b) are wrongly shaped.",
"Firstly, the shape inside the black square (around the signal points) is wrong due to overfitting; the training examples in Fig.",
"REF (a) can be approximately separated by many piecewise linear boundaries, including the ones shown in Fig.",
"REF (b).",
"Secondly, since all training examples are inside the square, the behavior outside the square is somewhat random; the neural network has learned to interpolate between training examples but not to extrapolate outside the square.",
"This is a practical issue since received signals far outside the square occasionally appear due to the long-tailed Gaussian distribution.",
"This is a general phenomenon; neural networks are good at handling typical inputs but may generalize poorly to atypical inputs.",
"We could have used prior domain knowledge (from digital communications) to preprocess the input signals.",
"In this example, the neural network had to rediscover where the constellation points are, how the noise is distributed, and how to make the right detection.",
"If we would instead compute the Euclidean distance between the received signal and each of the four signal constellation points, we could use that as input to a neural network.",
"This will give more accurate and reliable results since we have utilized our domain knowledge to ensure that the neural network has fewer characteristics to learn.",
"However, it still cannot beat the optimal detection."
],
[
"Is There a Role of Deep Learning in Communications?",
"Since signal detection in AWGN channels is easy to perform optimally, it makes little sense to utilize artificial neural networks for that purpose.",
"There are many similar tasks in communications where deep learning cannot make any meaningful improvements.",
"For example, the fundamental performance limits were derived by Shannon [4] and we can operate close to those limits using modern channel codes.",
"Moreover, it is known how to perform optimal channel estimation, multi-user multiple-input multiple-output (MIMO) processing, and transmit power allocation in many wireless communication scenarios [5].",
"The fact that the information signals are man-made gives us strong prior information that makes it easier to devise effective man-made algorithms than in many other fields, where the signals are created by nature.",
"There are nevertheless some important roles that deep learning can play in communications.",
"Firstly, there are many problems where a known algorithm finds the optimum but has prohibitively high complexity for real-time implementation.",
"Secondly, there are cases where the standard system models are inadequate or incomplete.",
"It is sufficient to replace the noise distribution in the previous example with an unknown one to find a case where learning can help.",
"We will elaborate on these two applications in the remainder of this lecture note.",
"But before that, we stress that errors are unavoidable in the physical layer of communication systems and are conventionally dealt with using retransmissions.",
"This built-in fault tolerance is positive when it comes to the utilization of deep learning.",
"It gives robustness to the strange behaviors that occasionally occur when an atypical signal is fed into a neural network that has been trained to work well for typical input signals.",
"However, adversaries can also exploit atypical signals to perform jamming more efficiently [6]."
],
[
"Application 1: Algorithmic Approximation",
"The first important application of deep learning in communications is to approximate a known but computationally complicated algorithm.",
"There are many examples of iterative algorithms that asymptotically find a global (or local) optimum to an optimization problem, but require very many iterations for convergence and/or complicated operations in each iteration [7].",
"Such algorithms might not be practically useful in communication systems where latency constraints require execution times below a millisecond.",
"Figure: A known algorithm ff can be approximated by training a neural network f ^\\hat{f} to make f(x)≈f ^(x;θ * )f(x) \\approx \\hat{f}(x;\\theta ^*) for all possible inputs, as shown in (a).",
"The training procedure will iteratively update θ\\theta to gradually reduce the approximation errors until it converges to some θ * \\theta ^*.If the neural network is designed to have sufficiently low complexity, then the trained neural network in (b) can be used in real-time applications.The general procedure for training a neural network for algorithmic approximation is illustrated in Fig.",
"REF .",
"Suppose we have a known algorithm, represented by the function $y=f(x)$ , which cannot be implemented in real time.",
"To address this problem using deep learning, we can first create a training set containing a large number $T$ of input signals $x_t^{\\mathrm {train}}$ , for $t=1,\\ldots ,T$ .",
"We then run the algorithm $T$ times to compute the outputs $ y_t^{\\mathrm {train}} = f(x_t^{\\mathrm {train}}).$ After having generated the training set, we can train an artificial neural network to provide approximately the same outputs for these inputs.",
"More precisely, we should find an optimized parameter vector $\\theta ^*$ in accordance to (REF ).",
"If the training is performed well, the neural network will generalize well (i.e., provide good outputs) to previously unseen input signals that were generated in the same way as the inputs used for training.",
"Simply speaking, this means that $f(x) \\approx \\hat{f}(x;\\theta ^*)$ for all inputs $x$ of practical interest.",
"There are many optimization problems to be solved in communication systems.",
"For example, at the transmitter, power allocation between concurrent transmissions is important to limit interference [5], [7].",
"At the receiver, non-linear signal detection problems must be solved to deal with interference in MIMO systems [8].",
"Some of these problems are convex and can be solved by off-the-shelf optimization software.",
"Other problems are non-convex but there exist iterative algorithms that converge to local or global optima.",
"In both cases, the computational complexity is often prohibitive for real-time applications, where similar optimization problems with different input data are solved repeatedly.",
"A neural network can then be trained to learn approximately how the solution depends on the input data.",
"This approximate input-output map can be evaluated with substantially lower computational cost, as exemplified in [7], [8].",
"Domain knowledge can be utilized to pre-process the input data, to focus the learning on the problem that the algorithm is solving and not on rediscovering known properties (e.g., that the desired signal lies in a certain subspace).",
"There are two main approaches.",
"One can learn the input-output mapping based on training data, as described above, while ignoring how it was produced.",
"Alternatively, the shape of the neural network can be selected so that each layer mimics one iteration of a known algorithm that converges asymptotically to an optimum.",
"This is called deep unfolding and exploits that many first-order iterative optimization methods have the same structure as a (recurrent) neural network [9].",
"The parameters of the neural network are then trained to give a nearly optimum solution after a predefined number of iterations, thereby speeding up the convergence.",
"In [8], the authors “unfold” a gradient-descent-like algorithm for MIMO detection to create a neural network where each layer performs similar operations but with optimized parameters.",
"When using this approach, (REF ) needs not to be determined in advance, which simplifies the training.",
"The practical benefit of this application is the complexity reduction it can provide; the neural network will essentially learn how to make algorithmic shortcuts to strike a good balance between accuracy and computational complexity.",
"Another important benefit is related to hardware implementation.",
"To solve a practical problem with real-time constraints, we conventionally would first need to design an algorithm and then develop a dedicated circuit based on it, which can be very time-consuming.",
"With the help of deep learning, we can instead predesign a general-purpose circuit that implements a neural network of a given maximum size (i.e., number of layers and neurons) with a predetermined run time.",
"We can then train a neural network to perform the algorithmic task we need and, finally, load the corresponding trained parameters (i.e., weights and biases) onto the circuit.",
"This new approach to hardware implementation can greatly reduce the time from that the algorithmic design begins to a product can hit the market.",
"A main issue with this application is the highly computationally demanding generation of desired outputs: the more complex the algorithm $f$ is, the longer time it takes to compute $f(x_t^{\\mathrm {train}})$ for $t=1,\\ldots ,T$ .",
"We are basically moving the complexity issue from the algorithmic run time to the design process.",
"There is a practical limit to which algorithms that we can approximate in this way.",
"If it takes 1 hour to generate one training example, it will take 11.4 years or extreme parallelism to generate 100,000 examples."
],
[
"Application 2: Inversion of an Unknown Function",
"The second important application is to invert an unknown function.",
"In particular, non-linear distortion can occur between the transmitter and receiver.",
"Three prominent examples are finite-resolution quantization in the receiver hardware, non-linear amplifiers in the transmitter hardware [10], and non-linear fiber-optical channels [11].",
"While quantizers typically are designed with known properties, the latter two examples can be represented by an unknown function $g$ that takes a signal $y$ as input and produces a distorted output $x=g(y)$ .",
"The conventional way to undo the distortion is to identify an appropriate parameterized model of the function, then estimate the parameters from measurements, and finally create an inverse function based on the estimates.",
"This three-step approach is suboptimal and prone to error-propagation.",
"An alternative is to train a neural network to directly invert the function, without explicit modeling or parameter estimation.",
"Whenever only suboptimal conventional algorithms exist, a learned algorithm can theoretically provide better performance and robustness, but only if the training is carried out successfully.",
"Figure: An unknown function gg with input yy is inverted using a neural network f ^\\hat{f} by training it to achieve f ^(g(y);θ * )≈y\\hat{f}(g(y);\\theta ^*) \\approx y, as shown in (a).The training procedure will iteratively update θ\\theta to gradually reduce the approximation errors until it converges to some θ * \\theta ^*.The trained neural network in (b) can be used to counteract the unknown function, without having to explicitly model it and estimate model parameters.The general procedure for training a neural network for function inversion is illustrated in Fig.",
"REF .",
"We need to generate a large number $T$ of possible communication signals $y_t^{\\mathrm {train}}$ and send them through the unknown function to measure $x_t^{\\mathrm {train}} = g(y_t^{\\mathrm {train}}) \\quad \\textrm {for } t=1,\\ldots ,T.$ It is then $x_t^{\\mathrm {train}}$ that is used as input to the neural network, while $y_t^{\\mathrm {train}}$ is the desired output.",
"Different from Application 1, the creation of a training set can be very computationally efficient in Application 2 because the outputs are man-made.",
"It is typically created to be statistically equivalent to the signals observed at run time, but one can also create a biased training set to emphasize typical or atypical examples.",
"Online learning when operating the communication system is possible by occasionally sending predefined reference signals to generate new training data.",
"This is useful when the function $g$ is time-varying (e.g., due to temperature variations in the hardware).",
"The key to successful utilization of deep learning is to identify tasks in communication systems that currently lack an optimal solution—there is then an opportunity to beat the state-of-the-art.",
"For example, a common way to deal with non-linear communication hardware is to apply the Bussgang decomposition [12] to write the output of the non-linear function $g$ as $g(y) = D y + n$ , where $D $ is a deterministic matrix and $n$ is distortion noise that is uncorrelated with $y$ but statistically dependent.",
"By pretending as if $n$ is independent noise, one can often develop communication algorithms (e.g., for channel estimation or data detection) that partially mitigate distortion, but such algorithms are suboptimal since the distortion is in fact dependent on the input.",
"As shown in [10], one can achieve substantially better performance by training neural networks instead."
],
[
"What We Have Learned",
"Although many parts of communication systems can be solved optimally, there are important cases where deep learning can give large improvements.",
"In particular, it can be used to reduce computational complexity of known algorithms or to deal with non-linear hardware or channels in an efficient way."
],
[
"Acknowledgment",
"This work was supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation."
],
[
"Authors",
"Emil Björnson ([email protected]) received the MSc degree in engineering mathematics from Lund University, Sweden, in 2007, and the PhD degree in telecommunications from the KTH Royal Institute of Technology, Sweden, in 2011.",
"He is now an associate professor at Linköping University, Sweden.",
"He has authored the textbooks Optimal Resource Allocation in Coordinated Multi-Cell Systems (2013) and Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency (2017).",
"He received the 2018 IEEE Marconi Prize Paper Award in Wireless Communications, the 2019 EURASIP Early Career Award, the 2019 IEEE Communications Society Fred W. Ellersick Prize, and the 2019 IEEE Signal Processing Magazine Best Column Award.",
"Pontus Giselsson ([email protected]) is an Associate Professor at the Department of Automatic Control at Lund University, Sweden.",
"His current research interests include mathematical optimization and its wide range of applications, e.g., in machine learning, control, signal processing, and wireless communication.",
"He received an MSc degree from Lund University in 2006 and a PhD degree from Lund University in 2012.",
"During 2013 and 2014, he held a postdoc position at Stanford University.",
"In 2012, he received the Young Author Price at the ADCHEM IFAC Symposium, in 2014, he received the Young Author Price at the IFAC World Congress, and in 2015, he received the Ingvar Carlsson Award from the Swedish Foundation for Strategic Research."
]
] | 2001.03350 |
[
[
"A Complex Networks Approach to Find Latent Clusters of Terrorist Groups"
],
[
"Abstract Given the extreme heterogeneity of actors and groups participating in terrorist actions, investigating and assessing their characteristics can be important to extract relevant information and enhance the knowledge on their behaviors.",
"The present work will seek to achieve this goal via a complex networks approach.",
"This approach will allow finding latent clusters of similar terror groups using information on their operational characteristics.",
"Specifically, using open access data of terrorist attacks occurred worldwide from 1997 to 2016, we build a multi-partite network that includes terrorist groups and related information on tactics, weapons, targets, active regions.",
"We propose a novel algorithm for cluster formation that expands our earlier work that solely used Gower's coefficient of similarity via the application of Von Neumann entropy for mode-weighting.",
"This novel approach is compared with our previous Gower-based method and a heuristic clustering technique that only focuses on groups' ideologies.",
"The comparative analysis demonstrates that the entropy-based approach tends to reliably reflect the structure of the data that naturally emerges from the baseline Gower-based method.",
"Additionally, it provides interesting results in terms of behavioral and ideological characteristics of terrorist groups.",
"We furthermore show that the ideology-based procedure tends to distort or hide existing patterns.",
"Among the main statistical results, our work reveals that groups belonging to opposite ideologies can share very common behaviors and that Islamist/jihadist groups hold peculiar behavioral characteristics with respect to the others.",
"Limitations and potential work directions are also discussed, introducing the idea of a dynamic entropy-based framework."
],
[
"Introduction",
"Complex networks have demonstrated their potential in many different domains.",
"Approaches that rely on dynamic, multi-mode, multi-partite and meta-networks have been fruitful in shedding light on a wide variety of phenomena, including social ones [3], [16], [56], [18].",
"In the last years, this process has indeed also touched areas as criminology, international security, and terrorism research [22], [6], [39] .",
"This methodological shift has been facilitated by the increasing availability of open access data sets, the sensibility and interest of social scientists towards novel empirical approaches and the dramatic popularity of statistical software and data-science oriented languages.",
"In spite of this shift, several critical points and pitfalls have been highlighted by scholars regarding the actual results of scientific inquiry in the field of terrorism research.",
"Sageman [51], for instance, noted that the lacking collaboration between intelligence and academia led to a stagnation that is mainly motivated by the scarcity of rich, detailed and precise data on terrorist groups and events, which makes it difficult for researchers to develop models that are actually useful in reducing or assessing the terrorist threat.",
"In fact, Sageman argued that the intelligence community should be more willing to share crucial and rich data sets to the academia, in order to exploit their methodological rigour and capabilities.",
"Recently, in an attempt to extensively review the field of terrorism research, Schuurman [52] noted that many longstanding weaknesses and issues have been either completely or partially solved (e.g.",
"scholars have expanded the range of data gathering techniques), while at the same time other issues are still in place.",
"Among the others, scarcity of international and interdisciplinary collaborations and the high number of one-time contributors are preventing the field to develop in a more structured direction, therefore limiting the probability for high-impact and practical research.",
"In spite of these structural limitations, we seek to demonstrate the potential capabilities of complex networks to highlight hidden patterns within the terrorist global scenario, with the final aim to stimulate the debate on the application of novel methodological frameworks to research on terrorism.",
"Hidden patterns could highlight operational similarities between groups that do not share any ideological background, peculiar attack-planning characteristics related to actors operating in certain areas, or even relevant behavioral differences between groups that fight for similar motivations but are settled in distinct regions.",
"Our intuition is that there is first and foremost a need for advancing and experimenting novel methodological approaches that, in case of promising results, might be employed and applied to other contexts with more reliable data, allowing to draw more useful and solid conclusions.",
"In fact, the unavailability of and search for better data should not stop the process of innovation within the field.",
"This works relies on data retrieved from the Global Terrorism Database (GTD henceforth) on terrorist attacks occurred at global level from 1997 to 2016.",
"The paper aims to propose a new algorithm for detecting latent clusters of terrorist groups expanding and extending the analytic approach we have provided in [15]: we demonstrate that, tested against our previous approach (which we will refer to as “baseline” throughout the paper) and a weak heuristic approach based on pure groups ideology, our novel algorithm confirms many results obtained with the baseline approach and also provides new interesting results on the hidden similarities across groups belonging to very different contexts and motivations.",
"The paper is organized as follows: the next section provides a review of network- and clustering-based approaches to the study of terrorism, trying to identify the main areas of application in which these methods have been experimented.",
"Following, the “Data” section will thoroughly present the information contained in the data employed to conduct our analyses.",
"The “Methodology” section will describe and explain the graph construction and algorithmic framework.",
"The “Results” section will then describe relevant outcomes and patterns found after the experiments.",
"Finally, “Discussion and Future Work” section will focus on the potential implications of this work, on its limits and on the possible directions that can be explored using this paper as a starting point.",
"In recent years, one of the methodological frameworks that have been tested and have attracted the attention of both scholars and policymakers in the social sciences is network science, broadly intended [8].",
"Network science has gained popularity in sociology [29], [33], [19], economics [53], [31], [59], [11], political science [32], [60], [27], [49] and criminology [43], [45], [1], [12], [23].",
"This also applies to terrorism research.",
"The first application of social network analysis to terrorism, the branch of network science that specifically seeks to map and study relation between human entities such as people or organizations, was the renowned paper by Krebs [36].",
"Krebs tried to understand the existing connections between hijackers and terrorists responsible for the 9/11 attacks using unstructured data retrieved from open access sources, as newspapers.",
"Despite its limited sophistication, the study opened a path towards the study of terrorism under this new perspective.",
"Following this strategy, other scholars have used relational data on individuals to reconstruct terrorist networks and investigate roles and key players [35], [10], [4].",
"Shifting from the pure physical and relational information gathered and structured to investigate the structure of groups, scholars have also tested and simulated the strength and resilience of terrorist networks.",
"As an example, works in this area have put a strong emphasis on the application for intelligence purposes.",
"They have relied on mathematical models focusing on network topology for either proposing methods for maximizing efficiency in disruption strategies [17], [40], [25], [48] or understand the most resilient topology structures to be learned from terrorism behavior and applied to other domains (e.g.",
"infrastructure networks) [30].",
"This interest towards increasingly complex questions regarding the nature and behavior of terrorist networks encourages scholars and scientists to integrate relational and topological information on networks with their spatial and temporal dynamics.",
"Spatial and temporal dynamics are crucial when aiming at understanding the evolution of a certain entity or phenomenon, therefore several works have focused on these aspects using either synthetic-generated data or real-world information on existing networks [42], [41].",
"In the meanwhile, the revolution of social media has provided an unprecedented and massive amount of data to study the online social behaviour of people.",
"As for the real physical world, individuals act criminally or violently also within the internet, and therefore researchers have started to be attracted by the potential consequences of criminal, and even terrorist, behaviors in the cyberspace.",
"Indeed, a recent stream of research has focused on the detection of terrorist or radical behaviors retrieving network-information from social media platforms.",
"Social media allow to go beyond pure relational information, integrating instead geographical, temporal features and many other profile attributes to infer patterns and dynamics of extremist users [9], [20], [34], [5].",
"The previous lines of research, though generally different in their data gathering techniques, modelling architectures and complexity scales, all mostly focus on mapping relations between individuals or, at most, organizations belonging to the same terrorist sphere (e.g.",
"the al Qaeda network).",
"However, a very recent sub-domain explored the power of the complex network landscape when dealing with event data and abstract meta-networks of attack characteristics, with the aim to predict future terrorist behaviours in terms of target or weapon selection, targeted locations and employed tactics [24], [58], [13] or, more broadly, to highlight operational similarities between different terrorist organizations [15], [14].",
"While network approaches for modelling terrorism have gained a certain degree of success and have tested and experimented techniques focusing on a variety of research questions, it is worth to note how this advancements have not been followed by the consequent combination of network science with unsupervised learning and, more specifically, cluster analysis.",
"In one of the first attempts at using cluster analysis to group terrorist organizations, Chenoweth and Lowham [21] used data on groups which targeted American citizens to explore alternative ways to conceive terrorist typologies.",
"Qi et al.",
"[47] used both social network analysis and unsupervised learning to group extremist web pages using an hierarchical multi-membership clustering algorithm based on the similarity score of these pages.",
"Finally, Lautenschlager et al [38] developed the Group Profiling Automation for Crime and Terrorism (GPACT) prototype that generates terrorist group profiling via a multi-step methodology that also includes clustering of terrorist events.",
"In light of this gap in research, following the intuition that network science may provide rich insights on the terror phenomenon, we modify our previous proposed methodology to test the performance of an automatic weighting scheme for the Gower's coefficient of similarity based on von Neumann's Entropy to preserve intrinsic qualities of the data that already emerge from our baseline approach.",
"This work relies on data retrieved from the Global Terrorism Database (GTD) [37], [57].",
"The GTD is the most comprehensive and detailed open access dataset on terrorist events at global scale, maintained by the START research center.",
"Information are gathered from different open sources, and events have to meet specific criteria to be included in the database.",
"These criteria are divided into two different levels.",
"The first level criteria are three and have all to be verified.",
"These mandatory ones are related to (1) intentionality of the incident, (2) presence of violence (or immediate threat of violence) of the incident and (3) to the sub-national nature of terrorist actors.",
"The second level criteria are three and at least two of them must be respected.",
"Second level criteria relate to (1) the specific political, economic, religious or social goal of each act, (2) the evidence of an intention to coerce, intimidate or convey messages to larger audiences than the immediate victims, (3) the context of action which has to be outside of legitimate warfare activities.",
"Finally, although an event respects these two levels, an additional filtering mechanism (variable doubter) controls for conflicting information or acts that may not be of exclusive terrorist nature [55].",
"For our analysis, we aggregated data (i.e., we did not separate by year or other time windows) from 1997 to 2016 on worldwide events and related perpetrators, excluding all the attacks which were of doubtful terrorist nature.We have also excluded attacks from 1970 to 1996 because, as reported in the official GTD codebook, many variables on attacks occurred prior to 1997 were not available or sufficiently reliable.",
"This methodological choice led from 106,114 events to a total of 88,513.",
"Furthermore, we have removed all the events plotted by “Unknown\" actors.",
"Considering the large amount of attacks with no identified perpetrator, we would have faced the risk of biased results.",
"We have thus kept only attacks of clear terrorist nature with an identified author, accounting for a total of 41,456 events.",
"The multi-partite network which has been created and employed for our study relied on six main terrorist dimensions, namely: Events ($N$ =41,456), Groups ($N$ =1,493), Targets ($N$ =22)Targets list includes: Abortion Related, Government (General), Private Citizens & Property, Business, Religious Figures/Institutions, Police, Airports & Aircraft, Utilities, Educational Institution, Unknown, Journalists & Media, Government (Diplomatic), Other, Military, Telecommunication, Tourists, Terrorists/Non-State Militia, Transportation, NGO, Violent Political Party, Maritime, Food or Water Supply., Weapons ($N$ =13)Weapons list includes: Incendiary, Explosives/Bombs/Dynamite, Firearms, Unknown, Melee, Fake Weapons, Chemical, Other, Sabotage Equipment, Vehicle (not to include vehicle-borne explosives, i.e., car or truck bombs), Biological, Tactics ($N$ =9)Tactics list includes: Facility/Infrastructure Attack, Bombing/Explosion, Armed Assault, Unknown, Assassination, Hostage Taking (Kidnapping), Unarmed Assault, Hijacking, Hostage Taking (Barricade Incident) and operating Regions ($N$ =12) Operating Region list covers the entire world and specifically includes: North America, South Asia, Middle East & North Africa, Sub-Saharan Africa, Western Europe, Eastern Europe, South America, Southeast Asia, East Asia, Central America & Caribbean, Australasia & Oceania, Central Asia.",
"These dimensions have been chosen because they represent the visible core of terrorist activity: the terror attack itself can indeed be represented by its perpetrator, the chosen target, the employed weapons and tactics and the geographic and political context in which it occurred.",
"These variables are thus helpful in gathering a rich knowledge structure that will then be crucial for our methodology.",
"In addition to this information which represent the basis of this work, other variables extracted from the GTD and other sources have been employed to detect and assess behavioral patterns of terrorist groups belonging to the same clusters ex post.",
"This information will include group-based attributes regarding terrorist activity such as ideology, success rate, suicide rate, fatality rate, casualty rate, multiplot rate, international rate and number of targeted countries.",
"The ideology of each group has been mapped using existing information present in two open access data sets (Big Allied and Dangerous 1 and an extraction of Big Allied and Dangerous 2) when that information was available within those sources [2], and by exception from other qualitative open access information sources.",
"This mapping led to include seven ideology categories: (i) Islamist/Jihadist groups, (ii) Far Left/Anarchist/Communist (FL), (iii) Far Right/Racist/Nazi (FR), (iv) Ethno-Nationalist, (v) Other/Unknown, (vi) Religious (Islam excluded), (vii) Animal-rights/Environmentalist.",
"A given group may belong to more than one category at a time (e.g.",
": the Popular Front for the Liberation of Palestine which contains at the same time elements of Marxism and Nationalism) (Table REF ).",
"It is worth to specify that these are labels that aim at giving context regarding the main motivations and ideological positions of the groups.",
"This of course does not imply that enviromentalism, for instance, has to be associated with terrorism per se.",
"These categories only mean that a given group that has plotted at least one attack included in the GTD had motivations and roots that can be matched with a given ideology.",
"The same applies to left-wing or right-wing organizations: having a particular political position does not automatically qualify an existing entity (either a person or a group) as terrorist.",
"However, there are diverted and extremist positions on both political sides that are tightly connected with groups and actors that have been responsible of terrorist acts.",
"Table: Descriptive Statistics of Group IdeologiesThe success share is given by the ratio between the successful attacks and the total number of events attributed to a given group.",
"The suicide share maps the ratio of suicide attacks over the total number of events plotted by the same group.",
"Fatality and casualty ratios are produced by the number of attacks with at least one dead victim (fatality) or one wounded victim (casualty) divided by the total number of events.",
"The international rate is simply the ratio between attacks with some international features (e.g.",
"logistic organization) and the total number of attacks.",
"Finally, multiplot share quantifies the share of attacks that were part of a coordinated strategy (e.g.",
"9/11 case), out of total attacks.",
"All these variables seek to enrich the knowledge associated to each group and to understand whether the identified clusters highlight certain patterned and eventually unexpected behaviors (Table REF ).",
"Table: Group-based Attributes on Terrorist Activity - Descriptive StatisticsAt the general level, the entropy-based approach that is presented and analyzed in this work is structured as follows: (i) calculation of the weights of each mode using the graph entropy of that mode; (ii) computation of the weighted Gower's Coefficient of Similarity between each of the terrorist groups using the entropy as the weight; (iii) extraction of the latent network from the pairwise Gower's Coefficient similarities and analyze its structural and intra-cluster properties.",
"The detailed process is described in the following subsections.",
"The entropy-based method will be then compared with the baseline model presented in [15] and with a heuristic method.",
"The baseline model uses a simplified version of Gower's method.",
"In this simplified version, no weights are applied to the different modes and we only consider the natural structure of the data deriving from the affinity matrix that originates by the pairwise Gower's coefficient of similarity.",
"In other words, instead of using the graph entropy of each mode as its weight, every mode is simply just given a weight of one.",
"The heuristic method we use for comparison only uses groups' ideologies as the clustering criterion.",
"In this method, we just use the dominant ideology of a particular terrorist group as its cluster label.",
"So, if two groups share a dominant ideology, like Ethno/Nationalist, then they are in the same cluster.",
"Since the variables of the modes of Targets, Weapons, Tactics, and Regions form a many-to-many relationship with the terrorist groups, we first model this data as a multi-partite network (Figure REF ) with each partition joined to the terrorist groups; this is often referred to as a `Star' structure with the partitions.",
"Figure: Example of a Multi-partite NetworkThe graphic illustrations shows a multi-partite network that includes two groups ii and jjMore specifically, we define: $\\mathfrak {G}^N:=\\langle \\left( V_{1},V_{2},\\cdots , V_{n} \\right), \\left( E_{1,2},E_{1,3}, \\cdots ,E_{m,n} \\right), \\left( W_{E1,2}, \\cdots ,W_{Em,n} \\right)\\rangle $ as a multi-partite graph that contains $N$ partitions describing relations between different sets of nodes $V_{m}$ and $V_{n}$ : these relations are formalized as edges $E_{m,n}$ that are weighted by $W\\in \\mathbb {\\mathbb {R}}_{\\ge 0}$ and each mode in the multi-partite network is represented as $G_{m,n}:=\\left\\langle \\left(V_{m},V_{n} \\right), E_{m,n}, W_{E_{m,n}} \\right\\rangle $ .",
"With this data structure we then employ Gower’s Coefficient of Similarity [28] to place the groups in a latent space, whereby we can create a latent network of the groups and assign groups to clusters based upon the multi-partite network.",
"In the latent network, the edges maps the similarity between group $i$ and $j$ , calculated using Gower’s Similarity Coefficient defined as: $S_{ij}=\\frac{\\sum _{k=1}^{n}w_{ijk}S^{(k)}_{ij}}{\\sum _{k=1}^{K}w_{ijk}}$ where $S_{ij}$ is the similarity between terrorist groups $i$ and $j$ on a variable (i.e.",
"Targets, Weapons, etc.",
"), $k$ , and $K$ is the total number of variables across all $N$ modes, and $w_{ijk}$ is the weight of the similarity between group $i$ and group $j$ for variable $k$ .",
"$S^{(k)}_{ij}$ is then dually defined as: $S^{(k)}_{ij}:\\left\\lbrace \\begin{matrix} 1, & if (x_{ik}=x_{jk}) \\ne \\emptyset \\\\ 0, & otherwise \\end{matrix}\\right.$ if the variable, $k$ , is categorical (to include binary) for node $i$ and $j$ 's responses, $x_{ik}$ , $x_{jk}$ , and: $S^{(k)}_{ij}: \\frac{\\left| x_{ik}-x_{jk} \\right|}{r_{k}}$ where $r_{k}$ is the range of $x_{k}$ , if $k$ is numerical.",
"For each variable, $k$ that is numerical the range is calculated as: $r^k = |max(x_k) - min(x_k)|$ which means that the range is given by the absolute value of the maximum value of the variable $k$ minus the minimum value of the variable $k$ .",
"Gower's coefficient of similarity provides a wide degree of flexibility as it can take various data types, like integer, binary, or continuous values and does so without the use of dummy variables.",
"So, with this coefficient of similarity we are able to incorporate various means of describing terrorist groups, which can be of nearly any data type, and do so in such a way that keeps the original structure of each of the modes as bipartite networks intact.",
"Another advantage of Gower's Coefficient is the weighting term.",
"As was noted at the beginning of this section, each of the possible variables used to find similarity between terrorist groups are not independent, but rather fall into various related modes.",
"For example, if group $i$ has operated in the Middle East, it is possible that it has also operated in North Africa or Southwest Asia.",
"Furthermore, since the relationships within the mode are many to many (e.g.",
"a terrorist groups can use many different weapons and vice-versa) each of these modes are a bipartite network.",
"Thus, our data is modeled as a collection of bipartite networks, where there are relationships between the entities within each of the mode networks.",
"So, to take advantage of this model of our data, we employed network entropy as the means to weight different modes for the weighting scheme in the Gower's Coefficient [46].",
"Von Neumann's network entropy is a spectral measure originating from Gibbs entropy that has been applied to the quantum realm and that provides information on the complexity of a graph and on the amount of information that a network contains.",
"In general, network entropy can be thought of as a measure of how heterogeneous a network is in terms of its connections [46], [26].",
"As such, network entropy has been used to characterize changes within dynamic graphs, as it is good as distinguishing different graph snapshots from each other [62].",
"Furthermore, network entropy has also been used a means of distinguishing certain graphs from each other, with those graphs that have a higher entropy having more complex structures like subgroups [46], [61].",
"So, we similarly employ network entropy to distinguish between the different modes, which are bipartite networks, in such a way that those modes that have more heterogeneous structures — and as such are likely to be better for separating terrorist groups into clusters — are considered as more important.",
"Since Gower's Coefficient allows for weighting for exactly the purpose of emphasizing more important features in data, we can use network entropy with the weighting term in Gower's Coefficient to automatically emphasize more useful modes of our data.",
"Following the derivations of the entropy of a network in [46] and [54], we define the entropies of each of our modes as: $H^{n} = -\\sum ^{|V|}_{i=1}\\frac{\\tilde{\\lambda _i}}{|V|} ln \\frac{\\tilde{\\lambda _i}}{|V|}$ where $\\tilde{\\lambda _i}$ are the eigenvalues of the normalized Laplacian of the graph of the particular mode.",
"So, for $n \\in N$ the normalized Laplacian is $\\tilde{L}^{n} = D^{-\\frac{1}{2}}(D-X^{n})D^{-\\frac{1}{2}}$ , where $(D-X^{n})$ is the unnormalized Laplacian and therefore $X^n$ is the adjacency matrix from a particular mode and $D$ is the degree matrix, which is created by: $D:\\left\\lbrace \\begin{matrix} \\sum _{j=1}^V X_{ij} & \\mathrm {if}\\; D_{ii} \\\\ 0 & \\mathrm {otherwise} \\\\ \\end{matrix}\\right.$ Following the findings on using network entropies to characterize heterogeneous graphs in [62] and [26], we let those modes with higher entropy have more impact on the similarity measurements through Gower's coefficient.",
"So, the weighting term in our Gower's coefficient is: $w_{ijk} = \\sum _n^N \\delta (k,n) \\times H^{n}$ where $\\delta (k,n)$ is an indicator function that returns 1 if variable $k$ is in mode $n$ , and 0 otherwise.",
"It should be noted that each variable within a mode will recieve the same weight.",
"So, those modes which have a more heterogeneous structure, which should be better for producing structures like clusters, will have a higher weight in the comparison of the various terrorist groups.",
"Having obtained pairwise similarities between all of the terrorist actors, we now move on to extracting a network from the data, which we refer to as the latent network.",
"Following our work in [15], we continue to use the kNN modularity maximization procedure proposed in [50].",
"At a a high level, once similarities have been computed for each of the terrorist groups, the method iterates through various possible numbers of neighbors for each node, $k$ , and selects that $k$ which produces the most modular graph, relative to a null-model, random graph produced on the same similarities.",
"Modularity in this case is the network modularity as described in [44]: $mod(G) = \\frac{1}{2m}\\sum _{ij}[ A_{ij} - \\frac{deg(i)\\times deg(j)}{2m} \\delta (c_i, c_j) ]$ where $m$ is the number of links in the network, and $c$ are the cluster assignments of the nodes.",
"A graph with high modularity is one which will have sub groups that have a lot of interconnections.",
"Since it is known that random graphs can give rise to modular structures, we also compare this modularity value to the modularity value obtained from a random graph with the same number of vertices and edges, and the same similarities between the vertices [50].",
"The general idea behind this method is that a kNN that is higher in modularity, relative to a null-model, is better for detecting community structure in the underlying data used to make the network.",
"It should be noted that this procedure applies only after a measure of similarity has been applied to the data.",
"We have, however, modified the algorithm slightly to better suit our data.",
"First, we use an asymetric kNN network.",
"More precisely, for each point $i$ , let $N_k(i)$ be the $k$ nearest neighbors of $i$ , then an asymmetric kNN network has links between two nodes $i$ and $j$ if $i \\in N_k(j)$ OR $j \\in N_k(i)$ .",
"Second, in the clustering step, we differ from the original algorithm proposed in [50], as we use a faster method of modularity maximization of unimodal networks, the Louvain Method [7], as opposed to the author's QCut algorithm.",
"The psuedo-code of our implementation of network construction by kNN modularity maximization is detailed in algorithm REF .",
"kNN Modularity Maximization Procedure input: Distance or Affinity Matrix, $S$ , (n x n) output: Optimal k-Nearest Neighbor Network, $G^*$ , and sub group assignments $C(G^*)$ .",
"$i=1 : \\left\\lfloor {log_2 (n)}\\right\\rfloor $ $k \\leftarrow 2^i$ $G_k \\leftarrow kNN(S,k)$ $C(G_k) \\leftarrow Louvain(G_k)$ $G^r_k \\leftarrow randomize(G_k)$ $C(G^r_k) \\leftarrow Louvain(G^r_k)$ $Modularity_k \\leftarrow Modularity(C(G_k)) - Modularity(C(G^r_k))$ $k^* \\leftarrow argmax_k Modularity_k$ $G^* \\leftarrow kNN(s, k^*)$ $C(G^*) \\leftarrow Louvain(G^*)$ $C(G^*)$ , $G^*$ In Algorithm REF , $G_k$ is a particular kNN graph, where each vertex connects to exactly k of its nearest neighbors.",
"The sub-step of $randomize(G_k)$ is to randomly re-wire all of the edges in $G_k$ .",
"This is equivalent to creating an Erdos-Renyi random graph that has the same number of edges and vertices as $G_k$ .",
"This step is performed in order to create a null-model of $G_k$ , so that we can get a better idea of the strength of the modularity of the proposed $G_k$ by comparing it to the modularity of its null counterpart, $G^r_k$ .",
"So, a good kNN should not just have high modularity, but also high modularity with respect to a randomized version of that kNN; the modular structures should not be just an artifact of the kNN's density or size.",
"Finally, we return $G^*$ which is that kNN which has the most modular structure.",
"A Python implementation of the code will be available on the author's GitHub page with publication of this article.",
"To compare the similarity of clustering assignments of the three grouping procedures we have calculated the Adjusted Mutual Information (AMI), which is a modified version of the ordinary mutual information adjusted for randomness, and it is calculated as $AMI(U,V)=\\frac{MI(U,V)-E\\left\\lbrace MI(U,V) \\right\\rbrace }{max\\left\\lbrace H(U), H(V) \\right\\rbrace -E\\left\\lbrace MI(U,V) \\right\\rbrace }$ where $E\\left\\lbrace MI(U,V) \\right\\rbrace $ is the expected mutual information between two random clusterings and $H(U)$ and $H(V)$ are the entropies associated to each partition $U$ and $V$ .",
"This is a standard way to measure how different or similar are the outcomes of clustering procedures.",
"The results are displayed in Figure REF .",
"Figure: Adjusted Mutual Information of three Subrgrouping Schemes.The ideology-based heuristic approach produces subgroupings which are extremely different when compared with the baseline and the entropy-based approachThe results clearly show how, with respect to the baseline unweighted model, subgrouping using entropy weighting is far more similar than the ideology-based subgrouping.",
"This, on one hand, suggests that relying only on this latter heuristic can extremely underestimate and distort the latent similarity that exist across groups when fully considering behavioral or operational variables.",
"While ideology is certainly important for contextualizing a certain terrorist actor, the multi-partite original network includes information that are not captured by this method.",
"On the other hand, the entropy-based weighting is able to capture a relatively high portion of the information associated with the unweighted baseline model.",
"While certainly introducing this data-driven discriminatory procedure affects the final clusters, our first conclusion is that this method is far more reliable if we want to preserve the original information structure of our data.",
"Moreover, this algorithmic approach might provide a more solid tool to analysts and policymakers if they need to go beyond the original data, exploiting the richness of the original data itself.",
"Given these results, we now proceed to compare the baseline and the entropy-based clusterings more in depth, to understand whether, besides pure group assignments, they also share stable similarities in terms of behavioral and ideological features.",
"Our algorithmic procedure to create the k-Nearest Neighbor network yielded two different graphs.",
"Focusing on global characteristics of both networks, we can highlight how these graphs hold distinct structural and topological characteristics (Table REF ).",
"Table: Network structural and topological characteristics for both approachesThe table above highlights how the $k$ -NN procedure provided two different optimal k's for the two networks.",
"The entropy-based network has a higher k, and this justifies the higher number of links (both overall and bi-directional), and the higher network density.",
"Nonetheless, the unweighted network proves to be higher in clustering coeficient, betweenness centralization and eigenvector centralization, while the entropy-based one yielded higher total degree centralization.",
"With regard to modularity, the unweighted network notably performs a higher value, suggesting that the emerging clusters are more defined than the ones yielded by the entropy-based approach.",
"Connected to this aspect, and with regard to the actual subgroupings, is the fact that the entropy-based procedure produced less clusters (21) compared to the unweighted one (37).",
"As Figure REF shows, the entropy-based approach produces a greater number of highly populated clusters, while in the unweighted case, a considerable amount of clusters includes a little number of groups (in fact, 25 clusters include less than 50 terrorist groups each).",
"These figures further justify the different scores in term of modularity, since a higher number of smaller clusters is highly likely to indicate a higher degree of diversity in the network itself, as captured by modularity.",
"Figure: Distribution of Cluster Assignments per Different ApproachThe unweighted approach produces a higher number of small clusters, highlighting a higher modularity.Focusing on the different node-level measure distributions, Figure REF displays histograms and 2D Kernel Density Estimations (KDE) of three selected metrics, namely Log Unscaled Total Degree Centrality, Log Betweenness Centrality and Clustering Coefficient.",
"Total Degree Centrality and Betweenness Centrality have been transformed in log scale in order to provide more intuitive graphic results, since the original distributions are extremely left-skewed and the bivariate visualizations would have been extremely difficult to interpret.",
"Figure: Histograms (20 bins) and 2D Kernel Density Estimation of Log Unscaled Total Degree Centrality, Log Betweenness Centrality and Clustering Coefficient of Terrorist Groups across both approaches2D Kernel Density Estimations are performed through regular grids made of 500 bins for each axis.Regarding Log Unscaled Total Degree Centrality, the histogram highlights how the groups in the Entropy-based approach generally have less connections that the ones in the Unweighted approach.",
"The 2D KDE displays a strong concentration of data points in the bottom-left side of the graph, with more dense concentrations.",
"With regards to Log Betweenness Centrality, the histogram shows relatively similar distributions for the considered approaches.",
"The Unweighted one displays a greater number of groups with betweenness equal to zero (log($n)\\approx -3).$When betweenness was equal to 0 we have transformed it to 0.001 in order to allow the value for log transformation.",
"This transformation did not affect the results since it was performed only to provide intuitive visualizations and interpretable bivariate relations The 2D KDE displays a very high concentration of nodes on the top right of the plot, showing a positive correlation of log betweenness centrality across nodes for both approaches.",
"However, it is worth noting that there is also an interesting small concentration of nodes that have very high value in the Entropy-based case but, conversely, very low ones in the Unweighted case.",
"Conversely, in relation to clustering coefficient, more evident differences emerge when looking at the histogram.",
"In fact, the Entropy-based approach shows a more concentrated distribution, while the Unweighted one highlights a very different behavior, with a considerably high number of extreme values, on both left and right side of the x-axis.",
"However, these differences are mitigated in the KDE plot.",
"Indeed, it shows the concentration of the majority of data points in the bottom left, almost indicating a linear relationship.",
"In spite of this, it is wort noting that there exist a portion of groups which obtain very high levels of clustering coefficient in the Unweighted case, while their corresponding values in the Entropy-based approach are significantly lower.",
"In light of the considerations on these detected differences in topology, structural and node-level measures of both networks and cluster formation, it is worth inspecting the types of groups in terms of operational and behavioral features and ideologies that are clustered together in both approaches.",
"The correlation results interestingly showcase that the majority of relations (either positive or negative) hold stably across both approaches, while only few have opposite directions from one approach to the other (Figure REF ).",
"Notably, correlations on events (first columns of both plots) are generally very different.",
"This may suggest that the raw number of events do not drive any consistent information flow regarding cluster assignments.",
"This would indicate that there are other types of features that actually capture similarities or differences across terror groups, and that the latent data structure is independent from the individual frequency of attacks of actors.",
"In terms of stable results, both approaches demonstrate how clusters with a high percentage of islamist or jihadist groups are associated with high levels of attack success, while this type of relation goes in the opposite direction for all the other ideologies.",
"This marks a distinctive feature of jihadism or islamism as terror motivation.",
"Expectedly, suicide attacks are also found to be positively correlated with jihadist or islamist ideology.",
"Furthermore, Islamist/jihadist ideology is again the only one positively associated with high levels of both casualties and fatalities in both approaches, while other ideologies seem to be less lethal.",
"In terms of the multiplot, which captures the extent to which a terrorist group is able to plot multiple coordinated attacks in the same day as part of a more complex logistic structure, FL groups, along with religious (non-Islamist) groups, are the only ideologies to display a stable, positive relation.",
"Figure: Correlation among behavioral and ideological variables Unweighted (left) and Entropy-based (right) approachesCorrelation shows what bivariate relations stay stable across both approaches (namely, they maintain the same direction).",
"Bottom plot highlights (in black) stable relations against non-stable (white).Clear results emerge when focusing on pairwise relations between ideologies (Figure REF ).",
"Overall, the majority of relations are stable across both algorithmic approaches.",
"Islamism has negative correlations in both cases with all the other ideologies, implying that groups belonging to or motivated by this ideology represent very distinguished entities in the global terrorist scenario.",
"With regard to FL groups, they strongly share similar cluster assignments with enviromentalist and animalist groups and, surprisingly, they share similar assignments also with FR actors.",
"This result suggest that, while these two ideologies are considered very different from one another and the motivations of groups belonging to these factions are extremely distant, from the operational point of view (namely, from the standpoint of employed weapons, hit targets, applied tactics and targeted regions) FL and FR terrorist groups are quite similar, and this result is corroborated by its stability across the two approaches.",
"FL groups are not the only actors that share cluster assignments with FR terrorists.",
"In fact, the analysis show that the higher the fraction of FR groups, the higher the number of religious (Islam excluded) and ethno/nationalist actors.",
"Figure: Kernel Density, Histogram and Scatter Plots of Cluster Distribution for Both Approaches - Ideological VariablesThe graphic visualizations demonstrates that more represented or well-defined ideologies tend to have stabler relations across both approaches.",
"Relation for environmental/animalist and religious (Islamist excluded) groups appear to be more volatile.These type of relations might be expected, considering that many FR groups include elements of bigotry or radical religious views and that many nationalist actors generally rely upon fascist or far-right discourse and political behaviours.",
"However, another surprising result is given by the negative relation between FL and ethno/nationalist groups, considering two factors: first, the positive stable correlation between FR and FL groups; second, the fact that radical leftist or communist ideologies are generally the other opposite driving force of certain nationalist or independence-driven actors, such as the Euskadi Ta Askatasuna (ETA) in Spain.",
"This means that, besides the potentially similar motivations and background, these two types of groups act and organize attacks that are generally dissimilar.",
"The comparative analysis across the two approaches demonstrated that the entropy-based approach preserves a relatively high amount of outcomes derived from the baseline model, also shedding light on additional mechanisms when focusing on the $ex$ -post analysis of behavioral and ideological features.",
"However, our experiment comes with a limitation that should be adressed in the future.",
"In fact, our original multi-partite network spans across twenty years and includes the cross-sectional information on the whole universe of active groups without taking into account time in a dynamic fashion.",
"Terrorism has undergone several changes in the last two decades: many new groups have appeared only recently, many have disappeared or have been dismantled, other actors have been active only for very short period.",
"In general, over the years, the trend of active actors has not shown a stable behavior.",
"As it is expected, this type of trend also regards the number of attacks and terrorist events (Figure REF ).",
"Figure: Number of Active Groups and Terror Attacks per Year (1997-2016).These changes and trends over the last twenty years may also be related to strategies and types of attack, besides mere frequency of attacks.",
"This is an aspect which intersects our analysis.",
"Since our original multi-partite network stores all the information of the past twenty years, these changes and trends may be underestimated or even vanish altogether in our algorithmic procedure.",
"As for the current data structure, groups that have plotted very few attacks in two distant years may be clustered together considering their similarity in the data space, however, it might be not useful for analysts or policy-makers to compare two groups that are too distant in time.",
"For this reason, our intuition is to introduce a potential solution to this issue via a dynamic Entropy-based approach.",
"Instead of constructing a single static multi-partite network, we build yearly multi-partite graphs to capture the variations in the entropy of each mode (Figure REF ).",
"Figure: Entropy variation for each mode (1997-2016).As shown by the plot above, there exists clear variations in the entropies of each mode from which we have built our multi-partite network.",
"The plot highlights how the Weapon mode generally follows completely different trends with respect to all the others.",
"At the same time, the entropies of Tactic and Target modes display very similar behavior in the last four years of our considered timespan.",
"Notably, there is an almost complete similarity of entropies for Region, Tactic and Target modes with respect to year 2001.",
"This graph clearly suggests that time should be considered in the algorithmic procedure, since it is highly likely that sensible changes in the subgrouping outcomes will emerge.",
"Besides the relevance of embedding temporal dynamics from the purely research point of view, this furthermore provides a richer tool for potential users and analysts interested in using the algorithm to detect, assess and study patterns in the data.",
"In fact, since our model seeks to provide a practical framework that can be easily deployed for the aforementioned purposes, we feel that time-aware results are able to exclude all the non-active groups in a given year and would increase the usability of previous years information and its efficiency for real-time objectives oriented to intelligence profiling.",
"This work has presented a novel algorithmic framework for detecting latent clusters of similar terrorist groups via a complex network approach.",
"We have created a multi-partite network for the entire known population of terror groups active worldwide from 1997 to 2016, where modes were Weapons, Tactics, Targets, operating Regions, and proposed a novel clustering architecture expanded from [15].",
"We have then compared our new entropy-based architecture with two alternative solutions: a weak-heuristic approach based on terror group clustering by ideology and our baseline unweighted approach.",
"The entropy-based approach modifies the baseline approach simply weighting each mode by its graph entropy, in order to provide a data-driven approach that takes into account the relevance of a certain mode with respect to the others.",
"The analysis has first demonstrated that subgrouping by ideology leads to cluster assignments very different to the ones obtained with our baseline method, where we let patterns emerge naturally from data with no a priori knowledge and, secondly, that the entropy-based and the baseline approaches have similar results both in terms of stability of cluster assignment for terrorist groups and behavioral and ideological intra-cluster association.",
"Both approaches corroborated interesting findings that go beyond the pure methodological intent of this work.",
"To investigate the meta-connections between groups resulting from our work, we have analyzed behavioral characteristics (e.g., share of successful attacks, international propensity, etc.)",
"and we have also focused on the ideological background of each actor, retrieving this information from BAAD version 1 and 2 and other open access qualitative sources.",
"Though labelling a group under few ideological categories may oversimplify certain complex components of terrorism, interesting relations emerged.",
"Besides several expected patterns (e.g.",
"Islamist/jihadist groups tend not to be associated with groups belonging to other ideologies), the algorithm reveals other results that may shed light on terrorism in terms of research and policy.",
"The clustering procedures highlighted a certain similarity between FL groups and FR groups, indicating that besides their divergent objectives and goals, these two types of groups share similar behaviors.",
"Furthermore, FL groups on one side are often associated with animal-rights and environmentalist actors, suggesting that some overlapping in terms of motives and aims is also connected to similar methods and ways of acting.",
"On the other side, FR groups tend to be clustered together with ethno/nationalist and religious groups, as it might be expected given that many FR groups hold nationalist or religion-related elements.",
"Overall, the entropy-based approach is a flexible tool for capturing the intrinsic and hidden knowledge included in the manifold via a data-driven procedure, rather than using subjective knowledge and weaker heuristics, which was one of the limitations of several experiments conducted in [15].",
"In spite of the aforementioned results, our approach may suffer from the fact that the original multi-partite network does not take time into account.",
"Indeed, the manifold includes the whole set of available data from 1997 and 2016.",
"This might be interpreted as a limitation, especially considering that our work is inherently policy-oriented.",
"The last twenty years have been susceptible of several dramatic changes in the ways terrorism manifests itself at global scale.",
"On one side, they have seen the rise of islamist and jihadist terrorism not only in Africa and Middle East, but also in Western and Eastern Europe and countries of the North America.",
"On the other side, politically motivated terrorism has showed shifts and different concentration over time and space.",
"Furthermore, the considered time span is relatively long and therefore includes groups that may have been already disappeared and dismantled or even actors that have plotted one or very few single attacks, therefore constituting a sort of \"noise\" in the whole scenario.",
"In light of this, we have opened the path for future work showing that, besides variations in the trends of active groups and actors and plotted attacks, there exist also significant variations in the entropies of each mode over time.",
"Entropies change sensibly over-time and we have highlighted the presence of some similarities in these trends across certain modes (e.g.",
"Tactic and Target), while others follow completely different behaviors (e.g.",
"Weapon).",
"For these reasons, future work should test the entropy-based setup within a dynamic framework.",
"While considering all the groups and being able of compare still active groups with those that have been already dismantled or disrupted is certainly useful, we feel that controlling for the noise in the manifold and including only groups that are still part of the global terrorist scenario will provide more insights and will help policy-makers or analysts in understanding to what extent certain groups are similar or different compared to others.",
"Additionally, leveraging upon the entropy-based structure automatically allows one to take into account the most relevant sources of information in real-time (e.g.",
"modes): this type of setup, for instance, would be capable of highlighting anomalous behaviours or strategical behavioral evolution of certain terror groups.",
"Future work will also seek to eventually exclude groups that are not strictly considerable as terrorists (e.g., Mexican drug cartels) although their actions are of terrorist nature: this operation would reduce potential noise and distortion of the results.",
"Finally, inherent limitations come from the data.",
"While the GTD is certainly the most reliable and solid open access dataset freely available for research-purposes on terrorist events, its structure poses issues of missing data and level of detail of the information.",
"Despite the fact that, as opposed to other criminal phenomena, terrorist attacks tracked and recorded do not face the risk of underestimation (generally, every terrorist attack is reported by newspapers or media agencies), not all details on the attacks might be consistently retrieved and included in the dataset.",
"This would therefore lead to a certain degree of bias or missing information regarding event characteristics, which are the core of our work.",
"Another linked type of limitation is the risk of too generic information, especially for terror attacks occurred outside Europe and North America (which are actually the majority of the events).",
"While our algorithmic framework has demonstrated a certain degree of potential using the GTD, the intent is to test it on more detailed databases in the future.",
"Additionally, in our algorithmic framework, we do not consider any correlation between the different modes of the data.",
"More specifically, it is possible that certain groups use certain weapons or tactics because of limited availability of alternative means and not, instead, as the product of a free choice.",
"Unfortunately, we are not able to assess whether this is the case for the groups under analysis, but this potential explanation shall be kept in mind.",
"Additionally, the ideology labelling process, though based on a scientifically recognized dataset, may oversimplify certain characteristics and motivations behind each group's actions.",
"Reducing the complexity of the causes and motives behind the decision to resort to terrorism is challenging and attention should be payed not to provide distorted or biased interpretation of the results.",
"GTD: Global Terrorism Database; BAAD1 and BAAD2: Bad, Allied and Dangerous Dataset versions 1 and 2; kNN: k-Nearest Neighbor; FL: Far Left/Communist/Anarchist; FR: Far Right/Racist/Nazi; AMI: Adjusted Mutual Information; KDE: Kernel Density Estimation The datasets analysed during the current study are available in the START repository.",
"The Global Terrorism Database is available at: https://www.start.umd.edu/gtd/ [57]; Bad Allied and Dangerous dataset version 1 is available at https://dataverse.harvard.edu/dataset.xhtml?persistentId=hdl%3A1902.1/16062 and Bad, Allied and Dangerous dataset version 2 is available for browsing at https://www.start.umd.edu/baad/database [2].",
"The authors declare that they have no competing interests.",
"Not applicable.",
"This work is supported in part by the Office of Naval Research under the Multidisciplinary University Research Initiatives (MURI) Program award number N000141712675, Near Real Time Assessment of Emergent Complex Systems of Confederates, the Minerva program under grant number N000141512797, Dynamic Statistical Network Informatics, a National Science Foundation Graduate Research Fellowship (DGE 1745016), and by the center for Computational Analysis of Social and Organizational Systems (CASOS).",
"The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the ONR or the U.S. government.",
"Gian Maria Campedelli, Iain Cruickshank and Kathleen M. Carley have developed together the theoretical setup of the study.",
"Gian Maria Campedelli and Iain Cruickshank have created the algorithmic framework, conducted the quantitative analyses and written the paper.",
"Kathleen M. Carley has supervised the entire project.",
"The authors wish to thanks the two anonymous reviewers for their comments and Bruce Desmarais, Cecilia Meneghini, Alberto Aziani and Pasquale De Meo for their precious suggestions on earlier versions of this manuscript."
]
] | 2001.03367 |
[
[
"Powers of two weighted sum of the first p divided Bernoulli numbers\n modulo p"
],
[
"Abstract We show that, modulo some odd prime p, the powers of two weighted sum of the first p-2 divided Bernoulli numbers equals the Agoh-Giuga quotient plus twice the number of permutations on p-2 letters with an even number of ascents and distinct from the identity.",
"We provide a combinatorial characterization of Wieferich primes, as well as of primes p for which p^2 divides the Fermat quotient q_p(2)."
],
[
"Introduction",
"Congruences involving Bernoulli numbers or divided Bernoulli numbers have drawn the attention of many mathematicians.",
"The list is so long that we may not venture citing any author here.",
"In this note, we study powers of two weighted sums of the first $p$ divided Bernoulli numbers modulo $p$ , where $p$ is a given odd prime number.",
"Bernoulli numbers are rational numbers and presently we view them as $p$ -adic numbers.",
"We define the divided Bernoulli numbers as $\\mathcal {B}_n=\\frac{B_n}{n}\\in \\mathbb {Q}_p\\;\\text{when $n\\ge 1$ and}\\;\\;\\mathcal {B}_0=-\\frac{1}{p}\\in \\mathbb {Q}_p,$ where $\\mathbb {Q}_p$ denotes the field of $p$ -adic numbers.",
"The convention for $\\mathcal {B}_0$ is our own convention, especially the minus sign which is simply there because it will allow for a nicer formula.",
"We recall that all the Bernoulli numbers with odd indices are zero, except $B_1$ which takes the value $-\\frac{1}{2}$ .",
"After excluding both the upper bound and the lower bound of the weighted sum, we are able to express the remaining sum in terms of the Agoh-Giuga quotient and of some Eulerian sum modulo $p$ .",
"The Agoh-Giuga quotient is the $p$ -adic integer $\\frac{p\\,B_{p-1}+1}{p}\\in \\mathbb {Z}_p$ Von Staudt-Clausen's theorem, a theorem dating from 1840 that was independently proven by von Staudt and Clausen, asserts that the Bernoulli numbers $B_{2k}$ sum to zero when added all the fractions $\\frac{1}{q}$ with $q$ prime such that $q-1$ divides $2k$ .",
"It is a consequence of von Staudt-Clausen's theorem that $p\\,B_{p-1}$ is a $p$ -adic integer of residue $-1$ modulo $p\\,\\mathbb {Z}_p$ .",
"Eulerian sums are sums involving Eulerian numbers.",
"The Eulerian number $E(m,n)$ is by definition the number of permutations on $n$ letters with $m$ ascents.",
"If $(i_1,i_2,\\dots ,i_n)$ denotes the permutation mapping the integer $j$ onto $i_j$ , an ascent is when $i_{k+1}>i_k$ .",
"Eulerian numbers originate in Euler's book dating from 1755 in which he investigates shifted forms of what are now called Eulerian polynomials: $E_n(t)=\\sum _{m=0}^n E(n,m)\\,t^m$ They get more attention a century later in the studies of Worpitzky, see [24].",
"Our main result is the following.",
"Main Theorem Let the $\\mathcal {B}_k$ 's denote the divided Bernoulli numbers, as defined earlier.",
"Then, we have $\\sum _{k=1}^{p-2}\\frac{\\mathcal {B}_k}{2^k}=\\frac{p\\,B_{p-1}+1}{p}+2\\left(\\begin{array}{l}\\#\\text{permutations on $(p-2)$ letters}\\\\\\text{with an even number of ascents}\\\\\\text{and distinct from the identity}\\end{array}\\right)\\;\\text{mod}\\,p$ The second component of the sum to the right hand side of $(1)$ is precisely twice the Eulerian sum $\\sum _{k=0}^{\\frac{p-5}{2}} E(p-2,2k)$ Eulerian numbers can be computed from the recursion formula $E(n,m)=(n-m)\\,E(n-1,m-1)+(m+1)\\,E(n-1,m)$ We prove more results including but not limited to the Fermat quotient $q_p(2)$ to the moduli $p$ and $p^2$ and congruences concerning sums of powers of odd integers and sums of divided Bernoulli numbers.",
"Some of the proofs in the current paper are based on using two equivalent definitions for the unsigned Stirling numbers of the first kind.",
"From these two definitions we can derive the following lemma.",
"Lemma 1 $p!&=&\\sum _{s=1}^p\\left[\\begin{array}{l} p\\\\s\\end{array}\\right]\\\\(p+1)!&=&\\sum _{s=1}^p\\left[\\begin{array}{l} p\\\\s\\end{array}\\right]2^s$ Equality $(2)$ comes from the fact that $\\left[\\begin{array}{l}p\\\\s\\end{array}\\right]$ counts the number of permutations of $p$ objects that decompose into a product of $s$ disjoint cycles.",
"Since any permutation of $p$ objects can be uniquely written as a product of cycles with disjoint supports, equality $(2)$ holds.",
"As for Equality $(3)$ , it arises from the other definition of the unsigned Stirling numbers of the first kind, namely, $\\left[\\begin{array}{l}p\\\\s\\end{array}\\right]$ is the unsigned coefficient of $x^s$ in the falling factorial $x(x-1)\\dots (x-(p-1))$ and so, one can write: $x(x-1)(x-2)\\dots (x-(p-1))=\\sum _{s=1}^p (-1)^{s-1}\\left[\\begin{array}{l}p\\\\s\\end{array}\\right]\\,x^s$ It now suffices to specialize $x=-2$ in Equality $(4)$ .",
"We thus get Equality $(3)$ of Lemma 1.",
"Modulo $p^2$ , the two factorials $p!$ and $(p+1)!$ of the lemma are equal.",
"By [11] or [6], we know how to calculate the Stirling numbers modulo $p^2$ in terms of Bernoulli numbers.",
"Then by using some results of Emma Lehmer [9], we are able to relate the investigated weighted sum to a sum of powers of integers modulo $p^2$ , just as stated in the following proposition.",
"Proposition 1 $\\sum _{a=1}^{\\frac{p-1}{2}}a^{p-1}=-1-\\frac{1}{2}\\,p\\,B_{p-1}+\\frac{p}{2}\\Bigg (\\frac{3}{2}+\\sum _{k=1}^{\\frac{p-3}{2}}\\frac{B_{2k}}{k\\,2^{2k}}\\Bigg )\\;\\text{mod}\\,p^2$ Independently, by working out the sum to the left hand side of $(5)$ modulo $p^2$ and by using some known formula relating an alternating sum of Eulerian numbers to the Bernoulli numbers, we obtain an intermediate result on the Fermat quotient $q_p(2)=\\frac{2^{p-1}-1}{p}$ The result is stated in the following theorem.",
"Theorem 1 $q_p(2)=2\\,\\left(\\begin{array}{l}\\#\\text{permutations on $(p-2)$ letters}\\\\\\text{with an even number of ascents}\\\\\\text{and distinct from the identity}\\end{array}\\right)+1\\;\\;\\text{mod}\\,p$ Forthcoming Theorem 9 of the introduction provides the next term in the $p$ -adic expansion of the Fermat quotient $q_p(2)$ , expressed combinatorially and in terms of the residue of the Agoh-Giuga quotient.",
"As a corollary obtained from Eisenstein's formula [4] for the Fermat quotient $q_p(2)=\\frac{1}{2}\\sum _{k=1}^{p-1}\\frac{(-1)^{k-1}}{k}\\;\\text{mod}\\,p,$ we have: Theorem 2 $\\sum _{k=1}^{p-1}\\frac{(-1)^k}{k}=2\\Bigg (1-2\\left(\\begin{array}{l}\\#\\text{permutations on $(p-2)$ letters}\\\\\\text{with an even number of ascents}\\end{array}\\right)\\Bigg )\\;\\;\\text{mod}\\,p$ And since we have $H_{p-1}=0\\;\\text{mod}\\,p$ by Wolstenholme's theorem [23], where $H$ denotes the harmonic number, we thus deduce Theorem 3 $q_p(2)=\\sum _{\\begin{array}{l}k=1\\\\\\text{k odd}\\end{array}}^{p-1}\\frac{1}{k}=2\\,\\left(\\begin{array}{l}\\#\\text{permutations on $(p-2)$ letters}\\\\\\text{with an even number of ascents}\\\\\\text{and distinct from the identity}\\end{array}\\right)+1\\;\\;\\text{mod}\\,p$ Remark 1 The harmonic sum of Theorem 3 with odd reciprocals is sometimes denoted by $H^{^{\\prime }}_{p-1}$ in the literature.",
"Back in 1900, Glaisher has obtained the congruence to the left in Theorem 3 to the modulus $p^2$ .",
"Namely, he showed in [7] that $q_p(2)=H^{^{\\prime }}_{p-1}+\\frac{1}{2}\\,p\\,(H^{^{\\prime }}_{p-1})^2\\;\\;\\text{mod}\\,p^2$ Vandiver provided a simpler proof for the latter congruence in [21].",
"At the end of his Remark $5.3$ of [18], Sun provides a congruence for $H^{^{\\prime }}_{p-1}$ to the modulus $p^3$ .",
"It is the following: $H^{^{\\prime }}_{p-1}=q_p(2)-\\frac{1}{2}\\,pq_p^2(2)+\\frac{1}{3}p^2q_p^3(2)-\\frac{1}{24}\\,p^2\\,B_{p-3}\\;\\;\\text{mod}\\,p^3$ Further, from Stern's formula [17] modulo $p$ for the Fermat quotient $q_p(2)$ , namely $q_p(2)=\\sum _{k=1}^{\\frac{p-1}{2}}\\frac{(-1)^{k-1}}{k}\\;\\text{mod}\\,p,$ joint with Lehmer's formula for the harmonic number $H_{\\frac{p-1}{2}}$ modulo $p^2$ , we derive Theorem 4 $\\sum _{\\begin{array}{l}k=1\\\\\\text{k odd}\\end{array}}^{\\frac{p-1}{2}}\\frac{1}{k}=\\frac{1}{2}\\Bigg (1-2\\,\\left(\\begin{array}{l}\\#\\text{permutations on $(p-2)$ letters}\\\\\\text{with an even number of ascents}\\end{array}\\right)\\Bigg )\\;\\text{mod}\\,p$ Coming back to the sketch of the proof of the main theorem, from Eq.",
"$(3)$ of Lemma 1 written modulo $p^2$ and Theorem 3, we derive the formula of the main theorem.",
"Stated under the form of the title of the current paper, it reads instead: Theorem 5 Under the convention that $\\mathcal {B}_0=-\\frac{1}{p}$ , we have: $\\sum _{k=0}^{p-1}\\frac{\\mathcal {B}_k}{2^k}=0\\;\\;\\;\\text{mod}\\,p$ Last, a consequence of Eq.",
"$(2)$ from Lemma 1 is that the sum of the first $(p-2)$ divided Bernoulli numbers is congruent to the residue of the Agoh-Giuga quotient modulo $p$ minus 1, that is congruent to the residue of the Wilson quotient.",
"We also gather more congruences in the theorem below.",
"Theorem 6 Let $\\mathcal {B}_i$ denote the $i$ th divided Bernoulli number.",
"Let $w_p$ denote the Wilson quotient and let $q_p(2)$ denote the Fermat quotient.",
"Then, we have: $\\sum _{k=1}^{p-2}\\mathcal {B}_k&=&\\frac{p\\,B_{p-1}+1}{p}-1\\;=\\;w_p\\qquad \\qquad \\;\\;\\,\\,\\,\\text{mod}\\,p\\\\\\sum _{k=1}^{p-2}\\frac{\\mathcal {B}_k}{2^k}&=& q_p(2)+w_p\\;\\;\\qquad \\qquad \\qquad \\qquad \\;\\;\\;\\;\\,\\,\\text{mod}\\,p\\\\\\sum _{k=1}^{p-2}\\frac{\\mathcal {B}_k}{2^k}&=&-\\frac{1}{2}H_{\\frac{p-1}{2}}+\\frac{p\\,B_{p-1}+1}{p}-1\\;\\;\\;\\,\\qquad \\text{mod}\\,p\\\\\\sum _{\\begin{array}{l}1\\le k\\le p-2\\\\\\text{k is even}\\end{array}}\\frac{B_k}{k\\,2^k}&=&\\frac{3}{4}+\\sum _{\\begin{array}{l}1\\le k\\le p-2\\\\\\text{k is even}\\end{array}}\\Big (\\frac{B_k-1}{k}\\Big )\\;\\;\\;\\;\\text{mod}\\,p$ In the statement above, $(10)$ and $(12)$ are obtained by combining several of the previous results.",
"Whereas $(11)$ is a rewriting of $(10)$ using the fact that $ H_{\\frac{p-1}{2}}=-2q_p(2)\\;\\;\\text{mod}\\,p$ This is for instance a special case of Vandiver's congruence dating from 1917.",
"His result of [21] is built out of Lerch's original formula dating from 1905 and asserts that $n\\,q_p(n)=-\\sum _{j=1}^{n-1} H_{\\lfloor \\frac{jp}{n}\\rfloor }\\;\\;\\text{mod}\\,p$ We will see along the paper that Lehmer has also obtained Congruence $(13)$ to the next modulus $p^2$ , and Sun in [18] even gets a congruence modulo $p^3$ .",
"Regarding $(10)$ , Beeger computed $w_p$ modulo $p$ for $p<300$ .",
"For this range of primes, $w_p=0\\;\\text{mod}\\,p$ only when $p=5$ and $p=13$ , see [1].",
"For these values of $p$ , we see with $(10)$ that the powers of two weighted sum of divided Bernoulli numbers equals the Fermat quotient modulo $p$ .",
"In particular, $\\sum _{k=1}^{11}\\frac{B_k}{k2^k}=q_{13}(2)\\;\\;\\text{mod}\\,13$ Additionally, we prove the following statements concerning sums of powers.",
"Theorem 7 $(i)$ The sum of the first $\\frac{p-1}{2}$ odd powers of the first $\\frac{p-1}{2}$ integers is congruent to $-\\frac{1}{2}$ modulo $p$ .",
"$(ii)$ The sum of the first $\\frac{p-1}{2}$ odd powers of the next $\\frac{p-1}{2}$ integers is congruent to $\\frac{1}{2}$ modulo $p$ .",
"For the next statement, we must introduce a few notations.",
"Notation 1 Denote by $N_{p-2}$ the number of permutations on $p-2$ letters with an even number of ascents.",
"Notation 2 Let $x$ be a $p$ -adic integer.",
"We denote by $(x)_k$ the $(k+1)$ -th coefficient in the $p$ -adic expansion of $x$ , that is $x=\\sum _{j=0}^{\\infty }(x)_j\\,p^j$ In particular, $(x)_0$ is the residue of $x$ modulo $p$ .",
"We will make an exception with $(pB_{p-1})_1$ which will denote the Agoh-Giuga quotient.",
"The following theorem addresses sums of powers of odd integers modulo $p^2$ .",
"Theorem 8 $(i)\\;\\; \\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1} x^{p-2}&=& (2N_{p-2})_0-1+p\\,\\Bigg ((p\\,B_{p-1})_1+(2N_{p-2})_1-\\big ((2N_{p-2})_0-1\\big )^2-2\\Bigg )\\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\;\\text{mod}\\,p^2\\\\(ii)\\;\\;\\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1} x^{p-1}&=& -\\frac{1}{2}+\\frac{p}{2}\\Bigg ((p\\,B_{p-1})_1-(2N_{p-2})_0+1\\Bigg )\\;\\;\\text{mod}\\,p^2$ By Beeger's result mentioned earlier, we know that $(13\\,B_{12})_1=1$ .",
"From the table below listing the Eulerian numbers with an even number of ascents computed from the Euler triangle, $\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline n\\textbackslash m&0&2&4&6&8&10\\\\\\hline 11&1&152\\,637&9\\,738\\,114&9\\,738\\,114&152\\,637&1\\\\\\hline \\end{tabular}$ we compute that $N_{11}=19\\,781\\,504$ and $(2N_{11})_0=4$ and $(2N_{11})_1=8$ .",
"We can check that Congruence $(15)$ is verified when $p=13$ .",
"Namely, we have: $1^{11}+3^{11}+5^{11}+\\dots +11^{11}=(2N_{11})_0-1+13\\Big ((2N_{11})_1-(2N_{11})_0^2+2(2N_{11})_0-2\\Big )\\;\\;\\text{mod}\\,13^2$ Finally, we confront our congruence $(15)$ of Theorem 8 and a result of Emma Lehmer in order to obtain the next term in the $p$ -adic expansion of the Fermat quotient $q_p(2)$ .",
"Our result is the following.",
"Theorem 9 $2^{p-1}=1+p\\Bigg ((2N_{p-2})_0-1\\Bigg )\\\\+\\;p^2\\Bigg ([(p\\,B_{p-1})_1][(2N_{p-2})_0]-(2N_{p-2})_0^2+(2N_{p-2})_0+(2N_{p-2})_1-2\\Bigg )\\;\\;\\text{mod}\\,p^3$ Confronting our Theorem 9 and Remark 1, we obtain: Theorem 10 Let $H^{^{\\prime }}_{p-1}=1+\\frac{1}{3}+\\frac{1}{5}+\\dots +\\frac{1}{p-2}$ .",
"We have modulo $p$ : $(2N_{p-2})_0=(H^{^{\\prime }}_{p-1})_0+1$ $\\begin{split}(2N_{p-2})_1+\\big ((2N_{p-2})_0-1)_1=&\\frac{3}{2}(H^{^{\\prime }}_{p-1})_0^2+(H^{^{\\prime }}_{p-1})_1+(H^{^{\\prime }}_{p-1})_0\\\\&-(p\\,B_{p-1})_1\\Big (1+(H^{^{\\prime }}_{p-1})_0\\Big )+2\\end{split}$ By computing $(2N_{p-2})_0$ in another way, we obtain the following interesting congruence.",
"Theorem 11 $\\sum _{\\begin{array}{l}m=1\\\\\\text{m odd}\\end{array}}^{p-2}H_m=\\frac{1}{2}\\big (1+(H^{^{\\prime }}_{p-1})_0\\big )=\\frac{1+q_p(2)}{2}\\;\\;\\text{mod}\\,p$ A prime $p$ is called a Wieferich prime if $q_p(2)=0\\;\\text{mod}\\,p$ .",
"In other words, a Wieferich prime satisfies to Fermat's little theorem at the next modulus of $p$ .",
"Wieferich had introduced these primes in 1909 while working on Fermat's last theorem.",
"He had shown that if there exist solutions to $x^p+y^p=z^p$ in integers $x,y,z$ and $p$ is an odd prime with $p$ coprime to $x$ , $y$ , $z$ , then $p$ is a Wieferich prime.",
"As of September 2018, the only known Wieferich primes are 1093 (found by Meissner [12] in 1913 and only such prime below 2000) and 3511 (found by Beeger [1] in 1922).",
"In [3], the authors show that there are no other Wieferich primes $p<6.7\\times 10^{15}$ , yet improving on the bounds provided by a dozen of other mathematicians (the latest bound had been reached in 2005).",
"Previously, Grave had conjectured that Wieferich primes did not exist.",
"In 1988, Joseph Silverman showed in [16] that if the abc conjecture holds, then there exist infinitely many non-Wieferich primes.",
"We now present a combinatorial characterization of a Wieferich prime.",
"In light of Theorem 3, we have: Theorem 12 Let $N_{p-2}$ denote the number of permutations on $p-2$ letters with an even number of ascents.",
"Let $(2N_{p-2})_0$ denote the residue modulo $p$ of twice this number.",
"Then, $\\text{$p$ is a Wieferich prime}&\\Leftrightarrow & (2N_{p-2})_0=1\\\\&\\Leftrightarrow & (H^{^{\\prime }}_{p-1})_0=0$ Definition 1 (super Wieferich prime) We will call a prime a \"super Wieferich prime\" if $q_p(2)=0\\;\\text{mod}\\,p^2$ In light of Theorems 9 and 10, we have: Theorem 13 Let $N_{p-2}$ denote the number of permutations on $p-2$ letters with an even number of ascents.",
"Let $(2N_{p-2})_0$ denote the residue modulo $p$ of twice this number.",
"Let $(2N_{p-2})_1$ denote the next residue in the $p$ -adic expansion of twice this number.",
"Then, $\\bigg (\\text{$p$ is a super Wieferich prime}\\bigg )&\\Leftrightarrow &\\left\\lbrace \\begin{array}{l}\\!\\!\\!",
"(2N_{p-2})_0=1\\;\\qquad \\qquad \\;\\;\\;\\;\\text{mod}\\,p\\\\\\!\\!\\!",
"(2N_{p-2})_1+(p\\,B_{p-1})_1=2\\!\\!\\!\\;\\;\\text{mod}\\,p\\end{array}\\right.\\\\&&\\\\&\\Leftrightarrow & (H^{^{\\prime }}_{p-1})_0=(H^{^{\\prime }}_{p-1})_1=0$ We deduce a combinatorial necessary condition for a prime to be a super Wieferich prime and a Wilson prime.",
"Theorem 14 (Search for Wilson and super Wieferich primes).",
"Let $N_{p-2}$ denote the number of permutations on $p-2$ letters with an even number of ascents.",
"Let $(2N_{p-2})_0$ denote the residue modulo $p$ of twice this number.",
"Let $(2N_{p-2})_1$ denote the next residue in the $p$ -adic expansion of twice this number.",
"If $p$ is a Wilson and super Wieferich prime, then $(2N_{p-2})_0=(2N_{p-2})_1=1$ We draw below a list of open questions.",
"Open problem 1 Does there exist any Wilson prime that is also a Wieferich prime ?",
"Open problem 2 Does there exist any super Wieferich prime ?",
"If the answer to both questions is yes, we ask: Open problem 3 Does there exist any Wilson prime that is a super Wieferich prime ?",
"In [2], Richard Crandall, Karl Dilcher and Carl Pomerance reported that there were no other Wilson primes than 5, 13 and 563 less than $5\\times 10^8$ .",
"The authors worked on both the Wieferich search and the Wilson search, but they did not study whether the two sets intersect or not.",
"It has been conjectured that infinitely many Wilson primes exist, as for Wieferich primes.",
"The paper is structured as follows.",
"In $§\\,2$ , we prove the main theorem as well as Theorems 1-6 concerning the residues of the Fermat quotient $q_p(2)$ (Theorem 1), resp the alternating harmonic sum and the harmonic sum of odd reciprocals corresponding to the harmonic number $H_{p-1}$ (Theorems 2 and 3), resp the harmonic sum of odd reciprocal corresponding to the harmonic number $H_{\\frac{p-1}{2}}$ (Theorem 4), resp the powers of two weighted sum of the first $p$ divided Bernoulli numbers (Theorem 5), resp a sum of divided Bernoulli numbers whose indices range between 1 and $p-2$ (Theorem 6).",
"The next paragraph $§\\,3$ deals with sums of powers of integers exclusively.",
"We prove Theorems 7 and 8.",
"In $§\\,4$ , we prove Theorem 9 which provides a congruence for the $p$ -adic expansion of $q_p(2)$ up to the modulus $p^3$ .",
"We deduce Theorem 10 by using Glaisher's expression for the Fermat quotient modulo $p^2$ , taken from [7].",
"The results of Theorems 12 and 13 can then be read out of Theorems 9 and 10.",
"At the end of this part, we also prove Theorem 11, providing a congruence relating modulo $p$ the Fermat quotient to a sum of harmonic numbers with odd indices.",
"The result follows from Cong.",
"$(18)$ of Theorem 10 and from a separate calculation of the Eulerian numbers modulo $p$ .",
"The latter computation is the purpose of forthcoming Proposition 4 in $§\\,4$ .",
"In the last part, we mention Zhi-Wei Sun's work dealing with similar powers of two weighted sums, concerning divided harmonic numbers this time.",
"We came across his work after this work was completed.",
"Tentatively connecting our work and Sun's work is left for future plans.",
"As part of his results, Z-W. Sun proves that the powers of two weighted sum of the first $(p-1)$ divided harmonic numbers is congruent to zero modulo $p$ .",
"Here we investigate when a similar statement holds if we consider a powers of two weighted sum of the first $(p-2)$ divided Bernoulli numbers.",
"Further, we generalize this investigation to powers of integers $k$ with $3\\le k\\le p-1$ .",
"Our conclusions are gathered in the following theorem.",
"Theorem 15 Let $p$ be an odd prime and let $k$ be an integer with $1\\le k\\le p-1$ .",
"$\\text{Let}\\;\\;\\Sigma _{p,k}:=\\sum _{l=1}^{p-2}\\frac{B_l}{l\\,k^l}$ The sum $\\Sigma _{p,k}$ is congruent to zero modulo $p$ if and only if the second residue in the $p$ -adic expansion of the root of $X^{p-1}+(p-1)!$ of residue $k$ is zero."
],
[
"Proof of the main theorem and the Fermat quotient $q_p(2)$ modulo {{formula:54796845-d95f-4ae7-9bb4-975a3554a58c}}",
"We start this part by introducing some expressions for various sums involving Bernoulli numbers.",
"These will be useful throughout of the paper.",
"We establish the following lemma which arises from writing Lemma 1 of the introduction modulo $p^2$ .",
"Lemma 2 $-p=p!=1+p\\Bigg (\\frac{B_{p-1}}{1}+\\frac{B_{p-3}}{3}+\\frac{B_{p-5}}{5}+\\dots +\\frac{B_2}{p-2}-\\frac{3}{2}\\Bigg )\\;\\;\\text{mod}\\,p^2$ $p\\,B_{p-1}=-1+p\\Bigg (1-2q_p(2)+\\sum _{s=1}^{\\frac{p-3}{2}}(1-2^{2s+1})\\frac{B_{p-(2s+1)}}{2s+1}\\Bigg )\\;\\;\\,\\,\\;\\;\\text{mod}\\,p^2$ Proof.",
"The expression for the Stirling numbers modulo $p^2$ is read out of Corollary 2 of [11].",
"We have by Eq.",
"$(1)$ of Lemma 1, $p!=p\\,B_{p-1}-p+\\sum _{s=1}^{\\frac{p-3}{2}}\\frac{p}{2s+1}\\,B_{p-2s-1}-\\frac{p}{2}+1\\;\\;\\text{mod}\\,p^2$ A simple rewriting of $(26)$ is $(24)$ .",
"Next, we write Eq.",
"$(2)$ of Lemma 1 modulo $p^2$ .",
"It allows to express the weighted sum of interest in terms of the Agoh-Giuga quotient and of the Fermat quotient.",
"We have, where we used again the expressions for the Stirling numbers modulo $p^2$ taken from Corollary 2 of [11], $-p=(p+1)!=2(p\\,B_{p-1}-p)+\\sum _{s^{^{\\prime }}=1}^{\\frac{p-3}{2}}\\frac{p}{2s^{^{\\prime }}+1}\\,B_{p-2s^{^{\\prime }}-1}\\,2^{2s^{^{\\prime }}+1}-\\frac{p}{2}+2^p\\;\\;\\text{mod}\\,p^2$ Congruence $(25)$ is obtained by subtracting $(27)$ and $(26)$ .",
"If in the sum of Congruence $(27)$ we make the change of indices corresponding to $2s=p-(2s^{^{\\prime }}+1)$ , we get $2p\\,\\sum _{s=1}^{\\frac{p-3}{2}}\\frac{B_{2s}}{2s\\,2^{2s}}=2(p\\,B_{p-1}-p)+\\frac{p}{2}+2^p\\;\\;\\text{mod}\\,p^2$ We deduce, using also that $B_1=-\\frac{1}{2}$ and $B_j=0$ for $j$ odd greater than 1, Lemma 3 $\\sum _{k=1}^{p-2}\\frac{B_k}{k\\,2^k}=\\frac{p\\,B_{p-1}+1}{p}+q_p(2)-1\\;\\;\\text{mod}\\,p$ The next goal is to prove Theorem 3 which provides a combinatorial interpretation for the Fermat quotient $q_p(2)$ modulo $p$ .",
"To that aim, we will need a result of Lehmer.",
"Theorem 16 (Due to Emma Lehmer [9], 1938) If $p-1\\lnot |\\,2k-2$ , then $p\\,B_{2k}\\equiv \\frac{1}{2^{2k-1}}\\sum _{a=1}^{\\frac{p-1}{2}}(p-2a)^{2k}\\;\\;\\text{mod}\\,p^3$ Apply Lehmer's result with $2k=p-1$ after noting that $p-1\\lnot |p-3$ .",
"We get: $p\\,B_{p-1}=\\frac{1}{2^{p-2}}\\sum _{a=1}^{\\frac{p-1}{2}}(p-2a)^{p-1}\\;\\;\\text{mod}\\,p^3$ Expanding the product modulo $p^2$ yields: $(p-2a)^{p-1}&=&2^{p-1}a^{p-1}-p(p-1)2^{p-2}a^{p-2}\\;\\;\\text{mod}\\,p^2\\\\&=&2^{p-2}(2a^{p-1}+p\\,a^{p-2})\\qquad \\;\\;\\;\\;\\,\\;\\;\\;\\text{mod}\\,p^2$ It follows that $p\\,B_{p-1}=2\\sum _{a=1}^{\\frac{p-1}{2}}a^{p-1}+p\\,\\sum _{a=1}^{\\frac{p-1}{2}}a^{p-2}\\;\\;\\text{mod}\\,p^2$ We stop here to introduce some notations which will be useful throughout the paper.",
"Notation 3 Denote by $S_k$ the sum of $k$ th powers $S_k=\\sum _{a=1}^{\\frac{p-1}{2}}a^{k}$ Using Notation 3, we summarize congruence $(34)$ in the following lemma.",
"Lemma 4 $p\\,B_{p-1}=2\\,S_{p-1}+p\\,H_{\\frac{p-1}{2}}\\;\\;\\text{mod}\\,p^2$ Remark 2 In [18], Sun provides a congruence for $H_{\\frac{p-1}{2}}$ modulo $p^3$ in terms of the Bernoulli number $B_{p-3}$ and of the Fermat quotient $q_p(2)$ .",
"It is the following: $H_{\\frac{p-1}{2}}=-2q_p(2)+p\\,q_p^2(2)-\\frac{2}{3}p^2\\,q_p^3(2)-\\frac{7}{12}\\,p^2\\,B_{p-3}\\;\\;\\text{mod}\\,p^3$ Previously in [9], Lehmer had obtained the same congruence modulo $p^2$ only.",
"Another useful result is the following.",
"Theorem 17 $S_k=\\sum _{r=1}^{\\frac{p-1}{2}}r^k={\\left\\lbrace \\begin{array}{ll}(2^{-k+1}-1)B_k\\frac{p}{2}\\;\\;\\text{mod}\\,p^2&\\text{if $k$ is even}\\;\\;(I)_k\\\\&\\\\\\big (\\frac{1}{2^{k+1}}-1\\big )\\,\\frac{2\\,B_{k+1}}{k+1}\\;\\;\\text{mod}\\,p^2&\\text{if $k$ is odd}\\;\\;(II)_k\\end{array}\\right.",
"}$ under the condition $p-1\\lnot |\\,k-1$ .",
"The first congruence due to Lehmer for the even $k$ 's appears in [9] while the second congruence for odd $k$ 's is due to Mirimanoff and appears in [15] at the bottom of page 299.",
"Mirimanoff's notations for the Bernoulli numbers are the same as Glaisher's notations in [6].",
"We provide here a global proof which gathers at once both cases: $k$ is even and $k$ is odd.",
"We will use the Bernoulli polynomials: $B_n(x)=\\sum _{k=0}^n\\binom{n}{k}B_{n-k}\\,x^k$ First and foremost, we have from the Bernoulli formula for the sums of powers: $\\sum _{r=1}^{\\frac{p-1}{2}}r^k=\\frac{1}{k+1}\\Bigg \\lbrace \\sum _{l=0}^{k+1}\\binom{k+1}{l}B_l\\bigg (\\frac{p-1}{2}\\bigg )^{k+1-l}-B_{k+1}\\Bigg \\rbrace $ Eq.",
"$(38)$ holds by considering Bernoulli numbers of the second kind, that is $B_1=\\frac{1}{2}$ instead of $B_1=-\\frac{1}{2}$ .",
"After the change of indices corresponding to $m=k+1-l$ , the difference in $(38)$ can be written with Bernoulli polynomials instead, as follows.",
"$\\sum _{r=1}^{\\frac{p-1}{2}}r^k=\\frac{B_{k+1}\\big (\\frac{p-1}{2}\\big )-B_{k+1}}{k+1}$ We will use the translation formula $B_n(x+y)=\\sum _{k=0}^n\\binom{n}{k}B_k(x)y^{n-k}$ We apply this formula with $x=-\\frac{1}{2}$ and $y=\\frac{p}{2}$ and obtain $B_{k+1}\\big (\\frac{p-1}{2}\\big )=\\sum _{m=0}^{k+1}\\binom{k+1}{m}B_m\\big (-\\frac{1}{2}\\big )\\frac{p^{k+1-m}}{2^{k+1-m}}$ When working with Bernoulli numbers of the second kind instead of the first kind in the expression for $B_l(-x)$ and conversely in the expression for $B_l(x)$ , we have $B_l(-x)=(-1)^l\\,B_l(x)$ Moreover, we have (see for instance [8]), $B_n\\Big (\\frac{1}{2}\\Big )=\\Big (\\frac{1}{2^{n-1}}-1\\Big )B_n$ Modulo $p^2$ , when $p-1\\lnot |\\,k-1$ (so that $p$ does not divide the denominator of $B_{k-1}$ by Von Staudt-Clausen's theorem), there are only two terms in the sum of $(41)$ , respectively obtained from $m=k+1$ and $m=k$ .",
"Using $(42)$ and $(43)$ , we get $B_{k+1}\\big (\\frac{p-1}{2}\\big )=(-1)^{k+1}\\bigg (\\frac{1}{2^k}-1\\bigg )B_{k+1}+(k+1)(-1)^k\\frac{p}{2}\\bigg (\\frac{1}{2^{k-1}}-1\\bigg )B_k$ When $k$ is even (resp odd), the first (resp second) term vanishes.",
"We obtain the congruences of Theorem 17.",
"We now apply $(I)_{p-1}$ .",
"It yields, $S_{p-1}=\\sum _{r=1}^{\\frac{p-1}{2}}r^{p-1}=\\bigg (\\frac{1}{2^{p-2}}-1\\bigg )B_{p-1}\\frac{p}{2}\\;\\;\\text{mod}\\,p^2$ We deduce, $2^{p-2}S_{p-1}=\\bigg (\\frac{1}{2}+\\frac{1}{2}-2^{p-2}\\bigg )B_{p-1}\\frac{p}{2}\\;\\;\\text{mod}\\,p^2$ Hence, $2^{p-1}S_{p-1}=\\frac{p\\,B_{p-1}}{2}+\\frac{p\\,B_{p-1}}{2}(1-2^{p-1})\\;\\;\\text{mod}\\,p^2$ We now derive $(1+pq_p(2))S_{p-1}=S_{p-1}-p\\,q_p(2)-\\frac{p}{2}\\frac{(p-1)}{2^{p-1}}\\sum _{m=0}^{p-3}(-1)^m\\,E(p-2,m)\\;\\;\\text{mod}\\,p^2,$ where we used Congruence $(35)$ of Lemma 4, Congruence $(36)$ of Remark 2, and a formula relating the alternating sum of Eulerian numbers to the Bernoulli numbers by $(2^{n+1}-1)\\frac{B_{n+1}}{n+1}=\\frac{1}{2^{n+1}}\\sum _{m=0}^{n-1}(-1)^m\\,E(n,m),$ applied here with $n+1=p-1$ .",
"Modulo $p$ , the sum $S_{p-1}$ is simply $-\\frac{1}{2}$ , hence we further derive $\\frac{1}{2}p\\,q_p(2)=\\frac{p}{2}\\sum _{m=0}^{p-3}(-1)^m\\,E(p-2,m)\\;\\;\\text{mod}\\,p^2$ And after simplifying by $p$ and multiplying by 2, we obtain $q_p(2)=\\sum _{m=0}^{p-3}(-1)^m\\,E(p-2,m)\\;\\;\\text{mod}\\,p$ In order to conclude to Theorem 1, it suffices to notice that the non alternating sum is the total number of permutations on $p-2$ letters.",
"Moreover, adding the alternating sum with the non alternating sum leads to twice the number of permutations on $p-2$ letters with an even number of ascents.",
"Then $(p-2)!+q_p(2)$ equals this latter number.",
"But $(p-2)!=1\\;\\text{mod}\\,p$ .",
"We thus obtain the result of Theorem 1.",
"The main Theorem follows from the conjunction of Theorem 1 and Lemma 3.",
"As for Theorem 5, it is derived from the Main Theorem, from the congruence $\\frac{pB_{p-1}+1}{p}-\\frac{1}{p}+\\frac{B_{p-1}}{(p-1)2^{p-1}}=1-q_p(2)\\;\\;\\;\\text{mod}\\,p$ and from Theorem 1.",
"Before closing this part, we say a word about Theorem 6.",
"Congruence $(9)$ can be derived from Congruence $(24)$ ; in the main theorem, modulo $p$ , the Agoh-Giuga quotient equals the Wilson quotient plus 1 and still modulo $p$ , the second term in the sum is the Fermat quotient minus 1.",
"We thus obtain $(10)$ .",
"Congruence $(11)$ was already discussed before.",
"As for Congruence $(12)$ , it comes from a combination of $(24)$ , $(25)$ , Theorem 3 and Wolstenholme's theorem.",
"Indeed, Theorem 3 and Wolstenholme's theorem imply that $q_p(2)=\\sum _{\\begin{array}{l}k=1\\\\\\text{k even}\\end{array}}^{p-1}\\frac{(-1)}{k}\\;\\;\\text{mod}\\,p$ Then, by $(25)$ , we have $2\\sum _{\\begin{array}{l}1\\le k\\le p-2\\\\\\text{k is even}\\end{array}}\\frac{B_k}{k2^k}=\\frac{p\\,B_{p-1}+1}{p}+1+\\sum _{\\begin{array}{l}1\\le k\\le p-2\\\\\\text{k is even}\\end{array}}\\Big (\\frac{B_k-2}{k}\\Big )\\;\\;\\text{mod}\\,p$ Finally, by $(24)$ , we have $\\frac{p\\,B_{p-1}+1}{p}=\\frac{1}{2}+\\sum _{\\begin{array}{l}1\\le k\\le p-2\\\\\\text{k is even}\\end{array}}\\frac{B_k}{k}\\;\\;\\text{mod}\\,p$ Congruence $(12)$ follows."
],
[
"Congruences concerning sums of sums of powers and sums of powers of odd integers",
"We begin this part by showing Proposition 1 which until now remained an independent result of the introduction.",
"Proposition 1 relates the sum of powers $S_{p-1}$ to the weighted sum studied in the paper.",
"We use a combination of $(25)$ and $(35)$ .",
"The terms in $-2q_p(2)$ vanish and we simply get $S_{p-1}=\\frac{p-1}{2}+\\frac{p}{2}\\sum _{s=1}^{\\frac{p-3}{2}}(1-2^{2s+1})\\frac{B_{p-(2s+1)}}{2s+1}\\;\\text{mod}\\,p^2$ It follows that $S_{p-1}=\\frac{p-1}{2}+\\frac{1}{2}(\\frac{p}{2}-1-p\\,B_{p-1})-p\\sum _{s=1}^{\\frac{p-3}{2}}2^{2s}\\frac{B_{p-2s-1}}{2s+1}\\;\\;\\text{mod}\\,p^2,$ where we used $(24)$ .",
"From there we easily derive the congruence of Proposition 1.",
"Note with $p\\,B_{p-1}=-1\\;\\text{mod}\\,p$ that the formula is correct modulo $p$ .",
"We can now derive the new Proposition 2 below.",
"Proposition 2 $\\sum _{\\begin{array}{l}1\\le a\\le p-1\\\\\\text{a odd}\\end{array}}a^{p-1}=\\frac{1}{2}+p\\,B_{p-1}-\\frac{3p}{8}-\\frac{p}{4}\\sum _{k=1}^{\\frac{p-3}{2}}\\frac{B_{2k}}{k2^k}\\;\\;\\text{mod}\\,p^2$ Proof.",
"We have, $\\sum _{\\begin{array}{l}1\\le a\\le p-1\\\\\\;\\,\\text{a odd}\\end{array}}a^{p-1}&=&\\sum _{a=1}^{p-1}a^{p-1}-\\sum _{a=1}^{\\frac{p-1}{2}}(2a)^{p-1}\\\\&=&p\\,B_{p-1}+2^{p-1}+2^{p-2}p\\,B_{p-1}-\\frac{p}{2}\\bigg (\\frac{3}{2}+\\sum _{k=1}^{\\frac{p-3}{2}}\\frac{B_{2k}}{k2^{2k}}\\bigg )\\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\;\\;\\text{mod}\\,p^2$ For the sum of $(p-1)$ -th powers of the first $p-1$ integers modulo $p^2$ , the reader is referred to [11] or to Congruence $(5.1)$ of [18].",
"Next, we have $2^{p-2}=\\frac{1}{2}+\\frac{1}{2}p\\,q_p(2),$ and so $2^{p-2}\\,p\\,B_{p-1}=\\frac{1}{2}p\\,B_{p-1}-\\frac{1}{2}p\\,q_p(2)\\;\\;\\text{mod}\\,p^2$ Before moving forward in the computation, we introduce a new notation.",
"Notation 4 $\\mathcal {S}_p:=\\sum _{k=1}^{\\frac{p-3}{2}}\\frac{B_{2k}}{k2^{2k}}$ Plugging $(57)$ and $(58)$ into $(56)$ yields after simplifying, $\\sum _{\\begin{array}{l}1\\le a\\le p-1\\\\\\;\\,\\text{a odd}\\end{array}}a^{p-1}=\\frac{3}{2}\\,pB_{p-1}+1+\\frac{1}{2}pq_p(2)-\\frac{p}{2}(\\frac{3}{2}+\\mathcal {S}_p)\\;\\;\\text{mod}\\,p^2$ Now, from Congruence $(25)$ we derive: $p\\,B_{p-1}-p+1=-2p\\,q_p(2)+\\frac{p}{2}-p\\,B_{p-1}-1+p\\,\\mathcal {S}_p\\;\\;\\text{mod}\\,p^2$ From there, we get $\\frac{1}{2}pq_p(2)=-\\frac{1}{2}-\\frac{1}{2}p\\,B_{p-1}+\\frac{3p}{8}+\\frac{1}{4}p\\,\\mathcal {S}_p\\;\\;\\text{mod}\\,p^2$ And so, plugging back into $(59)$ , we obtain Congruence $(54)$ of Proposition 2.",
"More generally, let $k$ be an integer with $1<k\\le \\frac{p-1}{2}$ .",
"We can express the sum $1^{2k-1}+3^{2k-1}+5^{2k-1}+\\dots +(p-2)^{2k-1}$ in terms of an alternating sum of Eulerian numbers modulo $p^2$ .",
"Indeed, we have: $\\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1}x^{2k-1}&=&\\sum _{x=1}^{p-1}x^{2k-1}-2^{2k-1}\\sum _{x=1}^{\\frac{p-1}{2}}x^{2k-1}\\\\&=&-2^{2k-1}\\sum _{x=1}^{\\frac{p-1}{2}}x^{2k-1}\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{mod}\\,p^2\\\\&=&\\big (2^{2k-1}-\\frac{2^{2k-1}}{2^{2k}}\\big )\\frac{2B_{2k}}{2k}\\;\\;\\text{mod}\\,p^2\\\\&=&(2^{2k}-1)\\frac{B_{2k}}{2k}\\;\\;\\;\\;\\qquad \\;\\;\\;\\;\\;\\text{mod}\\,p^2$ The first sum in the right hand side of $(62)$ is congruent to zero modulo $p^2$ .",
"This fact gets extensively discussed in [11].",
"Congruence $(64)$ is then obtained by applying $(II)_{2k-1}$ of Theorem 17 since $k$ was chosen so that $p-1\\lnot | 2k-2$ .",
"An application of $(49)$ now yields the result stated in the following proposition.",
"Proposition 3 Let $k$ be an integer with $1<k\\le \\frac{p-1}{2}$ .",
"Then, we have $\\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1}x^{2k-1}=\\frac{1}{2^{2k}}\\sum _{m=0}^{2k-2}(-1)^m\\,E(2k-1,m)\\;\\;\\text{mod}\\,p^2$ In particular, applying $(66)$ with $k=\\frac{p-1}{2}$ yields: $\\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1}x^{p-2}=\\frac{1}{2^{p-1}}\\sum _{m=0}^{p-3}(-1)^m\\,E(p-2,m)\\;\\;\\text{mod}\\,p^2$ It follows that, where we used Congruence $(51)$ , $(1+pq_p(2))\\,\\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1}x^{p-2}=(q_p(2))_0+p\\,\\Bigg (\\sum _{m=0}^{p-3}(-1)^m\\,E(p-2,m)\\Bigg )_1\\;\\;\\text{mod}\\,p^2$ Moreover, modulo $p$ , the sum we are computing is nothing else than $H^{^{\\prime }}_{p-1}$ which by Theorem 3 is also $q_p(2)$ modulo $p$ .",
"We thus derive: $\\begin{split}\\big (q_p(2)\\big )_0+p\\,\\Bigg \\lbrace \\Big (\\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1}x^{p-2}\\Big )_1&+\\big (q_p(2)\\big )_0^2\\Bigg \\rbrace =\\big (q_p(2)\\big )_0\\\\&+p\\,\\Bigg (\\sum _{m=0}^{p-3}(-1)^m\\,E(p-2,m)\\Bigg )_1\\;\\;\\text{mod}\\,p^2\\end{split}$ It comes: $\\Big (\\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1}x^{p-2}\\Big )_1=\\Bigg (\\sum _{m=0}^{p-3}(-1)^m\\,E(p-2,m)\\Bigg )_1-\\big (q_p(2)\\big )_0^2\\;\\;\\;\\;\\,\\;\\;\\text{mod}\\,p$ Moreover, we have $(p-2)!+\\sum _{m=0}^{p-3}(-1)^m\\,E(p-2,m)=2\\,\\sum _{k=0}^{\\frac{p-3}{2}}E(p-2,2k)$ And so, $\\begin{split}-(p+1)(p\\,B_{p-1}-p)&+(q_p(2))_0+\\\\&p\\Bigg (\\sum _{m=0}^{p-3}(-1)^m\\,E(p-2,m)\\Bigg )_1=2\\,N_{p-2}\\;\\;\\text{mod}\\,p^2\\end{split}$ Thus, $\\begin{split}2p-p\\,B_{p-1}&+(q_p(2))_0+\\\\&p\\Bigg (\\sum _{m=0}^{p-3}(-1)^m\\,E(p-2,m)\\Bigg )_1=(2N_{p-2})_0+p(2N_{p-2})_1\\;\\;\\text{mod}\\,p^2\\end{split}$ From there, we retrieve in particular the fact that $(q_p(2))_0=(2N_{p-2})_0-1\\,\\;\\;\\text{mod}\\,p,$ which is the result of Theorem 1.",
"In order to proceed further, we will need a simple lemma whose statement appears right below.",
"Lemma We have using Notation 2: $\\Big [(q_p(2))_0+1-(2N_{p-2})_0\\Big ]_1=-\\big ((2N_{p-2})_0-1\\big )_1\\;\\;\\text{mod}\\,p$ Proof of the lemma.",
"Indeed, we have: $\\Big [(q_p(2))_0+1-(2N_{p-2})_0\\Big ]_1&=&\\Bigg [-\\Big ((2N_{p-2})_0-1-\\big ((2N_{p-2})_0-1\\big )_0\\Big )\\Bigg ]_1\\\\&=&-\\Big ((2N_{p-2})_0-1-\\big ((2N_{p-2})_0-1\\big )_0\\Big )_1\\hspace{0.0pt}\\!\\qquad \\;\\;\\text{mod}\\,p\\\\&=&-\\big ((2N_{p-2})_0-1\\big )_1\\,\\qquad \\qquad \\qquad \\qquad \\qquad \\;\\;\\text{mod}\\,p$ By combining Congruences $(70)$ and $(73)$ and using the lemma, we obtain: $\\Big (\\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1}x^{p-2}\\Big )_1=(p\\,B_{p-1})_1+(2N_{p-2})_1+\\big ((2N_{p-2})_0-1\\big )_1-\\big ((2N_{p-2})_0-1\\big )^2-2\\,\\;\\;\\text{mod}\\,p$ From there, point $(i)$ of Theorem 8 follows directly.",
"For point $(ii)$ , we cannot use Proposition 3, hence we make a direct calculation.",
"We have, $\\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1}x^{p-1}&=&\\sum _{x=1}^{p-1}x^{p-1}-2^{p-1}\\sum _{x=1}^{\\frac{p-1}{2}}x^{p-1}\\\\&=&\\frac{1}{2}(p\\,B_{p-1}-pq_p(2))\\qquad \\qquad \\qquad \\qquad \\;\\;\\;\\;\\,\\,\\,\\;\\;\\text{mod}\\,p^2\\\\&=&-\\frac{1}{2}+\\frac{p}{2}\\bigg ((p\\,B_{p-1})_1-(2N_{p-2})_0+1\\bigg )\\qquad \\;\\;\\text{mod}\\,p^2$ Congruence $(77)$ is obtained by applying $(I)_{p-1}$ and $(78)$ follows from applying Theorem 1.",
"Theorem 8 is thus entirely proven.",
"We now deal with sums of sums of powers.",
"Using Notation 3, the sum of the first $\\frac{p-1}{2}$ odd powers of the first $\\frac{p-1}{2}$ integers is $S_1+S_3+\\dots +S_{p-4}+S_{p-2}$ First we sum $T:=S_1+S_3+\\dots +S_{p-4}$ , using $(II)_{2k-1}$ of Theorem 17.",
"It comes: $p\\sum _{k=1}^{\\frac{p-3}{2}}S_{2k-1}=p\\,\\sum _{k=1}^{\\frac{p-3}{2}}\\frac{B_{2k}}{k2^{2k}}-2p\\sum _{k=1}^{\\frac{p-3}{2}}\\frac{B_{2k}}{2k}\\;\\;\\text{mod}\\,p^2$ Since we work modulo $p^2$ , the second sum may be replaced with $-\\sum _{k=1}^{\\frac{p-3}{2}}\\frac{B_{2k}}{p-2k}$ instead, and so we get by using $(24)$ together with Proposition 1, $p\\sum _{k=1}^{\\frac{p-3}{2}}S_{2k-1}&=&2\\bigg (S_{p-1}+1+\\frac{1}{2}p\\,B_{p-1}\\bigg )-\\frac{3}{2}p+2\\bigg (\\frac{3}{2}p-p\\,B_{p-1}-1-p\\bigg )\\;\\;\\text{mod}\\,p^2\\\\&=&-\\frac{1}{2}p-p\\,B_{p-1}+2S_{p-1}\\;\\;\\text{mod}\\,p^2\\\\&=&-\\frac{1}{2}p+2p\\,q_p(2)\\;\\;\\;\\;\\;\\text{mod}\\,p^2$ The last congruence is obtained by using $(35)$ and $(36)$ which jointly provide $2S_{p-1}=p\\,B_{p-1}+2pq_p(2)\\;\\;\\;\\text{mod}\\,p^2$ Then, after simplifying by $p$ , we get: $T=-\\frac{1}{2}+2q_p(2)\\;\\;\\text{mod}\\,p$ Further, $S_{p-2}=H_{\\frac{p-1}{2}}=-2q_p(2)\\;\\;\\text{mod}\\,p$ So, adding all the terms in the considered sum now yields $S_1+S_3+\\dots +S_{p-4}+S_{p-2}=-\\frac{1}{2}=S_{p-1}\\;\\;\\text{mod}\\,p,$ as announced in the statement $(i)$ of Theorem 7.",
"Point $(ii)$ then simply follows from the fact that the sum of any odd power less than or equal to $p-2$ of the first $p-1$ integers is divisible by $p$ .",
"Hence, so is the sum over the first $\\frac{p-1}{2}$ odd powers."
],
[
"The Fermat quotient $q_p(2)$ modulo {{formula:e466296d-fb5b-45fe-bd39-b07daf9adb6a}} .",
"In this part, we show how our work and Emma Lehmer's work combined allow to get to the next $p$ -power in the $p$ -adic expansion of the Fermat quotient $q_p(2)$ .",
"The congruence which we will use and which is Congruence $(34)$ of [9] is the following.",
"Lehmer's notation for the Fermat quotient $q_p(2)$ is simply $q_2$ and her $w_p$ denotes the Wilson quotient.",
"Theorem 18 (Emma Lehmer [9], 1938) $\\sum _{r=1}^{\\frac{p-1}{2}}r^{p-2}=-2q_2(1-pw_p)+2pq_2^2\\;\\;\\text{mod}\\,p^2$ Since $\\sum _{x=1}^{p-1}x^{p-2}=p\\,B_{p-2}=0\\;\\;\\text{mod}\\,p^2,$ we have: $\\sum _{\\begin{array}{l}x=1\\\\\\text{x odd}\\end{array}}^{p-1}x^{p-2}&=&-\\sum _{x=1}^{\\frac{p-1}{2}}2^{p-2}x^{p-2}\\qquad \\qquad \\qquad \\qquad \\qquad \\;\\;\\;\\;\\,\\,\\;\\;\\text{mod}\\,p^2\\\\&=&-\\frac{1}{2}(1+p\\,q_p(2))(-2q_p(2)(1-p\\,w_p)+2p\\,q_p(2)^2)\\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\;\\;\\text{mod}\\,p^2\\\\&=&q_p(2)(1-p\\,w_p)\\qquad \\qquad \\qquad \\qquad \\qquad \\;\\;\\;\\;\\,\\,\\;\\;\\text{mod}\\,p^2$ By comparing Congruence $(89)$ with Congruence $(15)$ of Theorem 8 (point $(i)$ ), we conclude to Congruence $(17)$ of Theorem 9.",
"And since by Remark 1, we also have (cf Glaisher's result), $q_p(2)=H^{^{\\prime }}_{p-1}+\\frac{1}{2}p(H^{^{\\prime }}_{p-1})^2\\;\\;\\text{mod}\\,p^2,$ then Congruences $(18)$ and $(19)$ of Theorem 10 follow.",
"We end this part by showing that $(N_{p-2})_0$ can be expressed in terms of a sum of harmonic numbers with odd indices (cf Proposition 4 below) and we shall also get an expression to the next power of $p$ (cf forthcoming Proposition 8) Proposition 4 $N_{p-2}=\\sum _{\\begin{array}{l}m=1\\\\\\text{m odd}\\end{array}}^{p-2}H_m\\;\\;\\text{mod}\\,p$ Proof.",
"Let $m$ be an integer with $2m\\le p-3$ .",
"A classical formula on Eulerian numbers reads $E(p-2,2m)=\\sum _{k=0}^{2m}(-1)^k\\binom{p-1}{k}(2m+1-k)^{p-2}$ Lemma 5 The following congruence holds: $\\forall 1\\le k\\le p-1,\\,\\binom{p-1}{k}= (-1)^k\\;\\;\\text{mod}\\,p$ Proof of the lemma.",
"As is well known, $\\binom{p}{k}=\\binom{p-1}{k-1}+\\binom{p-1}{k}$ and $\\binom{p}{k}$ is divisible by $p$ for each integer $k$ with $1\\le k\\le p-1$ .",
"It follows that $\\binom{p-1}{k}=-\\binom{p-1}{k-1}\\;\\;\\text{mod}\\,p$ Then, working inductively, we obtain the result of the lemma.",
"We note that conversely, it was recently proven by Mestrovic in [13] that if $n>1$ and $q>1$ are integers such that $\\forall 0\\le k\\le n-1,\\,\\binom{n-1}{k}=(-1)^k\\,\\text{mod}\\,q,$ then $q$ is a prime and $n$ is a power of $q$ .",
"Applying Lemma 5 to Eq.",
"$(91)$ yields: $E(p-2,2m)&=&\\sum _{k=0}^{2m}\\;(2m+1-k)^{p-2}\\;\\;\\text{mod}\\,p\\\\&&\\\\&=&\\sum _{k=0}^{2m}\\;\\frac{1}{2m+1-k}\\qquad \\;\\;\\;\\text{mod}\\,p\\\\&&\\\\&=&H_{2m+1}\\qquad \\qquad \\qquad \\;\\;\\text{mod}\\,p$ The result of Proposition 4 follows.",
"By the same arguments as before, we have for each $m$ with $0\\le m\\le \\frac{p-5}{2}$ , $E(p-2,2m+1)=H_{2m+2}\\;\\;\\text{mod}\\,p$ It follows that $(p-2)!=\\sum _{k=0}^{p-3}E(p-2,k)&=&\\sum _{m=0}^{\\frac{p-3}{2}}E(p-2,2m)+\\sum _{m=0}^{\\frac{p-5}{2}}E(p-2,2m+1)\\\\&&\\\\&=&\\sum _{\\begin{array}{l}m=1\\\\\\text{m odd}\\end{array}}^{p-2}H_m+H_2+H_4+\\dots +H_{p-3}\\\\&&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\;\\;\\text{mod}\\,p$ And since by Wolstenholme's theorem we also have $H_{p-1}=0\\,\\text{mod}\\,p$ , we conclude to the congruence: Proposition 5 $\\sum _{k=1}^{p-1}H_k=1\\;\\;\\text{mod}\\,p$ This is Corollary $1.5$ of [14], which we retrieved here by a different method.",
"In [14], Mestrovic and Andjic prove as a special case of a more general result the stronger statement.",
"Theorem 19 (Mestrovic and Andjic [14], 2017) $\\sum _{k=1}^{p-1}H_k=1-p\\;\\text{mod}\\,p^3$ Proposition 4 joint with Eq.",
"$(18)$ of Theorem 10 imply the unexpected result of Theorem 11 from the introduction.",
"We now investigate $N_{p-2}$ modulo $p^2$ .",
"To that aim, we first show the following proposition.",
"Proposition 6 If $p\\ge 3$ is a prime, then $\\binom{p-1}{k}=(-1)^k(1-p\\,H_k)\\;\\;\\text{mod}\\,p^2$ Indeed, the result follows from the following two lemmas.",
"Lemma 6 $\\binom{p-1}{k}=\\binom{p}{k}-\\binom{p}{k-1}+\\binom{p}{k-2}-\\dots +(-1)^{k-1}\\binom{p}{1}+(-1)^k$ Lemma 7 $\\binom{p}{k}=p\\;\\frac{(-1)^{k+1}}{k}\\;\\;\\text{mod}\\,p^2$ We note that Proposition 6 is part of a stronger congruence by Zhi-Wei Sun.",
"Theorem 20 (Z-W. Sun, Lemma $2.1$ $(2.2)$ of [19], 2012) If $p\\ge 3$ is a prime, then $\\binom{p-1}{k}=(-1)^k(1-p\\,H_k+\\frac{p^2}{2}(H_k^2-H_{k,2}))\\;\\text{mod}\\,p^3$ We deduce an expression for $E(p-2,2m)$ modulo $p^2$ .",
"Proposition 7 $E(p-2,2m)=\\sum _{s=1}^{2m+1}s^{p-2}-p\\;\\sum _{K=p-(2m+1)}^{p-2}\\frac{H_K}{K+(2m+2)}\\;\\;\\text{mod}\\,p^2$ Proof.",
"We have by applying Proposition 6 in Eq.",
"$(91)$ , $E(p-2,2m)=\\sum _{s=1}^{2m+1}s^{p-2}-p\\,\\sum _{s=1}^{2m}\\frac{H_{2m+1-s}}{s}\\;\\;\\text{mod}\\,p^2$ Moreover, we have by using Wolstenholme's theorem, $H_{p-(2m+1-s+1)}=H_{2m+1-s}\\;\\;\\text{mod}\\,p$ Indeed, we proceed by induction.",
"We have, $H_{p-2}&=&H_{p-1}-\\frac{1}{p-1}=1=H_1\\;\\;\\text{mod}\\,p\\\\H_{p-3}&=&H_{p-2}-\\frac{1}{p-2}=1+\\frac{1}{2}=H_2\\;\\;\\text{mod}\\,p\\\\&\\vdots &\\\\H_{p-k}&=&H_{p-k+1}-\\frac{1}{p-k+1}=1+\\frac{1}{2}+\\dots +\\frac{1}{k-1}=H_{k-1}\\;\\;\\text{mod}\\,p$ Congruence $(102)$ follows immediately.",
"We derive Proposition 8 below.",
"Proposition 8 $N_{p-2}=\\sum _{m=0}^{\\frac{p-3}{2}}\\Bigg (S_{2m+1,\\,p-2}-p\\;\\sum _{K=p-(2m+1)}^{p-2}\\frac{H_K}{K+(2m+2)}\\Bigg )\\;\\;\\text{mod}\\,p^2$ We note that our formula is consistent with our Proposition 4 modulo $p$ .",
"Proposition 8 is only the starting point of further investigations needed.",
"Namely, expressing Proposition 4 and 8 in terms of Bernoulli numbers requests more work.",
"It is to expect that working out further the sums in Proposition 8 could lead to conclusions on the existence or non-existence of Wilson and super Wieferich primes."
],
[
"Ending notifications and proof of Theorem 15",
"When we were about to submit this paper, we learned that Zhi-Wei Sun, Zhi-Hong Sun's brother, and his co-author Li-Lu Zhao studied congruences concerning the same weighted sums, where the (non divided) Bernoulli numbers are replaced with harmonic numbers or generalized harmonic numbers instead.",
"Their result taken from [20] is the following, where notations are standard.",
"Theorem 21 (Zhi-Wei Sun and Li-Lu Zhao [20], 2013) Let $p$ be a prime with $p>3$ .",
"Then, $\\sum _{k=1}^{p-1}\\frac{H_k}{k\\,2^k}&=&\\frac{7}{24}\\,p\\,B_{p-3}\\;\\;(mod\\,p^2)\\\\\\sum _{k=1}^{p-1}\\frac{H_{k,2}}{k2^k}&=&-\\frac{3}{8}\\,B_{p-3}\\;\\;(mod\\,p)$ Previously, Zhi-Wei Sun in [19] had shown under the same conditions that $\\sum _{k=1}^{p-1}\\frac{H_k}{k2^k}=0\\;\\;\\text{mod}\\,p$ We note that if $(p,p-3)$ is an irregular pair, the powers of two weighted sum of the first $(p-1)$ divided harmonic numbers (resp generalized harmonic numbers) is divisible by $p^2$ (resp by $p$ ).",
"Such a prime is called a Wolstenholme prime as for these primes, Wolstenholme's theorem [23] holds to the next power of $p$ , that is $H_{p-1}=0\\;\\text{mod}\\,p^3$ and $H_{p-1,2}=0\\;\\text{mod}\\,p^2$ .",
"Here, our congruence $(10)$ implies that $\\sum _{k=1}^{p-2}\\frac{B_k}{k2^k}=0\\;\\text{mod}\\,p\\Leftrightarrow q_p(2)=-w_p\\;\\text{mod}\\,p$ We provide a polynomial interpretation of Congruence $(106)$ .",
"Our polynomial actor is $X^{p-1}+(p-1)!$ , whose factored form in the ring of $p$ -adic integers appears in the proposition below.",
"Proposition 9 We have the factorization in $\\mathbb {Z}_p[X]$ and subsequent equality $X^{p-1}+(p-1)!&=&(X-1-pt_1)(X-2-pt_2)\\dots (X-(p-1)-pt_{p-1})\\\\q_p(2)+w_p&=&-t_2(1-pt_1)\\Big [\\prod _{i=1}^{p-3}(i+p\\,t_{i+2})\\Big ]$ Proof.",
"By [11], we may factor the given polynomial in $\\mathbb {Z}_p[X]$ as in the statement, where the $t_i$ 's with $1\\le i\\le p-1$ are some adequate $p$ -adic integers.",
"Specializing $X=2$ leads to identity $(107)$ .",
"We retrieve from $(107)$ the fact that $t_2^{(0)}=2(q_p(2)+w_p)\\,\\;\\;\\text{mod}\\,p$ .",
"This is a special case of Lemma 1 of [11].",
"Thus, we see that Congruence $(106)$ holds if and only if the second residue in the $p$ -adic expansion of the root of $X^{p-1}+(p-1)!$ of residue 2 is zero.",
"In fact, we can generalize this result to powers of $k$ weighted sums of divided Bernoulli numbers, when $k$ is an integer with $3\\le k\\le p-1$ .",
"It suffices to establish the following two lemmas.",
"Lemma 8 Let $k$ be an integer with $1\\le k\\le p-1$ .",
"Then, $p=-\\sum _{s=1}^{p}\\left[\\begin{array}{l}p\\\\s\\end{array}\\right]\\,k^s\\;\\;\\text{mod}\\,p^2$ Lemma 9 Let $k$ be an integer with $1\\le k\\le p-1$ .",
"Let $q_p(k)$ denote the Fermat quotient to base $k$ , with standard notations.",
"Then, $\\sum _{l=1}^{p-2}\\frac{B_l}{l\\,k^l}=0\\;\\;\\text{mod}\\,p\\Leftrightarrow q_p(k)=-w_p\\;\\;\\text{mod}\\,p$ Proof.",
"Lemma 8 follows from specializing $x=-k$ in Eq.",
"$(4)$ and applying Wilson's theorem modulo $p$ .",
"First, we have from the specialization: $\\frac{(p-1+k)!}{(k-1)!",
"}=\\sum _{s=1}^p\\left[\\begin{array}{l}p\\\\s\\end{array}\\right]k^s$ Then, we write $(p-1+k)!&=&(p-1)!p(p+1)\\dots (p-1+k)\\\\&=&-p\\,(k-1)!\\;\\;\\text{mod}\\,p^2$ Then, Lemma 9 can be derived from Lemma 8 by using the expressions for the Stirling numbers modulo $p^2$ as provided for instance in [11].",
"The calculations are similar to those realized before and are left to the reader.",
"Moreover, we have by using the specialization $X=k$ in both factored and expanded forms of the polynomial $X^{p-1}+(p-1)!$ , $q_p(k)+w_p=(-1)^{k+1}\\,t_k^{(0)}\\,(k-1)!\\,(p-1-k)!\\;\\;\\text{mod}\\,p$ This implies by the Gauss lemma that $t_k^{(0)}=0$ .",
"Hence the result of Theorem 15 that was originally announced at the end of the introduction.",
"Note that the theorem holds even when $k=1$ .",
"Email address: [email protected]"
]
] | 2001.03471 |
[
[
"Tidal Disruptions of Main Sequence Stars -- I. Observable Quantities and\n their Dependence on Stellar and Black Hole Mass"
],
[
"Abstract This paper introduces a series of papers presenting a quantitative theory for the tidal disruption of main sequence stars by supermassive black holes.",
"Using fully general relativistic hydrodynamics simulations and MESA-model initial conditions, we explore the pericenter-dependence of tidal disruption properties for eight stellar masses ($0.15 \\leq M_*/M_\\odot \\leq 10$) and six black hole masses ($10^5 \\leq M_{BH}/M_\\odot \\leq 5 \\times 10^7$).",
"We present here the results most relevant to observations.",
"The effects of internal stellar structure and relativity decouple for both the disruption cross section and the characteristic energy width of the debris.",
"Moreover, the full disruption cross section is almost independent of $M_*$ for $M_*/M_\\odot \\lesssim 3$.",
"Independent of $M_*$, relativistic effects increase the critical pericenter distance for full disruptions by up to a factor $\\sim 3$ relative to the Newtonian prediction.",
"The probability of a direct capture is also independent of $M_*$; at $M_{BH}/M_\\odot \\simeq 5 \\times 10^6$ this probability is equal to that of a complete disruption.",
"The width of the debris energy distribution $\\Delta E$ can differ from the standard estimate by factors from 0.35 to 2, depending on $M_*$ and $M_{BH}$, implying a corresponding change in the characteristic mass-return timescale.",
"The \"frozen-in approximation\" is inconsistent with $\\Delta E$, and mass-loss continues over a long span of time.",
"We provide analytic forms, suitable for use in both event rate estimates and parameter inference, to describe all these trends.",
"For partial disruptions, we find a nearly-universal relation between the star's angular momentum and the fraction of $M_*$ remaining.",
"Within the \"empty loss-cone\" regime, partial disruptions must precede full disruptions.",
"These partial disruptions can drastically affect the rate and appearance of subsequent total disruptions."
],
[
"Introduction",
"Supermassive black holes (SMBHs) reside in the nuclei of virtually every massive galaxy [25].",
"The orbits of stars around the central BH are stochastically perturbed by weak gravitational encounters with other stars.",
"Occasionally these perturbations place stars on orbits taking them so close to the BH that they are tidally disrupted, losing part or all of their mass in a tidal disruption event (TDE).",
"Roughly half the mass torn off the star swings far out from the stellar orbit's pericenter and then returns.",
"The energy it releases as it falls deeper into the black hole potential generates a luminous flare.",
"Many examples of tidal disruption events have now been seen.",
"Since the detection of the first TDE candidates [24] in the ROSAT all-sky survey [55], greatly improved searches have been conducted, including X-ray surveys such as the XMM-$Newton$ slew survey [50] and UV/optical surveys, e.g., the GALEX Deep Imaging Survey [11], Pan-STARRS [6], PTF [29] and ASAS-SN [18].",
"From these, dozens of transients have been identified as TDE candidates [23], [56].",
"In the near future, this number is likely to grow rapidly with detections by ongoing surveys like the Zwicky Transient Facility [14] and upcoming surveys, e.g., the eROSITA All-Sky Survey [37] and the Large Synoptic Survey Telescope [33].",
"An order-of-magnitude estimate for the “tidal radius\" see Equation REF below) is commonly used as an indicator of when a star is torn apart.",
"Another order-of-magnitude argument is used to estimate the energy spread of the debris in order to set the timescale of the event.",
"Newtonian dynamics underly both of these estimates, even though typically these events take place no more than a few tens of gravitational radii from a black hole.",
"Although tidal effects are strongly dependent upon distance from the star's center-of-mass, often no consideration is given to the stars' internal density profiles, or the polytrope approximation is taken to be general.",
"The energy spread estimate is based upon conditions at a single point in the star's orbit (sometimes the “tidal radius\", sometimes the pericenter) although the star can travel a significant distance while it loses mass.",
"Finally, relatively little attention is paid to partial disruptions, although the rate at which they occur should be comparable to, or even larger than to the rate of total disruptions.",
"It is our goal to remove all these limitations.",
"To clarify the context in which we are working, it is useful to briefly expand upon the present state-of-the-art.",
"The term “tidal radius\" usually refers to an estimate of the pericenter distance at which tidal effects can be important to stars [17]: $r_{\\rm t}&=\\left(\\frac{M_{\\rm BH}}{M_{\\star }}\\right)^{1/3}R_{\\star }\\\\\\nonumber &\\simeq 47\\left(\\frac{M_{\\rm BH}}{10^{6}\\;\\mathrm {M}_{\\odot }}\\right)^{-2/3}\\left(\\frac{M_{\\star }}{1\\;\\mathrm {M}_{\\odot }}\\right)^{-1/3}\\left(\\frac{R_{\\star }}{1\\;\\mathrm {R}_{\\odot }}\\right)r_{\\rm g},$ where $M_{\\star }$ and $R_{\\star }$ are the stellar mass and radius, respectively.",
"$M_{\\rm BH}$ is the mass of the BH and $r_{\\rm g}$ is the gravitational radius of the BH, with $r_{\\rm g} \\equiv GM_{\\rm BH}/c^{2}$ .",
"Following [46], it is generally assumed that the energy distribution of the debris mass $dM/dE$ is non-zero for $-\\Delta \\epsilon \\le E \\le +\\Delta \\epsilon $ .",
"The characteristic energy spread $\\Delta \\epsilon $ is set to an order-of-magnitude estimate for the range in fluid binding energies [53] $\\Delta \\epsilon &= \\frac{GM_{\\rm BH}R_{\\star }}{r_{\\rm t}^{2}}= \\left(\\frac{R_{\\star }}{r_{\\rm g}}\\right)\\left(\\frac{r_{\\rm t}}{r_{\\rm g}}\\right)^{-2}c^2, \\nonumber \\\\&\\sim 2 \\times 10^{-4} \\left(\\frac{R_{\\star }}{\\rm {R}_\\odot }\\right)^{-1} \\left(\\frac{M_{\\rm BH}}{10^{6}\\;\\mathrm {M}_{\\odot }}\\right)^{1/3} \\left(\\frac{M_{\\star }}{1\\;\\mathrm {M}_{\\odot }}\\right)^{2/3} c^2.$ Sometimes $r_{\\rm t}$ is replaced with the actual pericenter of the orbit $r_{\\rm p}$ [32].",
"[43] was the first to recognize that $r_{\\rm t}$ is not exactly the maximum orbital pericenter for a complete tidal disruption, a distance we would like to name the “physical tidal radius\" (we assign it the symbol $\\mathcal {R}_{\\rm t}$ ).",
"To remedy the neglect of internal stellar structure, he suggested that $\\mathcal {R}_{\\rm t}$ could be estimated by applying to $r_{\\rm t}$ a correction factor based on the star's apsidal motion constant and its dimensionless binding energy.",
"For this reason, $r_{\\rm t}$ is sometimes reinterpreted to be ${\\cal R}_{\\rm t}$ , but without evaluating how it might differ from $r_{\\rm t}$ [53].",
"Several groups have tried to include stellar structure in the calculation of $\\mathcal {R}_{\\rm t}$ , but employing purely Newtonian dynamics on polytropic stars [34], [22], [16], [35].",
"Recently, there have been efforts beginning from genuine main sequence stellar structures, but still restricted to Newtonian dynamics, and examining a limited range of stellar masses (only $1\\;\\mathrm {M}_{\\odot }$ in [12], $1\\;\\mathrm {M}_{\\odot }$ and $3\\;\\mathrm {M}_{\\odot }$ at several ages in [30], $0.3\\;\\mathrm {M}_{\\odot }$ , $1\\;\\mathrm {M}_{\\odot }$ and $3\\;\\mathrm {M}_{\\odot }$ at three different ages in [13]).",
"Others have explored the dependence on black hole mass induced by relativistic effects, but without any reference to internal stellar structure or the hydrodynamics of disruption [20], [21], [51].",
"Earlier works employed a post-Newtonain approximation [4] or explored the use of relativistic hydrodynamics simulations for strong encounters of polytropic stars, but without stellar self-gravity [28].",
"In some cases, relativistic effects were approximated by a “generalized Newtonian potential\" [10], [9] or in terms of genuine relativistic dynamics [8], but assuming a polytropic structure for the star and computing stellar self-gravity in an entirely Newtonian fashion (in the last case, fixing it to its initial stellar surface value).",
"Many of these explorations of $\\mathcal {R}_{\\rm t}$ also computed the energy distribution $dM/dE$ and explored the relation between the remnant mass and orbital pericenter in partial disruptions; in one case [36], they also examined the remnants' orbital properties.",
"However, all this work was subject to the limitations already enumerated.",
"Figure: Successive moments in a full TDE.",
"The red line indicates the star's orbit around the black hole (black circle).",
"Each inset figure presents a snapshot of the density distribution in the orbital plane within our simulation box.The white circle in each snapshot shows the initial stellar radius.",
"Partway through the event, we replace the cubic box with a rectangular box (see Ryu2+2019); we draw a red square in the rectangular boxes to show the position and size of the original cubic box.",
"Note that the rectangular boxes are not drawn to the same scale as the cubic boxes, and the dotted curves marking r t r_{\\rm t}, 10r t 10~r_{\\rm t} and 20r t 20~r_{\\rm t} are likewise not drawn to scale.This is the first in a series of four papers in which we present the results of a large number of simulations designed to include all relevant physical processes.",
"Tidal stresses are treated in exact general relativity, as are the fluid dynamics of the disrupted star.",
"The stars' initial states are taken from the stellar evolution code MESA, with ages halfway through their main sequence lifetimes so as to correspond to the time-averaged state of main-sequence disruptions.",
"Stellar self-gravity is computed with the Newtonian Poisson equation, but in a tetrad frame comoving with the star whose metric (within the simulation volume) departs from Minkowski only by very small amounts (see Appendix in [47]).",
"For a fiducial black hole mass of $10^6 \\;\\mathrm {M}_{\\odot }$ , we treat stars of eight different masses $M_{\\star }$ , from $0.15\\;\\mathrm {M}_{\\odot }$ to $10\\;\\mathrm {M}_{\\odot }$ .",
"For three of these stellar masses ($0.3\\;\\mathrm {M}_{\\odot }$ , $1\\;\\mathrm {M}_{\\odot }$ , and $3\\;\\mathrm {M}_{\\odot }$ ), we consider black holes of six different masses, from $10^5 \\;\\mathrm {M}_{\\odot }$ to $5\\times 10^7\\;\\mathrm {M}_{\\odot }$ .",
"All our black holes, however, have zero spin.",
"In each simulation, the star's trajectory has eccentricity $e$ such that $1-e \\simeq 10^{-8}$ .",
"Both to closely determine $\\mathcal {R}_{\\rm t}$ and to map out how the properties of partial disruptions depend on $r_{\\rm p}/\\mathcal {R}_{\\rm t}$ , we simulated encounters for each $(M_{\\star },M_{\\rm BH})$ pair for a number of pericenters spaced by $\\simeq 0.05-0.2~r_{\\rm t}$ .",
"Figure: (Left panel) Ψ BH ≡Ψ(M BH ;M ☆ )/Ψ(10 6 ;M ☆ ) \\Psi _{\\rm BH} \\equiv \\Psi (M_{\\rm BH}; M_{\\star })/ \\Psi (10^6;M_{\\star })as a function of M BH M_{\\rm BH} for M ☆ =0.3M_{\\star }=0.3 (blue circles), 1 (red stars) and 3 (green triangles).",
"Ψ\\Psi is defined as ℛ t /r t \\mathcal {R}_{\\rm t}/r_{\\rm t}.The dashed line depicts our fit to Ψ BH \\Psi _{\\rm BH} (Equation (Right panel) Ψ(M ☆ ,10 6 )≡Ψ ☆ (M ☆ )≡ℛ t /r t \\Psi (M_\\star ,10^6) \\equiv \\Psi _\\star (M_\\star ) \\equiv \\mathcal {R}_{\\rm t}/r_{\\rm t} for M BH =10 6 M_{\\rm BH}=10^{6}.",
"Numerical values are shown by filled circles, the analytic fit given in Equation by a red dashed curve.",
"In both panels, the error bars indicate the uncertainties in ℛ t \\mathcal {R}_{\\rm t} originating from the finite sampling of r p /r t r_{\\rm p}/r_{\\rm t}.A schematic overview of the entire process can be viewed in Figure REF .",
"The star begins to be stretched when its distance to the black hole is $r_{\\rm t}$ , and is already quite distorted by the time it reaches its pericenter (here $\\sim r_{\\rm t}/2$ ).",
"However, substantial mass-loss continues until the star has traveled far from the black hole (see [47] for full details.)",
"In this first paper, we give an overview of this series' principal findings: the physical tidal radius as a function of stellar mass and black hole mass (Section REF ), the full disruption cross section (Section REF ), the energy scale of stellar debris from full disruptions and its scaling with $M_{\\star }$ and $M_{\\rm BH}$ (Section ) and the relationship between remnant mass and orbital angular momentum in partial disruptions (Section ).",
"We then discuss the implications these results have for predicted event rates for full and partial disruptions (Section REF ), and for the orbital properties of stellar debris from full disruptions (Section REF ).",
"We conclude with a summary of our findings (Section ).",
"The following three papers in this series provide details supporting and expanding upon the findings discussed in this paper.",
"In [47] (Ryu2+2019) and [48] (Ryu3+2019), we focus on the stellar mass dependence of disruption outcomes: In Ryu2+2019, we: describe our methodology, including hydrodynamic algorithms, relativistic self-gravity calculation, and grid-resolution; explain our MESA-based initial conditions and simulation setup; present our detailed results having to do full disruptions; and compare these results to related work by others.",
"Ryu3+2019 reports our results relevant to partial disruptions.",
"In [49] (Ryu4+2019), we explore how relativistic effects lead to dependence of TDE properties on black hole mass.",
"There we demonstrate how our study extends prior efforts concerning relativistic effects and evaluate the quality of various approximations to general relativistic physics in the TDE literature.",
"Hereafter all masses are measured in solar mass and stellar radii in solar radius.",
"The physical tidal radius $\\mathcal {R}_{\\rm t}$ , the maximum pericenter for a total disruption, plays a key role in determining the fate of a TDE.",
"We define $\\Psi $ as the ratio of $\\mathcal {R}_{\\rm t}$ to the “order of magnitude tidal radius\" $r_{\\rm t}$ defined in Equation REF , i.e., $\\Psi \\equiv \\mathcal {R}_{\\rm t}/r_{\\rm t}$ .",
"The physical radius $\\mathcal {R}_{\\rm t}$ depends on both stellar mass $M_{\\star }$ and black hole mass $M_{\\rm BH}$ .",
"Combining results from Ryu2+2019 and Ryu4+2019, we find, as illustrated in the left panel of Figure REF , that the $M_{\\rm BH}$ -dependences for different stellar masses are essentially identical.",
"This fact allows us to factor out the dependence of $\\Psi $ on $M_{\\star }$ from its dependence on $M_{\\rm BH}$ , $\\Psi (M_{\\rm BH},M_{\\star }) \\equiv \\frac{\\mathcal {R}_{\\rm t}}{r_{\\rm t}} = \\Psi _{\\rm BH}(M_{\\rm BH})\\Psi _{\\star } (M_{\\star }),$ where $\\Psi _{\\star }$ accounts for the stellar internal structure while $\\Psi _{\\rm BH}$ encapsulates the behavior due to relativistic effects.",
"We choose to describe these two functions with $\\Psi _{\\rm BH}$ normalized to unity for $M_{\\rm BH}=10^6$ .",
"Figure: The ratio of the relativistic full tidal disruption cross section to its Newtonian analog, (ℒ t 2 -L dc 2 )/(L N 2 -L N, dc 2 )(\\mathcal {L}_{\\rm t}^2 - L_{\\rm dc}^2)/(L_{\\rm N}^2-L_{\\rm N,dc}^2), as a function of M BH M_{\\rm BH} for three stellar masses.The error bars indicate the errors propagated from the uncertainties of ℛ t \\mathcal {R}_{\\rm t}.Note that the ratio for M ☆ =3M_{\\star }=3 (blue circles) and M BH =5×10 7 M_{\\rm BH}=5\\times 10^{7} is not plotted because L N <L N, dc L_{\\rm N}<L_{\\rm N,dc}.For all three values of $M_{\\star }$ we studied (i.e., $M_{\\star }=0.3$ , 1, and 3) the dependence of $\\Psi $ on $M_{\\rm BH}$ is very well described by $\\Psi _{\\rm BH}(M_{\\rm BH})=0.80 + 0.26~\\left(\\frac{M_{\\rm BH}}{10^{6}}\\right)^{0.5},$ as depicted by the dashed curve in the left panel of Figure REF .",
"From the Newtonian limit (represented by $M_{\\rm BH}=10^5$ ) to the highest black hole mass we examined ($M_{\\rm BH} = 5 \\times 10^7$ ), $\\Psi _{\\rm BH}$ increases by a factor of 3, with its slope continually steepening as $M_{\\rm BH}$ grows.",
"Just as the dashed curve in the left panel of Figure REF represents the black hole mass-dependence of $\\mathcal {R}_{\\rm t}$ for all stellar masses, the dashed curve in the right panel represents the shape of the stellar mass dependence of $\\Psi $ ($\\Psi _{\\star }$ ) for all black hole masses; it is well-described by the expression $\\Psi _{\\star }(M_{\\star })& = \\frac{1.47+ ~\\exp [(M_{\\star }-0.669 )/0.137]}{1 + 2.34~\\exp [(M_{\\star }-0.669)/0.137]}.$ Although $\\Psi _{\\star }$ becomes nearly constant at both mass extremes, it has a sharp transition across the range of masses $0.4\\lesssim M_{\\star } \\lesssim 1$ .",
"For low-mass stars For explanatory convenience, we categorize stars into “low-mass” ($M_{\\star }\\le 0.5$ ) and “high-mass” ($M_{\\star }\\ge 1)$ based on the properties of TDE outcomes.",
"These mass ranges may be different from those typically used in stellar evolution studies.",
"($M_{\\star } \\le 0.5$ ), which are predominantly convective, $\\Psi _{\\star } \\simeq 1-1.45$ .",
"For higher mass stars ($M_{\\star } \\ge 1$ ), which are predominantly radiative, $\\Psi _{\\star } \\simeq 0.45$ .",
"The large coefficient of $M_{\\star }$ in the exponentials of Equation REF conveys how sharp the transition is from low-mass to high-mass stars.",
"When $\\mathcal {R}_{\\rm t}$ is measured not in ratio to $r_{\\rm t}$ , but in physical units (e.g., $r_{\\rm g}=GM_{\\rm BH}/c^{2}$ ), its value depends on $M_{\\rm BH}$ , but is nearly independent of $M_{\\star }$ over most of the (logarithmic) range of possible stellar masses.",
"For $M_{\\rm BH} = 10^6$ , it is almost constant at $\\simeq ~27~r_{\\rm g}$ from $M_{\\star }\\simeq 0.15$ to $M_{\\star } \\simeq 3$ (the right panel of Figure REF ), with a maximum departure only 20% either up or down.",
"Such a small contrast in $\\mathcal {R}_{\\rm t}$ across nearly the entire range of stellar mass is very different from the strong dependence on $M_{\\star }$ predicted by the order-of-magnitude estimate $r_{\\rm t}$ (the dot-dashed curve in the right panel of Figure REF ), which rises as $\\propto M_{\\star }^{-1/3}R_{\\star }\\propto M_{\\star }^{0.55}$ (Ryu2+2019).",
"The fact that $\\mathcal {R}_{\\rm t}/r_{\\rm g}$ is nearly independent of stellar mass likely also explains the near $M_{\\star }$ -independence of $\\Psi _{\\rm BH}$ : $\\mathcal {R}_{\\rm t}$ lies at roughly the same location in the black hole potential for stars of all masses.",
"The value of $\\mathcal {R}_{\\rm t}$ is primarily determined by the star's central density $\\rho _{\\rm c}$ rather than by its mean density $\\bar{\\rho }_{\\star }(=3M_{\\star }/[4R_{\\star }^{3}])$ .",
"In Ryu2+2019 we demonstrate that $\\mathcal {R}_{\\rm t} \\simeq 2.13\\left(\\frac{\\bar{\\rho }_{\\star }}{\\rho _{\\rm c}}\\right)^{1/3} r_{\\rm t}\\simeq 1.32\\left(\\frac{M_{\\rm BH}}{\\rho _{\\rm c}}\\right)^{1/3}.$ The increase in $\\Psi _{\\rm BH}$ toward larger $M_{\\rm BH}$ results in a slower decrease in $\\mathcal {R}_{\\rm t}/r_{\\rm g}$ with $M_{\\rm BH}$ than would be predicted by Newtonian dynamics (the left panel of Figure REF ).",
"Whereas the Newtonian prediction is that $r_{\\rm t}/r_{\\rm g}\\propto M_{\\rm BH}^{-2/3}$ , for $M_{\\rm BH}\\gtrsim 5\\times 10^{6}$ , $-d\\ln (\\mathcal {R}_{\\rm t}/r_{\\rm g})/d \\ln M_{\\rm BH} \\simeq 0.3$ –0.4.",
"Nonetheless, the $M_{\\rm BH}$ -dependence of $\\mathcal {R}_{\\rm t}/r_{\\rm g}$ can still be well-described with a term $\\propto M_{\\rm BH}^{-2/3}$ , provided a weakly $M_{\\star }$ -dependent offset is added: $\\frac{\\mathcal {R}_{\\rm t}}{r_{\\rm g}} ={\\left\\lbrace \\begin{array}{ll}23.5 ~\\left(\\frac{M_{\\rm BH}}{10^{6}} \\right)^{-2/3} + 3.5 \\hspace{7.22743pt} \\text{for $M_{\\star }=0.3$,}\\\\19.1 ~\\left(\\frac{M_{\\rm BH}}{10^{6}}\\right)^{-2/3} + 3.9 \\hspace{7.22743pt} \\text{for $M_{\\star }=1.0$,}\\\\28.8 ~\\left(\\frac{M_{\\rm BH}}{10^{6}} \\right)^{-2/3} + 4.6\\hspace{7.22743pt} \\text{for $M_{\\star }=3.0$.}\\end{array}\\right.",
"}$"
],
[
"Event cross sections: comparison with estimates based on $r_{\\rm t}$ .",
"The cross section for tidal encounters with pericenter $\\le r_{\\rm p}$ is proportional to $L^{2}(r_{\\rm p})$ .",
"Here, $L$ is the specific angular momentum of the star's orbit.",
"For parabolic orbits, $L$ is related to $r_{\\rm p}$ by $L^2(r_{\\rm p}) = (r_{\\rm g}c)^2 \\frac{2 (r_{\\rm p}/r_{\\rm g})^2}{r_{\\rm p}/r_{\\rm g}- 2},$ which reduces to $[L(r_{\\rm p})/(r_{\\rm g}c)]^{2} = 2~r_{\\rm p}/r_{\\rm g}$ in the Newtonian limit.",
"Therefore, the cross section for full disruptions is $\\propto L^{2}(\\mathcal {R}_{\\rm t}) \\equiv \\mathcal {L}_{\\rm t}^2$ , while the Newtonian angular momentum associated with a pericenter of $r_{\\rm t}$ we call $L_{\\rm N}^2 = 2 G M_{\\rm BH}r_{\\rm t}$ .",
"To demonstrate how our calculations for full disruption cross sections compare to estimates made without regard to either internal stellar structure or relativity, we examine in Figure REF the ratio of our calculation of the total disruption cross section to the estimate neglecting relativity and stellar structure, i.e., $(\\mathcal {L}_{\\rm t}^2 - L_{\\rm dc}^2)/(L_{\\rm N}^2-L_{\\rm N,dc}^{2})$ for $M_{\\star }=0.3$ , 1 and 3.",
"Here, $L_{\\rm dc} = 4r_{\\rm g} c$ is the angular momentum of a parabolic direct capture orbit, and $L_{\\rm N,dc}=2r_{\\rm g}c$ .",
"Relativistic effects become important only for $M_{\\rm BH} \\gtrsim 10^7$ .",
"When they do enter significantly, the cross section for total disruptions of low-mass stars rises sharply relative to the simple prediction, reaching a factor $5\\times $ greater for $M_{\\rm BH}=3 \\times 10^7$ .",
"For these stars, the greater strength of relativistic tidal stress is the dominant mechanism.",
"However, for high-mass stars, the dominant effect is the growing importance of direct capture, and, relative to the simple estimate, the cross section for total disruptions falls."
],
[
"Energy distribution and characteristic energy width of stellar debris for full disruptions",
"In the conventional picture [46], the distribution function of debris mass with energy $dM/dE$ is approximated as flat from $-\\Delta \\epsilon $ to $+\\Delta \\epsilon $ , and vanishes outside that range.",
"The magnitude of $\\Delta \\epsilon = {GM_{\\rm BH}R_{\\star }}/{r_{\\rm t}^{2}}$ was estimated by [46] on the basis of tidal spin-up; the same estimate results from consideration of the spread of potential energy within the star when its distance from the black hole is some fiducial value [27], [32], [53], [51]; this fiducial value is chosen variously to be $r_{\\rm p}$ [32] or $r_{\\rm t}$ [53].",
"We use the latter in the definition of $\\Delta \\epsilon $ .",
"To determine the orbital energy of the debris from our simulation data, we transform the 4-velocity at each boundary cell from the simulation box coordinates following the star frame to the Schwarzchild coordinates associated with the black hole.",
"The energy $E$ is then given by $-1 - u_{\\rm t}$ evaluated in the Schwarzchild frame (we use a $-+++$ signature).",
"Figure REF shows the energy distribution $dM/dE$ for three different stellar masses (see Figure 8 in Ryu2+2019 for $dM/dE$ for five more values of $M_{\\star }$ ).",
"Although not far from flat, particularly for low-mass stars, it is not exactly flat, and in units of $\\Delta \\epsilon $ tends to be wider for higher $M_{\\star }$ .",
"Nonetheless, it still retains a sharp edge, clearly defining a characteristic value of the energy.",
"We define this characteristic width of the distribution function $\\Delta E$ such that the mass within $|E|<\\Delta E$ is 90% of the total debris mass.",
"The dependence of $\\Delta E/\\Delta \\epsilon $ upon both $M_{\\star }$ and $M_{\\rm BH}$ can be reasonably well-described by an analytic form related to the ones we have proposed for $\\mathcal {R}_{\\rm t}/r_{\\rm t}(\\equiv \\Psi )$ : $\\Xi (M_{\\star },M_{\\rm BH})\\equiv \\frac{\\Delta E}{\\Delta \\epsilon }\\simeq \\Psi _{\\rm BH}^{-1}\\left[\\frac{2}{1 + \\Psi _{\\star }^2}\\right] .$ As shown in Figure REF , this expression does a fairly good job matching the trends with $M_{\\rm BH}$ (the left panel) and $M_{\\star }$ (the right panel) found in our data.",
"Its form also implies that, like $\\mathcal {R}_{\\rm t}/r_{\\rm t}$ , $M_{\\rm BH}$ -dependence (expressed by $\\Psi _{\\rm BH}^{-1}$ ) and $M_{\\star }$ -dependence (the factor $2[1+\\Psi _{\\star }^{2}]^{-1}$ ) are separable.",
"Figure: The fractional remnant mass M rem /M ☆ M_{\\rm rem}/M_{\\star }.",
"(Left panel) M rem /M ☆ M_{\\rm rem}/M_{\\star } for M ☆ =3M_{\\star }=3 stars disrupted by BHs with various M BH M_{\\rm BH} as a function of the ratio of the cross-section for all disruptions to the cross section for full disruptions.",
"(Right panel) M rem /M ☆ M_{\\rm rem}/M_{\\star } as a function of r p /ℛ t r_{\\rm p}/\\mathcal {R}_{\\rm t} for stars of eight different masses disrupted by a 10 6 M ⊙ 10^{6}\\;\\mathrm {M}_{\\odot } BH.",
"In both panels, the shaded regions around the solid lines demarcate the ranges determined by the uncertainties of ℛ t \\mathcal {R}_{\\rm t}, filled with the same colors as the solid lines.",
"The uncertainty in ℛ t \\mathcal {R}_{\\rm t} is due to our discrete sampling of r p r_{\\rm p} (0.05-0.10.05-0.1 in r p /r t r_{\\rm p}/r_{\\rm t}).",
"The dotted horizontal lines show the 50% and 90% remnant mass-fraction levels.",
"The fitting formula is plotted using a black dashed line in each panel, Equation (left panel) and Equation (right panel).Equation REF also has a significant implication: the two key quantities defining the properties of TDEs, the physical tidal radius and the characteristic energy width, are related.",
"However, they are not related in a trivial way with $\\Delta E \\propto \\mathcal {R}_{\\rm t}^{-1}$ .",
"If one wishes to rewrite Equation REF in terms of a single radius, that radius is a complicated function of $r_{\\rm t}$ and $\\mathcal {R}_{\\rm t}$ .",
"Nonetheless, it is in this way that the energy width, which sets the characteristic fallback time and therefore the peak fallback rate (Section REF ), can be found directly from $r_{\\rm t}$ supplemented by $\\Psi _{\\star }$ and $\\Psi _{\\rm BH}$ .",
"The nature of the $M_{\\star }$ - and $M_{\\rm BH}$ -dependence of $\\Delta E/\\Delta \\epsilon $ is that, at fixed $M_{\\rm BH}$ , $\\Delta E$ is smaller in magnitude than $\\Delta \\epsilon $ for the low-mass, less centrally-concentrated stars, while it is larger in magnitude for the high-mass, more centrally-concentrated stars.",
"At fixed $M_{\\star }$ , $\\Delta E/\\Delta \\epsilon $ diminishes with increasing $M_{\\rm BH}$ .",
"If an even closer fit is required than the simple one defined in Equation REF , one can instead make use of two fitting formulae, of which the first substitutes for $\\Psi _{\\rm BH}^{-1}$ in Equation REF , while the second replaces $2/(1+\\Psi _{\\star }^2)$ : $\\Xi _{\\rm BH}(M_{\\rm BH}) &=1.27 - 0.300\\left(\\frac{M_{\\rm BH}}{10^{6}}\\right)^{0.242},\\\\ \\Xi _{\\star }(M_{\\star }) &= \\frac{0.620+\\exp {[(M_{\\star }-0.674)/0.212]}}{1 + 0.553~\\exp {[(M_{\\star }-0.674)/0.212]}}.$ These expressions replicate our numerical results with very small errors (the dashed lines in both panels of Figure REF )."
],
[
"Remnant mass in partial disruptions",
"Stars are partially disrupted and lose some fraction of their masses when $\\mathcal {R}_{\\rm t}<r_{\\rm p} < \\widehat{R}_{\\rm t}$ .",
"$\\widehat{R}_{\\rm t}$ refers to the largest pericenter distance yielding partial disruptions, which is a few times $\\mathcal {R}_{\\rm t}$ (Ryu3+2019, Ryu4+2019).",
"Substantial mass-loss (e.g., as defined by $M_{\\rm rem}/M_{\\star } < 0.7$ ) occurs for predominantly radiative stars for $r_{\\rm p}/\\mathcal {R}_{\\rm t}\\lesssim 1.5$ , but for a somewhat smaller range of pericenters for convective stars.",
"Remarkably, the most noteworthy parameter characterizing a partial disruption, the mass retained by the star at the end of the event, can be simply related to the star's angular momentum $L$ , making it easy to estimate the rate at which partial disruptions of a given character occur.",
"As illustrated in the left panel of Figure REF , we find that for partial tidal disruptions by a non-spinning black hole it is a good approximation—independent of both $M_{\\star }$ and $M_{\\rm BH}$ —to write $\\frac{M_{\\rm rem}}{M_{\\star }} = 1 - \\left[\\frac{L(r_{\\rm p})^{2}-L_{\\rm dc}^{2}}{\\mathcal {L}_{\\rm t}^{2}-L_{\\rm dc}^{2}}\\right]^{-3}.$ In the Newtonian limit, this expression reduces to $\\frac{M_{\\rm rem}}{M_{\\star }} = 1 - \\left(\\frac{r_{\\rm p}}{\\mathcal {R}_{\\rm t}}\\right)^{-3}.$ which is depicted in the right panel of Figure REF , showing the behavior of different mass stars when $M_{\\rm BH}$ is small enough that the Newtonian limit does not create a large error.",
"The full expression (Equation REF ) is surprisingly good considering the wide range of $M_{\\star }$ and $M_{\\rm BH}$ to which it can be applied.",
"For $M_{\\star }=3$ , the curve defined by Equation REF runs right down the middle of the $M_{\\rm rem}/M_{\\star }$ curves for specific black hole masses.",
"The largest departure (for the smallest $M_{\\rm rem}/M_{\\star }$ with $M_{\\rm BH}=10^{5}$ ) is a factor $\\simeq 1.5$ ; the second largest is only a factor $\\simeq 0.7$ , and the rest are small enough that the mean fractional error is $\\simeq 10\\%$ .",
"For other high-mass stars, the fit is almost as close.",
"On the other hand, for lower stellar masses, particularly $M_{\\star } < 0.4$ , the fit performance is poorer (the right panel of Figure REF ); for $M_{\\star }=0.15$ , the fit's functional dependence of $M_{\\rm rem}/M_{\\star }$ on $r_{\\rm p}/\\mathcal {R}_{\\rm t}$ is correct, but its magnitude is too small by a factor $\\sim 4$ ."
],
[
"Implications",
"$M_{\\star }$ -dependence, manifested through internal structure, and $M_{\\rm BH}$ -dependence, due to relativistic effects, lead to significant changes in observable quantities.",
"Changes in the range of pericenters producing tidal disruptions translate directly into changes in event cross sections.",
"Because the debris energy distribution determines the debris orbital period distribution, these changes alter the predicted fallback rate.",
"Our analytic fits to $\\Psi $ and $\\Xi $ enable us to transform the simple conventional formulae linking stellar mass and black hole mass to mass fallback properties into more accurate expressions.",
"The quantitative contrast with the older formalism can have significant implications for observations; we consider a few here."
],
[
"The “full loss-cone\" regime",
"The rate of full TDEs depends on $\\mathcal {R}_{\\rm t}$ or, equivalently, $L(\\mathcal {R}_{\\rm t}) \\equiv \\mathcal {L}_{\\rm t}$ (Equation REF ).",
"However, the nature of this dependence varies with circumstances [7], [31], [2], [38].",
"When the angular momentum distribution of stellar orbits is smooth across all directions, including those implying passage very close to the nuclear black hole, it is appropriate to use the cross section formalism, in which the rate of events with pericenters $\\le \\mathcal {R}_{\\rm t}$ is $\\propto \\mathcal {L}_{\\rm t}^{2}$ .",
"This is the case, for example, when a number of possible mechanisms operate, e.g., resonant relaxation [45], [44] or triaxiality in the stellar cluster [39].",
"On the other hand, if the only mechanism influencing the stellar angular momentum distribution is gravitational encounters with individual stars, the situation changes in a way that depends upon the magnitude of the ratio between $(\\Delta L)^2$ , the per-orbit mean-square change in angular momentum, and $\\mathcal {L}_{\\rm t}^2$ .",
"The stellar distribution function remains smooth in all directions only when $\\Delta L^2/\\mathcal {L}_{\\rm t}^2 > 1$ (the “full loss-cone\" or “pinhole\" regime).",
"Whenever it is appropriate to speak in terms of cross sections, our results translate directly into alterations to the event rates.",
"If the total disruption cross section were $\\propto L^2(r_{\\rm t})$ , the stellar mass-dependence of the rate would be $\\propto M_{\\star }^{-1/3}R_{\\star }\\propto M_{\\star }^{0.55}$ (see Ryu2+2019).",
"However, we find that $\\mathcal {L}_{\\rm t}^2$ is nearly independent of $M_{\\star }$ for all $M_{\\star }\\lesssim 3$ .",
"For all $M_{\\rm BH} < 10^7$ , the value of this cross section is $\\simeq 1/2 \\times $ the value derived from $r_{\\rm t}$ for $M_{\\star }=1$ , after allowance for direct captures, whose cross section is $\\propto L_{\\rm dc}^2$ .",
"[21].",
"At higher black hole masses, the cross section for total disruptions relative to the Newtonian prediction can either increase or decrease, depending on stellar mass, as two relativistic effects, greater tidal stress and direct capture, compete (Figure REF ).",
"However, the near-independence of $\\mathcal {R}_{\\rm t}$ with respect to $M_{\\star }$ combined with the decoupling between $\\Psi _{\\star }$ and $\\Psi _{\\rm BH}$ means that the absolute cross section for tidal disruption remains nearly independent of $M_{\\star }$ for all black hole masses.",
"This implies that the suppression of total tidal disruptions by competition with direct capture is nearly independent of $M_{\\star }$ .",
"For example, in Schwarzschild spacetime, more than 50% of all events with pericenter $\\lesssim 14~r_{\\rm g}$ produce direct captures; this corresponds to a factor of $>2$ suppression of the complete tidal disruption rate when $M_{\\rm BH} \\gtrsim 5 \\times 10^6$ .",
"The suppression rises to a factor $>10$ when $M_{\\rm BH} \\gtrsim 3 \\times 10^7$ (See Figure 7 in Ryu4+2019)."
],
[
"The “empty loss-cone\" regime",
"On the other hand, when $ (\\Delta L)^2 < \\mathcal {L}_{\\rm t}^2$ , the stellar distribution function is said to evolve in the “empty loss-cone\" or “diffusive\" regime.",
"It has been generally thought that in these circumstances, most stars approach the edge of the loss-cone only gradually, taking small steps up and down in $L^2$ [31], [38], while occasionally being scattered more strongly [58].",
"This regime is often associated with lower orbital energy [2], [54].",
"Once a star slips inside the loss-cone, it is destroyed at its first pericenter.",
"Consequently, the majority of the stars within the loss-cone have velocities directed very close to its edge, making the “cross section\" language inappropriate because the distribution of impact parameters is not uniform.",
"The event rate then depends only logarithmically on $\\mathcal {L}_{\\rm t}$ [31], [38], [58].",
"However, while stars move through the angular momentum range just outside the loss-cone, i.e., $\\mathcal {L}_{\\rm t}^2 < L^2 \\lesssim 3\\mathcal {L}_{\\rm t}^2$ (see Figure REF ), they suffer partial disruption every time they pass through pericenter.",
"Consequently, normal full disruptions do not happen in this regime.",
"Instead, several alternative pathways to destruction may be followed.",
"One possibility is that, as discussed at greater length in Ryu3+2019, some remnants created by partial disruptions may not relax to main sequence structure in a single orbit.",
"Remnants when first formed are considerably more distended than a main sequence star of that mass because they are hotter and rotate more rapidly.",
"If their relaxation time (in particular for thermal properties) is longer than an orbit, the value of $\\mathcal {L}_{\\rm t}$ corresponding to their distended structure would be rather larger than it would be for a main sequence star of mass $M_{\\rm rem}$ .",
"Because the specific angular momentum of the remnant is essentially unchanged, its pericenter would then be well inside its new $\\mathcal {R}_{\\rm t}$ , and it would be fully disrupted one orbit later.",
"The full disruption of such a perturbed remnant would be quite different from a full disruption of an ordinary main sequence star because the quantitative characteristics of tidal disruptions depend significantly on the star's internal structure.",
"Another possible outcome is that the orbital energy of a remnant could be sufficiently larger than the initial value that $(\\Delta L)^2$ on the new orbit becomes large relative to $\\mathcal {L}_{\\rm t}^2$ .",
"From this point on, such a star would evolve in the full loss-cone regime.",
"This process can also act in the opposite direction: a star may enter a partial disruption event with large $(\\Delta L)^2/\\mathcal {L}_{\\rm t}^2$ and exit it as a remnant whose specific energy is small enough to make $(\\Delta L)^2/\\mathcal {L}_{\\rm t}^2$ small.",
"In this manner, partial disruptions can act as a transfer channel between scattering regimes.",
"There is also a third way stars could evolve if the remnants' orbital energies are such as to keep them in the regime of small angular momentum change per orbit, and they relax to main sequence structure within an orbit.",
"In this case, their pericenters remain in the range associated with partial disruptions.",
"Consequently, their mass decreases at each pericenter passage [54].",
"Although the value of $\\mathcal {L}_{\\rm t}$ changes as each partial disruption diminishes the star's mass, it does not change much; as we have shown in Section REF , $\\mathcal {R}_{\\rm t}$ depends only weakly on $M_{\\star }$ for main sequence stars with $M_{\\star } \\lesssim 3$ .",
"The relation between $M_{\\rm rem}/M_{\\star }$ and $L^2$ we have uncovered shows that this mass-loss can be substantial: once $L^2 \\lesssim 1.4\\mathcal {L}_{\\rm t}^2$ , $M_{\\rm rem}/M_{\\star } < 1/2$ .",
"Moreover, because their progress through the partial disruption zone is stochastic, these stars are likely to suffer numerous partial disruptions before their angular momentum becomes small enough that they can be totally disrupted: if $(\\Delta L)^2/\\mathcal {L}_{\\rm t}^2$ is independent of $L$ , $\\simeq [(\\Delta L)^2/\\mathcal {L}_{\\rm t}^2]^{-1}$ orbital periods are required to wander from $\\simeq 3\\mathcal {L}_{\\rm t}^2$ to $\\mathcal {L}_{\\rm t}^2$ .",
"Thus, their mass upon total disruption will, in most cases, be a small fraction of their initial mass.",
"Only two effects limit this process.",
"One, which has probability $\\sim 10\\%$ , is that this chain of partial disruptions is interrupted by a strong scattering event, and the star either goes into the loss-cone or far outside the zone of partial disruptions [58].",
"The other is the trend for stars with $M_{\\star } \\lesssim 0.4$ to retain a larger fraction of their mass than more massive stars do (see the right panel of Figure REF ) .",
"The overall result of this third possibility would be to reduce sharply the rate of total disruptions of massive stars, while only slightly enlarging that of the least massive stars compared to that of high-mass stars because low-mass stars dominate the general stellar population.",
"Each of these scenarios has a different effect on the event rate.",
"When the first applies, it effectively changes the rate of total disruptions from the rate at which stars cross the loss-cone boundary by diffusion to the rate given by the cross section for a partial disruption severe enough to materially disturb the structure of the remnant created.",
"It also entails one substantial partial disruption for each (non-standard) total disruption.",
"The second effectively makes the angular momentum range associated with partial disruptions a venue for substantial change in the stars' specific energy.",
"The third does not alter the event rate predicted in the small $(\\Delta L)^2/\\mathcal {L}_{\\rm t}^2$ regime, but it materially changes the distribution in stellar mass for total disruptions.",
"It is also associated with a large number of partial disruptions, of varying severity, for each eventual total disruption of a star that has gone through this process."
],
[
"Testing the “frozen-in\" approximation",
"An heuristic argument dubbed “the frozen-in approximation\" has sometimes been used as a device to estimate the characteristic energy width of the debris $\\Delta E$ in terms of the spread in potential energy across the star near the beginning of the encounter [32], [53].",
"The basic idea is that from some special moment onward, individual fluid elements within the star travel on ballistic orbits.",
"When the point at which the energy spread is evaluated is independent of the actual orbital pericenter (e.g., $r_{\\rm t}$ or $\\mathcal {R}_{\\rm t}$ ), $\\Delta E$ is the same for all $r_{\\rm p} \\le \\mathcal {R}_{\\rm t}$ .",
"As we have shown in Section , applying the “frozen-in approximation\" at any of the plausible choices for the fiducial point ($r_{\\rm t}$ , $r_{\\rm p}$ , or $\\mathcal {R}_{\\rm t}$ ) always leads to errors in $\\Delta E$ at the factor $\\sim 2$ level.",
"This fact suggests that the debris energy is influenced by more than just the matter's initial location in the star and the instantaneous tidal potential at some special location.",
"Hydrodynamics, stellar self-gravity and the nature of the tidal stress over an extended range of distances from the BH are also important.",
"In fact, in Ryu2+2019, we show that stars undergoing tidal encounters lose mass over a wide range of orbital separations from the BH, from near $r_{\\rm p}$ to a distance an order of magnitude larger ([16] also reported a version of this behavior), a finding that also points toward no single location having a unique role in determining the debris energy distribution.",
"Moreover, the fluid elements of the star do not follow ballistic orbits as they pass near the black hole and then travel outward until finally shed.",
"During the entire period of mass-loss, the star's instantaneous distance from the black hole is quite close to its “instantaneous tidal radius\", the one computed on the basis of the star's density and distance from the black hole as functions of time (see Ryu2+2019 for a lengthier discussion of the instantaneous tidal radius).",
"Consequently, for the extended period of time while the star is torn apart, stellar self-gravity and black hole tidal gravity are comparable in magnitude (as are pressure forces as well).",
"Similar behavior was seen in the simulations of [52], who studied non-relativistic disruption dynamics for $r_{\\rm p}/r_{\\rm t}\\simeq 0.14 - 0.2$ .",
"All these results are contrary to the motivation of the frozen-in approximation, which [32] explicitly describe as an “impulse\" approximation.",
"Note, however, that this conclusion does not remove the possibility that $\\Delta E$ is independent of $r_{\\rm p}$ for $r_{\\rm p} \\le \\mathcal {R}_{\\rm t}$ [53].",
"Our simulation data are consistent with this possibility, but do not span a wide enough range in $r_{\\rm p}$ to test it credibly.",
"It is possible that some mechanism other than a freezing of fluid element energy enforces this outcome."
],
[
"Peak mass return: time and rate",
"The peak return rate and the time at which this peak is reached are quantities of particular interest to predictions of a TDE's light output.",
"The two corrections introduced here, for stellar internal structure and for general relativistic effects, both alter these parameters' dependence on $M_{\\star }$ and $M_{\\rm BH}$ .",
"These changes are encapsulated in the change to the debris energy, $\\Xi \\equiv \\Delta E/\\Delta \\epsilon $ .",
"However, as we shall see, the stellar internal structure also changes the fallback rate in a subtler way.",
"The mass fallback rate of stellar debris on ballistic orbits is [46], [43] $\\dot{M}_{\\rm fb}=\\frac{dM}{dE}\\left|\\frac{dE}{dt}\\right|=\\frac{(2G M_{\\rm BH})^{2/3}}{3}\\frac{dM}{dE}t^{-5/3}.$ The steepness of the rise to $\\dot{M}_{\\rm peak}$ (peak mass return rate) and the specific shape of that peak depend on the shape of $dM/dE$ for the most tightly-bound matter.",
"When $dM/dE$ has a sharp, steep edge at $E = - \\Delta E$ , as it does for low-mass stars (Ryu2+2019), independent of $M_{\\rm BH}$ (Ryu4+2019), the rise to the peak is also comparatively sharp.",
"On the other hand, when there is a noticeable wing extending to energies below $-\\Delta E$ , as is the case for high-mass stars disrupted by low-mass black holes, the rise is more gradual (Ryu2+2019, Ryu4+2019).",
"It is the impact of these effects on the time-dependence of mass fallback that we encapsulate in the factor $f$ in Equation REF .",
"Because of the differing shapes of the fallback rate curves, we define $t_{\\rm peak}$ as the time at which 5% of $M_{\\star }$ has returned to the black hole.",
"This time corresponds to the time of the absolute maximum when the peak is sharp (for all stars being disrupted by high-mass black holes, and low-mass stars for black holes of any mass), and the beginning of the maximum when the peak is relatively flat (for high-mass stars encountering low-mass black holes).",
"Figure: (Top) M ☆ M_{\\star }-dependence of t peak t_{\\rm peak} (red) and M ˙ peak \\dot{M}_{\\rm peak} (blue).",
"Solid curves use our most precise fit to Ξ ☆ \\Xi _{\\star } (Equation ); dotted use the traditional prediction.",
"(Middle) M BH M_{\\rm BH}-dependence of t peak t_{\\rm peak}, using our fit (Equation ) to Ξ BH \\Xi _{\\rm BH} (red solid) and the traditional form (black dots).",
"The M BH M_{\\rm BH}-dependence of M ˙ peak \\dot{M}_{\\rm peak} is the inverse of that of t peak t_{\\rm peak}.",
"(Bottom) Net correction factor for three stellar masses: 0.3M ⊙ 0.3\\;\\mathrm {M}_{\\odot } (red), 1M ⊙ 1\\;\\mathrm {M}_{\\odot } (black), and 3M ⊙ 3 \\;\\mathrm {M}_{\\odot } (blue); for t peak t_{\\rm peak} (solid lines) and for M ˙ peak \\dot{M}_{\\rm peak}, including the ff correction (Equation , dashed lines).In terms of $\\Xi $ , the peak fallback time $t_{\\rm peak}$ is $t_{\\rm peak}&=\\frac{}{\\sqrt{2}}\\frac{G M_{\\rm BH}}{\\Delta {E}^{3/2}},\\nonumber \\\\&\\simeq 0.11\\;\\mathrm {yr}~\\Xi ^{-3/2} M_{\\star }^{-1}R_{\\star }^{3/2}\\left(\\frac{M_{\\rm BH}}{10^{6}}\\right)^{1/2},$ and the peak fallback rate $\\dot{M}_{\\rm peak}$ at $t=t_{\\rm peak}$ is $\\dot{M}_{\\rm peak}&\\simeq f \\frac{M_{\\star }}{3t_{\\rm peak}},\\nonumber \\\\&\\simeq 1.49\\;\\mathrm {M}_{\\odot }\\;\\mathrm {yr}^{-1}\\left(\\frac{f}{0.5}\\right)\\Xi ^{3/2}M_{\\star }^{2}R_{\\star }^{-3/2} \\left(\\frac{M_{\\rm BH}}{10^{6}}\\right)^{-1/2},$ where the correction factor $f$ accounts for the different shape of the energy distribution near the tails (see discussion below).",
"We take $f=1$ for $M_{\\star }\\le 0.5$ and $0.5$ for $M_{\\star }>0.5$ .",
"To isolate the dependence of $t_{\\rm peak}$ and $\\dot{M}_{\\rm peak}$ on $M_{\\rm BH}$ and $M_{\\star }$ , we first split $\\Xi $ into its portions dependent on $M_{\\star }$ and $M_{\\rm BH}$ .",
"Next we insert the $M_{\\star }-R_{\\star }$ relation for our main sequence stellar models, i.e., $R_{\\star }\\propto M_{\\star }^{0.88}$ (Ryu2+2019).",
"Equations REF and REF then become $t_{\\rm peak}&\\propto \\left[\\Xi _{\\star }^{-3/2} \\;M_{\\star }^{0.32}\\right]\\left[ \\Xi _{\\rm BH}^{-3/2} M_{\\rm BH}^{1/2}\\right],\\\\\\dot{M}_{\\rm peak}&\\propto \\left[\\Xi _{\\star }^{3/2} \\;M_{\\star }^{0.68}\\right] ~\\left[\\Xi _{\\rm BH}^{3/2} M_{\\rm BH}^{-1/2}\\right].$ The top panel of Figure REF displays the effect on the $M_{\\star }$ -dependence due to our corrections.",
"Compared to the simple approximation in which $\\Delta E = \\Delta \\epsilon $ , $t_{\\rm peak}$ is later, by a little less than a factor of two, for low-mass stars and sooner, by a little more than factor of two, for high-mass stars.",
"Only for $M_{\\star } \\simeq 0.7$ do $t_{\\rm peak}$ and $\\dot{M}_{\\rm peak}$ coincide with the prediction of the simple approximation.",
"The scaling of both $t_{\\rm peak}$ and ${\\dot{M}}_{\\rm peak}$ with $M_{\\star }$ is close to the approximate scaling for both ends of the stellar mass spectrum, but differs strongly from $M_{\\star } \\simeq 0.5$ to $M_{\\star } \\simeq 1.5$ .",
"Overall, realistic internal structures cause $t_{\\rm peak}$ to have only a weak net trend as a function of $M_{\\star }$ and to make this dependence non-monotonic, in contrast with the simple estimate's prediction that $t_{\\rm peak} \\propto M_{\\star }^{0.32}$ .",
"For $\\dot{M}_{\\rm peak}$ , realistic internal structures steepen the global $M_{\\star }$ -dependence, with the sharpest dependence concentrated in the range $0.5 \\lesssim M_{\\star } \\lesssim 1$ .",
"The middle panel of Figure REF shows how our corrections alter $M_{\\rm BH}$ -dependence.",
"There is little departure from the traditional dependence until $M_{\\rm BH} > 5 \\times 10^{6}$ , but for more massive black holes, $t_{\\rm peak}$ occurs later, by as much as a factor of 3 for $M_{\\rm BH} = 5 \\times 10^7$ .",
"Because the relativistic correction factor for $t_{\\rm peak}$ (i.e., $\\Xi _{\\rm BH}^{-3/2}$ ) varies with $M_{\\rm BH}$ in the same sense that the Newtonian expression does, these effects result in stronger dependences of $t_{\\rm peak}$ and $\\dot{M}_{\\rm peak}$ on $M_{\\rm BH}$ than are predicted by Newtonian dynamics.",
"Moreover, relativistic corrections become dominant over the Newtonian $M_{\\rm BH}$ -dependence ($d\\ln \\Xi _{\\rm BH}^{-3/2}/d\\ln M_{\\rm BH}>0.5$ ) for $M_{\\rm BH}>3\\times 10^{7}$ .",
"These corrections can have a significant impact on parameter inference resting on mass-return rates.",
"For example, as shown in the upper panel of Figure REF , neglecting the internal structure correction factor for $M_{\\star }$ could lead to inferring $M_{\\star }=0.3$ from a measurement of $t_{\\rm peak}$ when the real $M_{\\star } \\simeq 0.8$ .",
"Similarly, when Newtonian analysis of $t_{\\rm peak}$ would indicate $M_{\\rm BH} = 10^7$ , a properly relativistic approach leads to $M_{\\rm BH} \\simeq 5 \\times 10^6$ .",
"Therefore, it is imperative to take into account both corrections (the bottom panel of Figure REF ) for more accurate inferences of $M_{\\star }$ and $M_{\\rm BH}$ .",
"Analyses neglecting both of these effects [40] incur systematic errors of factors of several in both the inferred stellar mass and the inferred black hole mass, on top of whatever additional systematic errors might be present due to other aspects of the lightcurve modeling (e.g., direct identification of light output with mass return rate)."
],
[
"Unbound debris energy and speed at infinity",
"The energy of the most highly-bound matter determines the time of peak mass-return; the energy of the most highly-unbound matter determines the fastest speed of the ejecta that never return to the black hole, as well as the total amount of energy available for deposition in surrounding gas.",
"We find that the total energy of unbound debris is $\\simeq 4\\times 10^{50}~\\Xi ~ M_{\\star }^{0.79}~(M_{\\rm BH}/10^{6})^{1/3}\\;\\mathrm {erg}$ (Equations REF , and REF ), and the greatest speed at infinity for the bulk of the ejecta mass is $\\simeq 6\\times 10^{3}~\\Xi ^{1/2}~M_{\\star }^{-0.11}~(M_{\\rm BH} / 10^{6})^{1/6}\\;\\mathrm {km}\\;\\mathrm {s}^{-1}$ ; these scalings take into account the main sequence mass-radius relation (Ryu2+2019).",
"Because $\\Xi ^{1/2}$ changes by at most a factor of 1.6 from low-mass stars to high-mass and a factor of 1.4 from the Newtonian limit to $M_{\\rm BH} = 5 \\times 10^7$ , this speed is exceedingly weakly dependent on both $M_{\\star }$ and $M_{\\rm BH}$ .",
"Both the energy and the mass of the unbound ejecta are comparable to those of a supernova remnant.",
"One might therefore expect that when the unbound debris shocks against whatever gas surrounds the black hole, there would be radio emission [15].",
"Such emission has, in fact, been seen in several cases, with particularly rich datasets obtained from ASASSN-14li and CNSS J0019+00 [1], [57], [3].",
"Using the equipartition formalism for relativistic synchrotron self-absorbed spectra [5], [26] found that the linear scale of the emission region in ASASSN-14li expanded at a constant speed $\\simeq (1.45 - 2)\\times 10^{4}$ km s$^{-1}$ (see also [1] for a comparable estimate), while [3] performed a very similar analysis on CNSS J0019+00 and likewise found a constant speed $\\approx 1.5 \\times 10^4$ km s$^{-1}$ .",
"Because these speeds are close to those expected for the fastest-moving unbound ejecta, [26] suggested that the unbound ejecta are, indeed, responsible.",
"Our results strengthen that conclusion for two reasons.",
"First, the characteristic energy spread we find for $M_{\\rm BH}=10^{6}$ is larger than the conventional estimate for all $M_{\\star } > 0.7$ , and larger by a factor $\\simeq 1.8$ for $M_{\\star } \\gtrsim 3$ .",
"The ratio $\\Xi $ of the characteristic energy width to the conventional estimate decreases as $M_{\\rm BH}$ increases (see the left panel of Figure REF ), but it nonetheless remains larger than unity over a wide range of $M_{\\rm BH}$ and $M_{\\star }$ , e.g., for $M_{\\star } \\ge 1$ and $M_{\\rm BH} \\lesssim 10^{7}$ .",
"Second, the amount of mass required is only $\\sim 10^{-4} - 10^{-2}\\;\\mathrm {M}_{\\odot }$ (in ASASSN-14li, [26] inferred $\\sim 10^{-4}\\;\\mathrm {M}_{\\odot }$ ; in CNSS J0019+00, [3] found $7 \\times 10^{-3}\\;\\mathrm {M}_{\\odot }$ ).",
"This matter could therefore come from the power-law tail in $dM/dE$ that extends beyond $\\Delta E$ when $M_{\\star }>0.7\\;\\mathrm {M}_{\\odot }$ (Ryu2+2019).",
"For $M_{\\rm BH} \\sim 10^6$ , there is $\\sim 3 \\times 10^{-4} M_{\\star }$ at energies $\\gtrsim 3\\Delta \\epsilon $ for all stars with $M_{\\star } > 0.7$ ; although the wings narrow at higher black hole mass, for $M_{\\rm BH} \\sim 10^7$ , the energy at this mass fraction is still $\\simeq 2 \\Delta \\epsilon $ (Ryu4+2019).",
"The speed of this material would then be $\\ge 10,000 M_{\\star }^{-0.11} \\;\\mathrm {km}\\;\\mathrm {s}^{-1}$ for black hole masses not too much more than $10^6$ , or $\\ge 12500 M_{\\star }^{-0.11}$ km s$^{-1}$ for $M_{\\rm BH} \\sim 10^7$ .",
"Alternatively, it may also be possible for a small amount of debris mass to reach such speeds in a highly-penetrating event [59]."
],
[
"Summary",
"In this paper we have presented the principal results from our suite of relativistic tidal disruption simulations using realistic main sequence stellar structures for stars of many different masses encountering black holes over a wide range of mass.",
"We have shown that $M_{\\star }$ -dependence in disruption properties (due to mass-dependent contrasts in internal structure) is largely decoupled from $M_{\\rm BH}$ -dependence (due to relativistic effects).",
"The $M_{\\star }$ - and $M_{\\rm BH}$ -dependence of both $\\mathcal {R}_{\\rm t}$ and $\\Delta E$ due to these effects can be described quite accurately by analytic formulae.",
"Subsequent papers in this series will fill in the details, both of our methods and of our results; here we focused on our results' implications for observable properties such as event rates and the time-dependence of mass fallback.",
"Several broad themes can be seen clearly.",
"The order-of-magnitude estimate $r_{\\rm t}$ for the maximum pericenter yielding a full disruption requires correction by factors $0.4-4$ , depending on $M_{\\star }$ and $M_{\\rm BH}$ .",
"One way to understand the $M_{\\star }$ -dependent part of this correction is a physically intuitive model that relates $\\mathcal {R}_{\\rm t}$ to the central density of the star rather than to its mean density.",
"Because the sense of the $M_{\\star }$ -dependent correction to $r_{\\rm t}$ runs opposite to the dependence of stellar radius on stellar mass, the net result is a physical tidal radius that is roughly constant over the range $M_{\\star }\\simeq 0.15-3$ , a range spanning the overwhelming majority of all stars.",
"On the other hand, for fixed $M_{\\star }$ , the ratio $\\mathcal {R}_{\\rm t}/r_{\\rm t}$ increases with $M_{\\rm BH}$ .",
"For $M_{\\rm BH}\\gtrsim 3\\times 10^{7}$ , the location of the physical tidal radius is more sensitive to relativistic effects than to the Newtonian physics embodied in $r_{\\rm t}$ .",
"These corrections figure directly into rate predictions.",
"A comparable correction is required for the characteristic energy spread of the tidal debris.",
"The simple estimate is too large for low-mass stars and too small for high-mass stars when black hole masses are low.",
"However, due to relativistic corrections, for higher black hole masses, the discrepancy between the simple estimate and the characteristic energy width becomes larger for low-mass stars, but smaller for high-mass stars.",
"Moreover, no version of the “frozen-in\" approximation can correctly predict this quantity, and total disruption of stars with $M_{\\star } > 0.7$ generically produces power-law tails in the distribution function $dM/dE$ except for events involving very high-mass black holes.",
"Alterations in the energy spread immediately imply differing fallback timescales, as the time of peak mass-return is $\\propto (\\Delta E)^{-3/2}$ .",
"Neglect of these corrections when inferring $M_{\\star }$ or $M_{\\rm BH}$ from fallback histories could lead to significant errors.",
"The cross section for total disruptions of low-mass stars is larger than the simple estimate, while it is more than a factor of two smaller for high-mass stars.",
"Partial disruptions should occur at a rate comparable to total disruptions.",
"The fraction of the star's mass lost in such an encounter can be described surprisingly accurately by a simple analytic expression (Equation REF ).",
"Partial disruptions also play a significant, but previously neglected, role in the approach of stars to disruption when stellar encounters perturb their angular momentum only weakly (the “empty loss-cone\" or “diffusive\" regime).",
"They can deflect stars into different parts of phase space, they can whittle down formerly massive stars before they are completely disrupted, and they can transform a main sequence star into a distended remnant that may be totally disrupted in a highly unconventional way upon its next return to pericenter."
],
[
"Acknowledgements",
"We are grateful to the anonymous referee for comments and suggestions that helped us to improve the paper.",
"We thank Nicholas Stone and Re'em Sari for helpful discussions.",
"This work was partially supported by NSF grant AST-1715032, Simons Foundation grant 559794 and an advanced ERC grant TReX.",
"S. C. N. was supported by the grants NSF AST 1515982, NSF OAC 1515969, and NASA 17-TCAN17-0018, and an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center administrated by USRA through a contract with NASA.",
"The authors acknowledge the analysis toolkit matplotlib [19] for making the plots in the paper.",
"This research project (or part of this research project) was conducted using computational resources (and/or scientific computing services) at the Maryland Advanced Research Computing Center (MARCC).",
"The authors would like to thank Stony Brook Research Computing and Cyberinfrastructure, and the Institute for Advanced Computational Science at Stony Brook University for access to the high-performance SeaWulf computing system, which was made possible by a $\\$1.4$ M National Science Foundation grant (#1531492).",
"matplotlib [19]; MESA[42]; Harm3d[41]."
]
] | 2001.03501 |
[
[
"The prototype star $\\gamma$ Doradus observed by TESS"
],
[
"Abstract $\\gamma$ Doradus is the prototype star for the eponymous class of pulsating stars that consists of late A-early F main-sequence stars oscillating in low-frequency gravito-inertial modes.",
"Being among the brightest stars of its kind (V = 4.2), $\\gamma$ Dor benefits from a large set of observational data that has been recently completed by high-quality space photometry from the TESS mission.",
"With these new data, we propose to study $\\gamma$ Dor as an example of possibilities offered by synergies between multi-technical ground and space-based observations.",
"Here, we present the preliminary results of our investigations."
],
[
"TESS photometry",
"TESS observed $\\gamma $ Dor during Sectors 3, 4, and 5, representing 80 days of nearly uninterrupted data.",
"Bright as it is, the star saturates the CCDs and leaves bleeding trails along CCD columns.",
"This is shown on the Full Frame Image (FFI) cutout on the left panel of Fig.",
"REF .",
"The lightcurve was extracted by performing simple aperture photometry on FFI cutouts with a custom mask chosen to include most of the target flux.",
"Data downlink or scattered light from the Earth or the Moon are responsible for gaps in the data but overall, a duty cycle of 87% is achevied.",
"Figure: Left: Example of TESS Full Frame Image (FFI) cutout for γ\\gamma Dor.",
"Right: Lightcurve of γ\\gamma Dor extracted from TESS FFI cutouts.Figure: Amplitude periodograms before (top) and after pre-whintening (bottom) of the first two frequencies of high amplitude.",
"Red circles represent frequencies extracted from the TESS lightcurve.",
"Upside down triangles indicate frequencies found by from ground-based photometry and spectroscopy."
],
[
"Asteroseismology from ground and space",
"We carried out the frequency analysis of the TESS lightcurve by iterative prewhitening.",
"We found 21 frequencies with S/N $ > 4$ (see Fig.",
"REF ).",
"The 6 previously known frequencies from ground-based observations [3] are all confirmed with TESS data.",
"Based on line profile variations, [3] identified the three frequencies at 15.3, 15.8 and 21.7 $\\mu $ Hz to be prograde gravity modes of $(\\ell ,m)=(1,-1)$ .",
"Assuming the frequency group at $\\sim $ 16 $\\mu $ Hz is mostly consisting of $(1,-1)$ modes, we applied the method of [4] to estimate the near-core rotation rate of $\\gamma $ Dor.",
"The mode frequencies are compatible with rotation rates in the 9-12 $\\mu $ Hz range but the limited resolution of the periodogram prevent us from obtaining a more precise value.",
"The second frequency group around $\\sim $ 32 $\\mu $ Hz can be interpreted as either combination frequencies or $(\\ell ,m)=(2,-2)$ modes."
],
[
"Surface rotation",
"In order to determine the surface rotation rate, we have need of the luminosity $L$ , the effective temperature $T_{\\rm eff}$ , the projected surface velocity $v\\sin i$ and the inclination angle $i$ .",
"The luminosity is adopted from Gaia DR2 [5].",
"We derive $T_{\\rm eff} \\approx 7145 \\pm 150$ K by averaging the photometric and spectroscopic estimates available in the literature.",
"The projected velocity $v\\sin i = 59.5 \\pm 3~{\\mathrm {km~s^{-1}}}$ is taken from [1].",
"$\\gamma $ Dor hosts a debris disc that has been observed with Herschel and modelled by [2], who found $i \\approx 70^\\circ $ .",
"Assuming the rotation axis of $\\gamma $ Dor is aligned with its disc axis, the surface rotation rate is estimated to be $8.4 \\pm 0.8$ $\\mu $ Hz.",
"This is close to the range of near-core rotation rates estimated from seismology suggesting a nearly uniform rotation profile.",
"This research made use of Lightkurve, a Python package for Kepler and TESS data analysis [8]; Period04 [7] and Astroquery [6].",
"S.C. acknowledges support from the Programme National de Physique Stellaire (PNPS) of the CNRS/INSU co-funded by CEA and CNES."
]
] | 2001.03497 |
[
[
"A Unified Conversational Assistant Framework for Business Process\n Automation"
],
[
"Abstract Business process automation is a booming multi-billion-dollar industry that promises to remove menial tasks from workers' plates -- through the introduction of autonomous agents -- and free up their time and brain power for more creative and engaging tasks.",
"However, an essential component to the successful deployment of such autonomous agents is the ability of business users to monitor their performance and customize their execution.",
"A simple and user-friendly interface with a low learning curve is necessary to increase the adoption of such agents in banking, insurance, retail and other domains.",
"As a result, proactive chatbots will play a crucial role in the business automation space.",
"Not only can they respond to users' queries and perform actions on their behalf but also initiate communication with the users to inform them of the system's behavior.",
"This will provide business users a natural language interface to interact with, monitor and control autonomous agents.",
"In this work, we present a multi-agent orchestration framework to develop such proactive chatbots by discussing the types of skills that can be composed into agents and how to orchestrate these agents.",
"Two use cases on a travel preapproval business process and a loan application business process are adopted to qualitatively analyze the proposed framework based on four criteria: performance, coding overhead, scalability, and agent overlap."
],
[
"Introduction",
"Automation made its way into manufacturing plants in the early 1900s, however comparatively, its integration into the service industry has been slow.",
"The service industry first took advantage of automation through customer care chatbots which routed incoming customer calls to the appropriate human operators.",
"Gradually, these robotic answering machines improved and began to replace services provided by some human operators by walking customers through simple steps to resolve problems and escalating to humans only when necessary.",
"As enabling technologies improved, so have these customer care chatbots which can now carry out more elaborate conversations with the customers.",
"Beyond chatbots, little automation was incorporated in the services industry despite the plethora of automation opportunities.",
"The emergence of Robotic Process Automation (RPA) led to the next breakthrough.",
"RPA developed robotic software agents capable of automating simple tasks within a business process such as form filling and data mining.",
"RPAs reduced the cost of deploying autonomous agents but covered a small portion of automatable tasks, namely simple repetitive tasks.",
"This led to the emergence of Business Process Automation (BPA) which supersedes RPA's scope.",
"In addition to task automation, BPA looked to automate decision making, data management, content digitization, and workflow improvement.",
"BPA relies on various artificial intelligence technologies including optical character recognition (OCR) systems and decision making models to automate business process tasks.",
"As state-of-the-art algorithms become more accurate and reliable - deep learning OCR has achieved super-human accuracy levels - automation in service industry will become more cost-effective.",
"Nevertheless, wide-spread deployment of autonomous systems has still lagged behind expectations.",
"Trustworthiness, coding overhead, scalability, algorithm life cycle, technical expertise and user experience have been a few challenges that have stood in the way of more successful integration into business process solutions.",
"In this work, we claim that a proactive multi-agent conversational framework can be that one stop shop that resolves many of these challenges.",
"However, multiple research questions at the intersection of automation and business processes must be address first.",
"A conversational interface reduces the necessary technical expertise to interact, monitor and customize the framework, but many algorithms do not have a conversational interface.",
"How can we effectively integrate such algorithms into a conversational framework with minimal coding overhead?",
"How can such independently developed (and potentially incompatible) functions coexist and cooperate within the same ecosystem?",
"How could the user access all of the automation options in the framework with minimal interface and protocol hoping?",
"The goal is not to create a natural language interaction for every part of the process but a single interactive interface.",
"If it is accessing different documents, running OCR on a new application, checking our database or setting up alerts on specific events in our system, one would prefer to interact with a single interface which is conveniently conversational in this scenario.",
"We believe an orchestrator and agent composition approach with a specific pipeline would provide solutions to these questions.",
"We present a multi-agent orchestration framework that consists of heterogeneous agents such as dialog, informational retrieval, task execution, and alerting agents and an orchestration layer that consists of three components (scorer, selector and sequencer) that would control the execution of the agents in the framework.",
"As a proof of concept, we create two instances of this assistant for a travel preapproval business process and a loan application business process – deployed in a sandbox environment – and discuss how this framework can address these challenges."
],
[
"Business Process Automation",
"Task automation has driven the digital transformation in enterprises with the emergence of RPA as a light-weight approach for automating repetitive tasks.",
"RPA is a paradigm that seeks to automate the “mouse-click\" in user interfaces to relieve human resources of repetitive tasks.",
"RPA adopts an outside-in approach that requires minimal changes to legacy software [21]; they have been applied to various sectors including accounting [13], auditing [2], human resources [15], banking [19], public administration [6] and energy sectors [9].",
"Multiple approaches have been adopted in the development of RPAs.",
"[4] proposed an RPA solution for document flow automation in a debt collector business process.",
"They used deep learning OCR and classification to control the path of the document through the business process.",
"User behavior was captured to train a self-learning agent capable of identifying relationships between tasks and deploying RPAs based on the learned rules [22].",
"This approach, known as form-to-rule, relied on first-order logic to deduce the rules.",
"Another branch of literature focused on identifying RPA-eligible tasks automatically.",
"Leopold et al.",
"relied on natural language processing of business process descriptions [10].",
"They adopted supervised machine learning to classify instances into one of three classes.",
"As the scope of RPA increases to include more complex tasks, RPA agents incorporate other technological advancements such as OCR for document digitization, and data mining for content collection and management to successfully automate complex tasks.",
"These advancements have achieved the best performance when adopting deep learning models [16].",
"However, these models are computationally expensive to train and have seen slow adoption despite their super-human accuracy due to their lack of transparency and explainability.",
"Beyond RPA, BPA also seeks for automated decision making within business processes.",
"The field of automated planning has been considered a prime candidate to enable more advanced levels of automation in business processes [11].",
"Planning algorithms can be used to automatically generate business process models, adapt the business process' behavior based on unforeseen circumstances, and autonomously perform conformance checking.",
"End-to-end BPA has yet to find success in real-world deployments despite the ubiquity of computing devices and the advancements in software technology.",
"Trust and reliability have been two of the main challenges preventing wide-spread adoption of automation approaches.",
"With business stakeholders lacking the technical skills to monitor autonomous agents and intervene when they do not operate as expected, they have been reluctant to adopt this technology.",
"Providing a natural language interface to allow such users to interact with autonomous agents may be the key to increase the adoption of automation in service enterprises."
],
[
"Conversational Agents",
"Natural language is the main communication modality in business enterprises.",
"From conversations to written documents, forms and emails, understanding and generating natural language can significantly facilitate the integration of various autonomous agents into domains where users do not speak the language of computers.",
"Conversational agents have been developed for a wide range of applications using a plethora of machine learning techniques that have resulted in agents with increasing degrees of sophistication.",
"In this section, we will focus on chatbots developed for enterprise applications and specifically those used in conjunction with RPA.",
"Two approaches to chatbot development have been mainly adopted in the literature: 1) building domain specific chatbots by relying on natural language understanding and generative machine learning models, and 2) training deep learning models on data samples [3].",
"While the former requires more overhead to develop, it has been the more widely adopted approach for enterprise chatbots due to its predictable and controllable behavior.",
"In enterprises, chatbots were first adopted for customer support which involved answering customers' questions about the services, products, location etc.",
"of businesses [3].",
"Beyond question answering bots, a shopping bot was developed to increase the sales of shops by answering questions from customers and even providing custom prices for products [5].",
"A delivery bot was also developed for food delivery services that reduced the effort required from customers to order pizza [5].",
"While only automating 30% of the processes with these bots, the bot developers have recently begun investigating the incorporation of RPA to increase the percentage of automation.",
"Matthies et al.",
"developed a chatbot for agile software development teams [12] which analyzed commits in version control systems to provide insights into the teams' performance.",
"Kalia et al.",
"proposed an approach to generate a conversational agent based on dialog trees given a business process flow [8].",
"Even though they provide a systematic way to develop chatbot that were sound, significant overhead and domain knowledge were necessary.",
"Furthermore, these approaches have focused on designing a single conversational agent capable of answering frequently asked questions and performing tasks in a process such as determining the price of products in the shopping bot.",
"Expanding the scope of their operation would require significant coding overhead by developers who understand the existing system.",
"Also, reusing existing agents may not be possible due to compatibility reasons.",
"In this work, we propose a conversation agent framework based on the multi-agent system paradigm as a potential solution to these issues.",
"Instead of designing a single powerful agent, multiple specialized agents with narrower scopes can be independently designed and then integrated into a multi-agent system to act as a single conversational agent from the user's perspective."
],
[
"Overall Framework",
"The framework for the proactive conversational assistant for BPA consists of three main components: skills, agents, and an orchestrator, as shown in Fig.",
"REF .",
"Skills perform well defined tasks; they require a set of inputs to produce a set of outputs.",
"Agents are composed of skills that fall into three main categories: understand skills, act skills, and respond skills.",
"An execution pattern for the skills must be provided; it can be as simple as a static sequential script or as complex as a full execution graph.",
"The orchestrator coordinates between agents and it determines which agent or agents must execute to successfully respond to a user.",
"When a user enters an utterance, this natural language sentence is forwarded to all the agents in the assistant by the orchestrator after potentially preprocessing the input.",
"The orchestrator requests a preview of each agent's response and their confidence score.",
"This score depicts how confident an agent is in its ability to respond to the user.",
"Depending on the adopted orchestration pattern, the orchestrator selects one or more agents to execute and return their responses to the user.",
"Figure: Multi-agent Orchestration Overview"
],
[
"Skills",
"Skills are atomic functions, the building blocks of agents.",
"They are categorized according to their role within an agent and their effect on the state of the world.",
"Skills can have one of three roles: understand, act and respond.",
"Furthermore, skills can either be world changing or non-world changing."
],
[
"Understand skills",
"Serving as the entry point to an agent, understand skills identify the user's intent.",
"In the conversational setting, understand skills are generally intent and entity recognition skills but can also be as simple as a key-phrase detection.",
"They may take a user's natural language utterance and classify it into one of many predefined intents that the agent can handle or check if a specific phrase designated to trigger the agent is there.",
"They may also annotate any entities they recognize within the input.",
"In a non-dialog setting, understand skills are event triggered skills that watch for specific events and state changes in the system.",
"These events could be a user logging in or a change in database."
],
[
"Act skills",
"Act skills process the user input to produce an output that can be used to generate a response.",
"We distinguish two main types of act skills: world changing and non-world changing.",
"World changing skills perform actions that modify the state of the world or have side-effects outside the confines of the orchestrator world.",
"Examples of such skills include credit score checking skills, email sending skills, etc.",
"Non-world changing skills do not have any side effects.",
"Examples include email reading skills, weather checking skills, etc.",
"Act skills in the business process world can perform various tasks or activities.",
"One task is gathering information by querying databases or making API calls to information sources.",
"Decision points are a set of activities that move the business process forward.",
"Another task that moves the business process forward is submitting applications or forwarding gathered information to the next activity."
],
[
"Respond skills",
"Respond skills create the response that will be returned to the user by the conversational assistant.",
"The skills take the output of act skills and process them to form natural language responses to the user; these could be full flesh natural language generation to template responses given different outputs of the other skills.",
"Other modalities of responses can also be considered such as visualization of the output in the form of plots or images."
],
[
"Agents",
"Individually, the skills may not be able to perform complex tasks required by the activities within a business process.",
"Instead of increasing the scope of individual skills by implementing more complex functionality within them, composing them into agents capable of completing these activities is more effective.",
"This modular approach simplifies debugging of skills and agents and increases the reusability of skills by making them as domain agnostic as possible.",
"We define a contract for the agents in order to facilitate their orchestration within a single assistant.",
"The agents expect to receive an input message or utterance and the current context which represents the state of the system.",
"The agent is capable of producing a preview response that does not cause any world-changing actions to be invoked, an execute response that results from fully executing its pipeline and may cause the world to change, an updated context once the agent has executed, and a confidence in its response.",
"Multiple approaches can be adopted to compose skills into agents.",
"Creating the execution pipeline can be done statically where a software developer writes a script that explicitly defines the order in which skills must be executed and directs the flow of data by routing inputs and outputs.",
"In goal-oriented non-deterministic planning based composition [14], [1], software developers declaratively define the inputs and outputs of skills and the end goal of the agent.",
"Then, a planning engine composes an execution tree using the subset of available skills that will lead to the goal.",
"While this approach allows independent developers to include their skills in a catalog, composing skills into agents in an effective way requires a minimum level of compatibility between the skills, including the adoption of a unified vocabulary for input/output variables.",
"Agents are composed of an understand skill, a set of act skills, and a respond skill.",
"Depending on the functionality of the agent, none, one or multiple act skills may be included in the composition.",
"Next, we discuss four main types of agents based on their functionality.",
"While many more agents can be created, we consider the ones that would be most relevant and commonly used in BPA."
],
[
"Dialog Agents",
"Users may just need to inquire about certain topics and do not require any action to be taken by the assistant.",
"In such scenarios, a dialog agent, composed of an understand skill for intent and entity recognition and a respond skill to answer the user's query.",
"Examples of such agents include chit-chat agents, FAQ or “help\" agents, etc.",
"They can be implemented in dialog tree type services.",
"These agents do not change the world; their preview mode and execute mode are the same."
],
[
"Information Retrieval Agents",
"Unlike dialog agents, information retrieval agents must connect to an external service (such as making an API call or querying a database) in order to produce their answer to a user's query as opposed to having the answering within the agent implementation.",
"As a result, such agent compositions consist of three types of skills: understand skill that determine the intent of the user and annotate any entities that may be useful, one or more act skills that queries the appropriate service to produce its answer, and a respond skill that appropriately constructs the response returned to the user.",
"In general, most agents within this category do not contain skills that change the world.",
"However, their preview and execute modes may vary for computational/latency reasons.",
"In some situations, querying does change the world; for example, retrieving the credit score of a loan applicant is considered as world changing as it will have a negative impact on the applicant if queried a large number of times.",
"In other cases, simply incurring a cost, for example monetary or in the form or using up API calls that are limited, to query a database can be modeled as world changing to minimize the cost incurred by the system."
],
[
"Task Execution Agents",
"Beyond querying for information, the business user may want the assistant to perform tasks within the business process.",
"Task execution agents are capable of moving the business process forward by submitting application, making decisions at decision points and such.",
"These agents will change the world and therefore may require a preview and execute mode based on the type of orchestrator adopted.",
"These agents are composed of an understand skill, a set of act skills and a respond skill."
],
[
"Alerting Agents",
"Setting up alerts to receive notifications triggered by the occurrence of specific events allows business users to monitor the business process in an asynchronous way.",
"Like the other agents, these agents are also composed of an understand skill, a set of act skills and a respond skill.",
"However, their understand skill does not necessarily have to be a natural language understanding skill.",
"However, setting up custom alerts would require an intent and entity recognition understand skill.",
"These agents consist of a watcher skill on the system that may trigger the alert, a notification generation skill and a communication channel skill that sends the notification over the appropriate communication channel (e.g.",
"email, SMS, slack...).",
"A natural language understanding skill can be added to allow the user to customize the trigger and settings through dialog.",
"If that is the case, a Natural Language to Query (NLQ) skill would be necessary to convert natural language trigger definitions to structured, well-defined queries on the database that may trigger the alert and is watched by the daemon.",
"If an NLQ skill is not available, hard-coded conversions must be present.",
"However, this approach restricts the number of alerts that can be created.",
"NLQ provides a broader range of triggers that can be set.",
"It also allows the creation of domain agnostic alert agents which would be more portable and require minimal coding overhead to deploy in new domains, a desirable feature in the BPA field.",
"In this paper, we distinguish between alerts and notifications.",
"The former is the occurrence of an event that causes the conditions of the trigger to be satisfied.",
"The latter is the actual message sent to the user when an alert happens."
],
[
"Orchestration",
"Combining these diverse agents into a single assistant requires a domain agnostic orchestration layer capable of determining which subset of agents are best suited to respond to the user's utterance.",
"Various orchestration frameworks with diverse characteristics can be adopted to achieve this goal.",
"Stateless orchestrators do not keep track of the state in a centralized location.",
"Instead the context variables (state) are passed between the orchestrator and agents.",
"Stateful orchestrators on the other hand, maintain the state of the system in a centralized location.",
"The computational cost of data transfer in the former approach increases with the number of context variables maintained in the state.",
"As a result, system designers tend to minimize the number of variables in their state resulting a decrease in the amount of information agents have access to.",
"Stateful orchestrators, however, require a central device to maintain the state which could be costly from a deployment and scalability perspective.",
"Apriori orchestrators decide which agents should execute by considering the user input and what they know about agent capabilities.",
"Posterior orchestrators pass the inputs to the agents requesting their responses, then decide which agents to select.",
"In this case, preview and execute modes of agents must be available for those agents that perform world changing actions.",
"Orchestrators also expect agents to provide a confidence in their responses which are used as a feature in their decision making algorithm.",
"Unlike dialog managers, orchestrators do not require knowledge of the dialog to select agents.",
"Dialog managers are assumed to be part of those agents that need it.",
"For example, dialog agents would have managers that keep track of which node of a tree the conversation is in or perform dialog disambiguation and co-reference resolution.",
"The 3S orchestrator is an example of a posterior orchestrator that relies on the agent's confidence, among other metrics, to determine which agents should respond.",
"Its pipeline consists of three main steps: scoring, selecting, and sequencing agents [23], [20].",
"Figure: 3S Orchestration Pipeline"
],
[
"Scorer",
"When the orchestrator receives the user's utterance, it forwards this input to all agents and requests a preview of their response.",
"Once the agents return their preview responses and their confidence score, the scorer performs additional computations to obtain the final scores per agent.",
"An identity function scorer simply forwards the scores as is to the selector.",
"A scorer may normalize scores, especially since they could have been produced by very diverse agents that compute their scores differently.",
"A more sophisticated scorer could enhance the agents' confidences with other measures such as conversation stickiness, dialog depth (within a dialog tree) and agent engagement.",
"A Bayesian model that adapts to user feedback could also be adopted."
],
[
"Selector",
"Once the scorer computes the final scores per agent, it forwards them to the selector that must select one or more agents that will respond to the user.",
"It can be as simple as a Top 1 (or Top K) score selector that picks the agent (or K agents) with the highest score.",
"More complex selection criteria could be adopted such as heuristic rules, learned selection models from labeled user data, online reinforcement learning models trained on user feedback, or a combination of multiple approaches.",
"Once the selector determines the subset of agents that must execute, the orchestrator requests their execution responses.",
"Various machine learning algorithms can be adopted to improve the robustness of the selector as the noise in the score increases.",
"In a supervised learning domain, labels can represent agents that are the output of classifiers and the input to the classifier would be a feature vector consisting of the agents' scores and/or user utterances.",
"In a reinforcement learning domain, actions can represent agents and the environment produces a reward when the correct agent is selected.",
"For both approaches, deep learning models can be trained if enough data is provided to further improve the selection model."
],
[
"Sequencer",
"After receiving the agents' execution responses, a sequencer determines the order in which these responses will be presented to the user.",
"If a Top 1 score selector was adopted in the previous stage, then no sequencing is necessary.",
"The response is directly outputted to the user through the conversational interface.",
"However, when a subset of agents has responded, a sequencer could adopt different approaches to order these responses: e.g.",
"descending score, heuristic rules (e.g.",
"greeting agent must always respond before other agents), supervised or reinforcement learning models trained on labeled data and user feedback."
],
[
"Use Case – 1: Travel Preapproval Application",
"We will now detail the implementation of a conversational assistant (in Slack) for two personas (employees, and managers/directors) in a simplified travel preapproval application process using the framework described so far.",
"Figure: Travel Preapproval ProcessTable: Conversation samples"
],
[
"Travel Preapproval Business Process",
"We adopt a simplified business process revolving around travel preapprovals generally used across many corporations.",
"We consider the scenario of researchers/employees requesting travel funds to attend research conferences.",
"The travel preapproval process, shown in Fig.",
"REF , was implemented in a business process management software.",
"Employees submit a travel preapproval request to attend a conference by filling in specific information into a form, shown in Fig.",
"REF (a).",
"Once submitted, the application is forwarded to the employee's manager who can approve/reject the application (Fig.",
"REF (b)) or request additional information/changes to the application (Fig.",
"REF (c)).",
"If approved by the manager, the application is forwarded to the director who makes the final decision on the travel request."
],
[
"Agents in ",
"Travelbot contains multiple agents that enable it to respond to a broad range of user requests.",
"The 3S orchestrator coordinates between the composed agents to respond to the user or execute any actions requested by the user.",
"Table REF includes a sample interaction between Travelbot and a manager."
],
[
"Chit-Chat Agent",
"A simple dialog agent, Chit-Chat is implemented as a question answering dialog tree that is capable of answering utterances related to greetings (e.g.",
"“hello”, “how are you”, “goodbye”, etc.).",
"It can also introduce the assistant answering questions such as “who are you” and “how can you help me”.",
"Additional intents and dialog trees are also included to make the assistant more human-like and improve the user experience – e.g.",
"“what are you doing”, “tell me a joke”, etc.",
"This agent does not have any act skills; it does not cause any side effects when executed and hence has identical preview and execute modes."
],
[
"Publication Query Agent",
"This information retrieval agent queries a database containing the publications of employees and conference information.",
"In the adopted use case, this agent can be used to auto-fill the information in the application form related to travel justification and conference location and dates.",
"It can also be used by managers and directors to obtain information necessary for making decisions.",
"For example, travel requests would be approved for employees presenting their work at the conference as first authors.",
"This agent makes this information accessible to managers through a conversational interface and eliminates the need for them to switch to a different interface.",
"The static composition of this agent consists of a Natural Language Understanding (NLU) understand skill, an act skill capable of retrieving employee information from an employee directory that would be used by the publication query skill (second act skill) to retrieve publication/conference information.",
"Finally, the respond skill generates the final response that will be returned to the user."
],
[
"Business Process Data Query Agent",
"Another informational retrieval agent, the business process data query agent allows managers and directors to query the existing travel request applications.",
"Managers and directors can ask questions like “How many pending travel requests are in my queue?” or “How many applications have been submitted by employee X?”.",
"Such queries factor into the decision making process and allow decision makers to make more informed decisions.",
"This agent is composed of three skills: an NLU understand skill, a data query act skill, and a respond skill."
],
[
"Business Process Task Execution Agent",
"This agent allows users to move the business process forward by requesting the assistant to submit an application on their behalf (e.g.",
"“Submit an application to AAAI 2020 for me.”) or approve/reject/send back an application (e.g.",
"“Reject Jack's application to ICML 2019.”).",
"The agent composition is more complex in this case and consists of four act skills that may or may not be invoked depending on the recognized intent.",
"Like the other agents, this agent also has an intent and entity recognition understand skill and a respond skill.",
"Its act skills are employee directory query, paper directory skill, travel pricing skill and business process task execute act skill.",
"The pricing skill is used auto-fill the “requested amount” field in Fig.",
"REF (a)."
],
[
"Travel Estimation Agent",
"This agent can respond to inquiries about travel such as – “What is the cheapest flight from BOS to SFO leaving on 2019/12/01 and returning on 2019/12/07?”.",
"This information retrieval agent consists of an intent and entity recognition understand skill, a travel pricing skill which uses the Amadeus API to estimate the cost of flights and hotels, and a respond skill that formulates the answer returned to the user.",
"Managers or directors can use this agent to verify the “requested amount” field submitted by the employee.",
"Figure: Employee Travel Preapproval User Interfaces: (a) Employee Application Form, (b) Manager/Director Review Form, (c) Employee/Manager/Director Revision Form"
],
[
"Visualization Agent",
"Beyond natural language responses, the visualization agent provides the assistant with the ability to return visual responses.",
"This is useful in situations where information retrieval agents return data that is easier to digest in a plot form.",
"Therefore, the user can ask the assistant to plot the results.",
"This agent is composed of an intent and entity recognition understand skill to if and which plot the user wants and what variables to use if multiple options exist.",
"A plotter skill converts the plottable data to an image of a bar graph or doughnut chart or other visualization format that is then forwarded to the user using the respond skill.",
"Due to the stateless implementation of the orchestrator, the data is persisted for a finite number of iterations in the context that is passed between agents the orchestrator.",
"If the visualization agent is invoked by a user utterance, it will look for plottable data in the context.",
"If this data does not exist, then a default response of “There is no data to be plotted.” is returned to the user."
],
[
"Alerting using Natural Language Query",
"TravelBot allows the user to create and customize alerts based on user's data-store.",
"The system accepts user input and if it is determined that intent is the creation of alert, the sentence is then passed through an NLQ module [18], [17], [7].",
"The NLQ module can express any natural language sentence into a data query which can run on top of many different types of data-stores (ElasticSearch, SQL, etc).",
"The query from NLQ module is then stored and a daemon runs in the background monitoring these queries.",
"If there is any change in the data store (Addition of new row, changes in values of an existing row) and the query output changes, an alert is generated and sent to the user on the communication channel.",
"The system determines the communication channel for a specific user through conversation.",
"Alerts could be configured to send notifications via Slack, email or SMS using the Twilio API."
],
[
"Use Case – 2: Loan Application",
"Using the same multi-agent framework, we also develop a loan officer chatbot for a loan application business process.",
"We include: 1) a data query agent capable of answering questions related to existing applications, 2) a business rules agent that can provide insights into whether an application will be accepted or rejected, 3) a visualization agent capable of representing data in different formats, 4) an alerts agent that sends notifications when changes to loan applications occur, and 5) a content analyzer that can process loan documents and extract information from them.",
"Sample conversations with this chatbot are included in Table REF and Fig.",
"REF .",
"Figure: Multi-modal responses from the LoanBot.Table: Sample conversation with LoanBot"
],
[
"Challenges",
"Multiple challenges have hindered the deployment of autonomous conversational agents in BPA domains.",
"We focus on a few of these challenges and discuss how the proposed framework can overcome them."
],
[
"Coding Overhead",
"The enterprises that need chatbots often do not have the technical expertise or resources to maintain and improve the framework.",
"Thus, it is important that the system requires minimal coding overhead to deploy in new domains, expand the scope of the system and improve its performance by adding new agents during its lifetime.",
"Our framework assumes that independent developers, whether domain experts or software developers, have created the autonomous agents but require that they abide by the adopted input-output contracts.",
"This reduces the coding overhead to integrate the agents.",
"Furthermore, the 3S orchestrator is robust enough to handle the noise in the agent confidence which is a byproduct of independent agent developers."
],
[
"Scalability",
"Expanding the scope and capabilities of a conversational agent requires significant developer effort and may result in performance degradation.",
"In the proposed framework, integration is trivial but naively adding agents may result in performance issues if the orchestrator incorrectly selects agents.",
"Adopting an orchestration model that learns from previous interactions and changes in the system would improve the overall performance of the assistant."
],
[
"Agent Overlap",
"As the number of agents in the framework increases, some agent functionality and knowledge may overlap.",
"This may cause conflicts among agents when attempting to respond to users and degrade overall performance.",
"As a result, it is important to model or quantify agent overlap and incorporate it within the orchestrator model to insure that the orchestrator will select the best agent to complete the task of responding to the user.",
"Adopting a learning selector in the orchestrator pipeline can lead to better selection models that can handle agent overlap and prevent performance degradation."
],
[
"Multi-user Support",
"Giving users the ability to compose their personalized assistants requires the framework to support multiple users.",
"While this can be resolved by creating a session per user, it provides an opportunity to increase the available data from which the orchestrator can learn better execution models.",
"Therefore, the proposed framework should consider approaches to learn from multiple users' data while maintaining privacy and security of the data."
],
[
"Access Control",
"With different personas in business processes, some agents may not be accessible to specific users.",
"For example, with the Travelbot, employees should not have access to the \"approve/reject\" functionality.",
"Our framework requires that these restrictions are implemented inside the agent.",
"When the user submits a request, the orchestrator forwards this request to all agents and expects them to decline execution if the user does not have the required access."
],
[
"Conclusion",
"In this paper, we presented a multi-agent framework to develop a conversational assistant supporting multiple capabilities such as querying data through natural language, autonomously executing tasks in a business process, alerting users of changes in the business process and visualizing data in various forms based on the user's requests.",
"An orchestrator capable of combining independently developed skills with minimal overhead is the key to effectively deploy more powerful conversational assistants.",
"While many challenges remain such as scalability and agent overlap, preliminary results motivate follow on research."
]
] | 2001.03543 |
[
[
"DSGN: Deep Stereo Geometry Network for 3D Object Detection"
],
[
"Abstract Most state-of-the-art 3D object detectors heavily rely on LiDAR sensors because there is a large performance gap between image-based and LiDAR-based methods.",
"It is caused by the way to form representation for the prediction in 3D scenarios.",
"Our method, called Deep Stereo Geometry Network (DSGN), significantly reduces this gap by detecting 3D objects on a differentiable volumetric representation -- 3D geometric volume, which effectively encodes 3D geometric structure for 3D regular space.",
"With this representation, we learn depth information and semantic cues simultaneously.",
"For the first time, we provide a simple and effective one-stage stereo-based 3D detection pipeline that jointly estimates the depth and detects 3D objects in an end-to-end learning manner.",
"Our approach outperforms previous stereo-based 3D detectors (about 10 higher in terms of AP) and even achieves comparable performance with several LiDAR-based methods on the KITTI 3D object detection leaderboard.",
"Our code is publicly available at https://github.com/chenyilun95/DSGN."
],
[
"Introduction",
"3D scene understanding is a challenging task in 3D perception, which serves as a basic component for autonomous driving and robotics.",
"Due to the great capability of LiDAR sensors to accurately retrieve 3D information, we witness fast progress on 3D object detection.",
"Various 3D object detectors were proposed [9], [25], [60], [28], [29], [35], [41], [55], [10] to exploit LiDAR point cloud representation.",
"The limitation of LiDAR is on the relatively sparse resolution of data with several laser beams and on the high price of the devices.",
"Figure: DSGN jointly estimates depth and detects 3D objects from a stereo image pair.",
"It intermediately generates a plane-sweep volume and 3D geometric volume to represent 3D structure in two different 3D space.In comparison, video cameras are cheaper and are with much denser resolutions.",
"The way to compute scene depth on stereo images is to consider disparity via stereo correspondence estimation.",
"Albeit recently several 3D detectors based on either monocular [38], [7], [6], [32], [50] or stereo [27], [47], [39], [58] setting push the limit of image-based 3D object detection, the accuracy is still left far behind compared with the LiDAR-based approaches.",
"Challenges One of the greatest challenges for image-based approaches is to give appropriate and effective representation for predicting 3D objects.",
"Most recent work [27], [38], [50], [39], [42], [2] divides this task into two sub ones, , depth prediction and object detection.",
"Camera projection is a process that maps 3D world into a 2D image.",
"One 3D feature in different object poses causes local appearance changes, making it hard for a 2D network to extract stable 3D information.",
"Another line of solutions [47], [58], [49], [32] generate intermediate point cloud followed by a LiDAR-based 3D object detector.",
"This 3D representation is less effective since the transformation is non-differentiable and incorporates several independent networks.",
"Besides, the point cloud faces the challenge of object artifacts [19], [49], [58] that limits the detection accuracy of the following 3D object detector.",
"Our Solution In this paper, we propose a stereo-based end-to-end 3D object detection pipeline (Figure REF ) – Deep Stereo Geometry Network (DSGN), which relies on space transformation from 2D features to an effective 3D structure, called 3D geometric volume (3DGV).",
"The insight behind 3DGV lies in the approach to construct the 3D volume that encodes 3D geometry.",
"3D geometric volume is defined in 3D world space, transformed from a plane-sweep volume (PSV) [11], [12] constructed in the camera frustum.",
"The pixel-correspondence constraint can be well learned in PSV, while 3D features for real-world objects can be learned in 3DGV.",
"The volume construction is fully differentiable and thus can be jointly optimized for learning of both stereo matching and object detection.",
"This volumetric representation has two key advantages.",
"First, it is easy to impose the pixel-correspondence constraint and encode full depth information into 3D real-world volume.",
"Second, it provides 3D representation with geometry information that makes it possible to learn 3D geometric features for real-world objects.",
"As far as we know, there was no study yet to explicitly investigate the way of encoding 3D geometry into an image-based detection network.",
"Our contribution is summarized as follows.",
"To bridge the gap between 2D image and 3D space, we establish stereo correspondence in a plane-sweep volume and then transform it to 3D geometric volume for capability to encode both 3D geometry and semantic cues for prediction in 3D regular space.",
"We design an end-to-end pipeline for extracting pixel-level features for stereo matching and high-level features for object recognition.",
"The proposed network jointly estimates scene depth and detects 3D objects in 3D world, enabling many practical applications.",
"Without bells and whistles, our simple and fully-differentiable network outperforms all other stereo-based 3D object detectors (10 points higher in terms of AP) on the official KITTI leaderboard [14]."
],
[
"Related Work",
"We briefly review recent work on stereo matching and multi-view stereo.",
"Then we survey 3D object detection based on LiDAR, monocular images, and stereo images.",
"Stereo Matching In the field of stereo matching on binocular images, methods of [22], [4], [59], [15], [45], [48] process the left and right images by a Siamese network and construct a 3D cost volume to compute the matching cost.",
"Correlation-based cost volume is applied in recent work [33], [57], [54], [15], [30], [44].",
"GC-Net [22] forms a concatenation-based cost volume and applies 3D convolution to regress disparity estimates.",
"Recent PSMNet [4] further improves the accuracy by introducing pyramid pooling module and stacks hourglass modules [34].",
"State-of-the-art methods already achieved less than 2% 3-pixel error on KITTI 2015 stereo benchmark.",
"Multi-View Stereo Methods of [5], [56], [20], [21], [18], [17] reconstruct 3D objects in a multi-view stereo setting [1], [3].",
"MVSNet [56] constructs plane-sweep volumes upon a camera frustum to generate the depth map for each view.",
"Point-MVSNet [5] instead intermediately transforms the plane-sweep volume to point cloud representation to save computation.",
"Kar [21] proposed the differentiable projection and unprojection operation on multi-view images.",
"LiDAR-based 3D Detection LiDAR sensors are very powerful, proven by several leading 3D detectors.",
"Generally two types of architectures, i.e., voxel-based approaches [60], [9], [26], [10] and point-based approaches [36], [37], [41], [55], [51], were proposed to process point cloud.",
"Image-based 3D Detection Another line of detection is based on images.",
"Regardless of monocular- or stereo-based setting, methods can be classified into two types according to intermediate representation existence.",
"3D detector with depth predictor: the solution relies on 2D image detectors and depth information extraction from monocular or stereo images.",
"Stereo R-CNN$_{\\rm Stereo}$ [27] formulates 3D detection into multiple branches/stages to explicitly resolve several constraints.",
"We note that the keypoint constraint may be hard to generalize to other categories like Pedestrian, and the dense alignment for stereo matching directly operating raw RGB images may be vulnerable to occlusion.",
"MonoGRNet$_{Mono}$ [38] consists of four subnetworks for progressive 3D localization and directly learning 3D information based solely on semantic cues.",
"MonoDIS$_{\\rm Mono}$ [42] disentangles the loss for 2D and 3D detection.",
"It achieves both tasks in an end-to-end manner.",
"M3D-RPN$_{\\rm Mono}$ [2] applies multiple 2D convolutions of non-shared weights to learn location-specific features for joint prediction of 2D and 3D boxes.",
"Triangulation$_{\\rm Stereo}$ [39] directly learns offset from predefined 3D anchors on bird's eye view and establishes object correspondence on RoI-level features.",
"Due to low resolutions, pixel correspondence is not fully exploited.",
"3D representation based 3D Detector: 3DOP$_{\\rm Stereo}$ [7], [8] generates point cloud by stereo and encodes the prior knowledge and depth in an energy function.",
"Several methods [47], [58], [49], [32] transform the depth map to Pseudo-LiDAR (point cloud) intermediately followed by another independent network.",
"This pipeline yields large improvement over previous methods.",
"OFT-Net$_{\\rm Mono}$ [40] maps image feature into an orthographic bird's eye view representation and detects 3D objects on bird's eye view."
],
[
"Our Approach",
"In this section, we first explore the proper representation for 3D space and motivate our network design.",
"Based on the discussion, we present our complete 3D detection pipeline under a binocular image pair setting.",
"Figure: Overview of Deep Stereo Geometry Network (DSGN).",
"The whole neural network consists of four components.",
"(a) A 2D image feature extractor for capture of both pixel- and high-level feature.",
"(b) Constructing the plane-sweep volume and 3D geometric volume.",
"(c) Depth Estimation on the plane-sweep volume.",
"(d) 3D object detection on 3D geometric volume."
],
[
"Motivation",
"Due to perspective, objects appear smaller with the increase of distance, which makes it possible to roughly estimate the depth according to the relative scale of objects sizes and the context.",
"However, 3D objects of the same category may still have various sizes and orientations.",
"It greatly increases the difficulty to make accurate prediction.",
"Besides, the visual effect of foreshortening causes that nearby 3D objects are not scaled evenly in images.",
"A regular cuboid car appears like an irregular frustum.",
"These two problems impose major challenges for 2D neural networks to model the relationship between 2D imaging and real 3D objects [27].",
"Thus, instead of relying on 2D representation, by reversing the process of projection, an intermediate 3D representation provides a more promising way for 3D object understanding.",
"The following two representations can be typically used in 3D world.",
"Point-based Representation Current state-of-the-art pipelines [47], [58], [32] generate intermediate 3D structure of point cloud by depth prediction approaches [13], [4], [22] and apply LiDAR-based 3D object detectors.",
"The main possible weakness is that it involves several independent networks and potentially loses information during intermediate transformation, making the 3D structure (such as cost volume) boiled down to point cloud.",
"This representation often encounters streaking artifacts near object edges [19], [49], [58].",
"Besides, the network is hard to be differentiated for multi-object scenes [5], [10].",
"Voxel-based Representation Volumetric representation, as another way of 3D representation, is investigated less intensively.",
"OFT-Net$_{\\rm mono}$ [40] directly maps the image feature to the 3D voxel grid and then collapses it to the feature on bird's eye view.",
"However, this transformation keeps the 2D representation for this view and does not explicitly encode the 3D geometry of data.",
"Our Advantage The key to establishment of an effective 3D representation relies on the ability to encode accurate 3D geometric information of the 3D space.",
"A stereo camera provides an explicit pixel-correspondence constraint for computing depth.",
"Aiming to design a unified network to exploit this constraint, we explore deep architectures capable of extracting both pixel-level features for stereo correspondence and high-level features for semantic cues.",
"On the other hand, the pixel-correspondence constraint is supposedly imposed along the projection ray through each pixel where the depth is considered to be definite.",
"To this end, we create an intermediate plane-sweep volume from a binocular image pair to learn stereo correspondence constraint in camera frustum and then transform it to a 3D volume in 3D space.",
"In this 3D volume with 3D geometric information lifted from the plane-sweep volume, we are able to well learn 3D features for real-world objects."
],
[
"Deep Stereo Geometry Network",
"In this subsection, we describe our overall pipeline – Deep Stereo Geometry Network (DSGN) as shown in Figure REF .",
"Taking the input of a binocular image pair ($I_L, I_R$ ), we extract features by a Siamese network and construct a plane-sweep volume (PSV).",
"The pixel-correspondence is learned on this volume.",
"By differentiable warping, we transform PSV to a 3D geometric volume (3DGV) to establish 3D geometry in 3D world space.",
"Then the following 3D neural network on the 3D volume learns necessary structure for 3D object detection."
],
[
"Image Feature Extraction",
"Networks for stereo matching [22], [4], [15] and object recognition [16], [43] have different architecture designs for their respective tasks.",
"To ensure reasonable accuracy of stereo matching, we adopt the main design of PSMNet [4].",
"Because the detection network requires a discriminative feature based on high-level semantic features and large context information, we modify the network for grasping more high-level information.",
"Besides, the following 3D CNN for cost volume aggregation takes much more computation, which gives us room to modify the 2D feature extractor without introducing extra heavy computation overhead in the overall network."
],
[
"Network Architecture Details ",
"Here we use the notations conv_1, conv_2, ..., conv_5 following [16].",
"The key modification for 2D feature extractor is as follows.",
"Shift more computation from conv_3 to conv_4 and conv_5, , changing the numbers of basic blocks of conv_2 to conv_5 from $\\lbrace 3, 16, 3, 3\\rbrace $ to $\\lbrace 3, 6, 12, 4\\rbrace $ .",
"The SPP module used in PSMNet concatenates the output layers of conv_4 and conv_5.",
"The output channel number of convolutions in conv_1 is 64 instead of 32 and the output channel number of a basic residual block is 192 instead of 128.",
"Full details of our 2D feature extraction network are included in the supplementary material."
],
[
"Constructing 3D Geometric Volume",
"To learn 3D convolutional features in 3D regular space, we first create a 3D geometric volume (3DGV) by warping a plane-sweep volume to 3D regular space.",
"Without loss of generality, we discretize the region of interest in 3D world space to a 3D voxel occupancy grid of size $(W_V, H_V, D_V)$ along the right, down and front directions in camera view.",
"$W_V, H_V, D_V$ denote the width, height and length of the grid, respectively.",
"Each voxel is of size $(v_w, v_h, v_d)$ .",
"Plane-Sweep Volume In binocular vision, an image pair $(I_L, I_R)$ is used to construct a disparity-based cost volume for computing matching cost, which matches a pixel $i$ in the left image $I_L$ to the correspondence in the right image $I_R$ horizontally shifted by an integral disparity value $d$ .",
"The depth is inversely proportional to disparity.",
"It is thus hard to distinguish among distant objects due to the similar disparity values [27], [47], [58].",
"For example, objects 40-meter and 39-meter away have almost no difference ($<0.25$ pix) on disparity on KITTI benchmark [14].",
"In a different way to construct the cost volume, we follow the classic plane sweeping approach [11], [12], [56] to construct a plane-sweep volume by concatenating the left image feature $F_L$ and the reprojected right image feature $F_{R->L}$ at equally spaced depth interval, which avoids imbalanced mapping of features to 3D space.",
"The coordinate of PSV is represented by $(u, v, d)$ , where $(u, v)$ represents $(u,v)$ -pixel in the image and it adds another axis orthogonal to the image plane for depth.",
"We call the space of $(u, v, d)$ grid camera frustum space.",
"The depth candidates $d_i$ are uniformly sampled along the depth dimension with interval $v_d$ following the pre-defined 3D grid.",
"Concatenation-based volume enables the network to learn semantic features for object recognition.",
"We apply 3D convolution to this volume and finally get a matching cost volume for all depth.",
"To ease computation, we apply only one 3D hourglass module, contrary to the three used in PSMNet [4].",
"We note that the resulting performance degradation can be compensated in the following detection network since the overall network is differentiable.",
"Figure: Illustration of volume transformation.",
"The image is captured at the image plane (red solid line).",
"PSV is constructed by projecting images at equally spaced depth (blue dotted lines) in left camera frustum, which is shown in the 3D world space (left) and camera frustum space (middle).",
"Car is shown to be distorted in the middle.",
"Mapping by the camera intrinsic matrix KK, PSV is warped to 3DGV, which restores the car.3D Geometric Volume With known camera internal parameters, we transform the last feature map of PSV before computing matching cost from camera frustum space $(u,v,d)$ to 3D world space $(x,y,z)$ by reversing 3D projection with $\\begin{split}\\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix}=\\begin{pmatrix}1/f_x & 0 & -c_u/f_x \\\\0 & 1/f_y & -c_v/f_y \\\\0 & 0 & 1\\end{pmatrix}\\begin{pmatrix}ud\\\\vd\\\\d\\end{pmatrix}\\end{split}$ where $f_x, f_y$ are the horizontal and vertical focal lengths.",
"This transformation is fully-differentiable and saves computation by eliminating background outside the pre-defined grid, such as the sky.",
"It can be implemented by warp operation with trilinear interpolation.",
"Figure REF illustrates the transformation process.",
"The common pixel-correspondence constraint (red dotted lines) is imposed in camera frustum while object recognition is learned in regular 3D world space (Euclidean space).",
"There obviously is difference in these two representations.",
"In the last feature map of plane-sweep volume, a low-cost voxel $(u, v, d)$ means the high probability of object existing at depth $d$ along the ray through the focal point and image point $(u,v)$ .",
"With the transformation to regular 3D world space, the feature of low cost suggests that this voxel is occupied in the front surface of the scene, which can serve as a feature for 3D geometric structure.",
"Thus it is possible for the following 3D network to learn 3D object features on this volume.",
"This operation is fundamentally different from differentiable unprojection [21], which directly lifts the image feature from 2D image frame to 3D world by bilinear interpolation.",
"Our goal is to lift geometric information from cost volume to 3D world grid.",
"We make pixel-correspondence constraint easy to be imposed along the projection ray.",
"The contemporary work [58] applies a similar idea to construct depth-cost-volume like plane-sweep volume.",
"Differently, we aim to avoid imbalanced warping from plane-sweep volume to 3D geometric volume, and deal with the streaking artifact problem.",
"Besides, our transformation keeps the distribution of depth instead of deducting it to a depth map.",
"Our strategy intriguingly avoids object artifacts."
],
[
"Depth Regression on Plane-Sweep Cost Volume",
"To compute the matching cost on the plane-sweep volume, we reduce the final feature map of plane-sweep volume by two 3D convolutions to get 1D cost volume (called plane-sweep cost volume).",
"Soft arg-min operation [22], [4], [59] is applied to compute the expectation for all depth candidates with probability $\\sigma (-c_d)$ as $\\hat{d} = \\sum _{d\\in \\lbrace z_\\text{min}, z_\\text{min} + v_d, ..., z_\\text{max}\\rbrace } d \\times \\sigma (-c_d)$ where the depth candidates are uniformly sampled within pre-defined grid $[z_\\text{min}, z_\\text{max}]$ with interval $v_d$ .",
"The softmax function encourages the model to pick a single depth plane per pixel."
],
[
"3D Object Detector on 3D Geometric Volume",
"Motivated by recent one-stage 2D detector FCOS [46], we extend the idea of centerness branch in our pipeline and design a distance-based strategy to assign targets for the real world.",
"Because objects of the same category are of similar size in 3D scene, we still keep the design of anchors.",
"Let $\\mathcal {V} \\in \\mathbb {R}^{W\\times H\\times D\\times C}$ be the feature map for 3DGV of size $(W, H, D)$ and denote the channels as $C$ .",
"Considering the scenario of autonomous driving, we gradually downsample along the height dimension and finally get the feature map $\\mathcal {F}$ of size $(W, H)$ for bird's eye view.",
"The network architecture is included in the supplementary material.",
"For each location $(x, z)$ in $\\mathcal {F}$ , several anchors of different orientations and sizes are placed.",
"Anchors $\\mathbf {A}$ and ground-truth boxes $\\mathbf {G}$ are represented by the location, prior size and orientation, , ($x_\\mathbf {A}, y_\\mathbf {A}, z_\\mathbf {A}, h_\\mathbf {A}, w_\\mathbf {A}, l_\\mathbf {A}, \\theta _A$ ) and ($x_\\mathbf {G}, y_\\mathbf {G}, z_\\mathbf {G}, h_\\mathbf {G}, w_\\mathbf {G}, l_\\mathbf {G}, \\theta _\\mathbf {G}$ ).",
"Our network regresses from anchor and gets the final prediction $(h_\\mathbf {A} e^{\\delta h}, w_\\mathbf {A} e^{\\delta w}, l_\\mathbf {A} e^{\\delta l}, x_\\mathbf {A} + \\delta x, y_\\mathbf {A} + \\delta y, z_\\mathbf {A} + \\delta z, \\theta _\\mathbf {A} + \\pi / N_\\theta \\tanh (\\delta \\theta ))$ , where $N_\\theta $ denotes the number of anchor orientations and $\\delta \\cdot $ is the learned offset for each parameter."
],
[
"Distance-based Target Assignment",
"Taking object orientation into consideration, we propose distance-based target assignment.",
"The distance is defined as the distance of 8 corners between anchor and ground-truth boxes as $distance(\\mathbf {A}, \\mathbf {G}) = \\frac{1}{8}\\sum _{i=1}^8{\\sqrt{(x_{\\mathbf {A}_i} - x_{\\mathbf {G}_i})^2 + (z_{\\mathbf {A}_i} - z_{\\mathbf {G}_i})^2)}} \\nonumber $ In order to balance the ratio of positive and negative samples, we let the anchors with top $N$ nearest distance to ground-truth as positive samples, where $N = \\gamma \\times k$ and $k$ is the number of voxels inside ground-truth box on bird's eye view.",
"$\\gamma $ adjusts the number of positive samples.",
"Our centerness is defined as the exponent of the negative normalized distance of eight corners as $\\begin{split}centerness(\\mathbf {A},\\mathbf {G}) = e^{-\\text{norm}(distance(\\mathbf {A},\\mathbf {G}))},\\end{split}$ where $\\text{norm}$ denotes min-max normalization."
],
[
"Multi-task Training",
"Our network with stereo matching network and 3D object detector is trained in an end-to-end fashion.",
"We train the overall 3D object detector with a multi-task loss as $Loss = \\mathcal {L}_{depth} + \\mathcal {L}_{cls} + \\mathcal {L}_{reg} + \\mathcal {L}_{centerness}.$ For the loss of depth regression, we adopt smooth $L_1$ loss [22] in this branch as $\\mathcal {L}_{depth} = \\frac{1}{N_D}\\sum _{i=1}^{N_D} smooth_{L_1}\\left(d_i - \\hat{d_i}\\right),$ where $N_D$ is the number of pixels with ground-truth depth (obtained from the sparse LiDAR sensor).",
"For the loss of classification, focal loss [31] is adopted in our network to deal with the class imbalance problem in 3D world as $\\mathcal {L}_{cls} = \\frac{1}{N_{pos}} \\sum _{(x,z)\\in \\mathcal {F}} \\textit {Focal\\_Loss}(p_{\\mathbf {A}_{(x,z)}}, p_{\\mathbf {G}_{(x,z)}}),$ where $N_{pos}$ denotes the number of positive samples.",
"Binary cross-entropy (BCE) loss is used for centerness.",
"For the loss of 3D bounding box regression, smooth $L_1$ Loss is used for the regression of bounding boxes as $\\begin{split}\\mathcal {L}_{reg} = \\frac{1}{N_{pos}}\\sum _{(x,z)\\in F_{pos}} {\\rm centerness}(\\mathbf {A},\\mathbf {G}) \\times \\\\{\\rm smooth}_{L_1}( l1\\_{\\rm distance}(\\mathbf {A}, \\mathbf {G}) )\\end{split}$ where $F_{pos}$ denotes all positive samples on bird's eye view.",
"We try two different regression targets with and without jointly learning all parameters.",
"Separably optimizing box parameters.",
"The regression loss is directly applied to the offset of ($x, y, z, h, w, l, \\theta $ ).",
"Jointly optimizing box corners.",
"For jointly optimizing box parameters, the loss is made on the average L1 distance of eight box corners between predicted boxes from 3D anchors and ground-truth boxes following that of [35].",
"In our experiments, we use the second regression target for Car and the first regression target for Pedestrian and Cyclist.",
"Because it is hard for even human to accurately predict or annotate the orientation of objects like Pedestrian from an image, other parameter estimation under joint optimization can be affected."
],
[
"Experiments",
"Datasets Our approach is evaluated on the popular KITTI 3D object detection dataset [14], which is union of $7,481$ stereo image-pairs and point clouds for training and $7,518$ for testing.",
"The ground-truth depth maps are generated from point clouds following [47], [58].",
"The training data has annotation for Car, Pedestrian and Cyclist.",
"The KITTI leaderboard limits the access to submission to the server for evaluating test set.",
"Thus, following the protocol in [9], [27], [47], the training data is divided into a training set (3,712 images) and a validation set (3,769 images).",
"All ablation studies are conducted on the split.",
"For the submission of our approach, our model is trained from scratch on the 7K training data only.",
"Evaluation Metric KITTI has three levels of difficulty setting of easy, moderate (main index) and hard, according to the occlusion/truncation and the size of an object in the 2D image.",
"All methods are evaluated for three levels of difficulty under different IoU criteria per class , , IoU $\\ge $ 0.7 for Car and IoU $\\ge $ 0.5 for Pedestrian and Cyclist for 2D, bird's eye view and 3D detection.",
"Following most image-based 3D object detection setting [47], [27], [39], [38], [2], [42], the ablation experiments are conducted on Car.",
"We also report the results of Pedestrian and Cyclist for reference in the supplementary file.",
"KITTI benchmark recently changes evaluation where AP calculation uses 40 recall positions instead of the 11 recall positions proposed in the original Pascal VOC benchmark.",
"Thus, we show the main test results following the official KITTI leaderboard.",
"We generate the validation results using the original evaluation code for fair comparison with other approaches in ablation studies."
],
[
"Implementation",
"Training Details By default, models are trained on 4 NVIDIA Tesla V100 (32G) GPUs with batch-size 4 – that is, each GPU holds one pair of stereo images of size $384\\times 1248$ .",
"We apply ADAM [23] optimizer with initial learning rate 0.001.",
"We train our network for 50 epochs and the learning rate is decreased by 10 at 50-th epoch.",
"The overall training time is about 17 hours.",
"The data augmentation used is horizontal flipping only.",
"Following other approaches [58], [47], [60], [52], [41], [10], another network is trained for Pedestrian and Cyclist, we first pre-train the network with all training images for the stereo network and then apply fine-tune with 3D box annotation for both branches because only about $1/3$ images have annotations of these two objects.",
"Implementation Details For constructing plane-sweep volume, the image feature map is shrunk to 32D and downsampled by 4 for both left and right images.",
"Then by reprojection and concatenation, we construct the volume of shape $(W_{I}/4, H_{I}/4, D_{I}/4, 64)$ , where the image size is $(W_{I}=1248, H_{I}=384)$ and the number of depth is $D_{I}=192$ .",
"It is followed by one 3D hourglass module [4], [34] and extra 3D convolutions to get the matching cost volume of shape $(W_{I}/4, H_{I}/4, D_{I}/4, 1)$ .",
"Then interpolation is used to upsample this volume to fit the image size.",
"To construct 3D geometric volume, We discretize the region in range $[-30.4, 30.4]\\times [-1, 3]\\times [2, 40.4]$ (meters) to a 3D voxel occupancy grid of size $(W_{V}=300, H_{V}=20, D_{V}=192)$ along the right ($X$ ), down ($Y$ ) and front ($Z$ ) directions in camera's view.",
"3D geometric volume is formed by warping the last feature map of PSV.",
"Each voxel is a cube of size $(0.2, 0.2, 0.2)$ (meter).",
"Other implementation details and the network architecture are included in the supplementary file."
],
[
"Main Results",
"We give comparison with state-of-the-art 3D detectors in Tables REF and REF .",
"Without bells and whistles, our approach outperforms all other image-based methods on 3D and BEV object detection.",
"We note that Pseudo-LiDARs [47], [58] is with pre-trained PSMNet [4] on a large-scale synthetic scene flow dataset [33] (with 30,000+ pairs of stereo images and dense disparity maps) for stereo matching.",
"Stereo R-CNN [27] uses ImageNet pre-trained ResNet-101 as backbone and has input images of resolution $600\\times 2000$ .",
"Differently, our model is trained from scratch only on these 7K training data with input of resolution $384\\times 1248$ .",
"Also, Pseudo-LiDARs [47], [58] approaches apply two independent networks including several LiDAR-based detectors, while ours is just one unified network.",
"DSGN without explicitly learning 2D boxes surpasses those applying strong 2D detectors based on ResNet-101 [27] or DenseNet-121 [2].",
"It naturally achieves duplicate removal by non-maximum suppression (NMS) in 3D space, which coincides with the common belief that there is no collision between regular objects.",
"More intriguingly, as shown in Table REF , DSGN even achieves comparable performance on BEV detection and better performance on 3D detection on KITTI easy regime with MV3D [9] (with LiDAR input only) – a classic LiDAR-based 3D object detector, for the first time.",
"This result demonstrates a promising future application at least in the scenario of low-speed autonomous driving.",
"The above comparison manifests the effectiveness of 3D geometric volume, which serves as a link between 2D images and 3D space by combining the depth information and semantic feature.",
"Inference Time On a NVIDIA Tesla V100 GPU, the inference time of DSGN for one image pair is 0.682s on average, where 2D feature extraction for left and right images takes 0.113s, constructing the plane-sweep volume and 3D geometric volume takes 0.285s, and 3D object detection on 3D geometric volume takes 0.284s.",
"The computation bottleneck of DSGN lies on 3D convolution layers."
],
[
"Ablation study of 3D Volume Construction",
"One of the main obstacles to construct an effective 3D geometric representation is the appropriate way of learning 3D geometry.",
"We therefore investigate the effect of following three key components to construct a 3D volume.",
"Input Data Monocular-based 3D volume only has the potential to learn the correspondence between 2D and 3D feature, while stereo-based 3D volume can learn extra 2D feature correspondence for pixel-correspondence constraint.",
"Constructing 3D Volume One straightforward solution to construct 3D volume is by directly projecting the image feature to 3D voxel grid [21], [40] (denoted as IMG$\\rightarrow $ 3DV).",
"Another solution in Figure REF transforms plane-sweep volume or disparity-based cost volume to 3D volume, which provides a natural way to impose pixel-correspondence constraint along the projection ray in camera frustum (denoted as IMG$\\rightarrow $ (PS)CV$\\rightarrow $ 3DV).",
"Supervising Depth Supervised with or without the point cloud data, the network learns the depth explicitly or implicitly.",
"One way is to supervise the voxel occupancy of 3D grid by ground-truth point cloud using binary cross-entropy loss.",
"The second is to supervise depth on the plane-sweep cost volume as explained in Section REF .",
"For fair comparison, the models IMG$\\rightarrow $ 3DV and IMG$\\rightarrow $ (PS)CV$\\rightarrow $ 3DV have the same parameters by adding the same 3D hourglass module for the model IMG$\\rightarrow $ 3DV.",
"In addition, several important facts can be revealed from Table REF and are explained in the following.",
"Table: Ablation study of depth encoded approaches.",
"“PSCV\" and “3DV\" with “Supervision” header represent that the constraint is imposed in (plane-sweep) cost volume and 3D volume, respectively.",
"The results are evaluated in moderate level.Supervision of point cloud is important.",
"The approaches under the supervision of the LiDAR point cloud consistently perform better than those without supervision, which demonstrates the importance of 3D geometry for image-based approaches.",
"Stereo-based approaches work much better than monocular ones under supervision.",
"The discrepancy between stereo and monocular approaches indicates that direct learning of 3D geometry from semantic cues is a quite difficult problem.",
"In contrast, image-based approaches without supervision make these two lines yield similar performance, which indicates that supervision only by 3D bounding boxes is insufficient for learning of 3D geometry.",
"Plane-sweep volume is a more suitable representation for 3D structure.",
"Plane-sweep cost volume ($54.27$ AP) performs better than disparity-based cost volume ($45.89$ AP).",
"It shows that balanced feature mapping is important during the transformation to 3D volume.",
"Plane-sweep volume, as an intermediate encoder, more effectively contains depth information.",
"The inconsistency between IMG$\\rightarrow $ PSCV$\\rightarrow $ 3DV and IMG$\\rightarrow $ 3DV manifests that plane-sweep volume as the intermediate representation can effectively help learning of depth information.",
"The observation explains that the soft arg-min operation encourages the model to pick a single depth plane per pixel along the projection ray, which shares the same spirit as the assumption that only one depth-value is true for each pixel.",
"Another reason can be that PSCV and 3DV have different matching densities – PSCV intermediately imposes the dense pixel correspondence over all image pixels.",
"In contrast, only the left-right pixel pairs through the voxel centers are matched on 3DV.",
"From above comparison of volume construction, we observe that the three key facts affect the performance of computation pipelines.",
"The understanding and recognition of how to construct a suitable 3D volume is still at the very early stage.",
"More study is expected to reach comprehensive understanding of the volume construction from the multi-view images."
],
[
"Influence on Stereo Matching",
"We conduct experiments for investigating the influence of depth estimation, which is evaluated on KITTI val set following [47].",
"The average and median value of absolute depth estimation errors within the pre-defined range of $[z_\\text{min}, z_\\text{max}]$ is shown in Table REF .",
"A natural baseline for our approach is PSMNet-PSV* modified from PSMNet [4] whose 2D feature extractor takes $0.041$ s while ours takes $0.113$ s. Trained with depth estimation branch only, DSGN performs slightly better than PSMNet-PSV* with the same training pipeline in depth estimation.",
"For joint training of both tasks, both approaches suffer from larger and similar depth error ($0.5586$ meter for DSGN vs. $0.5606$ meter for PSMNet-PSV*).",
"Differently, DSGN outperforms the alternatives by $7.86$ AP on 3D object detection and $6.34$ AP on BEV detection.",
"The comparison indicates that our 2D network extracts better high-level semantic features for object detection.",
"Table: Influence on depth estimation, evaluated on KITTI val images.",
"PSMNet-PSV* is a variant of PSMNet , which uses one 3D hourglass module instead of three of them for refinement considering limited memory space and takes the plane-sweep approach to construct cost volume."
],
[
"Conclusion",
"We have presented a new 3D object detector on binocular images.",
"It shows that an end-to-end stereo-based 3D object detection is feasible and effective.",
"Our unified network encodes 3D geometry via transforming the plane-sweep volume to a 3D geometric one.",
"Thus, it is able to learn high-quality geometric structure features for 3D objects on the 3D volume.",
"The joint training lets the network learn both pixel- and high-level features for the important tasks of stereo correspondence and 3D object detection.",
"Without bells and whistles, our one-stage approach outperforms other image-based approaches and even achieves comparable performance with a few LiDAR-based approaches on 3D object detection.",
"The ablation study investigates several key components for training 3D volume in Table REF .",
"Although the improvement is clear and explained, our understanding of how the 3D volume transformation works will be further explored in our future work."
],
[
"Correlation between stereo (depth) and 3D object detection accuracy.",
"Following KITTI stereo metric, we consider a pixel as being correctly estimated if its depth error is less than an outlier_thresh.",
"The percentage of non-outlier pixels inside the object box is deemed as depth estimation precision.",
"With several outlier_threshes (0.1, 0.3, 0.5, 1.0, 2.0 meters), scatter plots of stereo and detection precision are drawn.",
"We observed that when outlier_thresh is 0.3 meter, the strongest linear correlation is yielded.",
"Table: Pearson's correlation coefficients (PCC) for a set of outlier_threshes between depth estimation precision and detection accuracy.",
"Outlier_thresh=0.3m=0.3m yields the strongest linear correlation.Figure REF shows the scatter plots with the outlier_thresh $0.3$ m and $0.1$ m. Quite a few predictions get over $0.7$ detected precision within a certain range of depth estimation error.",
"This reveals that the end-to-end network gives rise to its ability to detect 3D objects even with a larger depth estimation error.",
"The following 3D detector enables compensation for the stereo depth estimation error by 3D location regression with back-propagation.",
"Figure: BEV Detection precision versus depth precision for all predicted boxes for Car on KITTI val set.",
"Only TPs with IoU>0.01>0.01 and score >0.1>0.1 are shown."
],
[
"Relationship between distance and detection accuracy.",
"Figure REF illustrates, as distance increases, all detection accuracy indicators have a shrinking tendency.",
"The average accuracy maintains 80%+ within 25 meters.",
"In all indicators, 3D AP decreases the fastest, followed by BEV AP and last 2D AP.",
"This observation suggests that the 3D detection accuracy is determined by the BEV location precision beyond 20 meters.",
"Figure: Detection accuracy versus distance.",
"We separate the range [0, 40] (meters) into 8 intervals, each with 5 meters.",
"All evaluations are conducted within each interval."
],
[
"Influence of different features for 3D geometry.",
"We discuss the efficiency of different geometric representations for volumetric structure.",
"Most depth prediction approaches [13], [22], [4], [15] apply the strategy of “depth classification” (such as cost volume) instead of “depth regression”.",
"Thus, we have several choices for encoding the depth information of cost volume into a 3D volume.",
"The intuitive one is to use a 3D voxel occupancy (denoted by “Occupancy”).",
"An advanced version is by keeping the probability of voxel occupancy (denoted by “Probability”).",
"They both have explicit meaning for 3D geometry and can be easily visualized.",
"Another one is by using the last feature map for cost volume as geometric embedding for 3D volume (denoted by “Last Features”).",
"Table: Ablation study on 3D geometric representation.",
"“Occupancy” indicates only using binary feature for 3D volume.",
"It is 1 for voxel of minimum cost along the projection ray and is 0 otherwise.",
"“Probability” denotes keeping the probability of voxel occupancy instead of quantizing it to 0 or 1.",
"“Last Features” represents transforming the last features of cost volume to 3D volume.Table REF reveals the performance gap between “Occupancy” / “Probability”.",
"“Last Features” indicates the latent feature embedding (64D) that enables the network to extract more 3D latent geometric information and even semantic cues than the explicit voxel occupancy.",
"It aids learning of 3D structure."
],
[
"Technical details",
"We explore several technical details used in DSGN and discuss their importance in the pipeline in Table REF .",
"Joint optimization of bounding box regression improves accuracy ($+4.80$ AP) than the separable optimization.",
"The intermediate 3D volume representation enables the network to naturally retrieve image feature for more 2D semantic cues.",
"However, 3DGV cannot directly benefit from the concatenation of mapped 32D image features and warped predicted cost volume.",
"Instead, the image feature weighted by the depth probability achieves $+1.01$ AP gain.",
"Further, Involving more computation by an extra hourglass module on 3D object detector and flip augmentation, DSGN finally achieves $54.27$ AP on 3D object detection.",
"Table: Comparison of results for Pedestrian and Cyclist on KITTI val set.",
"PL: F-PointNet* uses extra Scene Flow dataset to pretrain the stereo matching network.Table: Comparison of results for Pedestrian and Cyclist on KITTI test set (official KITTI Leaderboard)."
],
[
"The main challenges for detecting Pedestrian and Cyclist are the limited data (about only 1/3 of images are annotated) and the difficulty to estimate their poses in an image even for human.",
"As a result, most image-based approaches yield poor performance or are not validated on Pedestrian and Cyclist.",
"Since the evaluation metric is changed on the official KITTI leaderboard, We only report the available results from original papers and the KITTI leaderboard.",
"Experimental results in Table REF shows that our approach achieves better results on Pedestrian but worse ones on Cyclist compared with PL: F-PointNet.",
"We note that PL: F-PointNet used Scene Flow dataset [33] to pre-train the stereo matching network.",
"Besides, PL: F-PointNet achieves the best result on Pedestrian and the model PL: AVOD works best on Car and Cyclist.",
"Table REF shows the submitted results on the official KITTI leaderboard."
],
[
"Network Architecture.",
"We show the full network architecture in Table REF , including the networks for 2D feature extraction, constructing plane-sweep volume and 3D geometric volume, stereo matching and 3D object detection."
],
[
"Implementation Details of 3D Object Detector.",
"Given the feature map $\\mathcal {F}$ on bird's eye view, we put four anchors of different orientation angles $(0, \\pi /2, \\pi , 3\\pi /2)$ on all locations of $\\mathcal {F}$ .",
"The box sizes of pre-defined anchors used for respectively Car, Pedestrian, Cyclist are $(h_\\mathbf {A}=1.56, w_\\mathbf {A}=1.6, l_\\mathbf {A}=3.9)$ , $(h_\\mathbf {A}=1.73, w_\\mathbf {A}=0.6, l_\\mathbf {A}=0.8)$ , and $(h_\\mathbf {A}=1.73, w_\\mathbf {A}=0.6, l_\\mathbf {A}=1.76)$ .",
"The horizontal coordinate $(x_\\mathbf {A}, z_\\mathbf {A})$ of each anchor lies on the center of each grid in bird's eye view and its center along the vertical direction locates on $y_\\mathbf {A} = 0.825$ for Car and $y_\\mathbf {A} = 0.74$ for Pedestrian and Cyclist.",
"We set $\\gamma = 1$ for Car and $\\gamma = 5$ for Pedestrian and Cyclist for balancing the positive and negative samples.",
"The classification head of 3D object detector is initialized following RetinaNet [31].",
"NMS with IoU threshold 0.6 is applied to filter out the predicted boxes on bird's eye view."
],
[
"Implementation Details of Differentiable Warping from PSV to 3DGV.",
"Let $\\mathcal {U} \\in \\mathbb {R}^{H_I\\times W_I\\times D_I \\times C}$ be the last feature map of PSV, where $C$ is the channel size of features.",
"We first construct a pre-defined 3D volume $\\in \\mathbb {R}^{H_V \\times W_V \\times D_V \\times 3}$ to store the center coordinate $(x,y,z)$ of each voxel in 3D space (Section REF ).",
"Then we get the projected pixel coordinate $(u,v)$ by multiplying the projection matrix.",
"$z$ is directly concatenated to pixel coordinate to get $(u,v,z)$ in camera frustum space.",
"As a result, we get a coordinate volume $\\in \\mathbb {R}^{H_V \\times W_V \\times D_V\\times 3}$ , which stores the mapped coordinates in camera frustum space.",
"By trilinear interpolation, we fetch the corresponding feature of $\\mathcal {U}$ at the projected coordinates to construct the 3D volume $\\mathcal {V} \\in \\mathbb {R}^{H_V \\times W_V \\times D_V\\times C}$ , , 3D geometric volume.",
"We ignore the projected coordinates outside the image by setting these voxel features to 0.",
"In backward operations, the gradient is passed and computed using the same coordinate volume."
],
[
"A.3. Future Work",
"More further studies on stereo-based 3D object detection are recommended here.",
"The Gap with state-of-the-art LiDAR-based approaches.",
"Although our approach achieves comparable performance with some LiDAR-based approaches on 3D object detection, there remains a large gap with state-of-the-art LiDAR-based approaches [55], [41], [10], [52].",
"Besides, an obvious problem is the accuracy gap on bird's eye view (BEV) detection.",
"As shown in the table of main results, there is almost 12 AP gap on the moderate and hard level in BEV detection.",
"One possible solution is high-resolution stereo matching [53], which can help obtain more accurate depth information to increase the robustness for highly occluded, truncated and far objects.",
"3D Volume Construction.",
"Table REF shows basic comparison of volume construction in DSGN.",
"We expect a more in-depth analysis of the volume construction from multi-view or binocular images, which serves as an essential component design for 3D object understanding.",
"Besides, the effectiveness of 3D volume construction methods still requires more investigation since it needs to balance and provide both depth information and semantic information.",
"Computation Bottleneck.",
"The computation bottleneck of DSGN locates on the computation of 3D convolutions for computing cost volume.",
"Recent stereo matching work [57], [48] focused on accelerating the computation of cost volume.",
"Another significant aspect of constructing cost volume is that current cost volume [22], [4] is designed for regressing disparity but not depth.",
"Further research might explore more efficient feature encoding for the plane-sweep cost volume.",
"Network Architecture Design.",
"There is a trade-off between stereo matching network and 3D detection network for balancing the feature extraction of pixel- and high-level features, which can be conducted by recent popular Network Architecture Search (NAS).",
"Application on Low-speed Scenario.",
"Our approach shows comparable performance with the LiDAR-based approach on 3D and BEV detection in the close range in the KITTI easy set.",
"Most importantly, it is affordable even with one strong GPU Tesla V100 ($11,458 (USD)) compared with the price of a 64-beam LiDAR $75,000 [58].",
"It is a promising application of image-based autonomous driving system for low-speed scenarios.",
"We provide a video demo https://www.youtube.com/watch?v=u6mQW89wBbo for visualization of our approach, which shows both the detected 3D boxes on front view and bird's eye view.",
"The ground-truth LiDAR point cloud is shown on bird's eye view.",
"The detection results are obtained by DSGN trained on KITTI training split only.",
"The unit of the depth map is meter.",
"Some noise observed in the predicted depth map is mainly caused by the implementation details.",
"(1) Noise in the near and far part: 3D volumes are constructed in [2, 40.4] (meters).",
"(2) Noise and large white zone in the higher region ($>$ 3m): The stereo branch is trained with a sparse GT depth map (64 lines around [-1,3 (meters) along the gravitational $z$ -axis, captured by a 64-ray LiDAR).",
"Table: Full network architecture of DSGN.",
"The color of the table highlights different components."
]
] | 2001.03398 |
[
[
"Catalyst-Assisted Probabilistic Coherence Distillation for Mixed States"
],
[
"Abstract The remarkable phenomenon of catalyst tells us that adding a catalyst could help state transformation.",
"In this paper, we consider the problem of catalyst-assisted probabilistic coherence distillation for mixed states under strictly incoherent operations.",
"To this end, we first present the necessary and sufficient conditions for distilling a target pure coherent state from an initial mixed state via stochastic strictly incoherent operations and the maximal probability of obtaining the target pure state from the initial state.",
"With the help of these results, we present the necessary and sufficient conditions for the existence of a catalyst that increases the maximal transformation probability."
],
[
"Introduction",
"Quantum coherence is a fundamental quantum resource of quantum physics, describing the capability of a quantum state to exhibit quantum interference phenomena.",
"It is an essential component in quantum information processing [1], and plays a central role in emergent fields, such as quantum computing [2], [3], quantum cryptography [4], quantum metrology [5], [6], and quantum biology [7].",
"Hence, the resource theory of coherence has attracted a growing interest due to the development of quantum information science in recent years [8], [10], [9], [12], [11].",
"For a quantum resource theory, there are two fundamental ingredients: free states and free operations [13], [14], [15].",
"With regards to the resource theory of coherence, the free states are quantum states which are diagonal in a prefixed reference basis.",
"There is no general consensus on the set of free operations in the resource theory of coherence.",
"Based on various physical and mathematical considerations, several free operations of coherence were presented [12].",
"Here, we focus our attention on the strictly incoherent operations which was given in Ref.",
"[16].",
"It was shown that the strictly incoherent operations neither create nor use coherence and have a physical interpretation in terms of interferometry in Ref.",
"[17].",
"Thus, the strictly incoherent operations are a physically well-motivated set of free operations for coherence and a strong candidate for free operations.",
"When we perform a quantum information processing task, it is usually the pure coherent states playing the central role [1].",
"Unfortunately, as a quantum system is unavoidably affected by noise, a pure state easily becomes a mixed state.",
"Thus, a central problem in the resource theory of coherence is the coherence distillation, i.e., the process that extracts target pure coherent states from initial mixed states via incoherent operations.",
"Recently, this problem has generated a great deal of interest [16], [25], [27], [24], [20], [18], [22], [23], [26], [19], [28], [21].",
"The coherence distillation of mixed states via various incoherent operations was studied in the asymptotic limit in Refs.",
"[16], [18], [19].",
"Another scheme, the scheme of one-shot coherence distillation, i.e., the process that extracts pure coherent states from mixed states via various incoherent operations in the nonasymptotic limit, was proposed in Refs.",
"[25], [27], [24], [20], [22], [23], [26], [19], [28], [21].",
"When we consider the strictly incoherent operations, it shows that, for a class of mixed states, it can never be transformed into a target pure state via strictly incoherent operations with certainty or with high probability [25], [28], [18], [19].",
"Thus, an important problem is that how to increase the transformation probability in this situation.",
"Inspired by the catalyst-assisted transformations in the resource theory of quantum entanglement [29], [31], [32], [30], catalyst-assisted coherence transformations for pure states [34], [33] was proposed as an efficient method to increase the transformation probability of coherence distillation.",
"Specifically, if the initial state and the target state are all pure states, they show that an appropriately chosen catalyst can increase the maximal transformation probability and they present the necessary and sufficient conditions for the existence of a catalyst that can increase the maximal transformation probability.",
"However, in practical applications, because a quantum system is unavoidably affected by its environment, we would expect to deal with the coherence distillation of mixed states rather than with pure states.",
"Therefore, an immediate question arises: With the help of coherence catalysts, can we increase maximal transformation probability of coherence distillation for the initial states being mixed states?",
"In other words, for the initial states being mixed states, what are the necessary and sufficient conditions for the existence of a catalyst that can increase the maximal transformation probability of coherence distillation?",
"In this work, we address the above question by considering the problem of catalyst-assisted probabilistic coherence distillation for mixed states under strictly incoherent operations.",
"To solve this problem, we divide our discussions into three steps.",
"In the first step, we present the necessary and sufficient conditions for distilling a target pure coherent state from an initial mixed state via stochastic strictly incoherent operations.",
"In the second step, we present the maximal probability of obtaining the target pure state from the initial state.",
"In the last step, we present the necessary and sufficient conditions for the existence of a catalyst that can increase the maximal transformation probability.",
"This paper is organized as follows.",
"In Sec.",
"II, we recall some notions of the quantum resource theory of coherence.",
"In Sec.",
"III, we present the necessary and sufficient conditions for the existence of a catalyst that can increase the maximal transformation probability.",
"Section IV is our conclusions."
],
[
"Resource theory of coherence",
"Let $\\mathcal {H}$ be the Hilbert space of a $d$ -dimensional quantum system.",
"A particular basis of $\\mathcal {H}$ is denoted as $\\lbrace |i\\rangle , ~i=1,2,\\cdot \\cdot \\cdot ,d\\rbrace $ , which is chosen according to the physical problem under discussion.",
"Coherence of a state is then measured based on the basis chosen.",
"Specifically, a state is said to be incoherent if it is diagonal in the basis.",
"Any state which cannot be written as a diagonal matrix is defined as a coherent state.",
"For a pure state $|\\varphi \\rangle $ , we will denote $|\\varphi \\rangle \\langle \\varphi |$ as $\\varphi $ , i.e., $\\varphi :=|\\varphi \\rangle \\langle \\varphi |$ .",
"Next, let us recall strictly incoherent operations [16], [17].",
"A strictly incoherent operation is a completely positive trace-preserving map, expressed as $\\Lambda (\\rho )=\\sum _n K_n\\rho K_n^\\dagger ,$ where the Kraus operators $K_n$ satisfy not only $\\sum _n K_n^\\dagger K_n= I$ but also $K_n\\mathcal {I}K_n^\\dagger \\subset \\mathcal {I}$ and $K_n^\\dagger \\mathcal {I}K_n\\subset \\mathcal {I}$ for $K_n$ , i.e., each $K_n$ as well $K_n^\\dagger $ maps an incoherent state to an incoherent state.",
"Here, $\\mathcal {I}$ represents the set of incoherent states.",
"There is at most one nonzero element in each column (row) of $K_n$ , and such a $K_n$ is called a strictly incoherent Kraus operator.",
"With this definition, it is elementary to show that a projector is an incoherent operator if and only if it has the form $\\mathbb {P}_\\alpha =\\sum _{i\\in \\alpha }|i\\rangle \\langle i|$ with $\\alpha \\subset \\lbrace 1,...,d\\rbrace $ .",
"In what follows, we will denote $\\mathbb {P}_\\alpha $ as strictly incoherent projective operators.",
"The dephasing map, which we will denote it as $\\Delta (\\cdot )$ , is defined as $\\Delta (\\rho )=\\sum _{i=1}^{d}|i\\rangle \\langle i|\\rho |i\\rangle \\langle i|$ .",
"With the aid of strictly incoherent operations, we then recall the notion of stochastic strictly incoherent operations [25].",
"A stochastic strictly incoherent operation is constructed by a subset of strictly incoherent Kraus operators.",
"Without loss of generality, we denote the subset as $\\lbrace K_{1},K_{2},\\dots , K_{L}\\rbrace $ .",
"Otherwise, we may renumber the subscripts of these Kraus operators.",
"Then, a stochastic strictly incoherent operation, denoted $\\Lambda _s(\\rho )$ , is defined by $\\Lambda _s(\\rho )=\\frac{\\sum _{n=1}^L K_{n}\\rho K_{n}^{\\dagger }}{\\mathrm {Tr}(\\sum _{n=1}^LK_{n}\\rho K_{n}^{\\dagger })},$ where $\\lbrace K_{1},K_{2},\\dots , K_{L}\\rbrace $ satisfies $\\sum _{n=1}^L K_{n}^{\\dagger }K_{n}\\le I$ .",
"Clearly, the state $\\Lambda _s(\\rho )$ is obtained with probability $P=\\mathrm {Tr}(\\sum _{n=1}^LK_{n}\\rho K_{n}^{\\dagger })$ under a stochastic strictly incoherent operation $\\Lambda _s$ , while the state $\\Lambda (\\rho )$ is fully deterministic under a strictly incoherent operation $\\Lambda $ .",
"A functional $C$ can be taken as a measure of coherence if it satisfies the four postulates [10], [17], [12], [11]: (C1) the coherence being zero (positive) for incoherent states (all other states); (C2) the monotonicity of coherence under strictly incoherent operations; (C3) the monotonicity of coherence under selective measurements on average; and (C4) the nonincreasing of coherence under mixing of quantum states.",
"In accordance with the general criterion, several coherence measures have been put forward.",
"Out of them, we recall the following coherence measures, which are considered in this work.",
"The coherence rank $C_r$ [16], [35] of a pure state (not necessarily normalized), $|\\varphi \\rangle =\\sum _{i=1}^R\\varphi _i|i\\rangle $ with $\\varphi _i\\ne 0$ , is defined as the number of terms with $\\varphi _i\\ne 0$ , i.e., $C_r(\\varphi )=R$ .",
"For an arbitrary pure state $|\\varphi \\rangle =\\sum _{i=1}^{d}\\varphi _i|i\\rangle $ , we define $C_l(|\\varphi \\rangle )=\\sum _{i=l}^{d}\\vert \\varphi _i\\vert ^2$ , with $l=1,2,\\cdots ,d$ and $\\vert \\varphi _1\\vert \\ge \\cdots \\ge \\vert \\varphi _{d}\\vert $ .",
"These measures were put forward in Ref.",
"[34]."
],
[
"Catalyst-Assisted Probabilistic Coherence Distillation for Mixed States",
"We start by specifying the notion of catalyst-assisted probabilistic transformation under stochastic strictly incoherent operations.",
"For a given initial state $\\rho $ and a target state $\\varphi $ , we denote the maximal probability of obtaining the state $\\varphi $ from $\\rho $ by means of stochastic strictly incoherent operations as $P_{\\max }(\\rho \\rightarrow \\varphi )$ , i.e., $P_{\\max }(\\rho \\rightarrow \\varphi )=\\max _{\\Lambda _s}\\mathrm {Tr}\\left(\\Lambda _s(\\rho )\\right),$ with $\\Lambda _s(\\rho )\\propto \\varphi $ .",
"With the notion of $P_{\\max }(\\rho \\rightarrow \\varphi )$ , we say that the transformation from $\\rho $ to $\\varphi $ can be enhanced by using a catalyst if there exists some catalyst $c$ such that $P_{\\text{max}}(\\rho \\otimes c \\rightarrow \\varphi \\otimes c)>P_{\\text{max}}(\\rho \\rightarrow \\varphi ).$ Otherwise, we say that the maximal transformation probability $P_{\\max }(\\rho \\rightarrow \\varphi )$ cannot be increased via catalysts.",
"To give the necessary and sufficient conditions for the existence of a catalyst that can increase the maximal transformation probability, we present our result as the following three steps.",
"First, we present the necessary and sufficient conditions for distilling a target pure coherent state from an initial mixed state via stochastic strictly incoherent operations.",
"We state the condition as the following Theorem.",
"Theorem 1.",
"One can distill a target pure coherent state $\\varphi $ with its coherence rank $C_r(\\varphi )=m\\le d$ from an initial coherent state $\\rho $ via a stochastic strictly incoherent operation $\\Lambda _s$ if and only if there exists an incoherent projective operator $\\mathbb {P}$ such that $\\frac{\\mathbb {P}\\rho \\mathbb {P}}{\\mathrm {Tr}(\\mathbb {P}\\rho \\mathbb {P})}=\\psi , $ with the coherence rank of $\\psi $ being $n(\\ge m)$ .",
"We now prove the theorem.",
"For the if part, we have to show that if the state $\\rho $ fulfills the condition in the theorem, i.e., Eq.",
"(REF ), then we can always construct an explicit stochastic strictly incoherent operation such that $\\Lambda _s(\\rho )=\\varphi $ .",
"To this end, by using the condition in Eq.",
"(REF ), we can obtain an incoherent projective operator $\\mathbb {P}$ such that $\\frac{\\mathbb {P}\\rho \\mathbb {P}}{\\mathrm {Tr}(\\mathbb {P}\\rho \\mathbb {P})}=\\psi $ with the coherence rank of $\\psi $ being equal to or greater than that of $\\varphi $ .",
"Without loss of generality, we assume that there is $|\\varphi \\rangle =\\sum _{i=1}^m\\varphi _i|i\\rangle $ with $\\vert \\varphi _1\\vert \\ge \\vert \\varphi _2\\vert \\ge \\cdots \\ge \\vert \\varphi _m\\vert >0$ .",
"Furthermore, we assume that $|\\psi \\rangle =U\\sum _{i=1}^n\\psi _i|i\\rangle $ , where $U$ is a permutation matrix such that there is $\\vert \\psi _1\\vert \\ge \\vert \\psi _2\\vert \\ge \\cdots \\vert \\psi _n\\vert >0$ .",
"Then, the stochastic strictly incoherent operation $\\Lambda _s$ can be chosen as $\\Lambda _s(\\rho )=\\frac{K\\rho K^\\dag }{\\mathrm {Tr}(K\\rho K^\\dag )},$ where $K=k~\\text{diag}\\left(a_1,a_2,\\cdots ,a_n\\right)U^\\dag \\mathbb {P},\\nonumber $ with $a_i=\\frac{\\varphi _i}{\\psi _i}$ for all $i=1,\\cdots ,n$ and $k$ being a complex number for guaranteeing $\\vert k\\vert \\le \\frac{1}{\\max _i\\lbrace \\vert a_i\\vert \\rbrace }$ .",
"It is straightforward to show that $\\Lambda _s(\\rho )=\\mathrm {Tr}[\\Lambda _s(\\rho )]\\varphi $ .",
"This completes the if part of the theorem.",
"Next, we show the only if part of the theorem, i.e.",
"; if we can distill $\\varphi $ from $\\rho $ via a stochastic strictly incoherent operation, then there exists an incoherent projective operator $\\mathbb {P}$ such that $\\frac{\\mathbb {P}\\rho \\mathbb {P}}{\\mathrm {Tr}(\\mathbb {P}\\rho \\mathbb {P})}=\\psi $ with the coherence rank of $\\psi $ being equal to or larger than that of $\\varphi $ .",
"First, we show that when we want to judge whether there exists a stochastic strictly incoherent operation such that $\\Lambda _s(\\rho )=\\varphi $ , we only need to consider the stochastic strictly incoherent operation with the form of $\\Lambda _s^1(\\rho )=\\frac{K\\rho K^\\dag }{\\mathrm {Tr}(K\\rho K^\\dag )}.",
"$ To this end, on the one hand, we assume that we can distill a given pure coherent state $\\varphi $ from $\\rho $ via a stochastic strictly incoherent operation $\\Lambda _s$ , i.e., $\\nonumber \\Lambda _s(\\rho )=\\frac{\\sum _{n=1}^L K_{n}\\rho K_{n}^{\\dagger }}{\\mathrm {Tr}(\\sum _{n=1}^LK_{n}\\rho K_{n}^{\\dagger })}=\\varphi .$ Then, since pure states are extreme points of the set of states, there must be $\\frac{K_{n}\\rho K_{n}^{\\dagger }}{\\mathrm {Tr}(K_{n}\\rho K_{n}^{\\dagger })}=\\varphi $ for all $n=1,...,L$ .",
"On the other hand, we note that $\\nonumber \\Lambda _s^1(\\rho )=\\frac{K_n(\\rho )K_n^\\dag }{\\mathrm {Tr}(K_n(\\rho )K_n^\\dag )}$ is also a stochastic strictly incoherent operation.",
"Thus, we obtain that if we can distill $\\varphi $ from $\\rho $ via $\\Lambda _s^1(\\rho )$ , then there exists a stochastic strictly incoherent operation such that $\\Lambda _s(\\rho )=\\varphi $ .",
"Second, we show that if we can distill a pure state $\\varphi $ from a state $\\rho $ via a stochastic strictly incoherent operation, there must be $\\frac{\\mathbb {P}\\rho \\mathbb {P}}{\\mathrm {Tr}(\\mathbb {P}\\rho \\mathbb {P})}=\\psi ,$ with the coherence rank of $\\psi $ being equal to or larger than that of $\\varphi $ .",
"To this end, let us recall the structure of the strictly incoherent Kraus operator [28].",
"From the definition of the strictly incoherent Kraus operator, we obtain that any strictly incoherent Kraus operator $K$ can always be decomposed into $K=P_\\pi K_\\Delta \\mathbb {P}$ , where the operator $P_\\pi $ is a permutation matrix, $K_\\Delta =\\text{diag}(a_1,...,a_n,0,0,...)$ is a diagonal matrix with $a_i\\ne 0$ , and $\\mathbb {P}$ is a projective operator corresponding to $K_\\Delta $ , i.e., $\\mathbb {P}=\\text{diag}(1,...,1,0,0,...)$ .",
"By using $K=P_\\pi K_\\Delta \\mathbb {P}$ and Eq.",
"(REF ), we then obtain that $\\frac{P_\\pi K_\\Delta \\mathbb {P}\\rho \\mathbb {P}K_\\Delta ^\\dag P_\\pi ^\\dag }{\\mathrm {Tr}(P_\\pi K_\\Delta \\mathbb {P}\\rho \\mathbb {P}K_\\Delta ^\\dag P_\\pi ^\\dag )}=\\varphi .",
"$ Without loss of generality, we assume that there is $\\mathbb {P}\\rho \\mathbb {P}=p\\sum _\\alpha p_\\alpha \\varphi _\\alpha $ , with $p=\\mathrm {Tr}(\\mathbb {P}\\rho \\mathbb {P})$ .",
"Thus, we immediately obtain that $\\frac{\\sum _\\alpha p_\\alpha P_\\pi K_\\Delta \\varphi _\\alpha K_\\Delta ^\\dag P_\\pi ^\\dag }{\\mathrm {Tr}(\\sum _\\alpha p_\\alpha P_\\pi K_\\Delta \\varphi _\\alpha K_\\Delta ^\\dag P_\\pi ^\\dag )}=\\varphi .\\nonumber $ Again, by using the fact that pure states are extreme points of the set of states, we then get that $\\frac{P_\\pi K_\\Delta \\varphi _\\alpha K_\\Delta ^\\dag P_\\pi ^t}{\\mathrm {Tr}(P_\\pi K_\\Delta \\varphi _\\alpha K_\\Delta ^\\dag P_\\pi ^t)}=\\varphi ~\\text{or}~\\textbf {0}, $ for all $\\varphi _\\alpha $ , where $\\textbf {0}$ is a square matrix with all its elements being 0.",
"Here, we should note that $P_\\pi K_\\Delta \\varphi _\\alpha K_\\Delta ^\\dag P_\\pi ^t$ cannot be $\\textbf {0}$ at the same time.",
"This means that there is $\\frac{\\mathbb {P}\\rho \\mathbb {P}}{\\mathrm {Tr}(\\mathbb {P}\\rho \\mathbb {P})}=\\psi $ .",
"Next, by using the result presented in Refs.",
"[36], [34], [38], [16], we can obtain that a pure state $\\psi $ can be transformed into a given pure coherent state $\\varphi $ via a stochastic strictly incoherent operation if and only if the coherence rank of $\\psi $ is equal to or greater than that of $\\varphi $ .",
"By using this result and the fact that $\\Lambda _s^1(\\rho )$ is a strictly incoherent operation, we immediately obtain that the coherence rank of $\\psi $ is equal to or larger than that of $\\varphi $ .",
"This completes the only if part of the theorem.",
"$\\blacksquare $ Second, we present the maximal probability of obtaining the target pure state $\\varphi $ from the initial state $\\rho $ by using stochastic strictly incoherent operations.",
"Before going further, we give the definition of pure coherent-state subspace.",
"If there is an incoherent projector $\\mathbb {P}$ such that $\\frac{\\mathbb {P_\\mu }\\rho \\mathbb {P_\\mu }}{\\mathrm {Tr}(\\mathbb {P_\\mu }\\rho \\mathbb {P_\\mu })}=\\psi _\\mu $ with the coherence rank of $\\psi _\\mu $ being $n$ , then we say that $\\rho $ has an $n$ -dimensional pure coherent-state subspace.",
"Specifically, the maximally pure coherent-state subspaces of $\\rho $ means that the coherence rank of $\\psi _\\mu $ , i.e., the rank of the corresponding $\\mathbb {P}$ , cannot be enlarged.",
"From Theorem 1, to obtain $P_{\\max }(\\rho \\rightarrow \\varphi )$ , we should solve the problem of how to identify the pure coherent-state subspaces of a mixed state.",
"We can solve this problem with the aid of $\\mathcal {A}=(\\Delta \\rho )^{-\\frac{1}{2}}\\vert \\rho \\vert (\\Delta \\rho )^{-\\frac{1}{2}},$ which was given in Ref.",
"[25].",
"Here, for $\\rho =\\sum _{ij}\\rho _{ij}|i\\rangle \\langle j|$ , $|\\rho |=\\sum _{ij}|\\rho _{ij}||i\\rangle \\langle j|$ and $(\\Delta \\rho )^{-\\frac{1}{2}}$ is a diagonal matrix with elements $(\\Delta \\rho )^{-\\frac{1}{2}}_{ii}= \\left\\lbrace \\begin{array}{ll}\\rho _{ii}^{-\\frac{1}{2}}, &\\text{if} ~ \\rho _{ii}\\ne 0;\\\\0,&\\text{if}~ \\rho _{ii}= 0.\\end{array}\\right.$ It is straightforward to show that all the elements of $\\mathcal {A}$ are 1 if and only if $\\rho $ is a pure state [25].",
"From this, we obtain that, if there is an $n$ -dimensional principal submatrices $\\mathcal {A}_\\mu $ of $\\mathcal {A}$ with all its elements being 1, then the corresponding subspace of $\\rho $ is an $n$ -dimensional pure coherent-state subspace.",
"By using this result, one can easily identify the pure coherent-state subspaces of $\\rho $ .",
"We denote the principal submatrices $\\mathcal {A}_\\mu $ corresponding to the maximally pure coherent-state subspaces as the maximally dimensional principal submatrices of $\\mathcal {A}$ .",
"Let the corresponding Hilbert subspaces of principal submatrices $\\mathcal {A}_\\mu $ be $\\mathcal {H}_\\mu $ , which is spanned by $\\lbrace |i_\\mu ^1\\rangle ,|i_\\mu ^2\\rangle ,\\cdots ,|i_\\mu ^n\\rangle \\rbrace \\subset \\lbrace |1\\rangle , |2\\rangle ,\\cdots ,|d\\rangle \\rbrace $ .",
"Then, the corresponding incoherent projective operators are $\\mathbb {P}_\\mu =|i_\\mu ^1\\rangle \\langle i_\\mu ^1|+|i_\\mu ^2\\rangle \\langle i_\\mu ^2|+\\cdots +|i_\\mu ^{n}\\rangle \\langle i_\\mu ^{n}|.\\nonumber $ Performing $\\lbrace \\mathbb {P}_\\mu \\rbrace $ on the state $\\rho $ , we obtain $\\lbrace \\psi _\\mu \\rbrace $ , i.e., $\\frac{\\mathbb {P}_\\mu \\rho \\mathbb {P}_\\mu }{\\mathrm {Tr}(\\mathbb {P}_\\mu \\rho \\mathbb {P}_ \\mu )}=\\psi _\\mu .\\nonumber $ Equipped with the above notions and Theorem 1, we can present the following Theorem 2.",
"Theorem 2.",
"The maximal probability of obtaining a target coherent state $\\varphi $ from an initial state $\\rho $ via stochastic strictly incoherent operations is $P_{\\max }(\\rho \\rightarrow \\varphi )=\\sum _\\mu p_\\mu \\min _l\\frac{C_l(\\psi _\\mu )}{C_l(\\varphi )},$ where $\\psi _\\mu $ correspond to the maximally pure coherent-state subspaces of $\\rho $ .",
"We now prove the theorem.",
"From Theorem 1, we obtain that if we want to get a target coherent state $\\varphi $ from an initial state $\\rho $ via stochastic strictly incoherent operations, we only need to consider the pure coherent-state subspaces of $\\rho $ .",
"First, we show that, to obtain $P_{\\max }(\\rho \\rightarrow \\varphi )$ , we only need to study the maximal pure coherent-state subspace of $\\rho $ .",
"To this end, according to the result in Refs.",
"[37], [36], [38], [39], we can get that $P_{\\max }(\\psi _\\mu \\rightarrow \\varphi )=\\min _l\\frac{C_l(\\psi _\\mu )}{C_l(\\varphi )}.",
"$ Then, for all $\\psi $ , $\\psi ^\\prime \\in \\mathcal {H}$ and $p\\in [0,1]$ , we may assume that $1\\le l\\le 2n$ and $P_{\\max }\\left(p\\psi \\oplus (1-p)\\psi ^\\prime \\rightarrow \\varphi \\right)=\\frac{C_l(p\\psi \\oplus (1-p)\\psi ^\\prime )}{C_l(\\varphi )}\\nonumber $ Then, there exist $l_\\psi \\le n$ and $l_{\\psi ^\\prime }\\le n$ such that $C_l(p\\psi \\oplus (1-p)\\psi ^\\prime )=pC_{l_\\psi }(\\psi )+(1-p)C_{l_{\\psi ^\\prime }}(\\psi ^\\prime )$ .",
"Thus, there is $&&P_{\\max }\\left(p\\psi \\oplus (1-p)\\psi ^\\prime \\rightarrow \\varphi \\right)=\\frac{C_l\\left(p \\psi \\oplus (1-p) \\psi ^\\prime \\right)}{C_l(\\varphi )}\\nonumber \\\\&&=\\frac{pC_{l_\\psi }(\\psi )+(1-p)C_{l_{\\psi ^\\prime }}(\\psi ^\\prime )}{C_l(\\varphi )}\\nonumber \\\\&&=\\frac{pC_{l}(\\psi \\oplus 0)+(1-p)C_l(\\psi ^\\prime \\oplus 0)}{C_l(\\varphi )}\\nonumber \\\\&&\\ge pP_{\\max }(\\psi \\rightarrow \\varphi )+(1-p)P_{\\max }(\\psi ^\\prime \\rightarrow \\varphi ),\\nonumber $ where $\\psi \\oplus 0$ and $\\psi ^\\prime \\oplus 0$ means that we append extra zeros to obtain $l$ -dimensional matrices, respectively.",
"Second, we show that the maximal transformation probability is $P_{\\max }(\\rho \\rightarrow \\varphi )=\\sum _\\mu p_\\mu \\min _l\\frac{C_l(\\psi _\\mu )}{C_l(\\varphi )}.\\nonumber $ From Theorem 1 and the above discussions, we obtain that $P_{\\max }(\\rho \\rightarrow \\varphi )=\\sum _\\mu p_\\mu P_{\\max }(\\psi _\\mu \\rightarrow \\varphi ),\\nonumber $ where $p_\\mu =\\mathrm {Tr}(\\mathbb {P}_{\\mu }\\rho \\mathbb {P}_ {\\mu })$ .",
"Then, from Eq.",
"(REF ), we immediately obtain $P_{\\max }(\\rho \\rightarrow \\varphi )=\\sum _\\mu p_\\mu \\min _l\\frac{C_l(\\psi _\\mu )}{C_l(\\varphi )}.$ This completes the proof of the theorem.",
"By the way, we would like to point that the results of Theorem 1 and Theorem 2 are a generalization of the result in Ref.",
"[28], which provided the conditions for the transformation from $\\rho $ into $\\varphi $ with certainty.",
"Furthermore, we note that the phenomenon of bound coherence under strictly incoherent operations was uncovered in Refs.",
"[19], [20], [18], i.e., there are coherent states from which no coherence can be distilled via strictly incoherent operations in the asymptotic regime.",
"The necessary and sufficient conditions for a state being bound state was presented in Refs.",
"[20], [18].",
"Their result shows that a state is a bound state if and only if it cannot contain any rank-one submatrix.",
"From the result presented in Theorem 1, we obtain a similar result when only one-copy of the state is supplied: Corollary 1.",
"We can distill an arbitrary pure coherent state from a mixed state $\\rho $ via stochastic strictly incoherent operations if and only if $\\rho $ contains at least one rank-2 pure coherent state subspace.",
"$\\blacksquare $ Finally, we present the necessary and sufficient conditions for the existence of a catalyst that can increase the maximal transformation probability.",
"To present the necessary and sufficient conditions for $P_{\\text{max}}(\\rho \\otimes c \\rightarrow \\varphi \\otimes c)>P_{\\text{max}}(\\rho \\rightarrow \\varphi )$ , we need the following lemma, which was presented in Refs.",
"[33], [30].",
"Lemma 1.—Let $\\textbf {p}$ and $\\textbf {q}$ be two $d$ -dimensional probability distributions arranged in nonincreasing order.",
"Then, there exists a probability distribution $\\textbf {c}$ such that $P_{\\text{max}}(\\textbf {p}\\otimes \\textbf {c}\\rightarrow \\textbf {p}\\otimes \\textbf {c})>P_{\\max }(\\textbf {p}\\rightarrow \\textbf {q})$ if and only if there are $P_{\\max }(\\textbf {p}\\rightarrow \\textbf {q})<\\min \\lbrace \\frac{p_d}{q_d},1\\rbrace $ .",
"With the Lemma 1and Theorems 1, and 2, we immediately obtain the following theorem which provides the necessary and sufficient conditions for the enhancement of $P_{\\max }(\\rho \\rightarrow \\varphi )$ : Theorem 3.—Suppose the states corresponding to maximally dimensional pure subspaces of $\\rho $ are $\\psi _\\mu $ and, without loss of generality, $|\\psi _\\mu \\rangle =\\sum _{i=1}^{n_1}\\psi _i^\\mu |i\\rangle $ with $\\vert \\psi _1^\\mu \\vert \\ge \\vert \\psi _2^\\mu \\vert \\ge \\cdots \\ge \\vert \\psi _{n_1}^\\mu \\vert >0$ for all $\\mu $ and $|\\varphi \\rangle =\\sum _{i=1}^{n_2}\\varphi _i|i\\rangle $ with $\\vert \\varphi _1\\vert \\ge \\vert \\varphi _2\\vert \\ge \\cdots \\ge \\vert \\varphi _{n_2}\\vert >0$ , respectively.",
"Then $P_{\\text{max}}(\\rho \\otimes c \\rightarrow \\varphi \\otimes c)>P_{\\text{max}}(\\rho \\rightarrow \\varphi )$ if and only if there exist at least one $\\psi _\\mu $ such that $P_{\\max }(\\psi _\\mu \\rightarrow \\varphi )<\\min \\lbrace \\frac{\\psi _n^\\mu }{\\varphi _n},1\\rbrace ,$ where $n=\\text{max}\\lbrace n_1,n_2\\rbrace $ .",
"In particular, we consider the catalyst-assisted deterministic coherence transformation.",
"For a given initial state $\\rho $ and a target state $\\varphi $ , we say that the transformation from $\\rho $ into $\\varphi $ can be catalyzed if there exists some catalyst $c$ such that $P_{\\text{max}}(\\rho \\otimes c \\rightarrow \\varphi \\otimes c)=1,$ with $P_{\\max }(\\rho \\rightarrow \\varphi )<1$ .",
"To present the necessary and sufficient conditions for the catalytic coherence transformations between the initial state $\\rho $ and the target state $\\varphi $ , we need the following lemma, which was presented in Refs.",
"[33], [32], [31].",
"Lemma 2.—Let $\\textbf {p}$ and $\\textbf {q}$ be two $d$ -dimensional probability distributions arranged in nonincreasing order with $\\textbf {p}$ having nonzero elements.",
"Then, there exists a probability distribution $\\textbf {c}$ such that $\\textbf {p}\\otimes \\textbf {c}\\prec \\textbf {p}\\otimes \\textbf {c}$ , where $\\textbf {p}\\prec \\textbf {q}$ means that $\\sum _{i=1}^lp_i^\\downarrow \\le \\sum _{i=1}^lq_i^\\downarrow $ , for all $1\\le l\\le n$ if and only if there are $A_\\alpha (\\textbf {p})>A_\\alpha (\\textbf {q}) ~\\text{for}~\\alpha \\in (-\\infty ,1)$ , $A_\\alpha (\\textbf {p})<A_\\alpha (\\textbf {q}) ~\\text{for}~\\alpha \\in (1,\\infty )$ , $S(\\textbf {p})>S(\\textbf {q})$ , where $A_\\alpha (\\textbf {p}):=(\\frac{1}{d}\\sum _{i=1}^dp_i^\\alpha )^{1/\\alpha }$ , $S(\\textbf {p})=-\\sum _{i=1}^dp_i\\ln p_i$ , and $A_0(\\textbf {p}):=(\\prod p_i)^{1/d}$ .",
"With the Lemma 2, Theorems 1, and 2, we immediately obtain the following theorem which provides the necessary and sufficient conditions for the deterministically catalytic coherence transformations between the initial state $\\rho $ and the target state $\\varphi $ .",
"Theorem 4.—For an initial state $\\rho $ with the state corresponding to maximally dimensional pure subspaces of $\\rho $ being $\\psi _\\mu $ and a target state $\\varphi $ , if $P_{\\max }(\\rho \\rightarrow \\varphi )<1$ , then there exists a catalyst $c$ such that $P_{\\max }(\\rho \\otimes c\\rightarrow \\varphi \\otimes c)=1$ if and only if there are $A_\\alpha (\\overrightarrow{\\Delta \\psi _\\mu })&&>A_\\alpha (\\overrightarrow{\\Delta \\varphi }) ~\\text{for}~\\alpha \\in (-\\infty ,1);\\nonumber \\\\A_\\alpha (\\overrightarrow{\\Delta \\psi _\\mu })&&<A_\\alpha (\\overrightarrow{\\Delta \\varphi }) ~\\text{for}~\\alpha \\in (1,\\infty );\\nonumber \\\\S(\\overrightarrow{\\Delta \\psi _\\mu })&&>S(\\overrightarrow{\\Delta \\varphi }),$ for all $\\mu $ .",
"Here, for the pure states $|\\psi _\\mu \\rangle =\\sum _{i=1}^n\\psi _i^\\mu |i\\rangle $ , $\\overrightarrow{\\Delta \\psi _\\mu }$ are defined as $(\\vert \\psi ^\\mu _1\\vert ^2,\\vert \\psi ^\\mu _2\\vert ^2,\\cdots ,\\vert \\psi ^\\mu _n\\vert ^2)$ .",
"$\\blacksquare $"
],
[
"Conclusions and Remarks",
"To summarize, we have considered the problem of catalyst-assisted probabilistic coherence distillation for mixed states under strictly incoherent operations.",
"Our main findings are presented as three theorems.",
"Theorem 1 presents the necessary and sufficient conditions for distilling a target pure coherent state from an initial mixed state via stochastic strictly incoherent operations.",
"Theorem 2 presents the maximal probability of obtaining the target pure state from the initial state.",
"With the help of these results, we have presented the necessary and sufficient conditions for the existence of a catalyst that can increase the maximal transformation probability in Theorem 5.",
"In passing, there are many open problems to which Theorem 5 may be of relevance.",
"It would be of great interest to determine the maximally achievable transformation probability by the presence of any catalyst state, which is left as an open issue for further investigation."
],
[
"Acknowlegements",
"We acknowledge fruitful discussions with Prof. Dian-Min Tong and Xiao-Dong Yu.",
"The authors acknowledge financial supported from NSF of China (Grant No.11775300), the National Key Research and Development Program of China (2016YFA0300603), and the Strategic Priority Research Program of Chinese Academy of Sciences No.",
"XDB28000000."
]
] | 2001.03508 |
[
[
"Formation and detection of Majorana modes in quantum spin Hall trenches"
],
[
"Abstract We propose a novel realization for a topologically superconducting phase hosting Majorana zero-modes on the basis of quantum spin Hall systems.",
"Remarkably, our proposal is completely free of ferromagnets.",
"Instead, we confine helical edge states around a narrow defect line of finite length in a two-dimensional topological insulator.",
"We demonstrate the formation of a new topological regime, hosting protected Majorana modes in the presence of s-wave superconductivity and Zeeman coupling.",
"Interestingly, when the system is weakly tunnel-coupled to helical edge state reservoirs, a particular transport signature is associated with the presence of a non-Abelian Majorana zero-mode."
],
[
"Introduction",
"The theoretical prediction[1], [2], [3] and experimental realization[4] of two-dimensional topological insulators marked the beginning of immense research activities in view of their functionalities in spintronics[5], [6], [7], [8], [9] and topological quantum computation[10].",
"In particular, the formation and detection of topological superconductivity on the basis of topological systems attracted a lot of attention [11], [12], [13], [14], [15], [16] and the emergence of topologically protected Majorana bound states came to the forefront of research [17].",
"The interest in those excitations is both fundamental and practical, since they obey non-Abelian statistics [18], [19], [20] and, hence, can potentially be used for topological quantum computation.",
"Regarding the realization of topologically confined Majoranas using topological insulators, the possibility of inducing superconducting pairing [21] is promising.",
"However, most proposals rely on the coexistence of ferromagnetic ordering [12], [22], [23], [24], which turns out to be difficult to achieve in the laboratory.",
"In parallel, another platform for topological superconductivity was found by the prediction of Majorana zero-modes in spin-orbit coupled quantum wires[25], [26].",
"Subsequently, several experimental works were able to confirm some of the proposed signatures [27], [28], [29].",
"However, the ultimate proof of the existence of Majoranas is probably still missing.",
"In this work, we propose a hybrid structure that combines the features of topological edge states and spin-orbit coupled quantum wires.",
"The system we investigate – a quantum spin Hall (QSH) anti-wire – defines itself through a narrow slit in a two-dimensional topological insulator (see Fig.",
"REF ).",
"This system shares similarities with QSH quantum point contacts, recently realized in the laboratory [30], for which the formation of Kramers pairs of Majorana fermions and other complex anyons were proposed [31], [32], [33], [34], [35].",
"We demonstrate below that the QSH anti-wire, in the presence of s-wave pairing and Zeeman coupling, possesses a topological phase hosting Majorana end-modes.",
"This phase emerges if the slit is narrow enough such that the edge states at opposite sides overlap.",
"Figure: Quantum spin Hall anti-wire.",
"(a) Schematic illustration of the system: A QSH anti-wire, covered by a s-wave superconductor under the influence of a magnetic field weakly coupled to helical edge states at the boundary of the QSH stripe.",
"(b) Sketch of the QSH constriction with the appearing scattering terms.This setup offers key advantages with respect to other platforms.",
"Indeed, the emergence of Majorana modes within a two-dimensional topological insulator makes it straightforward to couple them to topological edge channels, whose helical nature allows for richer transport signatures than a standard tunneling probe.",
"In particular, in the multi-terminal conductance $G_{1\\rightarrow 2}=\\mathrm {d}I_2/\\mathrm {d}V_1$ , between contacts 1 and 2 of Fig.",
"REF (a), we identify a qualitative Majorana signature beyond the well-known zero-bias peak: The presence of a Majorana-like state at zero energy gives rise to a negative $G_{1\\rightarrow 2}$ , which is otherwise positive.",
"In addition to that, our setup can be easily scaled-up by carving several slits within the same topological insulator.",
"The resulting collection of localized Majorana modes, which can be manipulated by tuning their pair-wise couplings via top gates, would represent a convenient playground for topological quantum computation applications.",
"The article is organized as follows.",
"In Sec.",
"we discuss the topological properties of narrow QSH trenches.",
"Subsequently, in Sec.",
", we investigate the formation of topologically protected Majorana modes associated with the topological phase.",
"This is followed by a discussion of possible transport signatures in Secs.",
", and .",
"Finally, we conclude in Sec.",
", where we summarize the results."
],
[
"Topological phase transition in the anti-wire",
"The setup we propose is sketched in Fig.",
"REF (a).",
"Its innovative ingredient is a long quantum constriction between two metallic edges of a quantum spin Hall insulator depicted in Fig.",
"REF (b).",
"To compute its topological properties, we first consider the limit of an infinitely long constriction.",
"The kinetic energy can be described by the effective Hamiltonian density $(\\hbar = 1)$ $\\mathcal {H}_p = \\sum _{\\nu ,\\sigma }\\hat{\\psi }_{\\nu ,\\sigma }^{\\dagger }(x) (-iv_F\\sigma \\nu \\partial _x-\\mu ) \\hat{\\psi }_{\\nu ,\\sigma }(x),$ where $\\hat{\\psi }_{\\nu ,\\sigma }(x)$ are annihilating fermionic fields carrying spin-index $\\sigma \\in \\lbrace \\uparrow ,\\downarrow \\rbrace =\\lbrace +,-\\rbrace $ and edge-index $\\nu \\in \\lbrace 1,2\\rbrace =\\lbrace +,-\\rbrace $ ; $\\mu $ acts as a chemical potential and $v_F$ is the Fermi velocity (estimated to be $(10^{5}-10^{6})\\mathrm {m/s}$ for QSH systems based on Hg(Cd)Te quantum wells [36]).",
"We assume a finite overlap of wave functions from states at different sides of the anti-wire.",
"In presence of time-reversal (TR) symmetry, two single particle terms emerge [37], [38], [33], [39], [40] $\\mathcal {H}_{t_0}&=&t_0 \\sum _{\\sigma }\\left[\\hat{\\psi }^{\\dagger }_{1,\\sigma }(x)\\hat{\\psi }_{2,\\sigma }(x)+\\mathrm {h.c.}\\right],\\\\\\mathcal {H}_{t_c}&=&t_c \\sum _{\\nu }\\left[\\nu \\hat{\\psi }_{\\nu ,\\uparrow }^{\\dagger }(x)\\hat{\\psi }_{-\\nu ,\\downarrow }(x)+\\mathrm {h.c.}\\right].$ While Eq.",
"(REF ) describes a hybridization of fermionic states with the same spin associated to different sides of the slit and does not require further symmetry breaking with respect to $\\mathcal {H}_p$ , Eq.",
"() is only finite if axial spin symmetry is absent and takes the role of an effective spin-orbit coupling across the slit [41].",
"The spectrum associated with $H_0=\\int _{-\\infty }^{+\\infty } dx\\, [\\mathcal {H}_p+\\mathcal {H}_{t_0}+\\mathcal {H}_{t_c}]$ is shown in Fig.",
"REF (a).",
"The additional application of a Zeeman field perpendicular to the $x$ direction opens a partial gap around $k=0$ .",
"For concreteness, we consider a field along the $z$ direction $\\mathcal {H}_B =B_z\\sum _{\\nu ,\\sigma }\\sigma \\hat{\\psi }_{\\nu ,\\sigma }^{\\dagger }(x)\\hat{\\psi }_{\\nu ,\\sigma }(x).$ The gyro-magnetic factor for the edge states is predicted to be $g\\sim 10$ [42] for typical QSH materials.",
"Moreover, the typical values for the effective electron mass in HgTe quantum wells [43] indicate that indeed a situation similar to hybrid systems based on spin-orbit nanowires is met [44].",
"This implies required magnetic fields of the order of few $\\mathrm {mT}$ , compatible with the presence of superconductivity.",
"The resulting band structure shares similarities with spin-orbit nanowires under the influence of magnetic fields.",
"It can hence be expected that topological physics emerges when s-wave superconductivity is taken into account via $\\mathcal {H}_{\\Delta }=\\Delta \\sum _{\\nu }\\left[\\hat{\\psi }_{\\nu ,\\uparrow }^{\\dagger }(x)\\hat{\\psi }_{\\nu ,\\downarrow }^{\\dagger }(x)+\\mathrm {h.c.}\\right].$ Typical values for the proximity induced superconducting order parameter $\\Delta $ are given by $\\Delta \\sim 40 \\mathrm {\\mu eV}$ in HgTe-based systems[21].",
"Indeed, the infinitely long anti-wire described by $H_0+\\int _{-\\infty }^{+\\infty }dx\\, [\\mathcal {H}_{\\Delta }+\\mathcal {H}_B]$ undergoes a topological phase transition, indicated by a gap-closing and reopening depending on the control parameters $\\mu $ and $B_z$ (see Fig.",
"REF (b)).",
"Since the coupling strength $t_c$ in Eq.",
"() effectively takes the role of a spin-orbit coupling, as long as it is non-zero, it hardly affects the topological parameter regime [see Fig.",
"REF (c)].",
"However, it controls the magnitude of the gaps in the topological regime and therefore the decay length of possible low-energy bound states in the presence of boundaries.",
"By contrast, Eq.",
"(REF ) has less influence on the magnitude of the gaps, but strongly affects the shape of the topological regime (Fig.",
"REF (d)).",
"While a concrete estimation of the magnitude of $t_0$ is difficult, it is clear that it can be tuned, up to the magnitude of the bulk gap, by reducing the width of the slit [45].",
"Figure: Topological phase diagram of the proximitized anti-wire.",
"(a) Eigenenergy spectrum of H 0 H_0.",
"The different colors represent states with orthogonal spin with t c =t 0 t_c=t_0.",
"(b) Phase diagram as function of μ\\mu and B z B_z (under the choice t 0 =t c =1t_0=t_c=1, Δ/t 0 =0.3\\Delta /t_0=0.3, v F =1v_F=1).",
"(c) Dependence of the topological phase on t c t_c.",
"The different curves correspond to gap closures for t c /t 0 =0.2,0.4,0.6,0.8,1.0t_c/t_0=0.2,0.4,0.6,0.8,1.0 (red to blue), Δ=0.3t 0 ,t 0 =1\\Delta =0.3 t_0,~t_0=1, v F =1v_F=1.",
"(d) Dependence of the topological phase on t 0 t_0.",
"The curves correspond to to gap closures for t 0 /t c =0.2,0.4,0.6,0.8,1t_0/t_c=0.2,0.4,0.6,0.8,1 (red to blue), Δ/t c =0.3,t c =1\\Delta /t_c=0.3,~t_c=1, v F =1v_F=1."
],
[
"Topologically protected Majoranas",
"To investigate the presence of topological bound states, we now focus on a slit with a finite length $L$ .",
"It is convenient to consider the additional Hamiltonian density $\\mathcal {H}_{T}\\!= T\\left[\\delta (x)\\!+\\!\\delta (x\\!-\\!L)\\right]\\sum _{\\sigma }\\left[\\hat{\\psi }^{\\dagger }_{1,\\sigma }(x)\\hat{\\psi }_{2,\\sigma }(x)\\!+\\!\\mathrm {h.c.}\\right],$ which describes the presence of barriers at $x=0$ and $x=L$ .",
"Indeed, in the limit $T\\rightarrow \\infty $ , the Hamiltonian $H_{\\textrm {A}W} = \\lim _{T\\rightarrow \\infty } \\int _0^L dx\\, [ \\mathcal {H}_p+\\mathcal {H}_{t_0}+\\mathcal {H}_{t_c} +\\mathcal {H}_B + \\mathcal {H}_\\Delta + \\mathcal {H}_T]$ defines an isolated antiwire in the region $x\\in [0,L]$ , whose fermionic fields obey the open boundary conditions (BCs) (see also App. )",
"[46] $\\begin{array}{lcr}\\hat{\\psi }_{1,\\uparrow }(x) = i \\hat{\\psi }_{2,\\uparrow }(-x), \\\\\\hat{\\psi }_{2,\\downarrow }(x) = i \\hat{\\psi }_{1,\\downarrow }(-x),\\end{array}$ where $\\hat{\\psi }_{\\nu ,\\sigma }(x)=\\sum _q \\psi _{\\nu ,\\sigma ,q}(x)\\hat{c}_q$ with annihilation operators $\\hat{c}_q$ and the quantization condition $q = (\\pi /L)(n-1/2)$ .",
"Figure: Majorana wavefunctions at the anti-wire ends.",
"(a) λ M \\lambda _M (yellow) and δΓ λ M \\delta \\Gamma _{\\lambda _M} (blue) as a function of LL.",
"(b) |U 0 (x)| 2 \\vert U_{0}(x)\\vert ^2 according to Eq.",
"() with U 0 (0)=ν λ U_{0}(0)=\\nu _{\\lambda }.",
"(c) Schematic illustration of the probability distribution in the (folded) anti-wire.",
"The parameters of the calculation are: B/t 0 =0.6B/t_0=0.6, μ/t 0 =2\\mu /t_0 = \\sqrt{2}, Δ/t 0 =0.3\\Delta /t_0=0.3, t c =t 0 =1t_c=t_0=1, v F =1v_F=1.We hence obtain $H_{\\mathrm {AW}}\\!",
"&=& \\!\\int _{-L}^L\\!\\mathrm {dx}\\, \\hat{\\Phi }^{\\dagger }(x)\\big [-iv_F\\partial _x+\\tau _z\\sigma _z B_z+\\tau _z\\sigma _0\\mu \\nonumber \\\\&+&\\tau _z\\sigma _x~\\mathrm {sign}(x)t_c\\big ]\\hat{\\Phi }(x)\\nonumber \\\\&-&\\!\\int _{-L}^L\\!\\mathrm {dx}\\, \\Phi ^{\\dagger }(x)\\big [\\tau _x\\sigma _y \\Delta +i~\\mathrm {sign}(x)t_0\\big ]\\hat{\\Phi }(-x),$ where $\\tau _j$ , $\\sigma _j$ ($j \\in \\lbrace x,y,z\\rbrace $ ) are Pauli matrices acting on particle-hole, spin-space, respectively, and $\\hat{\\Phi }(x)= (\\hat{\\psi }_{1,\\uparrow }(x),\\hat{\\psi }_{2,\\downarrow }(x),\\hat{\\psi }_{1,\\uparrow }^{\\dagger }(x),\\hat{\\psi }_{2,\\downarrow }^{\\dagger }(x))^T$ .",
"Our goal is to determine the eigenfunctions $U_{\\epsilon }(x)$ of the Hamiltonian density in Eq.",
"(REF ).",
"We can overcome its non-locality with the ansatz $U_{\\epsilon }(x)=u_{\\epsilon }(x)\\theta (x)+v_{\\epsilon }(-x)\\theta (-x),$ where $u_{\\epsilon }(x)$ and $v_{\\epsilon }(x)$ are spinors in the given basis.",
"From the continuity of the solutions $U_{\\epsilon }(x)$ at $x=0$ as well as from the anti-periodicity of the system with respect to $2L$ , the solution needs to obey the BCs $u_{\\epsilon }(0)=v_{\\epsilon }(0)$ and $u_{\\epsilon }(L)=-v_{\\epsilon }(L)$ .",
"The single particle problem associated with Eq.",
"(REF ) becomes equivalent to the set of equations for the functions $u_{\\epsilon }(x)$ and $v_{\\epsilon }(x)$ and the eigenenergies $\\epsilon $ $\\big [&-&iv_F\\partial _{x}s_z\\tau _0\\sigma _0+s_0\\tau _z\\sigma _z B_z+s_0\\tau _z\\sigma _0 \\mu + s_z\\tau _z\\sigma _x t_c\\nonumber \\\\&-&s_x\\tau _x\\sigma _y \\Delta + s_y\\tau _0\\sigma _0 t_0\\big ] \\chi _{\\epsilon }(x) = \\epsilon \\chi _{\\epsilon }(x),$ where we define the basis function $\\chi _{\\epsilon }(x)=(u_{\\epsilon }(x),v_{\\epsilon }(x))^T$ and the Pauli matrices $s_j$ acting on the space spanned by $u_{\\epsilon }(x)$ and $v_{\\epsilon }(x)$ .",
"The general solution of Eq.",
"(REF ) can be found by integration $\\chi _{\\epsilon }(x)&=&M_{\\epsilon }(x,x_0)\\chi _{\\epsilon }(x_0),$ where $\\begin{split}&M_{\\epsilon }(x,x_0)=\\exp \\!\\bigg [\\int _{x_0}^{x} \\mathrm {dx^{\\prime }}\\frac{i}{v_F}s_z\\tau _0\\sigma _0\\big (\\epsilon -(s_0\\tau _z\\sigma _z B_z+\\\\&\\quad +s_0\\tau _z\\sigma _0 \\mu + s_z\\tau _z\\sigma _x t_c-s_x\\tau _x\\sigma _y \\Delta + s_y\\tau _0\\sigma _0 t_0)\\big )\\!\\bigg ].\\end{split}$ Not every energy $\\epsilon $ is compatible with the BCs.",
"For the topological phase, however, in the limit $L\\rightarrow \\infty $ there should be a decaying solution for $\\epsilon \\rightarrow 0$ of the form $\\Gamma (0)=(\\zeta (0),\\zeta (0))$ (fulfilling the BCs at $x=0$ ).",
"Thus, in this limit, Eq.",
"(REF ) turns into an eigenvalue problem for $\\zeta (0)$ of the form $\\lim _{L\\rightarrow \\infty } M_0(L,0)\\Gamma (0) \\stackrel{!",
"}{=} 0.$ If we further demand the solution to be a Majorana, we require $\\zeta (0)=\\big (f(0),g(0),f^*(0),g^*(0)\\big )^T$ .",
"Note that demanding a Majorana from of $\\Gamma (0)$ implies this form to remain for any other point $x$ because of the particle-hole symmetry of $M_0(x,x^{\\prime })$ .",
"For finite $L$ , Eq.",
"(REF ) does not hold anymore.",
"However, we find that an approximate Majorana solution exists, i.e.",
"$M_0(L,0)$ possesses an eigenvalue $\\lambda _M \\sim \\exp (-\\alpha L)$ whose corresponding eigenvector $\\nu _{\\lambda _M}$ fulfils the BC at $x=0$ and deviates by $\\delta \\Gamma _{\\lambda _M} = \\frac{1}{2}\\big \\vert \\!\\big \\vert s_0(\\mathbb {1}-\\tau _x\\sigma _0)\\mathrm {Re}[\\nu _{\\lambda _M}]\\!+\\!s_0(\\mathbb {1}+\\tau _x\\sigma _0)\\mathrm {Im}[\\nu _{\\lambda _M}]\\big \\vert \\!\\big \\vert \\sim \\exp (-\\beta L)$ ($\\alpha , \\beta \\in \\mathbb {R}$ ) from the Majorana form (see Fig.",
"REF (a)).",
"The probability density associated to the wavefunction is shown in Fig.REF (b,c).",
"Figure: Transport measurements.",
"(a) Two-terminal conductance as function of energy ϵ\\epsilon and Zeeman field B z B_z.",
"(b-c) Multi-terminal conductance between contacts 1 and 2 with respect to Fig.",
"(a), as a function of μ\\mu and B z B_z (b), ϵ\\epsilon and B z B_z (c), respectively.",
"In (b), all values G 1→2 >0G_{1\\rightarrow 2}>0 are colored in blue.",
"In (c), all values G 1→2 <0G_{1\\rightarrow 2}<0 are colored in red.",
"Other parameters of the plots are: L=20ℏv F /t 0 L=20 \\hbar v_F/t_0, Δ/t 0 =0.3\\Delta /t_0=0.3, μ/t 0 =2\\mu /t_0=\\sqrt{2} ((a) and (c)), ϵ=0\\epsilon = 0 (b), t 0 =t c =1t_0=t_c=1 v F =1v_F=1.",
"For computational reasons, the delta distribution separating the anti-wire from the leads is replaced with its step function approximation δ a (x)= rect (x/a)/a\\delta _{a}(x) = \\mathrm {rect}(x/a)/a with a=0.1a=0.1.",
"Moreover, T=1.5T=1.5 for (a-b) and T=2T=2 for (c)."
],
[
"Transport characterization",
"Since the Majorana modes are naturally embedded into a two-dimensional topological insulator, it is straightforward to bring them in proximity to other boundaries of the sample.",
"In particular, as shown in Fig.",
"REF (a), it is possible to develop a weak tunnel coupling between the ends of the anti-wire and gapless helical edges.",
"The latter, which feature up to micrometer-size mean free paths in high-quality HgTe-based QSH systems [47], can be used as probes to perform particular transport measurements, taking advantage of their helical nature.",
"In order to study the transport, we consider the amplitude $T$ in Eq.",
"(REF ) to be finite.",
"The Hamiltonian of the whole system (i.e.",
"anti-wire and helical probes) thus reads $H_{\\mathrm {set}}=\\int _{-\\infty }^{+\\infty } dx [\\mathcal {H}_p+\\mathcal {H}_T ] + \\int _{0}^{L}dx\\, [\\mathcal {H}_{t_0}+\\mathcal {H}_{t_c}+\\mathcal {H}_{\\Delta }+\\mathcal {H}_{B}],$ where the kinetic terms for $x<0$ and $x>L$ describe the two outer helical edges.",
"We discuss two distinct transport schemes.",
"The first one aims at obtaining the two-terminal conductance.",
"In this scenario, contact 1 and 2 (3 and 4) of Fig.",
"REF (a) are treated as one lead, say $1\\!",
"2$ (34).",
"Then, we have $G^{2t} = \\frac{\\mathrm {d}I_{1\\!2}}{\\mathrm {d}V_{1\\!2}},$ where $I_{1\\!2}$ is the current exiting terminals 1 and 2 [see Fig.",
"REF (a)] and $V_{1\\!2}$ is their common bias with respect to the grounded superconductor.",
"In this scheme, the two terminals and the helical edge connecting them thus act as a single tunneling probe.",
"For small bias, we calculate $G^{2t}$ in terms of elements of the corresponding scattering matrix [48] $G^{2t}=\\frac{e^2}{2\\pi }\\left[2+\\!\\sum _{j\\in 1,2} \\big [\\vert r^{\\mathrm {eh}}_{1\\!2,j}\\vert ^2 - \\vert r^{\\mathrm {ee}}_{1\\!2,j}\\vert ^2 \\big ]\\right],$ where $r_{1\\!2,j}^{\\mathrm {e}\\nu }$ are normal ($\\nu = \\mathrm {e}$ ) and Andreev reflection amplitudes ($\\nu = \\mathrm {h}$ ) in lead $1\\!2$ in edge $j$ .",
"The elements of the scattering matrix are computed by integration of $H_{\\mathrm {set}}$ .",
"Fig.",
"REF (a) shows the two-terminal conductance $G^{2t}$ as a function of excitation energy $\\epsilon $ and applied Zeeman field $B_z$ .",
"Whenever an anti-wire bound state is on resonance, a peak in the two-terminal conductance emerges.",
"As expected, the Majorana clearly manifests itself with a strong zero-energy peak, whose properties have been extensively studied in the literature.",
"Importantly, such a signature is not exclusively associated with the presence of Majoranas and it is thus not sufficient as a proof for their existence [49], [50], [51], [52], [53], [54], [55].",
"In order to go beyond the simple zero-bias peak, we devise a different transport scheme which exploits the helical nature of our tunneling probe.",
"In particular, we consider the multi-terminal conductance between contacts 1 and 2 [see Fig.",
"REF (a)] which reads $G_{1\\rightarrow 2}=\\frac{\\mathrm {d}I_2}{\\mathrm {d}V_1}=\\frac{e^2}{2\\pi }\\left[\\vert t^{\\mathrm {ee}}_2\\vert ^2-\\vert t^{\\mathrm {eh}}_2\\vert ^2\\right].$ Importantly, $G_{1\\rightarrow 2}$ can either take positive or negative values, depending on which scattering process dominates: electron tunneling or crossed Andreev reflection.",
"In the following, we demonstrate that a negative signal at zero energy can be unambiguously associated with the presence of a Majorana bound state.",
"This statement is supported by Figs.",
"REF (b-c) which show that, when the anti-wire is in the topological phase and features Majoranas at its ends, the multi-terminal conductance $G_{1\\rightarrow 2}$ at zero-energy is indeed negative.",
"Moreover, Fig.",
"REF (c) shows that the negative signal (highlighted in red) is prominently seen at zero energy.",
"There are, however, also isolated scattering events at non-zero energy with the same property.",
"To better understand which additional information about the system can be deduced from the multi-terminal conductance, with respect to two-terminal transport, we investigate a simpler (toy) model which still describes the essential physics.",
"This allows us to properly clarify the meaning of a negative multi-terminal conductance."
],
[
"Negative multi-terminal conductance and the existence of Majorana modes",
"Our goal is two-fold: (i) We want to prove that the presence of a Majorana scatterer always leads to a negative multi-terminal conductance $G_{1\\rightarrow 2}$ .",
"(ii) We want to clarify under which circumstances the measurement of a negative $G_{1\\rightarrow 2}$ represents an unambiguous signature of the existence of a Majorana mode.",
"Figure: Majorana scatterer on the helical edge.",
"(a) Schematic illustration of a Majorana mode γ 1 \\gamma _1 side-coupled to a helical edge.",
"(b) Multi-terminal conductance as a function of energy ϵ\\epsilon with v F =1v_F=1, t ↑ =1.2t ↓ t_{\\uparrow }=1.2 t_{\\downarrow } and t ↓ =0.2t_{\\downarrow }=0.2 and ϵ d /t ↓ =0,1\\epsilon _d/t_{\\downarrow } =0,1 (blue, orange).We consider the simple model sketched in Fig.",
"REF (a).",
"It consists of a single helical edge described by the Hamiltonian density $\\mathcal {H}_p^{(\\nu =1)} = \\sum _{\\sigma }\\hat{\\psi }_{1,\\sigma }^{\\dagger }(x) (-iv_F\\sigma \\partial _x-\\mu ) \\hat{\\psi }_{1,\\sigma }(x),$ which connects the leads 1 and 2.",
"At $x=0$ , it is tunnel coupled with a single Majorana scatterer $\\hat{\\gamma }_1 = \\hat{d} + \\hat{d}^\\dagger $ via $H_c=\\sum _{\\sigma }t_{\\sigma }\\big [\\hat{\\gamma }_1\\hat{\\psi }_{1\\sigma }(0)+\\mathrm {h.c.}\\big ].$ The spin-dependent coupling constants $t_\\sigma $ accounts for the spin-texture of the Majorana mode [56], [57].",
"We consider a second Majorana mode $\\hat{\\gamma }_2 = i\\hat{d}-i \\hat{d}^\\dagger $ which is not directly coupled to the helical edge but can (weakly) hybridize with $\\hat{\\gamma }_1$ via $H_{d}=-i\\epsilon _d\\hat{\\gamma }_1\\hat{\\gamma }_2$ .",
"To determine the transport properties according to Eq.",
"(REF ), we need to compute the scattering matrix of the system [58] (see App.",
").",
"We obtain the analytical results $t_2^{eh} &= - \\frac{t_{\\uparrow }^2}{t_{\\uparrow }^2+t_{\\downarrow }^2\\!+\\!iv_F(\\epsilon _d^2-\\epsilon ^2)/\\epsilon },\\\\t_2^{ee} &=-1-t^{eh}_2$ where $\\epsilon $ is the energy at which the scattering process takes place.",
"For $\\epsilon $ sufficiently close to $\\pm \\epsilon _d$ , we find that $t_\\uparrow >t_\\downarrow $ implies $G_{1\\rightarrow 2}<0$ .",
"By contrast, we can show that $t_\\uparrow <t_\\downarrow $ leads to $G_{1\\rightarrow 2}>0$ but $G_{2\\rightarrow 1}<0$ (see Apps.",
"and ).",
"Hence, as long as the Majorana has a spin texture which is not polarized perpendicular to the spin quantization axis $z$ , one of the two multi-terminal conductances $G_{1\\rightarrow 2}$ or $G_{2\\rightarrow 1}$ have to be negative.",
"This is confirmed by Fig.",
"REF (b), which shows $G_{1\\rightarrow 2}$ for $t_{\\uparrow }=1.2 t_{\\downarrow }$ .",
"Without hybridization (blue line) the negative signal is centered around the Majorana energy $\\epsilon =\\epsilon _d=0$ .",
"The width of the dip is controlled by the magnitude of the coupling constant.",
"Even in presence of a finite hybridization energy $\\epsilon _d>0$ (orange line), the negative conductance is still present and centered around $\\epsilon = \\pm \\epsilon _d$ .",
"Importantly, we observe that in the anti-wire, the interplay between the competing Zeeman field $\\mathcal {H}_B$ and the spin-flipping scattering $\\mathcal {H}_{t_c}$ guarantees that the Majoranas do not feature a spin-texture perpendicular to the $z$ -axis.",
"Therefore, we conclude that the presence of an isolated Majorana in the anti-wire necessarily leads to a negative multi-terminal conductance.",
"Figure: Generic scatterer on the helical edge.",
"(a) Schematic illustration of the coupling between the helical edge and the particle-hole-symmetric system SS.",
"(b) Conductance G 1→2 G_{1\\rightarrow 2} for ϵ α =0\\epsilon _{\\alpha }=0, as a function of ξ\\xi and ϵ\\epsilon .",
"Only negative values of G 1→2 G_{1\\rightarrow 2} are shown.",
"The parameters are t ↑ =1.2t ↓ t_{\\uparrow }=1.2 t_{\\downarrow } with t ↓ =0.2t_{\\downarrow }=0.2.",
"(c) Multi-terminal conductance G 1→2 G_{1\\rightarrow 2} for the coupling to a generic eigenstate of a PHS system on resonance ϵ=ϵ α \\epsilon =\\epsilon _{\\alpha }, as a function of ξ\\xi .",
"The different lines correspond to ϵ α /t ↑ =2,0.2,0.02,\\epsilon _{\\alpha }/t_{\\uparrow }=2,~0.2,~0.02, and 0.0020.002 (blue to red).",
"Further parameters are t ↓ =3/5t ↑ t_{\\downarrow }=3/5 t_{\\uparrow } with t ↑ =0.5t_{\\uparrow }=0.5.We now discuss the opposite implication, eventually showing that a negative signal at zero-energy represents an unambiguous signature of a Majorana mode.",
"To this end, we need to consider the coupling of the helical edge with a more general particle-hole-symmetric system $S$ .",
"The latter, described by the Hamiltonian $H_S$ , features several single-particle eigenstate $|\\zeta _j\\rangle $ with energy $\\epsilon _j$ .",
"As sketched in Fig.",
"REF (a), we consider the point-like tunneling at $x=0$ between the edge and a specific fermionic site of $S$ , which we denote $c_1$ .",
"If we restrict our attention to a specific energy level $\\epsilon _{\\alpha }$ , its effect on the multi-terminal conductance can be computed by considering the effective system Hamiltonian $H_S^{(\\alpha )} = \\epsilon _\\alpha d^\\dagger _\\alpha d_\\alpha $ and the effective tunneling Hamiltonian $H_t = \\sum _\\sigma t_\\sigma \\left[ (\\zeta _{\\alpha ,1}^{(e)*} d_\\alpha ^\\dagger + \\zeta _{\\alpha ,1}^{{(h)}} d_\\alpha ) \\psi _{1\\sigma }(0) + \\text{h.c.}\\right],$ where the coefficient $\\zeta _{\\alpha ,1}^{(e)}$ ($\\zeta _{\\alpha ,1}^{(h)}$ ) represents the particle (hole) component of the state $|\\zeta _\\alpha \\rangle $ on site $c_1$ .",
"As before, the spin-dependent tunneling amplitudes $t_\\sigma $ effectively take into account the (possible) spin-texture of the state $|\\zeta _\\alpha \\rangle $ .",
"A careful demonstration of the validity of Eq.",
"(REF ) is provided in App.",
", where we explicitly consider the system $S$ as a Kitaev chain.",
"We parametrize $\\zeta _{\\alpha ,1}^{(e)} &= \\Upsilon _1 \\cos (\\xi )\\\\\\zeta _{\\alpha ,1}^{(h)} &= \\Upsilon _1 \\sin (\\xi ),$ neglecting a possible complex phase which has no effect on the results.",
"The parameter $\\Upsilon _1$ , characterizes which fraction of the eigenstate $|\\zeta _\\alpha \\rangle $ is localized on the site $c_1$ and its only effect is to renormalize the coupling constants.",
"As for $\\xi $ , it controls whether such a fraction is more electron- or hole-like.",
"In particular, for $\\xi =0$ , $H_t$ describes the coupling of the helical edge with an electronic state while, for $\\xi =\\pi /4$ , it describes the coupling with the Majorana considered in Eq.",
"(REF ).",
"The multi-terminal conductance $G_{1\\rightarrow 2}$ associated with the effective tunneling Hamiltonian $H_t$ is plotted in Fig.",
"REF (b,c).",
"Close to resonance $\\epsilon \\simeq \\epsilon _\\alpha $ , the multi-terminal conductance is negative provided that $\\xi $ is sufficiently close to the Majorana case, i.e.",
"$|\\xi - \\pi /4 \\; (\\text{mod } \\pi )|\\le \\bar{\\xi }$ .",
"The threshold $\\bar{\\xi }$ , depends on the detuning $(\\epsilon -\\epsilon _\\alpha )/t_\\downarrow $ as well as on the energy of the eigenstate $\\epsilon _{\\alpha }/t_\\downarrow $ .",
"In general, $\\bar{\\xi }$ is not particularly small and the multi-terminal conductance can be negative even for values of $\\xi $ which significantly differ from the Majorana case.",
"See, for example, the blue lines in Fig.",
"REF (c).",
"This justifies the presence of isolated red spots in Fig.",
"REF (c) at high energies, even when the presence of Majorana is not expected.",
"Importantly, however, for $\\epsilon _{\\alpha } = \\epsilon \\rightarrow 0$ , the threshold goes to zero $\\bar{\\xi }\\rightarrow 0$ .",
"In this case, a negative multi-terminal conductance provides a unambiguous signature of the Majorana mode."
],
[
"Influence of time-reversal breaking terms and robustness against backscattering",
"As the formation of Majorana zero modes in the anti-wire requires the presence of a Zeeman field, let us discuss its effects on the helical edges that serves as probes for transport measurements.",
"Importantly, the extension of the Zeeman coupling $\\mathcal {H}_B$ [see Eq.",
"(REF )] to the gapless helical regions outside the anti-wire (i.e.",
"for $x<0$ and $x>L$ ) does not modify the entries of the scattering matrix.",
"In App.",
", we explicitly show this for the scattering amplitudes in Eqs.",
"(REF ) and ().",
"In general, however, the lack of TR symmetry spoils the topological protection of the edges and can result in the presence of backscattering, for example caused by a magnetic field along the x axis or by local impurities.",
"This raises the question to what extent the existence of backscattering within the helical edge affects transmission and reflection amplitudes and questions the universality of the Majorana signature.",
"To rule out possible detrimental effects due to the breaking of TR, we investigate a slightly modified version of the toy model, discussed in the latter section, where the only modification that we apply is the addition of TR breaking backscattering terms in the helical edge which is side coupled to a Majorana, a generic BdG state, respectively (see Fig.",
"REF (a)).",
"For this model, we compute the scattering matrix and, from that, we obtain the conductance $G_{1\\rightarrow 2}$ (see App.",
").",
"Figure: Conductance G 1→2 G_{1\\rightarrow 2} for the discussed toy-models in the presence of TR breaking backscattering: (a) Schematic of the discussed system.",
"(b) Comparison between TR invariant (purple) and TR breaking (blue to green) transport in a helical edge, side-coupled to a Majorana.",
"The TR breaking parameter is given by B x /t ↑ =1.25B_x/t_\\uparrow =1.25, 2.5 2.5, 3.753.75, 6 (blue to green) with μ/t ↑ =5\\mu /t_\\uparrow =5 and ϵ d =0\\epsilon _d=0.",
"Note the for B x /t ↑ =6B_x/t_\\uparrow =6, we have B x >μB_x>\\mu which implies that no propagating modes are present for small ϵ\\epsilon .",
"(c) G 1→2 G_{1\\rightarrow 2} for side-coupling a generic BdG state at energy ϵ α =0\\epsilon _\\alpha =0 in dependence of the parametrization parameter ξ\\xi and energy ϵ\\epsilon .",
"(d) G 1→2 G_{1\\rightarrow 2} on resonance (ϵ=ϵ α \\epsilon =\\epsilon _\\alpha ), where ϵ α /t ↑ =5×10 -2 ,5×10 -3 ,5×10 -4 \\epsilon _\\alpha /t_\\uparrow = 5\\times 10^{-2}, 5\\times 10^{-3}, 5\\times 10^{-4} (green to blue).",
"Further parameters of the plot are: x i =-5v F /t ↑ x_i=-5 v_F/t_\\uparrow , x f =5v F /t ↑ x_f=5 v_F/t_\\uparrow , μ/t ↑ =5\\mu /t_\\uparrow =5, B x /t ↑ =2.5B_x/t_\\uparrow =2.5 ((c) and (d)), t ↓ =3/5t ↑ t_\\downarrow =3/5t_\\uparrow , t ↑ =0.2t_\\uparrow =0.2.Fig.",
"REF shows the resulting $G_{1\\rightarrow 2}$ for both scenarios.",
"Notably, finite TR breaking backscattering does not qualitatively modify the negative $G_{1\\rightarrow 2}$ which represents the universal signature of a Majorana zero mode (Fig.",
"REF (b)).",
"Moreover, also for the more generic case of coupling to a general BdG state, addition of TR breaking backscattering terms does not lead to qualitative different signatures in $G_{1\\rightarrow 2}$ as compared to the case without backscattering (compare Fig.",
"REF (c-d) and Fig.",
"REF (b-c)).",
"This is reasonable as backscattering acts in the same way to hole-like states as it does for electron-like states.",
"In fact, any imperfection with this property is not expected to degrade the universality of the proposed signature."
],
[
"Discussion",
"The requirements to construct isolated Majorana bound states at the helical edge, without the use of ferromagnetic barriers, are hence two pairs of helical edge modes brought into proximity with a connection in two points.",
"As helical edge modes develop in two-dimensional topological insulators at boundaries between topological and trivial regimes, there are two ways of constructing such a system.",
"First, cutting narrow slits in an elsewhere homogeneous two-dimensional topological insulator (Fig.",
"REF ).",
"This results in what has been coined anti-wire so far and has the advantage that, once it is possible to construct a single slit, the positioning of many slits is straightforward.",
"Therefore, the system possesses a natural scalability, that could be of importance when it comes to quantum computations.",
"Since different anti-wires emerge from the same underlying two-dimensional system, it is possible to tune their coupling via external gate voltages applied between two anti-wires (Fig.",
"REF ).",
"Hence, the link between the two anti-wires might be changed from insulating (chemical potential inside the bulk gap of the 2D TI) to conducting (chemical potential position in conduction band), allowing for controllable fusion of the Majoranas at the end of different anti-wires.",
"A second possibility to design a topologically superconducting phase is based on quantum constrictions.",
"This setup can be obtained from the anti-wire by interchanging topological and trivial regime.",
"Figure: Coupling of six anti-wires using gate potentialsapplied to the embedding quantum spin Hall insulator (orange regions).To summarize our findings, we have proposed a novel topological phase transition taking place in quantum spin Hall systems without the need of ferromagnets.",
"This topological phase hosts topologically protected Majorana modes localized at the two ends of the anti-wire.",
"The system we propose, being naturally hosted in a two-dimensional environment, is flexible towards scalability.",
"Moreover, the straightforward employment of helical probes allows for more in-depth analyses of the transport properties of the system.",
"In particular, it makes it possible to identify a novel and qualitative Majorana signature which goes beyond the standard observation of a (quantized) zero bias peak: (i) the multi-terminal conductance in the given setup carries a qualitative information based on its sign which is (ii) not expected to be influenced by particle-hole symmetric imperfections, such as backscattering processes, that are indeed detrimental for zero-bias peaks.",
"The experimental realization of our proposal comes with some potential challenges, in particular the realization of trenches narrow enough to allow a significant inter-edge tunneling and the coexistence of proximity-induced superconductivity with external magnetic field.",
"However, given the recent technological developments in both directions, we believe our system to be within experimental reach."
],
[
"Acknowledgements",
"This work was supported by the DFG (SPP1666, SFB1170 “ToCoTronics”), the Würzburg-Dresden Cluster of Excellence ct.qmat, EXC2147, project-id 390858490, the Elitenetzwerk Bayern Graduate School on “Topological insulators” and the Studienstiftung des Deutschen Volkes.",
"In this Appendix, we present further analysis of the calculations related to our proposal of a QSH anti-wire as a novel Majorana platform.",
"In particular, in Sec.",
", we derive the boundary conditions of the anti-wire; in Sec.",
"we compute the scattering matrix of the toy model, introduced in the main text; in Sec.",
", we compare the toy model with the numerical results.",
"In Sec.",
", we justify the form of the coupling Hamiltonian used in the main text and compare our results numerically with an extended toy model on the basis of coupling to a Kitaev chain.",
"Finally, in Sec.",
", we evaluate the scattering matrix of the toy model including TR breaking terms."
],
[
"Derivation of the boundary conditions for the QSH anti-wire",
"The kinetic Hamiltonian including impurity scattering at $x=0$ and $x=L$ can be written as $\\tilde{H}_{p}\\!&=& \\int \\mathrm {dx} \\sum _{\\nu ,\\sigma }\\hat{\\psi }_{\\nu ,\\sigma }^{\\dagger }(x) (-iv_F\\sigma \\nu \\partial _x) \\hat{\\psi }_{\\nu ,\\sigma }(x)\\\\&+&\\!T\\!\\int \\!\\mathrm {dx}\\left[\\delta (x)\\!+\\!\\delta (x\\!-\\!L)\\right]\\sum _{\\sigma }\\left[\\hat{\\psi }^{\\dagger }_{1,\\sigma }(x)\\hat{\\psi }_{2,\\sigma }(x)\\!+\\!\\mathrm {h.c.}\\right]\\nonumber $ with the fermionic fields $\\hat{\\psi }_{\\nu ,\\sigma }(x)$ annihilating a $\\nu ,\\sigma $ fermion at position $x$ .",
"We can formally diagonalize the Hamiltonian (REF ) with eigenfunctions from the associated single particle problem $\\tilde{h}_p(x)\\Psi (x)=E\\Psi (x),$ where $\\tilde{h}_p(x)=-iv_F\\eta _z\\sigma _z\\partial _x+T\\left[\\delta (x)+\\delta (x-L)\\right]\\eta _x\\sigma _0$ with Pauli matrices $\\eta _j$ , $\\sigma _j$ ($j\\in \\lbrace x,y,z\\rbrace $ ) acting on edge-, spin-space, respectively, and $\\Psi (x)=(\\psi _{1,\\uparrow }(x),\\psi _{1,\\downarrow }(x),\\psi _{2,\\uparrow }(x),\\psi _{2,\\downarrow }(x))^T$ .",
"In vicinity $\\delta x$ close to the impurities with $\\delta x \\rightarrow 0$ , Eq.",
"(REF ) is solved by $\\Psi (\\!-\\delta x)=e^{\\frac{T}{v_F} \\eta _y\\sigma _z}\\Psi (\\delta x),~~\\Psi (L\\!+\\!\\delta x)=e^{-\\frac{T}{v_F} \\eta _y\\sigma _z}\\Psi (L\\!-\\!\\delta x).$ In the limit $T\\rightarrow \\infty $ , this results in the boundary conditions $\\begin{array}{lcr}\\psi _{1,\\uparrow }(0) &=& i \\psi _{2,\\uparrow }(0),~~\\psi _{1,\\uparrow }(L)=-i\\psi _{2,\\uparrow }(L),\\\\\\psi _{2,\\downarrow }(0) &=& i \\psi _{1,\\downarrow }(0),~~\\psi _{2,\\uparrow }(L)=-i\\psi _{1,\\downarrow }(L).\\end{array}$ Note that in our notation the functions $\\psi _{1,\\uparrow }(x)$ and $\\psi _{2,\\downarrow }(x)$ (and as well $\\psi _{1,\\downarrow }(x)$ and $\\psi _{2,\\uparrow }(x)$ ) describe states of the same chirality.",
"Thus, we find that they obey $\\begin{array}{lcr}\\psi _{1,\\uparrow ,q}(x) &=& i \\psi _{2,\\uparrow ,q}(-x),\\\\\\psi _{2,\\downarrow ,q}(x) &=& i \\psi _{1,\\downarrow ,q}(-x)\\end{array}$ with the plane waves $\\psi _{\\nu ,\\sigma ,q}(x)=(1/\\sqrt{L})\\exp [i \\nu \\sigma q x]$ ($\\nu =(1,2)=(+,-)$ and $\\sigma =(\\uparrow ,\\downarrow )=(+,-)$ ) with quantized momenta $q = (\\pi /L)(n-1/2)$ .",
"By applying an expansion of the fermionic fields in terms of the functions $\\psi _{\\nu ,\\sigma ,q}(x)$ , namely $\\hat{\\psi }_{\\nu ,\\sigma }(x)=\\sum _q \\psi _{\\nu ,\\sigma ,q}(x)\\hat{c}_q$ , we obtain the boundary condition for the fields $\\begin{array}{lcr}\\hat{\\psi }_{1,\\uparrow }(x) &=& i \\hat{\\psi }_{2,\\uparrow }(-x),\\\\\\hat{\\psi }_{2,\\downarrow }(x) &=& i \\hat{\\psi }_{1,\\downarrow }(-x).\\end{array}$ Clearly, from the quantization of $q$ , the fields need to be anti-periodic with respect to $2L$ $\\hat{\\psi }_{\\nu ,\\sigma }(L)=- \\hat{\\psi }_{\\nu ,\\sigma }(-L).$ Eq.",
"(REF ) is stated in the main as Eq.",
"(REF )."
],
[
"Derivation of the scattering matrix",
"The system for which we aim to construct the scattering matrix is sketched in Fig.",
"REF (a) of the main text.",
"It is composed of three parts.",
"The helical edge passing by the anti-wire $(\\hbar = 1)$ is described by $H_p = \\int \\mathrm {dx} \\sum _{\\sigma }\\hat{\\psi }_{\\sigma }^{\\dagger }(x) (-iv_F\\sigma \\partial _x-\\mu ) \\hat{\\psi }_{\\sigma }(x),$ where $\\hat{\\psi }_{\\sigma }(x)$ are annihilating fermionic fields carrying an index $\\sigma \\in \\lbrace \\uparrow ,\\downarrow \\rbrace =\\lbrace +,-\\rbrace $ and $\\mu $ is a chemical potential.",
"Since the formation of Majorana zero modes in the anti-wire requires the presence of Zeeman fields, it is a reasonable assumption to also include it in the nearby helical edge states $H_B = \\int \\mathrm {dx} B_z \\sum _{\\sigma } \\sigma \\hat{\\psi }^{\\dagger }_{\\sigma }(x)\\hat{\\psi }_{\\sigma }(x).$ Further, we assume a point-like coupling of the fields $\\hat{\\psi }_{\\sigma }(x)$ to a Majorana mode $\\hat{\\gamma }_1$ of the anti-wire $H_c=\\int \\mathrm {dx}~\\delta (x)\\hat{\\gamma }_1\\sum _{\\sigma }t_{\\sigma }\\big [\\hat{\\psi }_{\\sigma }(x)-\\hat{\\psi }_{\\sigma }^{\\dagger }(x)\\big ]$ with coupling constant $t_{\\sigma }$ that might depend on $\\sigma $ .",
"Since TR symmetry is absent in the anti-wire, the coupling does not obey corresponding symmetry constraints.",
"Moreover, even though hybridization of the Majoranas is exponentially suppressed in the length of the anti-wire, they might acquire a small hybridization energy $H_{d}=-i\\epsilon _d\\hat{\\gamma }_1\\hat{\\gamma }_2.$ The two Majoranas $\\hat{\\gamma }_1$ and $\\hat{\\gamma }_2$ can be rewritten in terms of fermionic operators $\\hat{d}$ and $\\hat{d}^{\\dagger }$ with $\\begin{array}{lcr}\\hat{\\gamma }_1=\\hat{d}+\\hat{d}^{\\dagger },\\\\\\hat{\\gamma }_2=i\\hat{d}-i\\hat{d}^{\\dagger }.\\end{array}$ Using (REF ), $H=H_p+H_B+H_c+H_{d}$ can also be represented as $H=\\frac{1}{2}\\!\\int \\!\\mathrm {dx}\\tilde{\\Psi }^{\\dagger }(x)\\!\\begin{pmatrix}-iv_F\\partial _x\\!-\\!\\mu \\!+\\!",
"B_z & 0 & 0 & 0 & t_{\\uparrow }(x) & t_{\\uparrow }(x) \\\\0 & +iv_F\\partial _x\\!",
"- \\!\\mu \\!",
"-\\!",
"B_z & 0 & 0 & t_{\\downarrow }(x) & t_{\\downarrow }(x) \\\\0 & 0 & -iv_F\\partial _x \\!+\\!",
"\\mu \\!-\\!",
"B_z & 0 & -t_{\\uparrow }(x) & -t_{\\uparrow }(x) \\\\0 & 0 & 0 & +iv_F\\partial _x\\!",
"+\\!",
"\\mu \\!+\\!",
"B_z & -t_{\\downarrow }(x) & -t_{\\downarrow }(x) \\\\t_{\\uparrow }(x) & t_{\\downarrow }(x) & -t_{\\uparrow }(x) & -t_{\\downarrow }(x) & \\epsilon _d & 0\\\\t_{\\uparrow }(x) & t_{\\downarrow }(x) & -t_{\\uparrow }(x) & -t_{\\downarrow }(x) & 0 & -\\epsilon _d \\\\\\end{pmatrix}\\tilde{\\Psi }(x)~~~~$ with $\\tilde{\\Psi }(x)= \\big (\\hat{\\psi }_{\\uparrow }(x),\\hat{\\psi }_{\\downarrow }(x),\\hat{\\psi }_{\\uparrow }^{\\dagger }(x),\\hat{\\psi }_{\\downarrow }^{\\dagger }(x),\\hat{d},\\hat{d}^{\\dagger }\\big )^T$ and $t_{\\sigma }(x)=t_{\\sigma }\\delta (x)$ .",
"To diagonalize Eq.",
"(REF ), we expand $\\tilde{\\Psi }(x)$ in eigenfunctions of the Hamiltonian density $\\tilde{\\Psi }(x)=\\sum _{k,d} U_{k,d}(x)\\hat{\\chi }_{k,d}$ with matrices $U_{k,d}(x)$ and fermionic annihilation operators $\\hat{\\chi }_{k,d}=(\\hat{C}_k,\\hat{C}_d)^T$ with $\\hat{C}_k = (\\hat{c}_{\\uparrow ,k},\\hat{c}_{\\downarrow ,k},\\hat{c}_{\\uparrow ,k}^{\\dagger },\\hat{c}_{\\downarrow ,k}^{\\dagger })$ and $\\hat{C}_d=(\\hat{c}_d,\\hat{c}_d^{\\dagger })$ .",
"Inserting Eq.",
"(REF ) in (REF ), this yields $H=\\frac{1}{2}\\sum _{k,k^{\\prime },d,d^{\\prime }}\\hat{\\chi }_{k^{\\prime },d^{\\prime }}\\int \\mathrm {dx}~U_{k^{\\prime },d^{\\prime }}^{\\dagger }(x)\\Xi (x)~U_{k,d}(x)\\hat{\\chi }_{k,d},$ where we defined $\\Xi (x)=\\begin{pmatrix}A(x) & \\eta \\delta (x) \\\\\\eta ^{\\dagger }\\delta (x) & \\epsilon _d\\sigma _z \\\\\\end{pmatrix}$ with $A(x)=-iv_F\\partial _x\\tau _0 \\sigma _z-\\mu \\tau _z \\sigma _0 + B_z \\tau _z\\sigma _z$ and $\\eta = \\begin{pmatrix}t_{\\uparrow } & t_{\\downarrow } & -t_{\\uparrow } & -t_{\\downarrow } \\\\t_{\\uparrow } & t_{\\downarrow } & -t_{\\uparrow } & -t_{\\downarrow } \\\\\\end{pmatrix}^T.$ When the columns of $U_{k,d}(x)$ are formed by orthogonal eigenfunctions of $\\Xi (x)$ the problem becomes diagonal.",
"Hence, we need to search for functions $(\\Phi _k(x),\\Phi _d)$ , such that $\\begin{pmatrix}A(x)\\Phi _k(x) + \\eta \\delta (x)\\Phi _d \\\\\\eta ^{\\dagger }\\Phi _k(0) + \\epsilon _d\\sigma _z\\Phi _d \\\\\\end{pmatrix}=\\epsilon \\begin{pmatrix}\\Phi _k(x)\\\\\\Phi _d\\\\\\end{pmatrix},$ where in the second row, we performed the integration of Eq.",
"(REF ) right away as it contains no differential forms.",
"From Eq.",
"(REF ), we obtain an equation for the solutions $\\Phi _k(x)$ by solving the second row for $\\Phi _d$ and inserting the result in the first one $A(x)\\Phi _k(x)\\!",
"+ \\!\\delta (x)\\eta \\begin{pmatrix}\\frac{1}{\\epsilon - \\epsilon _d}\\!",
"& 0 \\!\\\\0\\!",
"& \\!\\frac{1}{\\epsilon +\\epsilon _d} \\\\\\end{pmatrix} \\eta ^{\\dagger }\\Phi _k(0) =\\epsilon \\Phi _k(x).~~$ This equation might be solved in the following way [58].",
"When $x\\ne 0$ the equation reduces to $A(x)\\Phi _{k}(x)=\\epsilon \\Phi _k(x)$ which is solved by plane waves.",
"Moreover, the $\\delta $ -distribution implies a discontinuous jump of the solutions at $x=0$ .",
"Hence, for $x>0$ , $x<0$ and $x=0$ , the solution takes different values.",
"This can be incorporated by the ansatz $\\Phi _k(x)=\\big (\\Phi ^e_k(x),\\Phi _k^h(x)\\big )$ with $\\Phi ^{e}_k(x)&=&\\begin{pmatrix}\\big (\\bar{\\phi }_{\\uparrow }^{e}+\\mathrm {sign}(x)\\delta \\phi _{\\uparrow }^{e}\\big )e^{i(k+B_z-\\mu ) x} \\\\\\big (\\bar{\\phi }_{\\downarrow }^{e}+\\mathrm {sign}(x)\\delta \\phi _{\\downarrow }^{e}\\big )e^{-i(k+B_z+\\mu )x}\\end{pmatrix}\\\\\\Phi ^{h}_k(x)&=&\\begin{pmatrix}\\big (\\bar{\\phi }_{\\uparrow }^{h}+\\mathrm {sign}(x)\\delta \\phi _{\\uparrow }^{h}\\big )e^{i(k-B_z+\\mu ) x} \\\\\\big (\\bar{\\phi }_{\\downarrow }^{h}+\\mathrm {sign}(x)\\delta \\phi _{\\downarrow }^{h}\\big )e^{-i(k-B_z-\\mu )x}\\end{pmatrix}$ where $\\bar{\\phi }_{\\uparrow /\\downarrow }^{e/h}=\\big (\\phi _{\\uparrow /\\downarrow ,-}^{e/h}+\\phi _{\\uparrow /\\downarrow ,+}^{e/h}\\big )/2,\\\\\\delta \\phi _{\\uparrow /\\downarrow }^{e/h}=\\big (\\phi _{\\uparrow /\\downarrow ,+}^{e/h}-\\phi _{\\uparrow /\\downarrow ,-}^{e/h}\\big )/2.$ Integration of Eq.",
"(REF ) using Eqs.",
"(REF -), this results in $-iv_F\\begin{pmatrix}\\sigma _z & 0 \\\\0 & \\sigma _z \\\\\\end{pmatrix}\\begin{pmatrix}\\phi ^e_{\\uparrow ,+}-\\phi _{\\uparrow ,-}^e \\\\\\phi ^e_{\\downarrow ,+}-\\phi _{\\downarrow ,-}^e \\\\\\phi ^h_{\\uparrow ,+}-\\phi _{\\uparrow ,-}^h \\\\\\phi ^h_{\\downarrow ,+}-\\phi _{\\downarrow ,-}^h \\\\\\end{pmatrix}+\\frac{1}{2}\\eta \\begin{pmatrix}\\frac{1}{\\epsilon - \\epsilon _d}\\!",
"& 0 \\!\\\\0\\!",
"& \\!\\frac{1}{\\epsilon +\\epsilon _d} \\\\\\end{pmatrix} \\eta ^{\\dagger }\\begin{pmatrix}\\phi ^e_{\\uparrow ,+}+\\phi _{\\uparrow ,-}^e \\\\\\phi ^e_{\\downarrow ,+}+\\phi _{\\downarrow ,-}^e \\\\\\phi ^h_{\\uparrow ,+}+\\phi _{\\uparrow ,-}^h \\\\\\phi ^h_{\\downarrow ,+}+\\phi _{\\downarrow ,-}^h \\\\\\end{pmatrix}=0.$ Eq.",
"(REF ) can be reorganized such that we obtain the scattering matrix $S$ $\\begin{pmatrix}\\phi _{\\downarrow ,-}^e \\\\\\phi _{\\downarrow ,-}^h \\\\\\phi _{\\uparrow ,+}^e \\\\\\phi _{\\uparrow ,+}^h \\\\\\end{pmatrix}= S\\begin{pmatrix}\\phi _{\\uparrow ,-}^e \\\\\\phi _{\\uparrow ,-}^h \\\\\\phi _{\\downarrow ,+}^e \\\\\\phi _{\\downarrow ,+}^h \\\\\\end{pmatrix}$ with $S= \\begin{pmatrix}R_{--} & T_{+-} \\\\T_{-+} & R_{++} \\\\\\end{pmatrix}$ and $R_{--}&=&\\begin{pmatrix}r_{--}^{ee} & r_{--}^{he} \\\\r_{--}^{eh} & r_{--}^{hh} \\\\\\end{pmatrix}, ~~ R_{++}=\\begin{pmatrix}r_{++}^{ee} & r_{++}^{he} \\\\r_{++}^{eh} & r_{++}^{hh} \\\\\\end{pmatrix},\\nonumber \\\\T_{+-} &=&\\begin{pmatrix}t_{+-}^{ee} & t_{+-}^{he} \\\\t_{+-}^{eh} & t_{+-}^{hh} \\\\\\end{pmatrix}, ~~ T_{-+} = \\begin{pmatrix}t_{-+}^{ee} & t_{-+}^{he} \\\\t_{-+}^{eh} & t_{-+}^{hh} \\\\\\end{pmatrix}.$ For the scattering amplitudes we find $R_{--}=R_{++}$ with $r_{--}^{ee}&=&r_{--}^{hh}=-r_{--}^{eh}=-r_{--}^{he}\\nonumber \\\\&=&\\frac{t_{\\uparrow }t_{\\downarrow }\\epsilon }{\\epsilon (t_{\\uparrow }^2+t_{\\downarrow }^2-iv_F\\epsilon )+iv_F\\epsilon _d^2},\\\\t_{-+}^{ee}&=& t_{-+}^{hh} = \\frac{t_{\\uparrow }^2\\epsilon }{\\epsilon (t_{\\uparrow }^2+t_{\\downarrow }^2-iv_F\\epsilon )+iv_F\\epsilon _d^2}-1,\\\\t_{-+}^{eh} &=& t_{-+}^{he}= -\\frac{t_{\\uparrow }^2\\epsilon }{\\epsilon (t_{\\uparrow }^2+t_{\\downarrow }^2-iv_F\\epsilon )+iv_F\\epsilon _d^2},\\\\t_{+-}^{ee}&=& t_{+-}^{hh} = \\frac{t_{\\downarrow }^2\\epsilon }{\\epsilon (t_{\\uparrow }^2+t_{\\downarrow }^2-iv_F\\epsilon )+iv_F\\epsilon _d^2}-1,\\\\t_{+-}^{eh} &=& t_{+-}^{he}= -\\frac{t_{\\downarrow }^2\\epsilon }{\\epsilon (t_{\\uparrow }^2+t_{\\downarrow }^2-iv_F\\epsilon )+iv_F\\epsilon _d^2}.$ With Eqs.",
"(REF -), it is easy to check that the scattering matrix of Eq.",
"(REF ) is unitary.",
"The elements of Eqs.",
"() and () are used in the main text.",
"For ease of notation, in the main text, we set $T_{-+}\\equiv T_{2}$ (and accordingly for its elements).",
"The results for the scattering amplitudes in Eqs.",
"(REF -) are independent of the values of $\\mu $ and $B_z$ as both parameters do not open spectral gaps within the helical edge states passing by the anti-wire and the $\\delta $ -scatterer discards any dependence on the momentum of incident particles.",
"Note that when the scatterer is modeled with a finite width $w$ , for instance by replacing the $\\delta $ with a Gaussian, a momentum dependence is indeed expected.",
"Yet, this will only be significant on energy scales $v_F/w$ .",
"Thus, for small $w$ (i.e.",
"large $v_F/w$ ) we expect no change in the low energy physics of our model.",
"In the presence of $B_z$ , the symmetry protection against impurity scattering is lost as the Zeeman term breaks TR symmetry.",
"This, however, does not influence the universality of our result as impurity scattering should affect electronic states in the same way as hole-like states.",
"Hence, even though the transmission amplitudes might be reduced due to impurity scattering, the ratio $\\vert t^{ee}_{\\bar{\\nu }\\nu }\\vert /\\vert t^{eh}_{\\bar{\\nu }\\nu }\\vert $ is expected to be (on average) constant.",
"Hence, also the multi-terminal conductance $G_{1\\rightarrow 2}=\\frac{e^2}{2\\pi }(\\vert t^{ee}_{-+}\\vert ^2-\\vert t^{eh}_{-+}\\vert ^2)$ , defined in the main text, is not expected to loose its qualitative information (based on its sign) in the presence of impurity scattering.",
"Moreover, long mean free path have been reported in the new generation of QSH systems [47].",
"This implies a low level of impurity scattering.",
"We consolidate this statement more in App.",
"."
],
[
"Numerical validation of the toy model",
"As discussed in the main text, for $t_{\\downarrow }> t_{\\uparrow }$ in the above model, we find a multi-terminal conductance $G_{1\\rightarrow 2}<0$ .",
"Likewise, the conductance $G_{2\\rightarrow 1}$ is then expected to satisfy $G_{2\\rightarrow 1}>0$ .",
"We can test the full model against the latter statement by numerically computing the multi-terminal conductances $G_{1\\rightarrow 2}$ and $G_{2\\rightarrow 1}$ using the Hamiltonian $H_{\\mathrm {open}}$ , defined in the main text.",
"The results are shown in Fig.",
"REF .",
"While for $G_{1\\rightarrow 2}$ there is a dominant negative signal around $\\epsilon =0$ , for $G_{2\\rightarrow 1}$ no such signal is obtained, but instead $G_{2\\rightarrow 1}>0$ .",
"This confirms the validity of the employed toy model for low energies."
],
[
"Coupling to a p-wave superconductor",
"The toy model can also be extended for higher energies, when we do not only couple to an isolated Majorana, but to a spin-less p-wave superconductor, which, in the 1D case, can be modeled by a Kitaev chain [17] $H_d=\\sum _{j=1}^N \\mu \\hat{c}_j^{\\dagger }\\hat{c}_j+\\sum _{j=1}^{N-1} \\big [(-t)\\hat{c}^{\\dagger }_j\\hat{c}_{j+1}+\\Delta \\hat{c}_j^{\\dagger }\\hat{c}_{j+1}^{\\dagger }+\\mathrm {h.c.}~\\big ]$ with fermionic fields $c_j$ , ($\\hat{c}_j^{\\dagger }$ ) annihilating (creating) a fermion at site $j$ .",
"The corresponding tunneling Hamiltonian can be written as $H_c =\\int \\mathrm {dx}\\sum _{\\sigma =\\uparrow ,\\downarrow }t_{\\sigma }\\delta (x)\\hat{\\psi }_{\\sigma }^{\\dagger }(x)\\hat{c}_1+\\mathrm {h.c.},$ where the fermions of the helical edge couple to the first site of the p-wave superconductor.",
"Repeating the calculations of Sec.",
", with Eqs.",
"(REF ) and (REF ) instead of Eq.",
"(REF ) and (REF ), this results in an equation for the eigenstates of the helical edge $\\!-\\!iv_F\\!\\begin{pmatrix}\\sigma _z & 0 \\\\0 & \\sigma _z \\\\\\end{pmatrix}\\!\\!\\begin{pmatrix}\\phi ^e_{\\uparrow ,+}-\\phi _{\\uparrow ,-}^e \\\\\\phi ^e_{\\downarrow ,+}-\\phi _{\\downarrow ,-}^e \\\\\\phi ^h_{\\uparrow ,+}-\\phi _{\\uparrow ,-}^h \\\\\\phi ^h_{\\downarrow ,+}-\\phi _{\\downarrow ,-}^h \\\\\\end{pmatrix}\\!+\\!\\frac{1}{2}\\Gamma G \\Gamma ^{\\dagger }\\!\\!\\begin{pmatrix}\\phi ^e_{\\uparrow ,+}+\\phi _{\\uparrow ,-}^e \\\\\\phi ^e_{\\downarrow ,+}+\\phi _{\\downarrow ,-}^e \\\\\\phi ^h_{\\uparrow ,+}+\\phi _{\\uparrow ,-}^h \\\\\\phi ^h_{\\downarrow ,+}+\\phi _{\\downarrow ,-}^h \\\\\\end{pmatrix}\\!=\\!0,\\nonumber \\\\$ where $G=[\\epsilon -H_d]^{-1}$ .",
"$\\Gamma $ is the Hamiltonian density of the coupling Hamiltonian $H_c$ , which can be written as $H_c=\\int \\mathrm {dx} \\delta (x)\\big (\\hat{\\psi }_{\\uparrow }^{\\dagger }(x),\\hat{\\psi }_{\\downarrow }^{\\dagger }(x),\\hat{\\psi }_{\\uparrow }(x),\\hat{\\psi }_{\\downarrow }(x)\\big ) \\Gamma \\begin{pmatrix}\\hat{c}_1 \\\\\\hat{c}_1^{\\dagger } \\\\\\vdots \\\\\\hat{c}_N^{\\dagger }\\end{pmatrix}$ with $\\Gamma = \\begin{pmatrix}t_{\\uparrow } & 0 & 0 & \\dots & \\dots & 0\\\\t_{\\downarrow } & 0 & 0 & \\ddots & & 0\\\\0 & -t_{\\uparrow } & 0 & & \\ddots & 0\\\\0 & -t_{\\downarrow } & 0 & \\dots & \\dots & 0 \\\\\\end{pmatrix}.$ Figure: (a) Multi-terminal conductance G 1→2 G_{1\\rightarrow 2} for a Kitaev chain, side-coupled to a helical edge as a function of the chains chemical potential μ\\mu and energy ϵ\\epsilon .",
"Negative values are colored red.",
"(b) Eigenstates of the Kitaev chain as a function of the system parameter μ\\mu and ϵ\\epsilon .",
"The colorcode represents the absolute difference of electronic (ζ α,1 (e) \\zeta _{\\alpha ,1}^{(e)}) and hole-like wavefunction (ζ α,1 (h) \\zeta _{\\alpha ,1}^{(h)}) at the first site of the chain normalized to the maximum value reached for all eigenstates indexed by α\\alpha .",
"Further parameters of the plots are: t=Δ=0.5t=\\Delta =0.5, the number of sites is N=15N=15.From Eq.",
"(REF ), we can compute the scattering matrix for the modes $\\phi _{\\uparrow /\\downarrow ,\\pm }^{e/h}$ , from which we obtain the conductance $G_{1\\rightarrow 2}$ .",
"The results are depicted in Fig.",
"REF (a).",
"In accordance with the main text and the toy model of Sec.",
", we find for the topological regime $\\mu <2\\vert t\\vert $ a prominent negative signal around $\\epsilon =0$ , signaling the presence of the Majorana.",
"However, even higher energy states (in particular close to $\\mu =0$ ) can return a negative signal.",
"To understand this result, we investigate again Eq.",
"(REF ), which, for the present case, takes the form $H=\\frac{1}{2}\\int \\mathrm {dx}\\tilde{\\Psi }^{\\dagger }(x)\\begin{pmatrix}h_p & \\Gamma \\delta (x)\\\\\\Gamma ^{\\dagger } \\delta (x) & h_d \\\\\\end{pmatrix}\\tilde{\\Psi }(x)$ with $h_p$ and $h_d$ the Hamiltonian density of the helical edge and the Kitaev chain and $\\tilde{\\Psi }(x)=\\big (\\hat{\\psi }_{\\uparrow }(x),\\hat{\\psi }_{\\downarrow }(x),\\hat{\\psi }_{\\uparrow }^{\\dagger }(x),\\hat{\\psi }_{\\downarrow }^{\\dagger }(x), ~\\hat{c}_1,\\hat{c}_1^{\\dagger },\\dots ,\\hat{c}_N^{\\dagger }\\big )$ [59].",
"We can now apply a unitary transformation to Eq.",
"(REF ) that diagonalizes $h_d$ $F=\\begin{pmatrix}\\mathbb {1} & 0 \\\\0 & U_d \\\\\\end{pmatrix}.$ Then, Eq.",
"(REF ) becomes $H=\\frac{1}{2}\\int \\mathrm {dx}\\tilde{\\Psi }^{\\dagger }(x)F\\begin{pmatrix}h_p & \\Gamma U_d \\delta (x) \\\\U_d^{\\dagger }\\Gamma ^{\\dagger } \\delta (x) &U_d^{\\dagger } h_d U_d\\\\\\end{pmatrix}F^{\\dagger }\\tilde{\\Psi }(x).$ Since $U_d$ diagonalizes $h_d$ , it is formed from the eigenstates of $h_d$ $U_d = \\big (\\zeta _1, \\zeta _2, \\dots \\zeta _{2N}\\big ),$ where $\\zeta _{\\alpha }=(\\zeta _{\\alpha ,1}^{(e)},\\zeta _{\\alpha ,1}^{(h)},...,\\zeta _{\\alpha ,N}^{(e)},\\zeta _{\\alpha ,N}^{(h)})^T$ are column vectors with the property $h_d\\zeta _{\\alpha }=\\epsilon _{\\alpha }\\zeta _{\\alpha }$ .",
"The transformed coupling Hamiltonian thus contains the elements of the eigenfunctions at the first site.",
"Consequently, in a low energy approximation around an eigenenergy $\\epsilon _{\\alpha }$ of $h_d$ , the coupling only happens to the first site of the corresponding eigenstate $\\zeta _{\\alpha }$ .",
"If we want to preserve particle-hole symmetry, it also has to connect to its particle-hole partner at $-\\epsilon _{\\alpha }$ , $\\hat{P}\\zeta _{\\alpha }$ with the particle-hole operator $\\hat{P}=\\mathbb {1}_{N\\times N}\\otimes \\sigma _x\\hat{K}$ , where $\\hat{K}$ denotes complex conjugation.",
"The effective Hamiltonian thus reads $H_{\\alpha }=\\frac{1}{2}\\int \\mathrm {dx}\\tilde{\\Psi }^{\\dagger }_{\\alpha }(x)\\begin{pmatrix}h_p & \\Gamma _{\\alpha } \\delta (x)\\\\\\Gamma _{\\alpha } \\delta (x) &\\epsilon _{\\alpha }\\sigma _z\\\\\\end{pmatrix}\\tilde{\\Psi }_{\\alpha }(x)$ with the basis $\\tilde{\\Psi }_{\\alpha }= \\big (\\hat{\\psi }_{\\uparrow }(x),\\hat{\\psi }_{\\downarrow }(x),\\hat{\\psi }_{\\uparrow }^{\\dagger }(x),\\hat{\\psi }_{\\downarrow }^{\\dagger }(x),\\hat{d}_{\\alpha }, \\hat{d}_{\\alpha }^{\\dagger }\\big )$ where $\\hat{d}_{\\alpha }^{\\dagger }$ creates a fermion at energy $\\epsilon _{\\alpha }$ .",
"The coupling matrix $\\Gamma _{\\alpha }$ is given by $\\Gamma _{\\alpha }\\!\\!=\\!\\!",
"\\begin{pmatrix}t_{\\uparrow } \\zeta _{\\alpha ,1}^{(e)} & t_{\\downarrow }\\zeta _{\\alpha ,1}^{(e)} & -t_{\\uparrow } \\zeta _{\\alpha ,1}^{(h)} & -t_{\\downarrow } \\zeta _{\\alpha ,1}^{(h)} \\\\t_{\\uparrow } \\zeta _{\\alpha ,1}^{(h)*} & t_{\\downarrow }\\zeta _{\\alpha ,1}^{(h)*} & -t_{\\uparrow } \\zeta _{\\alpha ,1}^{(e)*} & -t_{\\downarrow } \\zeta _{\\alpha ,1}^{(e)*} \\\\\\end{pmatrix}^T.$ As discussed in the main text, this effectively corresponds to the coupling to a particle $\\chi ^{\\dagger }=\\zeta _{\\alpha ,1}^{(e)*}d_{\\alpha }^{\\dagger }+\\zeta _{\\alpha ,1}^{(h)}\\hat{d}_{\\alpha }$ .",
"In particular, for $\\zeta _{\\alpha ,1}^{(e)}\\equiv \\zeta _{\\alpha ,1}^{(h)}=1$ , it corresponds to the toy model of Sec. .",
"On the basis of the effective model of Eq.",
"(REF ) we find (as discussed in the main text) two main results: (i) away from zero-energy a negative signal in the multi-terminal conductance $G_{1\\rightarrow 2}$ is reached whenever the form of the particle $\\chi $ deviates less than a threshold $\\bar{\\xi }$ from the Majorana from, i. e. whenever $\\delta \\zeta =\\vert \\vert \\zeta _{\\alpha ,1}^{(e)}\\vert -\\vert \\zeta _{\\alpha ,1}^{(h)}\\vert \\vert \\le \\bar{\\xi }$ and, more importantly, (ii) as $\\epsilon _{\\alpha }\\rightarrow 0$ also the threshold $\\bar{\\xi }\\rightarrow 0$ .",
"We can numerically confirm our analysis when analyzing the situation of the side-coupled Kitaev chain.",
"Fig.",
"REF (b) visualizes the (numerically) obtained values of $\\delta \\zeta $ for each eigenstate (on the first site).",
"At $\\mu = 0$ , each eigenstate of the Kiteav chain satisfies the Majorana condition at the first site.",
"Hence, we expect to find a negative multi-terminal conductance for all eigenstates, which coincides with the numerical results in Fig.",
"REF (a).",
"Away from $\\mu =0$ , eigenstates at $\\epsilon \\ne 0$ successively loose the Majorana condition and the dominant negative signal in the multi-terminal conductance is as well lost for those states.",
"At zero-energy, however, the Majorana form is kept throughout the whole topological phase and likewise also the negative signal persists."
],
[
"Stability against time-reversal breaking scattering",
"In App.",
", we have already seen that a TR symmetry breaking Zeeman field $B_z$ does not influence the universality of the obtained conductance signature (i.e.",
"negative $G_{1\\rightarrow 2}$ in the presence of the Majorana at zero energy).",
"This suggests that TR symmetry is not among the determinative symmetries to eventually obtain negative $G_{1\\rightarrow 2}$ .",
"Yet, one may wonder if this stems from the observation that $B_z$ does not induce TR breaking backscattering.",
"To rule out this possibility, we now discuss the influence of such backscattering terms.",
"The model we analyze is given by Eq.",
"(REF ), i.e.",
"$\\tilde{A}(x)\\Phi _k(x)\\!",
"+ \\!\\delta (x)\\eta \\begin{pmatrix}\\frac{1}{\\epsilon - \\epsilon _d}\\!",
"& 0 \\!\\\\0\\!",
"& \\!\\frac{1}{\\epsilon +\\epsilon _d} \\\\\\end{pmatrix} \\eta ^{\\dagger }\\Phi _k(0) =\\epsilon \\Phi _k(x),~~$ where $\\tilde{A}(x)=-iv_F\\tau _0\\sigma _z\\partial _x -\\mu \\tau _z\\sigma _0 + \\tau _z\\sigma _x B_x $ now contains TR breaking backscattering contributions $B_x$ .",
"Away from $x=0$ , Eq.",
"(REF ) is solved by integration $\\Phi _k(x_b)\\!=T_B(x_b,x_a)\\Phi _k(x_a),~$ where $T_B(x_b,x_a)=\\!\\exp \\left[\\!\\frac{i}{v_F}\\!\\int _{x_a}^{x_b}\\!\\mathrm {d}x~ \\tau _0\\sigma _z\\!\\left(\\!\\epsilon \\!-\\!",
"(B_x\\tau _z\\sigma _x\\!-\\!\\mu \\tau _z\\sigma _0)\\!\\right)\\!\\right].$ The $\\delta $ scattering event at $x=0$ requires more care.",
"As the eigenfunctions are not expected to always possess a pure plane-wave character, the ansatz of Eq.",
"(REF ) might no longer be valid.",
"Still, integration from $x=-\\varepsilon $ to $x=\\varepsilon $ and taking the limit $\\varepsilon \\rightarrow 0$ yields a defining equation for scattering at the $\\delta $ -barrier, given by $-i v_F \\tau _0\\sigma _z \\left[\\Phi _k(0^+)-\\Phi _k(0^-)\\right]+\\eta G_d \\eta ^{\\dagger }\\Phi _k(0)=0,~~~$ where we introduced the shorthand notation $G_d= \\begin{pmatrix}\\frac{1}{\\epsilon - \\epsilon _d}\\!",
"& 0 \\!\\\\0\\!",
"& \\!\\frac{1}{\\epsilon +\\epsilon _d} \\\\\\end{pmatrix}.$ Similar to Eq.",
"(REF ), Eq.",
"(REF ) can be solved with a symmetric ansatz $\\Phi _k(0)=1/2\\left(\\Phi _k(0^+)+\\Phi _k(0^-)\\right)$ .",
"This automatically leads to the transfer matrix, associated with the $\\delta $ -barrier $\\Phi _k(0^+)=T_{\\delta }\\Phi _k(0^-),$ where $T_{\\delta }=\\left[-iv_F \\tau _0\\sigma _z +\\frac{1}{2}\\eta G_d \\eta ^{\\dagger }\\right]^{-1}\\left[-iv_F\\tau _0\\sigma _z-\\frac{1}{2}\\eta G_d \\eta ^{\\dagger }\\right].$ The transmission in a helical edge from $x_i<0$ to $x_f>0$ , including backscattering by $B_x$ , side-coupled to a Majorana, is then described by the compiled transfer matrix $T(x_f,x_i)= T_B(x_f,0)T_\\delta T_B(0,x_i).$ From $T(x_f,x_i)$ , it is straightforward to compute the associated scattering matrix and, subsequently, the conductance $G_{1\\rightarrow 2}$ .",
"Moreover, it is straightforward to generalize Eq.",
"(REF ) to the generic case just by replacing $\\eta \\rightarrow \\Gamma _\\alpha $ ."
]
] | 2001.03475 |
[
[
"Standing autoresonant plasma waves"
],
[
"Abstract Formation and control of strongly nonlinear standing plasma waves (SPWs) from a trivial equilibrium by a chirped frequency drive is discussed.",
"If the drive amplitude exceeds a threshold, after passage through the linear resonance in this system, the excited wave preserves the phase-locking with the drive yielding a controlled growth of the wave amplitude.",
"We illustrate these autoresonant waves via Vlasov-Poisson simulations showing formation of sharply peaked excitations with local electron density maxima significantly exceeding the unperturbed plasma density.",
"The Whitham's averaged variational approach applied to a simplified water bag model yields the weakly nonlinear evolution of the autoresonant SPWs and the autoresonance threshold.",
"If the chirped driving frequency approaches some constant level, the driven SPW saturates at a target amplitude avoiding the kinetic wave breaking."
],
[
"Introduction",
"Plasmas can sustain laser intensities many orders of magnitude larger than typical solid state optical components.",
"This makes plasmas attractive for manipulation and control of intense laser beams, but requires formation of large and stable electron density structures in the plasma to impact the propagation of the laser light significantly.",
"A typical approach to this problem is via ponderomotive forces using additional driving laser beams.",
"Several applications of plasma photonics based on light scattering off electron density structures have been proposed including short pulse amplification via the resonant excitation of plasma waves [12], transient plasma gratings [10], crossed-beam energy transfer for symmetry control in ICF [14], and, more recently, plasma-based polarization control [13], [17], [16], [11].",
"The efficiency is an important factor in all these applications, as the goal is to create the largest amplitude plasma density perturbation using smallest possible driver intensities.",
"Recently, it was shown that very large amplitude travelling or standing ion acoustic waves (SIAW) can be excited via the autoresonance (AR) approach [8], [9], [7].",
"The latter exploits the salient property of nonlinear waves to stay in resonance with driving perturbations if some parameter in the system (e.g., the driving frequency) varies in time [for a review of several AR applications see [4], [5]].",
"Such systems preserve the resonance with the drive via self-adjustment of the driven wave amplitude leading to a large amplitude amplification even when the driver is relatively weak.",
"In this paper, we will discuss the AR approach to excitation of very large amplitude standing plasma waves (SPW) using both the kinetic and a simplified water bag models.",
"The nonlinear SIAWs [7] and SPWs [see a related analysis of nonlinear Trivelpiece-Gould waves [3]] are much more complex than their traveling wave counterparts.",
"For example, the autoresonant SIAW driven to large amplitudes by a weak, chirped frequency standing ponderomotive wave is a two-phase solution, each phase locked to one of the traveling waves comprising the drive [7].",
"The peak electron density in these waves can reach several times the initial plasma density.",
"We will show that similar giant plasma structures can be excited in the form of autoresonant SPWs.",
"The paper is organized as follows.",
"In Sec.",
"2, we will illustrate autoresonant SPWs in Vlasov-Poisson simulations and compare the results to those from the a simplified water bag model presented in Sec.",
"3.",
"We will discuss weakly nonlinear SPWs in Sec.",
"4 and use their form as an ansatz in developing a weakly nonlinear theory of driven-chirped SPWs based on the Whitham's averaged variational principle [18] in Sec.",
"5.",
"In Sec.",
"6, we reduce a system of slow coupled amplitude-phase mismatch equations yielding the autoresonance threshold on the driving amplitude for excitation of autoresonant SPWs.",
"In the same section, we will also discuss the control of large amplitude autoresonant SPW by tailoring the time dependence of the driving frequency.",
"Finally, Sec.",
"7 will present our conclusions."
],
[
"Standing autoresonant plasma waves in Vlasov-Poison simulations",
"Our study of autoresonant SPWs is motivated by numerical simulations of the following one-dimensional Vlasov-Poisson (VP) system describing an externally driven plasma wave $f_{t}+uf_{x}+(\\varphi +\\varphi _{d})_{x}f_{u}=0,\\varphi _{xx}=\\kappa ^{2}\\varphi +\\int fdu-1.",
"$ Figure: (color online) The colormap of the waveforms of the electrondensity nn (left panel) and fluid velocity vv (right panel) in the finaltime interval δτ=0.125\\protect \\delta \\protect \\tau =0.125 of the simulation.",
"Theslopes of the white lines are the phase velocities of the two travelingwaves comprising the drive.Figure: (color online) The snapshots of the electron distribution functionat two times τ=2.69\\protect \\tau =2.69 [panel (a)] and 2.72.7 [panel (b)] duringthe last period of the wave oscillation in Fig.",
"1.",
"The fluid velocityvanishes in panel (a), while the electron density reaches its maximal valueof 2.52.5.",
"In contrast, in panel (b), the fluid velocity |v||v| is at itsmaximum at two locations in xx.",
"The straight white lines are at the locationsof the phase velocities ±ω d /k\\pm \\protect \\omega _{d}/kHere we assume constant ion density, $f$ and $\\varphi $ are the electron velocity distribution and the electric potential, and $\\varphi _{d}=2\\varepsilon \\cos (kx)\\cos \\theta _{d}$ , where $\\theta _{d}=kx-\\int \\omega _{d}dt$ is a small amplitude standing wave-like ponderomotive potential having a slowly varying frequency $\\omega _{d}(t)$ .",
"All dependent and independent variables in (REF ) are dimensionless, such that the position, time, and velocity are rescaled with respect to $1/k$ , the inverse plasma frequency $\\omega _{p}^{-1}=\\sqrt{m/(4\\pi e^{2}n_{0})}$ ($n_{0}$ being the ion density), and $\\omega _{p}/k$ .",
"Then, in the dimensionless form of the driving phase, $k=1$ and $\\omega _{d}$ is rescaled by $\\omega _{p}$ .",
"The distribution function and the potentials in (REF ) are rescaled with respect to $kn_{0}/\\omega _{p}$ , and $m\\omega _{p}^{2}/(ek^{2})$ , respectively.",
"We have also added an effective screening term $\\kappa ^{2}\\varphi $ in Poisson equation modeling the typical finite radial extent of the ponderomotive potential [3].",
"We assume $2\\pi $ periodicity in $x$ and solve the time evolution problem, subject to simple initial equilibrium $\\varphi (x,0)=0$ and $f(u,x,0)=(2\\pi \\sigma ^{2})^{-1/2}\\exp (-u^{2}/2\\sigma ^{2})$ , where $\\sigma =k\\lambda _{D}$ ($\\lambda _{D}=u_{e}/\\omega _{p}$ being the Debye length and $u_{e}$ the initial electron thermal velocity).",
"Note that $\\sigma $ , $\\kappa $ and the driving parameters fully define our rescaled, dimensionless problem.",
"Finally, we are interested in the driving frequency of the order of the plasma frequency and, consequently, assume $\\sigma \\ll 1$ to avoid the kinetic Landau resonance ($\\omega _{d}/k\\approx u$ ) initially.",
"We have applied our VP code [6] for solving this problem numerically and show some results of the simulations in Figs.",
"1-4 for $\\sigma =0.1$ , $\\kappa =0$ (the effect of the nonzero $\\kappa $ will be discussed later), and driving frequency (dimensionless) $\\omega _{d}=\\omega _{0}+\\alpha t$ , where $\\omega _{0}=\\sqrt{(1+\\kappa ^{2})^{-1}+3\\sigma ^{2}}$ is the normalized plasma wave frequency in the linearized warm fluid model [see Eq.",
"(REF )], while $\\alpha =0.00005$ and $\\varepsilon =0.0021$ .",
"We present the results of the simulations versus slow time $\\tau =\\alpha ^{1/2}t$ (a convenient representation in AR problems), as the system evolves between initial $\\tau _{0}=-5$ and final $\\tau _{1}=2.7$ times (the total dimensionless time in this simulation is $\\Delta t=7.7/\\alpha ^{1/2}=1089$ , i.e., 173 periods of plasma oscillations).",
"Figure 1 shows the waveform of the electron density $n=\\int fdu$ and fluid velocity $v=\\int ufdu$ in a small time window $\\delta \\tau =0.125\\,$ of the evolution just before reaching $\\tau _{1}$ .",
"One can see a very large amplitude SPW with its peak electron density reaching $2.5$ times the initial plasma density (unity in our dimensionless problem).",
"Furthermore, as expected, for a given time, the solutions are $2\\pi $ periodic in $x$ .",
"But there also exist two directions shown by white lines in the figure with slopes $dx/d\\tau =\\pm \\omega _{d}/\\alpha ^{1/2}$ (these slopes are the phase velocities of the two traveling waves comprising the drive) along which the solutions are $\\pi $ periodic.",
"This suggests that $u,n$ are periodic functions of three arguments $\\Theta _{1},\\Theta _{2},x$ , where $\\Theta _{1,2}=x\\pm \\int \\omega _{d}t$ and the solutions are $2\\pi $ periodic in $\\Theta _{1,2}$ and $\\pi $ periodic in $x.$ Similar two-phase solutions, with each phase $\\Theta _{1,2}$ locked to one of the phases of the waves comprising the drive were also observed in autoresonant SIAWs [7].",
"The occurrence of multi-phase solutions is known in the theory of some partial differential equations [15].",
"Additional results of the simulations are presented in Fig.",
"2, showing the snapshots of the electron distribution function at two different times inside $\\delta \\tau $ .",
"Figure 2a corresponds to the time when the fluid velocity vanishes, while the electron density reaches its maximal value of $2.5$ , while at the time in Fig.",
"2b the fluid velocity $|v|$ is at its maximum at two locations in $x$ .",
"The white horizontal lines in Fig.",
"2 show the locations of the phase velocities $\\pm \\omega _{d}/k$ associated with the traveling waves comprising the drive.",
"One can see in Fig.",
"2b that a small fraction of the electrons in the tail of the distribution reaches the Landau resonance, indicating proximity to the kinetic wave breaking.",
"Figure: (color online) The excitation of the autoresonant SPW for drivingamplitude above the AR threshold.",
"Panel (a): the envelope of the maxima ofthe electron fluid velocity vv versus slow time τ=α 1/2 t\\protect \\tau =\\protect \\alpha ^{1/2}t in VP simulations (blue solid line) are compared to water bagmodel simulations (dotted red line).",
"The dashed line represents the phasevelocity ω d /k\\protect \\omega _{d}/k of the driving wave.",
"Panel (b) shows thefrequencies of the driving (straight line) and driven (dotted line) waves.The continuous AR frequency locking is seen starting τ≈-3\\protect \\tau \\approx -3 .Figure: (color online) The numerical simulations as in Fig.",
"3, but with thedriving amplitude below the AR threshold.",
"Panel (a) illustrates saturationof the excited wave amplitude, while panel (b) shows that the frequencylocking discontinues leading to saturation shortly after passage through thelinear resonance.The evolution of the system leading to the final stage shown in Figs 1 and 2 is illustrated in Fig.",
"3, where panel (a) presents the envelope of the maxima of fluid velocity $v$ during the evolution versus slow time.",
"In the same figure, we also show a similar envelope (dashed line) obtained by using a simplified water bag model described in the next section.",
"We observe an excellent agreement between the two models even for very large excitations.",
"This agreement is important since the water bag model will be used below in calculating the threshold for AR excitation of SPWs.",
"Also shown in Fig.",
"3a by dashed line is the driving wave phase velocity $u_{d}=\\omega _{d}/k=\\omega _{d}$ .",
"One can see that at the final times of the excitation the fluid velocity approaches the phase velocity, indicating again the proximity to the Landau resonance.",
"Finally, Fig.",
"3b shows the driving and driven wave frequencies (the latter is calculated from the time differences between successive peaks of the fluid velocity) versus $\\tau $ .",
"It illustrates the characteristic signature of autoresonant waves, i.e.",
"the frequency (phase) locking between the driven and driving waves, which starts even prior passage trough the linear resonance $\\omega _{d}=\\omega _{0}$ (at $\\tau =0$ ) and continues to the fully nonlinear stage.",
"We complete this sections by Fig.",
"4, showing the results of the simulations similar to those in Fig.",
"3 (the same parameters and initial conditions), but for a smaller driving amplitude $\\varepsilon =0.0011$ .",
"One can see in the figure that the wave excitation saturates in this case (see panel a), as the phase locking between the driven and driving wave discontinues (see panel b) shortly after passage through the linear resonance.",
"We find that the peak electron density in this case reaches 1.7 of the initial density, as compared to 2.5 in the AR case in Fig.",
"3.",
"These results illustrate the existence of the characteristic AR threshold $\\varepsilon _{th}$ on the driving amplitude [$\\varepsilon _{th}=0.0015$ for the parameters of the above simulations, see Eq.",
"(REF ) below].",
"The autoresonance threshold is a weakly nonlinear phenomenon studied in many applications [5].",
"We have seen in Figs.",
"3 and 4 that a simplified water bag model can be used in describing the weakly nonlinear evolution of the driven SPWs instead of using the full VP system.",
"The next four sections will present the theory of weakly nonlinear autoresonant SPWs based on this model.",
"Figure: (color online) The water bag model.",
"The electron distribution isconfined between two limiting trajectories u 1,2 u_{1,2}."
],
[
"The water bag model of driven-chirped plasma waves",
"The water bag model [1] assumes that the electron distribution function remains constant, $f(u,x,t)=1/(2\\Delta )$ , between two limiting trajectories $u_{1,2}(x,t)$ in phase space and vanishes outside these trajectories (see Fig.",
"5).",
"Then the electron density is $n(x,t)=\\int fdu=(u_{1}-u_{2})/(2\\Delta )$ and our kinetic SPW problem is governed by the following set of the momentum and Poisson equations $u_{1t}+u_{1}u_{1x}=(\\varphi +\\varphi _{d})_{x}, \\nonumber \\\\u_{2t}+u_{2}u_{2x}=(\\varphi +\\varphi _{d})_{x}, \\\\\\displaystyle \\varphi _{xx}=\\kappa ^{2}\\varphi +(u_{1}-u_{2})/(2\\Delta )-1, \\nonumber $ where we use the same normalization of $x,t,f,u_{1,2},\\varphi ,$ and $\\varphi _{d}$ as in the previous section.",
"We will be solving this system subject to $2\\pi $ periodicity in $x$ for a trivial initial equilibrium $\\varphi =0,$ $u_{1,2}=\\pm \\Delta $ .",
"Note that if one also defines $v(x,t)=(u_{1}+u_{2})/2$ , Eqs.",
"() yield $n_{t}+(vn)_{x}=0, \\nonumber \\\\v_{t}+vv_{x}=(\\varphi +\\varphi _{d})_{x}-\\Delta ^{2}nn_{x}, \\\\\\varphi _{xx}=\\kappa ^{2}\\varphi +n-1.",
"\\nonumber $ Therefore, the water bag model is isomorphic to the usual warm fluid limit of the driven plasma waves with the adiabatic electron pressure scaling $p\\sim n^{3}$ and $\\Delta ^{2}=3\\sigma ^{2}$ .",
"The results shown by the dotted red lines in Figs.",
"3 and 4 were obtained by solving Eqs.",
"() numerically and show excellent agreement between the VP and water bag simulations until approaching the Landau resonance.",
"These water bag simulations used a code similar to that for SIAWs [9] and based on a standard spectral method [2].",
"In analyzing the AR threshold in the problem, we use a Lagrangian approach.",
"We introduce new potentials $\\psi _{1,2}$ via $u_{1,2}=\\pm \\Delta +\\partial _{x}\\psi _{1,2}$ and rewrite Eqs.",
"() as $\\psi _{1tx}+\\Delta \\psi _{1xx}+\\psi _{1x}\\psi _{1xx}=(\\varphi +\\varphi _{d})_{x}, \\nonumber \\\\\\psi _{2tx}-\\Delta \\psi _{2xx}+\\psi _{2x}\\psi _{2xx}=(\\varphi +\\varphi _{d})_{x}, \\\\\\varphi _{xx}=\\kappa ^{2}\\varphi +(\\psi _{1x}-\\psi _{2x})/2\\Delta .",
"\\nonumber $ This system can be derived from the variation principle $\\delta (\\int Ldxdt)=0$ , with the three-field Lagrangian density $L=\\frac{1}{2}\\varphi _{x}^{2}+\\frac{\\kappa ^{2}\\varphi ^{2}}{2}-\\frac{\\psi _{1x}^{2}+\\psi _{2x}^{2}}{4}-\\frac{\\psi _{1x}\\psi _{1t}-\\psi _{2x}\\psi _{2t}}{4\\Delta } \\nonumber \\\\-\\frac{\\psi _{1x}^{3}-\\psi _{2x}^{3}}{12\\Delta }+\\frac{(\\psi _{1x}-\\psi _{2x})(\\varphi +\\varphi _{d})}{2\\Delta }.",
"$ Our next goal is to apply the Whitham's averaged variational principle [18] and derive weakly nonlinear slow evolution system describing driven-chirped SPWs governed by this Lagrangian.",
"We proceed by discussing weakly nonlinear driven and phase-locked, but not chirped SPWs."
],
[
"Weakly nonlinear SPWs",
"If one starts in the trivial equilibrium $\\varphi =0,$ $n=1$ , $v=0$ ($u_{1,2}=\\pm \\Delta $ ), Eqs.",
"(REF ) after averaging over one spatial period, yield constant-in-time averaged density and fluid velocity $\\left\\langle n\\right\\rangle =1$ , $\\left\\langle v\\right\\rangle =0$ ($\\left\\langle u_{1,2}\\right\\rangle =\\pm \\Delta $ ).",
"We consider the linear stage of driven SPWs first, i.e., write $n=1+\\delta n$ and linearize (REF ) to get $(\\delta n)_{t}+v_{x}=0, \\nonumber \\\\v_{t}=\\varphi _{x}-\\Delta ^{2}\\delta n_{x}-2\\varepsilon \\cos \\theta _{d}\\sin x, \\\\\\varphi _{xx}=\\kappa ^{2}\\varphi +\\delta n. \\nonumber $ In the case of a constant driving frequency, this set yields phase-locked standing wave solutions of frequency $\\omega =\\omega _{d}$ for all dependent variables $\\delta n=a\\cos \\theta \\cos x, \\nonumber \\\\v=b\\sin \\theta \\sin x, \\\\\\varphi =c\\cos \\theta \\cos x, \\nonumber $ where $\\theta =\\theta _{d}=\\omega t$ , $a=-\\frac{2\\varepsilon }{\\omega ^{2}-\\omega _{0}^{2}}, \\nonumber \\\\b=\\omega a, \\\\c=-\\frac{a}{1+\\kappa ^{2}}.",
"\\nonumber $ and $\\omega _{0}^{2}=\\frac{1}{(1+\\kappa ^{2})}+\\Delta ^{2}.",
"$ Note that $\\omega _{0}$ is the natural frequency of a linear SPW in the problem.",
"Equations (REF ) yield the linear solutions $u_{1,2}=v\\pm \\Delta (1+\\delta n)=\\pm \\Delta +b\\sin \\theta \\sin x\\pm \\Delta a\\cos \\theta \\cos x $ and, thus, $\\psi _{1,2}=-b\\sin \\theta \\cos x\\pm \\Delta a\\cos \\theta \\sin x.",
"$ Interestingly, in contrast to $n,u$ , and $\\varphi $ , the linear solutions for $u_{1,2}$ and $\\psi _{1,2}$ are not standing waves.",
"Our next goal is to include a weak nonlinearity in the problem, but still for a constant driving frequency case.",
"To this end, we use spatial periodicity and write truncated Fourier expansions $\\psi _{1,2}=b_{1,2}\\cos x+a_{1,2}\\sin x+d_{1,2}\\cos (2x)+e_{1,2}\\sin (2x), \\\\\\varphi =c_{1}\\cos x+c_{2}\\cos (2x), $ where $a_{1,2}$ , $b_{1,2}$ , and $c_{1}$ are time dependent amplitudes viewed as small first order objects, while $d_{1,2}$ , $e_{1,2}$ , and $c_{2}$ are due to the nonlinearity and, thus, are of second order.",
"The time dependence of the first order amplitudes is assumed to be that of the linear solutions (REF ) and (REF ), i.e.",
"$a_{1,2}=A_{1,2}\\cos \\theta $ , $b_{1,2}=B_{1,2}\\sin \\theta $ , and $c_{1}=C_{1}\\cos \\theta $ .",
"But what is the time dependence of the second order amplitudes?",
"Instead of working with the original system () for answering this question, we use a simpler Lagrangian approach in Appendix A.",
"The final result is [see Eqs.",
"(REF )] $d_{1,2}=D_{1,2}\\sin (2\\theta ) \\nonumber \\\\e_{1,2}=F_{1,2}+E_{1,2}\\cos (2\\theta ) \\\\c_{2}=B+C\\cos (2\\theta ) \\nonumber $ where $D_{1,2}$ , $F_{1,2}$ , $E_{1,2}$ , $B$ , and $C$ are constants.",
"Thus, the potentials characterizing a weakly nonlinear SPW phase-locked to the drive are: $\\psi _{i}=B_{i}\\sin \\theta \\cos x+A_{i}\\cos \\theta \\sin x+D_{i}\\sin (2\\theta )\\cos (2x)+[F_{i}+E_{i}\\cos (2\\theta )]\\sin (2x), $ $\\varphi =C_{1}\\cos \\theta \\cos x+[B+C\\cos (2\\theta )]\\cos (2x).",
"$ Here $i=1,2$ , all the coefficients (amplitudes) are constant and could be related to the driving amplitude $\\varepsilon $ by using the approach of Appendix A.",
"Nevertheless, we will not follow this route because, knowing the form of Eqs.",
"(REF ) and (REF ) is sufficient for addressing our original driven-chirped problem via the Whitham's averaged variational principle [18], as described next."
],
[
"Whitham's averaged variational principle for driven SPWs",
"As in the case of the standing IAWs [7], the Whitham's approach uses the ansatz of form (REF ) and (REF ) for the solution of our driven-chirped problem, but now all the amplitudes are viewed as slow functions of time and $\\theta $ is the fast wave phase having a slowly varying frequency $\\omega (t)=d\\theta /dt$ generally different from the frequency $\\omega _{d}(t)$ of the chirped driving wave.",
"This ansatz is then substituted into the Lagrangian density (REF ), where the driver is written as $\\varphi _{d}=2\\varepsilon \\cos (\\theta +\\Phi )\\sin x$ and the phase mismatch $\\Phi (t)=\\theta _{d}-\\theta $ is viewed as a slow function of time.",
"Then, all the slow variables are frozen in time and the Lagrangian density is averaged over $2\\pi $ in both $x$ and $\\theta $ (fast scales).",
"This results in a new averaged Lagrangian density $\\Lambda $ , which depends on slow objects only [i.e., $A_{1,2}$ , $B_{1,2}$ , $C_{1}$ (first order amplitudes), $D_{1,2}$ , $E_{1,2}$ , $F_{1,2}$ , $B$ , $C$ (second order amplitudes), the wave frequency $\\omega $ , and the phase mismatch $\\Phi $ ].",
"To fourth order in amplitudes we obtain [via Mathematica [19]] $\\Lambda =\\Lambda _{2}+\\Lambda _{4}+\\Lambda _{d}$ , where $\\Lambda _{2}=2C_{1}^{2}(1+\\kappa ^{2})-A_{1}^{2}-A_{2}^{2}-B_{1}^{2}-B_{2}^{2} \\nonumber \\\\-\\frac{2}{\\Delta }[\\omega (A_{1}B_{1}-A_{2}B_{2})-C_{1}(A_{1}-A_{2})],$ $\\Lambda _{4}=-4(2F_{1}^{2}+2F_{2}^{2}+D_{1}^{2}+D_{2}^{2}+E_{1}^{2}+E_{2}^{2})\\nonumber \\\\+2(4+\\kappa ^{2})(2B^{2}+C^{2})-\\frac{4}{\\Delta }[2\\omega (D_{1}E_{1}-D_{2}E_{2}) \\nonumber \\\\+2B(F_{1}-F_{2})+C(E_{1}-E_{2})]-\\frac{1}{\\Delta }[A_{1}B_{1}D_{1}-A_{2}B_{2}D_{2}] \\nonumber \\\\+\\frac{1}{2\\Delta }[B_{1}^{2}(2F_{1}-E_{1})-A_{1}^{2}(2F_{1}+E_{1})-B_{2}^{2}(2F_{2}-E_{2})+A_{2}^{2}(2F_{2}+E_{2})],$ $\\Lambda _{d}=\\frac{4\\varepsilon }{\\Delta }(A_{1}-A_{2})\\cos \\Phi .$ Note that $\\Lambda $ has an algebraic form with the wave phase $\\theta $ entering via $\\Phi =$ $\\theta _{d}-\\theta $ and $\\omega (t)=d\\theta /dt$ .",
"Therefore, the variations with respect to all amplitudes yield a set of eight algebraic equations $\\frac{\\partial \\Lambda }{\\partial Z_{i}}=0,i=1,..,8, $ where $Z_{i}$ represent each of the 8 slow amplitudes in the problem.",
"In addition, the variation with respect to $\\theta $ yields an ordinary differential equation (ODE) $\\frac{d}{dt}\\left( \\frac{\\partial \\Lambda }{\\partial \\omega }\\right) =-\\frac{\\partial \\Lambda }{\\partial \\Phi }.",
"$ To lowest order, the last equation becomes $\\frac{d}{dt}(A_{1}B_{1}-A_{2}B_{2})=-2\\varepsilon (A_{1}-A_{2})\\sin \\Phi .$ and, as expected, describes the slow evolution, provided $\\varepsilon $ is sufficiently small.",
"The plan for analyzing our slow driven system is as follows.",
"First, we will use the algebraic equations (REF ) for expressing seven of the eight amplitudes and $\\omega $ in terms of $\\Phi $ and the eighth amplitude (chosen to be $C_{1}$ ), i.e.",
"obtain relations $Z_{i}=Z_{i}(C_{1},\\Phi )$ and $\\omega =\\omega (C_{1},\\Phi )$ .",
"Then $\\frac{d\\Phi }{dt}=\\omega _{d}(t)-\\omega (C_{1},\\Phi ), $ which in combination with Eq.",
"(REF ) comprise a closed set of two ODEs describing the slow evolution in our driven system.",
"This plan is algebraically complex, but can be performed using Mathematica [19].",
"We describe the intermediate steps of this calculation in Appendix B and here present the final results.",
"To lowest significant order, the amplitudes in Eq.",
"(REF ) are as in the linear problem [see Eqs.",
"(REF ), ()] $A_{1}=-A_{2}\\approx -\\frac{C_{1}\\Delta }{\\omega _{0}^{2}-\\Delta ^{2}}=-C_{1}\\Delta (1+\\kappa ^{2}), \\\\B_{1}=B_{2}\\approx \\frac{C_{1}\\omega _{0}}{\\omega _{0}^{2}-\\Delta ^{2}}=C_{1}\\omega _{0}(1+\\kappa ^{2}), $ while [see Eqs.",
"(REF ) and (REF ) in Appendix B] $\\omega ^{2}(C_{1},\\Phi )\\approx \\omega _{0}^{2}+NC_{1}^{2}+\\frac{2\\varepsilon \\cos \\Phi }{(1+\\kappa ^{2})C_{1}}, $ where $N=\\frac{9\\kappa ^{2}(3+\\kappa ^{2})+R}{96[1+(4+\\kappa ^{2})\\Delta ^{2}]},$ and $R=(1+\\kappa ^{2})\\Delta ^{2}[3(60+56\\kappa ^{2}+11\\kappa ^{4})+20(1+\\kappa ^{2})(4+\\kappa ^{2})(5+2\\kappa ^{2})\\Delta ^{2}+16(1+\\kappa ^{2})^{2}(4+\\kappa ^{2})^{2}\\Delta ^{4}]$ ."
],
[
"Dynamics and control of chirped-driven SPWs",
"At this stage, we discuss the slow evolution in our driven-chirped SPWs system.",
"Upon substitution of Eqs.",
"(REF ) and () into Eq.",
"(REF ), we obtain $\\frac{dC_{1}}{dt}=-\\frac{\\varepsilon }{\\omega _{0}(1+\\kappa ^{2})}\\sin \\Phi .$ Next, for a small frequency deviation in the vicinity of the linear resonance, $\\omega =\\omega _{0}+\\delta \\omega $ , Eq.",
"(REF ) yields $\\delta \\omega \\approx \\frac{N}{2\\omega _{0}}C_{1}^{2}+\\frac{\\varepsilon \\cos \\Phi }{\\omega _{0}(1+\\kappa ^{2})C_{1}}, $ where the two terms on the right represent the wave frequency shifts due to the nonlinearity and interaction, respectively.",
"Then, from Eq.",
"(REF ) and using the driving frequency $\\omega _{d}=\\omega _{0}+\\alpha t$ , we obtain $\\frac{d\\Phi }{dt}=\\alpha t-\\frac{N}{2\\omega _{0}}C_{1}^{2}-\\frac{\\varepsilon \\cos \\Phi }{\\omega _{0}(1+\\kappa ^{2})C_{1}}.",
"$ Figure: (color online) The rescaled threshold driving amplitude ε th /α 3/4 \\protect \\varepsilon _{th}/\\protect \\alpha ^{3/4} versus σ=kλ D \\protect \\sigma =k\\protect \\lambda _{D} for different values of the screening parameter κ\\kappa Equations (REF ) and (REF ) comprise a complete set of amplitude-phase mismatch ODEs describing the passage through the linear resonance in our problem.",
"These equations involve several parameters, but the number of parameters can be reduced to just one by rescaling the problem.",
"Indeed, if we use the slow time $\\tau =\\alpha ^{1/2}t$ and define new amplitude $Q$ via $Q^{2}=\\frac{N}{2\\omega _{0}\\alpha ^{1/2}}C_{1}^{2}$ , our system reduces to $\\frac{dQ}{d\\tau }=-\\mu \\sin \\Phi , \\\\\\frac{d\\Phi }{d\\tau }=\\tau -Q^{2}-\\frac{\\mu \\cos \\Phi }{Q}.",
"$ where $\\mu =\\frac{\\varepsilon N^{1.2}}{\\sqrt{2}\\alpha ^{3/4}\\omega _{0}^{3/2}(1+\\kappa ^{2})}.",
"$ Note that, if one defines a complex variable $\\Psi =Qe^{i\\Phi }$ , our system is further reduced to a single complex ODE $i\\Psi _{\\tau }+(\\tau -|\\Psi |^{2})\\Psi =\\mu $ characteristic to AR problems in many different physical systems and studied in numerous application [5].",
"For example, if initially $\\Psi =0$ and one starts at sufficiently large negative $\\tau $ (i.e.",
"far from the linear resonance), this equation predicts transition to AR at large positive $\\tau $ if $\\mu $ is above the threshold $\\mu _{th}\\approx 0.41$ , or returning to our original parameters, $\\varepsilon >\\varepsilon _{th}=0.58\\frac{\\alpha ^{3/4}\\omega _{0}^{3/2}(1+\\kappa ^{2})}{N^{1.2}}.",
"$ The threshold (REF ) assumes its simplest form when either $\\Delta $ or $\\kappa $ vanish.",
"Indeed, in the cold plasma case, $\\Delta =0$ and $N=\\frac{3\\kappa ^{2}(3+\\kappa ^{2})}{32}.",
"$ Therefore, for small $\\kappa $ , $\\varepsilon _{th}$ scales as $\\varepsilon _{th}\\sim \\alpha ^{3/4}/\\kappa $ and one needs a non-vanishing screening factor $\\kappa $ to get a sufficiently small $\\varepsilon _{th}$ for the AR excitation.",
"In the case of $\\kappa =0$ and $\\Delta \\ll 1$ , $N\\approx \\frac{15\\Delta ^{2}}{8}, $ so $\\varepsilon _{th}$ scales as $\\varepsilon _{th}\\sim \\alpha ^{3/4}/\\Delta $ (recall that $\\Delta =3^{1/2}k\\lambda _{D}$ and measures the electron thermal spread in the problem).",
"We illustrate these results in Fig.",
"6 showing $\\varepsilon _{th}/\\alpha ^{3/4}$ vs $\\sigma $ (this is $k\\lambda _{D}$ in dimensional notations) for several values of $\\kappa $ .",
"Figure: (color online) The control of the autoresonant SPW by tapering the drivingfrequency.",
"Panel (a): The envelope of the maxima ofthe electron fluid velocity vv versus slow time τ=α 1/2 t\\protect \\tau =\\protect \\alpha ^{1/2}t. The dashed line represents the phase velocity ω d /k\\protect \\omega _{d}/k of the driving wave and the dotted red line represents theenvelope of the maxima of the upper limiting velocity of the water bagmodel.",
"Panel (b) shows the frequencies of the driving (red line) and driven(blue line) waves.",
"The waves are frequency locked starting τ≈-3\\protect \\tau \\approx -3.Finally, we address the possibility of autoresonant control of large amplitude SPWs.",
"This control uses a one-to-one correspondence between the amplitude and frequency of nonlinear waves.",
"Since in autoresonance the driven wave frequency is locked to the driving frequency, the wave amplitude can be controlled by simply varying the driving frequency.",
"For example, we have seen previously (see Fig.",
"3), that a continuous linear time variation of the driving frequency leads to the excited wave approaching the wave breaking limit.",
"But what if one wants to avoid this limit and excite a wave having some given target amplitude?",
"This goal can be achieved by tailoring the time variation of the driving frequency appropriately.",
"We illustrate such control in Fig.",
"7 showing the results of the simulations using the water bag model for the case of $\\sigma =0.05$ , $\\kappa =0.75$ , $\\varepsilon =0.0011$ ($\\varepsilon _{th}=0.00084$ in this case) and the driving frequency $\\omega _{d}=\\omega _{0}+\\alpha t$ for $t<0$ , but $\\omega _{d}=\\omega _{0}+\\frac{2d}{\\pi }arctg(t/T)$ for $t>0$ , where $T=$ $\\frac{2d}{\\pi \\alpha }$ .",
"This driving frequency approaches a fixed value of $\\omega _{0}+d$ at large times (we used $d=0.033$ in the simulations).",
"Fig.",
"7a shows the envelopes of the maxima of the electron fluid velocity $v$ (solid line) and of the upper limiting velocity $u_{1}$ of the water bag model (dotted line).",
"In the same figure, we also show the driving wave phase velocity (dashed line).",
"Separate simulations show that if the driving frequency in this example is chirped linearly in time, the upper limiting velocity of the water bag approaches the Landau resonance ($u_{1}\\approx \\omega _{d}/k$ ) at $\\tau =6$ , where wave breaking is expected.",
"In contrast, for the saturated driving frequency case shown in Fig.",
"7, the excited fluid velocity also saturates at larger times and performs slow oscillations around some fixed value.",
"The frequency of these oscillations scales as $\\varepsilon ^{1/2}$ [5] and their presence illustrates stability of the AR evolution.",
"The velocity $u_{1}$ also saturates and, thus, the kinetic wave breaking is avoided.",
"This saturation is caused by the autoresonant frequency locking in the system as illustrated in Fig.",
"7b, showing the evolution of the frequencies of the driven and driving waves."
],
[
"Conclusions",
"We have studied excitation and control of large amplitude SPWs by a chirped frequency driving wave.",
"The process involved passage through the linear resonance in the problem and transition to autoresonant stage of excitation, where the driven SPW self-adjusts its amplitude to stay in a continuous resonance with the drive.",
"The method allowed reaching extreme regimes, where the electron density developed a sharply peaked spatial profile with the maximum electron density exceeding the initial plasma density significantly (see Fig.",
"1).",
"These results were illustrated in both Vlasov-Poisson and water bag simulations.",
"The simpler water bag model [Eqs.",
"()] was used for developing the adiabatic theory using the Whitham's averaged variational principle for studying the weakly nonlinear stage of formation of autoresonant SPWs.",
"In this regime, the problem was reduced to the standard set of coupled amplitude-phase mismatch equations (REF ) and () characteristic to many other autoresonantly driven problems.",
"The reduction also allowed finding the threshold driving amplitude [see Eq.",
"(REF )] for the transition to autoresonance.",
"By slowly decreasing the chirp rate of the driving frequency and reaching some fixed frequency level, one could arrive at a given target amplitude of the autoresonant SPW and avoid the kinetic wave breaking (see Fig.",
"7).",
"The self-consistent inclusion of the variation of the driving amplitude and of 3D effects in the process of autoresonant excitation of SPWs seem to be an important goals for future research.",
"The form of the autoresonant SPW suggests that it comprises a two-phase solution, with each phase locked to one of the traveling waves comprising the drive.",
"A better understanding of such waveforms, analyzing other autoresonant multi-phase plasma waves, and studying details of the kinetic wave breaking process in application to autoresonant SPWs also comprise interesting goals for the future.",
"This work was supported by the US-Israel Binational Science Foundation grant No.",
"6079 and the Russian state program AAAA-A18- 118020190095-4.",
"The authors are also grateful to J.S.",
"Wurtele, P. Michel, and G. Marcus for helpful comments and suggestions."
],
[
"Second order amplitudes",
"For calculating the second order amplitudes, we substitute Eqs.",
"(REF ) and () into the Lagrangian (REF ), write the result to second nonlinear order in amplitudes and average over one spatial period.",
"This spatially averaged Lagrangian governs the time dependence of all the amplitudes.",
"Its variations with respect to the second order amplitudes yield a system of ODEs for these amplitudes versus time.",
"Algebraically, this reduction is a tedious process done here using Mathematica [19].",
"The resulting set of equations is $(e_{1,2})_{t}=\\frac{1}{2}(a_{1,2}b_{1,2}\\pm 4\\Delta d_{1,2}), \\nonumber \\\\(d_{1,2})_{t}=\\frac{1}{4}(4c_{2}-a_{1,2}^{2}+b_{1,2}^{2}\\mp 8\\Delta e_{1,2}), \\\\c_{2}=\\frac{e_{2}-e_{1}}{\\Delta (4+\\kappa ^{2})}, \\nonumber $ or after substituting the first order dependencies $a_{1,2}\\sim \\cos \\theta $ , $b_{1,2}\\sim \\sin \\theta $ , and $c_{1}\\sim \\cos \\theta $ $(e_{1,2})_{t}=p_{1,2}\\sin (2\\theta )\\pm 2\\Delta d_{1,2}, \\nonumber \\\\(d_{1,2})_{t}=q_{1,2}+r_{1,2}\\cos (2\\theta )+c_{2}\\mp 2\\Delta e_{1,2}, \\\\c_{2}=\\frac{e_{2}-e_{1}}{\\Delta (4+\\kappa ^{2})}, \\nonumber $ where $p_{1,2}$ , $q_{1,2}$ , $r_{1,2}$ are constants.",
"These equations have the following time periodic solutions $d_{1,2}=D_{1,2}\\sin (2\\theta ), \\nonumber \\\\e_{1,2}=F_{1,2}+E_{1,2}\\cos (2\\theta ), \\\\c_{2}=B+C\\cos (2\\theta ).",
"\\nonumber $ where $D_{1,2}$ , $F_{1,2}$ , $E_{1,2}$ , $B$ , and $C$ are constants."
],
[
"The reduction of the variational system",
"All algebraic manipulations in this Appendix use Mathematica [19].",
"We employ the averaged Lagrangian $\\Lambda $ [see Eqs.",
"(REF )-(REF )] and proceed from the variations with respect to the first order amplitudes $A_{1,2}$ , $B_{1,2}$ , $C_{1}$ : $-2C_{1}+2A_{1}F_{1}+B_{1}D_{1}+A_{1}E_{1}+2B_{1}\\omega +2A_{1}\\Delta =4\\varepsilon \\cos \\Phi , \\nonumber \\\\-2C_{1}+2A_{2}F_{2}+B_{2}D_{2}+A_{2}E_{2}+2B_{2}\\omega -2A_{2}\\Delta =4\\varepsilon \\cos \\Phi , \\nonumber \\\\2F_{1}B_{1}-A_{1}(D_{1}+2\\omega )-B_{1}(E_{1}+2\\Delta )=0, \\\\-2F_{2}B_{2}+A_{2}(D_{2}+2\\omega )+B_{2}(E_{2}-2\\Delta )=0, \\nonumber \\\\A_{1}-A_{2}+2C_{1}(1+\\kappa ^{2})\\Delta =0.",
"\\nonumber $ The first four equations in this set yield the linear approximation in terms of $C_{1}$ $A_{1}^{\\prime }=-A_{2}^{\\prime }=-\\frac{C_{1}\\Delta }{\\omega ^{2}-\\Delta ^{2}}, \\\\B_{1}^{\\prime }=B_{2}^{\\prime }=\\frac{C_{1}\\omega }{\\omega ^{2}-\\Delta ^{2}}.$ Then the fifth equation in Eq.",
"(REF ) gives the linear dispersion relation $\\omega _{0}^{2}=\\frac{1}{1+\\kappa ^{2}}+\\Delta ^{2}.",
"$ Next, we take variations with respect to the second order amplitudes $D_{1,2} $ , $E_{1,2}$ , $F_{1,2}$ , $B$ , $C$ (to lowest significant order using the linear result for the first order amplitudes and $\\omega $ ) to get $A_{1}^{\\prime }B_{1}^{\\prime }+8E_{1}\\omega _{0}+8D_{1}\\Delta =0, \\nonumber \\\\A_{2}^{\\prime }B_{2}^{\\prime }+8E_{2}\\omega _{0}-8D_{2}\\Delta =0, \\nonumber \\\\-8C0+A_{1}^{\\prime 2}+B_{1}^{\\prime 2}+16D_{1}\\omega _{0}+16E_{1}\\Delta =0,\\nonumber \\\\-8C0+A_{2}^{\\prime 2}+B_{2}^{\\prime 2}+16D_{2}\\omega _{0}-16E_{2}\\Delta =0, \\\\8B-A_{1}^{\\prime 2}+B_{1}^{\\prime }-16F_{1}\\Delta =0, \\nonumber \\\\8B-A_{2}^{\\prime 2}+B_{2}^{\\prime 2}+16F_{2}\\Delta =0, \\nonumber \\\\F_{1}-F_{2}+B(4+\\kappa ^{2})\\Delta =0, \\nonumber \\\\E_{1}-E_{2}+C(4+\\kappa ^{2})\\Delta =0, \\nonumber $ This system is now solved for the second order amplitudes via $C_{1}^{2}$ : $D_{1}=\\frac{C_{1}^{2}\\omega _{0}[2+\\omega _{0}^{2}(4+\\kappa ^{2})+3\\Delta ^{2}(4+\\kappa ^{2})]}{16(\\omega _{0}^{2}-\\Delta ^{2})^{2}[1-\\omega _{0}^{2}(4+\\kappa ^{2})+\\Delta ^{2}(4+\\kappa ^{2})]}, \\nonumber \\\\E_{1}=-\\frac{C_{1}^{2}\\Delta [3\\omega _{0}^{2}(4+\\kappa ^{2})+\\Delta ^{2}(4+\\kappa ^{2})]}{16(\\omega _{0}^{2}-\\Delta ^{2})^{2}[1-\\omega _{0}^{2}(4+\\kappa ^{2})+\\Delta ^{2}(4+\\kappa ^{2})]}, \\nonumber \\\\F_{1}=\\frac{C_{1}^{2}(4+\\kappa ^{2})\\Delta }{16(\\omega _{0}^{2}-\\Delta ^{2})[1+\\Delta ^{2}(4+\\kappa ^{2})]}, \\\\D_{2}=D_{1}, E_{2}=-E_{1}, F_{2}=-F_{1}, \\nonumber \\\\B=-\\frac{C_{1}^{2}}{8(\\omega _{0}^{2}-\\Delta ^{2})[1+\\Delta ^{2}(4+\\kappa ^{2})]}, \\nonumber \\\\C=\\frac{C_{1}^{2}(3\\omega _{0}^{2}+\\Delta ^{2})}{8(\\omega _{0}^{2}-\\Delta ^{2})^{2}[1-\\omega _{0}^{2}(4+\\kappa ^{2})+-\\Delta ^{2}(4+\\kappa ^{2})]}.\\nonumber $ Finally, we return to the first two equations in (REF ) and solve these equations for $A_{1,2}$ to higher (third) order in $C_{1}$ $A_{1}^{\\prime \\prime }=-\\frac{C_{1}\\Delta }{\\omega ^{2}-\\Delta ^{2}}+\\frac{C_{1}G_{1}-4\\varepsilon \\Delta (\\omega _{0}^{2}-\\Delta ^{2})\\cos \\Phi }{2(\\omega _{0}^{2}-\\Delta ^{2})^{2}}, $ $A_{2}^{\\prime \\prime }=\\frac{C_{1}\\Delta }{\\omega ^{2}-\\Delta ^{2}}+\\frac{C_{1}G_{2}+4\\varepsilon \\Delta (\\omega _{0}^{2}-\\Delta ^{2})\\cos \\Phi }{2(\\omega _{0}^{2}-\\Delta ^{2})^{2}}, $ where $G_{1}=\\omega _{0}^{2}(2F_{1}-E_{1})+2D_{1}\\omega _{0}\\Delta -\\Delta ^{2}(2F_{1}+E_{1})$ and $G_{2}=\\omega _{0}^{2}(2F_{2}-E_{2})-2D_{2}\\omega _{0}\\Delta -\\Delta ^{2}(2F_{2}+E_{2})$ .",
"Finally, we use Eqs.",
"(REF ) in Eqs.",
"(REF ) and (REF ) and substitute the resulting $A_{1,2}^{\\prime \\prime }$ into the last equation in (REF ).",
"This yields a higher order approximation for the frequency $\\omega $ of the wave: $\\omega ^{2}=\\omega _{0}^{2}+NC_{1}^{2}+\\frac{2\\varepsilon \\cos \\Phi }{(1+\\kappa ^{2})C_{1}}, $ where $N=\\frac{9\\kappa ^{2}(3+\\kappa ^{2})+R}{96[1+(4+\\kappa ^{2})\\Delta ^{2}]},$ and $R=(1+\\kappa ^{2})\\Delta ^{2}[3(60+56\\kappa ^{2}+11\\kappa ^{4})+20(1+\\kappa ^{2})(4+\\kappa ^{2})(5+2\\kappa ^{2})\\Delta ^{2}+16(1+\\kappa ^{2})^{2}(4+\\kappa ^{2})^{2}\\Delta ^{4}]$ ."
]
] | 2001.03364 |
[
[
"VERITAS Detection of LS 5039 and HESS J1825-137"
],
[
"Abstract With 8 hours of observations, VERITAS confirms the detection of two very high energy gamma-ray sources.",
"The gamma-ray binary LS 5039 is detected with a statistical significance of $8.8\\sigma$.",
"The measured flux above 1 TeV is $(2.5 \\pm 0.4) \\times 10^{-12} \\rm \\, cm^{-2} \\, s^{-1}$ near inferior conjunction and $(7.8 \\pm 2.8) \\times 10^{-13} \\rm \\, cm^{-2} \\, s^{-1}$ near superior conjunction.",
"The pulsar wind nebula HESS J1825-137 is detected with a statistical significance of $6.7\\sigma$ and a measured flux above 1 TeV of $(3.9 \\pm 0.8) \\times 10^{-12} \\rm \\, cm^{-2} \\, s^{-1}$."
],
[
"Introduction",
"We report observations of two known, bright, very-high-energy (VHE; E $>$ 50 GeV) gamma-ray emitting objects: the gamma-ray binary LS 5039 and the very extended pulsar wind nebula (PWN) HESS J1825-137.",
"These two objects are separated by about $1^\\circ $ on the sky and can both be observed in a single field of view with the Very Energetic Radiation Imaging Telescope Array System (VERITAS)http://veritas.sao.arizona.edu.",
"The observations are described in Section .",
"Background, spectra, timing, and morphological studies are described for each source in Section ."
],
[
"Observations",
"VERITAS is a four-telescope array exploiting the imaging atmospheric Cherenkov technique to detect VHE gamma rays and is most sensitive to energies between 0.085–30 TeV.",
"The array is located at the Fred Lawrence Whipple Observatory (FLWO) in southern Arizona ($\\rm 31^\\circ 40^{\\prime } N, 110^\\circ 57^{\\prime } W,$ 1.3 km above sea level).",
"For a performance summary, see [1].",
"VERITAS observed the field in late spring of 2013 and 2014 with an average zenith angle of $48^\\circ $ .",
"Standard analysis was performed [2] with background extracted from regions near the sources excluding a $0.35^\\circ $ radius region around LS 5039 and a $0.75^\\circ $ radius region around HESS J1825-137.",
"With 7.9 hours of usable observations after quality cuts, we confirm the HESS detections of these sources [3], [4].",
"Figure REF shows the VERITAS skymap made with an integration region radius of $\\theta = 0.089^\\circ $ .",
"The average low-energy threshold for VERITAS observations is 420 GeV due to the relatively large zenith angle of the observations."
],
[
"LS 5039",
"LS 5039 was identified as a high mass X-ray binary in the ROSAT Galactic plane survey [5] and radio emission was detected using the VLA [6].",
"It was also detected as a gamma-ray binary with EGRET [7] and at VHE with HESS [3].",
"Periodic emission is also detected by Fermi-LAT above 300 MeV [8].",
"The companion star is spectral type O6.5V [9], while the mass and nature of the compact object is undetermined [10].",
"However, some evidence exists for a neutron star being present in the system.",
"Orbital modulation in TeV gamma-rays peaks when the star and compact object are aligned relative to the observer which can be explained by the outward flow of relativistic particles from a PWN boosting emission from the star [11].",
"We detect VER J1826-148, the VHE source associated with LS 5039, at a significance of 8.8 standard deviations ($\\sigma $ ) [12] with 101 excess counts within a region of radius $\\theta = 0.089^\\circ $ centered at the known position from HESS $(\\alpha ,\\delta ) = (276.563^\\circ ,-14.848^\\circ )$ (J2000) [13].",
"Each observation was a single $\\sim 20-30$ minute exposure, and consecutive exposures were separated by 1-2 nights.",
"This provided approximately daily coverage of the 3.9 day orbital period [13] for 8 days in 2013 and 12 days in 2014.",
"Spectra separated into two orbital phase intervals, see Fig.",
"REF , agree within the uncertainties of those measured with HESS [13].",
"The flux above 1 TeV is $\\rm (2.5 \\pm 0.4_{stat} \\pm 0.5_{sys}) \\times 10^{-12} \\rm \\, cm^{-2} \\, s^{-1}$ near inferior conjunction (orbital phase $0.45< \\phi \\le 0.9$ , 232 minutes exposure) and $\\rm (7.8 \\pm 2.8_{stat} \\pm 1.6_{sys}) \\times 10^{-13} \\rm \\, cm^{-2} \\, s^{-1}$ near superior conjunction ($0.0 \\le \\phi \\le 0.45$ and $0.9 < \\phi \\le 1.0$ , 230 minutes).",
"This difference in flux demonstrates that the previously observed orbital flux variability [3] is detected by VERITAS as well.",
"Fitting a power law to the spectrum for each orbital phase interval gives a photon index near inferior conjunction of $\\rm \\Gamma =2.1 \\pm 0.2_{stat} \\pm 0.2_{sys}$ with $\\chi ^2 /\\rm DoF =0.19/1$ and $\\rm \\Gamma =2.4 \\pm 0.5_{stat} \\pm 0.2_{sys}$ with $\\chi ^2 /\\rm DoF =0.57/1$ near superior conjunction.",
"The VERITAS flux points match the best fitted spectral models from HESS with $\\chi ^2 /\\rm DoF =1.60/3$ and $\\chi ^2 /\\rm DoF =0.55/3$ for the phase intervals near inferior and superior conjunction, respectively, and support the PWN interpretation [11]."
],
[
"HESS J1825-137",
"The first indication of a PWN in this region was due to ROSAT observations of an extended ($\\sim 5^{\\prime }$ ) X-ray nebula associated with the young (21 kyr) and energetic ($\\dot{E} = 2.8\\ \\times 10^{36} \\rm \\, ergs \\, s^{-1}$ ) pulsar PSR B1823-13 [14].",
"HESS J1825-137 has a radius at least 5 times larger in VHE than X-ray, and the larger extent is consistent with the longer lifetime of TeV emitting electrons [4].",
"The large angular size of HESS J1825-137 makes it ideal for resolved VHE study of PWN to better understand the particle transport within[15].",
"Using the same observations and analysis described for LS 5039, VERITAS detected HESS J1825-137, at the center of the PWN reported by HESS $(\\alpha ,\\ \\delta ) = (276.421^\\circ $ , $-13.839^\\circ )$ (J2000) [15] with a significance of $5.3\\sigma $ ; the peak significance of $6.7\\sigma $ within a $0.4^\\circ $ radius region.",
"Fitting a spectrum extracted from this region to a power law model gave a normalization at 1 TeV of $\\rm A = (5.0 \\pm 0.7_{stat} \\pm 1.0_{sys}) \\times 10^{-12} \\rm \\, cm^{-2} \\, s^{-1} \\, TeV^{-1}$ , a photon index $\\Gamma =2.28 \\pm 0.15_{stat} \\pm 0.2_{sys}$ , and $\\chi ^2 /\\rm DoF = 2.27/1$ .",
"We found the PWN (named VER J1825-138) to be centered at $(\\alpha ,\\ \\delta ) = \\rm (276.37^\\circ \\pm 0.02^\\circ _{stat} \\pm 0.01^\\circ _{sys}, -13.83^\\circ \\pm 0.02^\\circ _{stat} \\pm 0.01^\\circ _{sys})$ (J2000) using a weighted centroid.",
"The region of interest, HESS centroid and VERITAS centroid are marked in Fig.",
"REF .",
"Summing in azimuth around the PWN, the radial profile is defined by a Gaussian with $\\sigma = 0.27^\\circ $ (1$\\sigma $ confidence interval of $\\ 0.22^\\circ - 0.28^\\circ $ ).",
"HESS reports a power-law fit from a $0.4^\\circ $ region of $\\rm A = (6.81 \\pm 0.07_{stat} \\pm 0.2_{sys}) \\times 10^{-12} \\rm \\, cm^{-2} \\, s^{-1} \\, TeV^{-1}$ with a photon index $\\Gamma =2.28 \\pm 0.01_{stat} \\pm 0.02_{sys}$ [15].",
"The VERITAS flux normalization is consistent with that found by HESS.",
"Similarly, the position of the PWN measured with VERITAS is within $0.04^\\circ $ of that detected by HESS.",
"The radial extent of the nebula measured by VERITAS is consistent with the original HESS observations of $\\sigma = 0.24^\\circ \\pm 0.02^\\circ $ with a 52hr exposure [16], but the exceptionally deep observations from HESS [15] yields a different, larger, energy dependent extent."
],
[
"Conclusion",
"VERITAS has confirmed the TeV gamma-ray detections of LS 5039 and HESS J1825-137 and analysis agrees with previous findings about the energetics and morphology."
],
[
"Acknowledgements",
"This research is supported by grants from the U.S. Department of Energy Office of Science, the U.S. National Science Foundation and the Smithsonian Institution, and by NSERC in Canada.",
"This research used resources provided by the Open Science Grid, which is supported by the National Science Foundation and the U.S. Department of Energy's Office of Science, and resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No.",
"DE-AC02-05CH11231.",
"We acknowledge the excellent work of the technical support staff at the Fred Lawrence Whipple Observatory and at the collaborating institutions in the construction and operation of the instrument.",
"Figure: Excess counts skymap of the region containing LS 5039 and HESS J1825-137.",
"LS 5039 is marked with a star and HESS J1825-137 is enclosed within a 0.4 ∘ 0.4^\\circ radius circle.",
"A cross and x respectively mark the VERITAS and HESS centroids of the PWN.Figure: Spectra of LS 5039 separated by orbital phase.",
"The filled circles and dashed line are at inferior conjunction.",
"The x's and solid lines are at superior conjunction.",
"Points are from this analysis and the fit spectra with shaded errors are from ."
]
] | 2001.03522 |
[
[
"Spectral gap and exponential mixing on geometrically finite hyperbolic\n manifolds"
],
[
"Abstract Let $\\mathcal{M}=\\Gamma\\backslash\\mathbb{H}^{d+1}$ be a geometrically finite hyperbolic manifold with critical exponent exceeding $d/2$.",
"We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in $L^2(\\mathrm{T}^1(\\mathcal{M}))$, with exponential error term essentially as good as the one given by the spectral gap for the Laplace operator on $L^2(\\mathcal{M})$ due to Lax and Phillips.",
"Combined with the work of Bourgain, Gamburd, and Sarnak and its generalization by Golsefidy and Varju on expanders, this implies uniform exponential mixing for congruence covers of $\\mathcal{M}$ when $\\Gamma$ is a Zariski dense subgroup contained in an arithmetic subgroup of $\\mathrm{SO}^{\\circ}(d+1,1)$."
],
[
"Introduction",
"Let ${\\mathcal {M}}=\\Gamma \\backslash \\mathbb {H}^{d+1}$ be a geometrically finite hyperbolic manifold, where $\\Gamma $ is a discrete subgroup of $G=\\operatorname{SO}^\\circ (d+1,1)$ .",
"We suppose that the critical exponent $\\delta $ of $\\Gamma $ is strictly bigger than $d/2$ .",
"Denoting by $\\Delta $ the negative of the Laplace operator on ${\\mathcal {M}}$ , the bottom eigenvalue of $\\Delta $ on $L^2({\\mathcal {M}})$ is known to be simple and given as $\\lambda _0=\\delta (d-\\delta )$ by Patterson [30] and Sullivan [40].",
"Lax and Phillips [24] showed that there are only finitely many eigenvalues of $\\Delta $ on $L^2({\\mathcal {M}})$ in the interval $[\\lambda _0,\\frac{d^2}{4})$ .",
"We denote by $\\lambda _1$ the smallest eigenvalue of $\\Delta $ in $(\\lambda _0, \\frac{d^2}{4})$ , and define $s_1=\\frac{d}{2}+\\sqrt{(\\frac{d}{2})^2-\\lambda _1}$ .",
"Then $s_1$ is the unique number in $(\\frac{d}{2},\\delta )$ such that $0<\\delta (d-\\delta )=\\lambda _0< \\lambda _1=s_1 (d-s_1)< d^2/4.",
"$ If there is no eigenvalue in the open interval $(\\lambda _0, \\frac{d^2}{4})$ , we set $\\lambda _1=d^2/4$ , i.e., $s_1=d/2$ .",
"Denote by $\\mathcal {G}^t$ the geodesic flow on the unit tangent bundle $\\operatorname{T{}}^1({\\mathcal {M}})$ and by $dx$ the Liouville measure on $\\operatorname{T{}}^1({\\mathcal {M}})$ ; this is a $\\mathcal {G}^t$ -invariant Borel measure which is infinite when $\\delta <d$ .",
"The main result of this paper is that the Lax-Philips spectral gap, that is, $\\lambda _1-\\lambda _0$ , or equivalently $\\delta -s_1$ , controls rather precisely the exponential convergence rate of the correlation function for the geodesic flow $\\mathcal {G}^t$ acting on $L^2(\\operatorname{T{}}^1({\\mathcal {M}}), dx)$ : Theorem 1.1 Let $\\delta >d/2$ , and set $\\eta :=\\min \\lbrace \\delta -s_1, 1\\rbrace $ .",
"There exists $m>d(d+1)/2$ such that for any $\\varepsilon >0$ and functions $\\psi _1, \\psi _2$ on $\\operatorname{T{}}^1({\\mathcal {M}})$ with ${\\mathcal {S}}^{m} (\\psi _1),{\\mathcal {S}}^{m} (\\psi _2)<\\infty $ , we have, as $t\\rightarrow +\\infty $ , $e^{(d-\\delta )t}\\int _{\\operatorname{T{}}^1({\\mathcal {M}})}\\psi _1\\big ({\\mathcal {G}}^t(x)\\big )\\psi _2(x)\\,dx =& \\frac{1}{m^{\\operatorname{BMS}}(\\operatorname{T{}}^1({\\mathcal {M}}))} m^{\\operatorname{BR}}(\\psi _1)m^{\\operatorname{BR}_*}(\\psi _2)\\\\&+ O_{\\varepsilon }\\left( e^{(-\\eta +\\varepsilon ) t}{\\mathcal {S}}^m(\\psi _1){\\mathcal {S}}^m(\\psi _2)\\right),$ where $m^{\\operatorname{BMS}}$ is the Bowen-Margulis-Sullivan measure on $\\operatorname{T{}}^1({\\mathcal {M}})$ ; $m^{\\operatorname{BR}}$ and $m^{\\operatorname{BR}_*}$ are, respectively, the unstable and stable Burger-Roblin measures on $\\operatorname{T{}}^1({\\mathcal {M}})$ , which are defined compatibly with the choice of $dx$ and $dm^{\\operatorname{BMS}}$ ; ${\\mathcal {S}}^m(\\psi _i)$ denotes the $L^2$ -Sobolev norm of $\\psi _i$ of degree $m$ and the implied constant depends only on $\\varepsilon $ .",
"We note that $|m^{\\operatorname{BR}}(\\psi _i)|<\\infty $ when ${\\mathcal {S}}^{m} (\\psi _i)<\\infty $ (see Lemma REF ), and hence the main term above is well-defined.",
"For $\\psi _i$ compactly supported, the asymptotic formula (without error term) holds for any $\\delta >0$ , as was obtained by Roblin [33].",
"Remark 1.1 When ${\\mathcal {M}}$ has finite volume, i.e., when $\\delta =d$ , exponential mixing of the geodesic flow is a classical result due to Ratner for $d=1$ and to Moore [28] for general $d\\ge 1$ .",
"Combining Moore's proof with Shalom's trick [37] yields a rate of mixing $\\eta =\\min \\lbrace d-s_1,1\\rbrace $ as in Theorem REF above.",
"This proof makes explicit use of the fact [16] that there are no non-spherical complementary series representations ${\\mathcal {U}}(\\upsilon ,s)$ with $s>d-1$ (see below for notation).",
"Our proof does not require this statement.",
"However, combining the aforementioned fact with our proof improves the rate of mixing for the geodesic flow on finite-volume hyperbolic manifolds to $\\eta =\\min \\lbrace d-s_1, 2\\rbrace $ .",
"In addition, if there is no non-spherical complementary series representation ${\\mathcal {U}}(\\upsilon ,s)$ with $s>s_1+1$ appearing in $L^2(\\Gamma \\backslash G)$ , then $\\eta $ can be taken to be $d-s_1$ (see Remark REF for details).",
"For $d=1$ and 2, Theorem REF can be deduced from [7] and [43], respectively.",
"For a general $d\\ge 1$ , the case of $\\delta >\\max \\lbrace d-1, d/2\\rbrace $ was obtained in [27] for some $\\eta >0$ which is not explicit.",
"The main novelties of Theorem REF are that it addresses all geometrically finite groups with $\\delta >d/2$ (even those with cusps), and that it gives an optimal value of $\\eta $ with respect to the dependency on the spectral gap $\\delta -s_1$ .",
"The order $m$ of Sobolev norm required is in principle completely computable; it may be chosen independently of $\\Gamma $ , and satisfies $m=O(d^2)$ .",
"The BMS measure $m^{\\operatorname{BMS}}$ is known to be the unique measure of maximal entropy (which is $\\delta $ ) for the geodesic flow ([40], [31]).",
"Babillot showed that the geodesic flow is mixing with respect to $m^{\\operatorname{BMS}}$ [2].",
"Theorem REF is known to imply the following exponential mixing for the BMS measure (see [27] for compactly supported functions, and [18] for general bounded functions): Theorem 1.2 There exist $\\beta >0$ (explicitly computable, depending only on $\\eta $ in Theorem REF ) and $m> d(d+1)/2$ such that for all bounded functions $\\psi _1, \\psi _2 $ on $\\operatorname{T{}}^1({\\mathcal {M}})$ supported on the one-neighbourhood of $\\mathrm {supp}({m^{\\mathrm {BMS}}})$ , we have, as $t\\rightarrow \\infty $ , $\\int _{\\operatorname{T{}}^1({\\mathcal {M}})}\\psi _1\\big ({\\mathcal {G}}^t(x)\\big )\\psi _2(x)\\,d{m^{\\mathrm {BMS}}}(x)&= \\frac{1}{m^{\\operatorname{BMS}}(\\operatorname{T{}}^1({\\mathcal {M}}))} {m^{\\mathrm {BMS}}}(\\psi _1){m^{\\mathrm {BMS}}}(\\psi _2)\\\\&+ O\\left( e^{-\\beta t}\\Vert \\psi _1\\Vert _{C^m} \\Vert \\psi _2\\Vert _{C^m}\\right)$ where $\\Vert \\psi _i\\Vert _{C^m}$ denotes the $C^m$ -norm of $\\psi _i$ .",
"Remark 1.2 When $\\Gamma $ is convex cocompact, Theorem REF , for some $\\beta >0$ (which is not explicit), follows from the work of Stoyanov [38], which is based on symbolic dynamics and Dolgopyat operators (see also [35]).",
"This result in turn implies Theorem REF with implicit $\\eta >0$ (see [29], [35]).",
"Let $\\Gamma $ be a Zariski dense subgroup of an arithmetic subgroup $G(\\mathbb {Z})$ of $G$ .",
"Denote by $\\Gamma _q$ the congruence subgroup of $\\Gamma $ of level $q$ : $\\Gamma _q=\\lbrace \\gamma \\in \\Gamma : \\gamma =e\\text{ mod $q$}\\rbrace $ .",
"When $\\pi ^{-1}(\\Gamma )$ satisfies the strong approximation property for the spin covering map $\\pi :\\operatorname{Spin}(d+1,1)\\rightarrow G$ , the work of Bourgain-Gamburd-Sarnak [6] and its generalization by Golsefidy-Varju [41] on expanders imply that there exists a finite set $S$ of primes such that the family $\\mathcal {F}:=\\lbrace \\Gamma _q: \\text{$q$ is square-free with no factors in $S$}\\rbrace $ has a uniform spectral gap in the sense that $ \\inf _{\\Gamma _q\\in \\mathcal {F}} \\lbrace \\delta -s_1(q)\\rbrace >0,$ where $s_1(q) (d-s_1(q))$ is the second smallest eigenvalue of the Laplacian $\\Delta $ on $L^2(\\Gamma _q\\backslash \\mathbb {H}^{d+1})$ In view of the recent preprint [15] and an upcoming work by He and de Saxcé, the square-free condition in $\\mathcal {F}$ may be removed..",
"This uses the transfer property from the combinatorial spectral gap to the archimedean spectral gap due to [5] (see also [19]).",
"For certain families of subgroups the uniform spectral gap (REF ) may be explicitly bounded from below, see [8], [9], [13], [25], [36].",
"We then have: Corollary 1.3 In Theorems REF and REF , the exponents $\\eta $ and $\\beta $ can be chosen uniformly over all congruence covers $\\operatorname{T{}}^1(\\Gamma _q\\backslash \\mathbb {H}^{d+1})$ , $\\Gamma _q\\in \\mathcal {F}$ .",
"When $\\Gamma $ is convex cocompact, Corollary REF was obtained by the second-named author and Winter [29] for $d=1$ and by Sarkar [34] for a general $d\\ge 1$ , combining Dolgopyat's methods with the expander theory ([6], [41]).",
"Theorems REF -REF and Corollary REF are known to have many immediate applications in number theory and geometry.",
"To name a few, see ([11], [12], [3], [5], [27]) for effective counting and affine sieve, [26] for the prime geodesic theorem, and [18] for shrinking target problems.",
"Fix $o\\in \\mathbb {H}^{d+1}$ and $v_o\\in \\operatorname{T{}}^1(\\mathbb {H}^{d+1})$ based at $o$ .",
"Setting $K=\\operatorname{Stab}_G(o)\\simeq \\operatorname{SO}(d+1)$ and $M=\\operatorname{Stab}_G(v_o)\\simeq \\operatorname{SO}(d)$ , we can identify ${\\mathcal {M}}=\\Gamma \\backslash G/K$ and $\\operatorname{T{}}^1({\\mathcal {M}})=\\Gamma \\backslash G/M$ .",
"We let $\\lbrace a_t\\rbrace $ denote the one-parameter diagonalizable subgroup of $G$ commuting with $M$ whose right translation on $\\Gamma \\backslash G/M$ corresponds to the geodesic flow $\\mathcal {G}^t$ on $\\operatorname{T{}}^1({\\mathcal {M}})$ .",
"As $L^2(\\operatorname{T{}}^1({\\mathcal {M}}))$ can be identified with the space of $M$ -invariant functions in $L^2(\\Gamma \\backslash G)$ , Theorem REF amounts to understanding the asymptotic behaviour of the matrix coefficients $\\langle a_t \\psi _1, \\psi _2\\rangle $ for $M$ -invariant functions $\\psi _1, \\psi _2\\in L^2(\\Gamma \\backslash G)$ .",
"The non-tempered part of the unitary dual $\\hat{G}$ consists of the trivial representation and the complementary series representations ${\\mathcal {U}}(\\upsilon ,s)$ , parameterized by a representation $\\upsilon $ in the unitary dual $ \\hat{M}$ and a real number $s\\in {\\mathcal {I}}_{\\upsilon }$ , where ${\\mathcal {I}}_{\\upsilon }\\subset (d/2, d)$ is an interval depending on $\\upsilon $ .",
"In this parameterization, the complementary series representation ${\\mathcal {U}}(\\upsilon ,s)$ is spherical if and only if $\\upsilon \\in \\hat{M}$ is trivial, and for $\\upsilon $ non-trivial, ${\\mathcal {I}}_{\\upsilon }$ is contained in $(d/2, d-1)$ [16]; this was the main reason for the hypothesis $\\delta >d-1$ in [27].",
"Our main work for the proof of Theorem REF lies in the detailed analysis of the behavior of matrix coefficients of the complementary series representations ${\\mathcal {U}}(\\upsilon ,s)$ .",
"We remark that Harish-Chandra's work ([46], [47]) does not give an asymptotic expansion for the matrix coefficients of ${\\mathcal {U}}(\\upsilon ,s)$ for all $s$ , as it excludes finitely many (unknown) parameters $s$ .",
"Even for those ${\\mathcal {U}}(\\upsilon , s)$ for which Harish-Chandra's expansion is given, it is hard to use his expansion directly, as it relies on various parameters and recursive formulas.",
"Denoting by $\\langle \\cdot , \\cdot \\rangle _{{\\mathcal {U}}(\\upsilon ,s)}$ the unitary inner product, for each $m\\in {\\mathbb {N}}$ we define a Sobolev norm $\\Vert \\cdot \\Vert _{ {{\\mathcal {S}}^m(\\upsilon ,s)}}$ on ${\\mathcal {U}}(\\upsilon ,s)$ as in (REF ), and let ${\\mathcal {S}}^m(\\upsilon ,s)$ denote the space of vectors in ${\\mathcal {U}}(\\upsilon ,s)$ with finite $\\Vert \\cdot \\Vert _{ {{\\mathcal {S}}^m(\\upsilon ,s)}}$ -norm.",
"Write ${\\mathcal {U}}(\\upsilon ,s)=\\oplus _{\\tau \\in \\hat{K}} \\; {\\mathcal {U}}(\\upsilon ,s)_\\tau $ for the decomposition into different $K$ -types and for $d/2<s < d$ , set $\\eta _s:=\\min \\lbrace 2s-d, 1\\rbrace >0.$ We show the following concrete asymptotic expansion of matrix coefficients with an optimal rate.",
"Theorem 1.4 There exists $m\\in {\\mathbb {N}}$ such that for any complementary series representation ${\\mathcal {U}}(\\upsilon ,s)$ containing a non-trivial $M$ -invariant vector, for all ${\\text{$u$}},{\\text{$v$}}\\in {\\mathcal {S}}^m(\\upsilon ,s)$ and $t\\ge 0$ , $\\langle {\\mathcal {U}}(\\upsilon , s) (a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)} =&e^{(s-d)t}\\left(\\sum _{\\tau _1,\\tau _2\\in {\\hat{K}}}\\langle T_{\\tau _1}^{\\tau _2}C_+(s)\\mathsf {P}_{\\tau _1}{\\text{$u$}},\\mathsf {P}_{\\tau _2}{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}\\right)\\\\&\\qquad \\qquad +O_s\\big (e^{(s-d-\\eta _s)t}\\Vert {\\text{$u$}}\\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)} \\big )$ and the sum $\\sum _{\\tau _1,\\tau _2\\in {\\hat{K}}}\\langle T_{\\tau _1}^{\\tau _2}C_+(s)\\mathsf {P}_{\\tau _1}{\\text{$u$}},\\mathsf {P}_{\\tau _2}{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}$ converges absolutely.",
"Here $ T_{\\tau _1}^{\\tau _2}: {\\mathcal {U}}(\\upsilon ,s)_{\\tau _1} \\rightarrow {\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}$ is given by (REF ), $C_+(s)$ is the Harish-Chandra $c$ -function given in (REF ), and $\\mathsf {P}_{\\tau }$ is the orthogonal projection onto the $K$ -type $\\tau $ .",
"Moreover, the implied constant is uniformly bounded over $s$ in compact subsets of the interval ${\\mathcal {I}}_{\\upsilon }$ .",
"One of the key observations made in this paper is that if $ {\\mathcal {U}}(\\upsilon ,s)$ is non-spherical and ${\\text{$u$}}\\in {\\mathcal {U}}(\\upsilon ,s)$ is $M$ -invariant, then the main term (REF ) vanishes (Corollary REF ): for all $\\tau _1, \\tau _2\\in \\hat{K}$ , $ T_{\\tau _1}^{\\tau _2}C_+(s)\\mathsf {P}_{\\tau _1}{\\text{$u$}}=0.$ Furthermore, there is additional uniformity with respect to the $s$ -variable when considering only $M$ -invariant vectors.",
"Corollary 1.5 There exists $m\\in {\\mathbb {N}}$ such that if ${\\mathcal {U}}(\\upsilon ,s)$ is non-spherical, then for all $M$ -invariant vectors ${\\text{$u$}},{\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)$ , $|\\langle {\\mathcal {U}}(\\upsilon , s) (a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}|\\ll _s e^{(s-d-\\eta _s)t}\\Vert {\\text{$u$}}\\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)}.$ Moreover, the implied constant is uniformly bounded over $s$ in compact subsets of $(\\frac{d}{2},d)$ .",
"Organization.",
"We start by recalling in Section some key background facts and notation.",
"In Section , we define the models of the complementary series that we will work in and the Eisenstein integrals that are key to understanding their matrix coefficients.",
"The main technical work is done in Section ; establishing the asymptotic expansion and bounds for matrix coefficients in complementary series representations.",
"Section is devoted to extending Roblin's mixing result [33] from compactly supported functions to arbitrary Sobolev functions.",
"Finally, the proof of Theorem REF is given in Section , the main part of which consists of decomposing the regular representation of $G$ on $L^2(\\Gamma \\backslash G)$ into irreducible representations and applying the bounds obtained in Section to each individual component."
],
[
"Preliminaries",
"Let $d\\ge 1$ and $G=\\operatorname{SO}^\\circ (d+1, 1)$ be the group of orientation-preserving isometries of $\\mathbb {H}^{d+1}$ .",
"Let $\\Gamma <G$ be a torsion-free discrete subgroup of $G$ , and let ${\\mathcal {M}}=\\Gamma \\backslash \\mathbb {H}^{d+1}$ be the associated hyperbolic manifold."
],
[
"Structure and subgroups of $G$ .",
"Let $K=\\operatorname{SO}(d+1) <G$ be a maximal compact subgroup, and let $A=\\lbrace a_t:t\\in \\mathbb {R}\\rbrace $ be a one-parameter diagonalizable subgroup and $M=\\operatorname{SO}(d)$ the centralizer of $A$ in $K$ .",
"We can identify ${\\mathcal {M}}$ with $\\Gamma \\backslash G/K$ and the unit tangent bundle $\\operatorname{T{}}^1({\\mathcal {M}})$ with $\\Gamma \\backslash G/M$ in the way that the geodesic flow on $\\operatorname{T{}}^1({\\mathcal {M}})$ is given by the right translation action of $a_t$ on $\\Gamma \\backslash G/M$ .",
"We denote by $N$ and $\\overline{N}$ the contracting and expanding horospherical subgroups, respectively; i.e.",
"$N=\\lbrace g\\in G: a_{-t} g a_{t}\\rightarrow e\\text{ as $t\\rightarrow +\\infty $}\\rbrace ;$ $\\overline{N}=\\lbrace g\\in G: a_{t} g a_{-t}\\rightarrow e\\text{ as $t\\rightarrow +\\infty $}\\rbrace .$ Then $\\overline{N}=\\omega N\\omega ^{-1}$ , where $\\omega \\in N_K(A)$ is such that $\\omega a \\omega ^{-1}=a^{-1}$ for all $a\\in A$ .",
"We note that $N$ and $\\overline{N}$ are both abelian subgroups, isomorphic to $\\mathbb {R}^d$ via the logarithm map, which are normalized by $AM$ .",
"Under this isomorphism $N\\simeq \\mathbb {R}^d$ , conjugation by an element $m\\in M=\\operatorname{SO}(d)$ is an isometry and conjugation by $a_t$ corresponds to scaling by $e^{t}$ on $N$ , and $e^{-t}$ on $\\overline{N}$ .",
"As usual, the Lie algebras of $G$ , $K$ , $A$ , $N$ , $\\overline{N}$ are denoted by $\\fg $ , $\\mathfrak {k}$ , $\\mathfrak {a}$ , $\\mathfrak {n}$ and $\\overline{\\mathfrak {n}}$ , respectively.",
"This gives $\\mathrm {Ad}({a_t})|_{\\mathfrak {n}}=e^t \\times \\mathrm {Id}\\quad \\text{ and } \\quad \\mathrm {Ad}({a_t})|_{\\overline{\\mathfrak {n}}}=e^{-t} \\times \\mathrm {Id}.$ We have an Iwasawa decomposition $G=KAN$ , where the product map $K\\times A\\times N\\rightarrow G$ is a diffeomorphism and the projection to each individual factor is a smooth map [21].",
"For $g\\in G$ , we write $g=\\kappa (g)\\exp (H(g)) n_g,$ where $\\kappa (g)\\in K$ , $H(g)\\in \\mathfrak {a}$ , and $n_g\\in N$ .",
"Letting $\\alpha \\in \\mathfrak {a}_{\\mathbb {C}}^*$ be the unique element such that $\\alpha (H(a_t))=t,$ the map $s\\mapsto s\\cdot \\alpha $ defines an identification ${\\mathbb {C}}\\simeq \\mathfrak {a}_{{\\mathbb {C}}}^*$ .",
"In view of this, we write $e^{sH(g)}=e^{s\\cdot \\alpha (H(g))} \\quad \\text{for $g\\in G$ and $s\\in {\\mathbb {C}}$.",
"}$"
],
[
" Various measures on $\\operatorname{T{}}^1({\\mathcal {M}})$ .",
"Let $\\Lambda \\subset \\partial \\mathbb {H}^{d+1}$ denote the limit set of $\\Gamma $ .",
"We assume that $\\Gamma $ is non-elementary, or equivalently, $\\Lambda $ has at least 3 points.",
"Throughout the paper, we assume that ${\\textit {\\Gamma is geometrically finite}},$ that is, the unit neighborhood of the convex core of ${\\mathcal {M}}$ , given by $\\Gamma \\backslash \\text{hull}(\\Lambda )$ , has finite Riemannian volume.",
"We let $\\delta $ denote the critical exponent of $\\Gamma $ , which is known to be equal to the Hausdorff dimension of $\\Lambda $ .",
"We remark that $\\delta =d$ if and only if $\\partial \\mathbb {H}^n=\\Lambda $ if and only if $\\Gamma $ is a lattice in $G$ .",
"We fix $o\\in \\mathbb {H}^{d+1}$ whose stabilizer is given by $K$ , and $v_o\\in \\operatorname{T{}}_o(\\mathbb {H}^{d+1})$ the unit vector whose stabilizer is $M$ .",
"Let $\\nu _o$ be the Patterson-Sullivan measure on $\\Lambda $ which is unique up to a constant multiple; this is characterized by the condition that $\\gamma ^*\\nu _o=|\\gamma ^{\\prime }|^{\\delta }\\cdot \\nu _o$ for all $\\gamma \\in \\Gamma $ , where $|\\gamma ^{\\prime }|$ denotes the derivative of $\\gamma $ in the spherical metric on $\\partial \\mathbb {H}^{d+1}$ with respect to $o$ .",
"We also let $m_o$ denote the $K$ -invariant probability measure on $\\partial \\mathbb {H}^{d+1}$ .",
"Let $\\pi :\\operatorname{T{}}^1(\\mathbb {H}^{d+1})\\rightarrow \\mathbb {H}^{d+1}$ be the base point projection.",
"For $u\\in \\operatorname{T{}}^1(\\mathbb {H}^{d+1})$ , we denote by $u^{\\pm }\\in \\partial \\mathbb {H}^{d+1}$ the forward and the backward endpoints of the geodesic determined by $u$ .",
"Consider the Hopf parameterization of $\\operatorname{T{}}^1(\\mathbb {H}^{d+1})$ given by: $u \\mapsto (u^+, u^-, s=\\beta _{u^-} (o,\\pi (u))),$ where $\\beta $ denotes the Busemann function.",
"Using the Hopf coordinates, the Bowen-Margulis-Sullivan measure $m^{\\operatorname{BMS}}$ , the Liouville measure $du$ , and the Burger-Roblin measure $m^{\\rm {BR}}$ on $\\operatorname{T{}}^1(\\mathbb {H}^{d+1})$ are respectively given as follows: $dm^{\\operatorname{BMS}}(u)= e^{\\delta \\beta _{u^+}(o,\\pi (u))}\\;e^{\\delta \\beta _{u^-}(o, \\pi (u)) }\\; d\\nu _o(u^+) d\\nu _o(u^-) ds;$ $ du=dm^{\\operatorname{Liouville}}(u)= e^{d\\beta _{u^+}(o,\\pi (u))}\\;e^{d\\beta _{u^-}(o, \\pi (u)) }\\; dm_o(u^+) dm_o(u^-) ds;$ $ dm^{\\rm {BR}}(u)= e^{d \\beta _{u^+}(o,\\pi (u))}\\;e^{\\delta \\beta _{u^-}(o, \\pi (u)) }\\; dm_o(u^+) d\\nu _o(u^-) ds.$ The BR measure $m^{\\operatorname{BR}}$ is a Lebesgue measure on each $\\overline{N}$ -leaf, so we call it the unstable BR measure.",
"The stable BR-measure $m^{\\operatorname{BR}_*}$ is defined similarly to $m^{\\operatorname{BR}}$ by exchanging the roles of $u^+$ and $u^-$ .",
"These measures are all left $\\Gamma $ -invariant, and hence induce Borel measures on $\\operatorname{T{}}^1({\\mathcal {M}})=\\Gamma \\backslash \\operatorname{T{}}^1(\\mathbb {H}^{d+1})$ for which we use the same notation.",
"We remark that $m^{\\operatorname{BMS}}$ is a finite measure, invariant under the geodesic flow.",
"We will normalize the Patterson-Sullivan measure $\\nu _o$ so that $m^{\\operatorname{BMS}}(\\operatorname{T{}}^1({\\mathcal {M}}))=1$ .",
"The other three measures are infinite, unless $\\delta =d$ .",
"We will sometimes consider these measures as measures on $\\Gamma \\backslash G$ by putting: for $\\psi \\in C_c(\\Gamma \\backslash G)$ , and for $\\star =\\operatorname{BMS},\\operatorname{Liouville}, \\operatorname{BR}$ , $m^{\\star }(\\psi )=m^{\\star } \\left(\\int _M \\psi \\; dm\\right),$ where $dm$ denotes the Haar probability measure on $M$ .",
"Note that the Liouville measure, considered as a measure on $\\Gamma \\backslash G$ , is $G$ -invariant; we will denote this by $dg$ ."
],
[
" The base eigenfunction $\\phi _0$ .",
"Throughout the article, we assume $\\delta >\\frac{d}{2} ,$ which is a necessary and sufficient condition for the existence of an eigenvalue of $\\Delta $ on $L^2({\\mathcal {M}})$ .",
"The smallest eigenvalue of $\\Delta $ on $L^2({\\mathcal {M}})$ is given by $\\lambda _0=\\delta (d-\\delta )$ , and is known to be simple.",
"We denote by $\\phi _0$ the the unit eigenfunction in $L^2({\\mathcal {M}})$ with eigenvalue $\\lambda _0$ , and call it the base eigenfunction.",
"Up to a constant multiple, $\\phi _0$ is given by: for $x\\in {\\mathcal {M}}$ , $\\phi _0(x)=\\int _{\\xi \\in \\Lambda } e^{-\\delta \\beta _{\\xi } (o, x)} d\\nu _o(\\xi ).$ The fact that the base eigenfunction $\\phi _0$ is square-integrable when $\\delta >d/2$ is a key result, which allows the unitary representation theory of $G$ , specifically the right translation action of $G$ on $L^2(\\Gamma \\backslash G)$ , to be used to prove dynamical results for the geodesic flow.",
"Theorem 2.1 (Lax-Phillips [24]) The intersection of the interval $[0,\\frac{d^2}{4})$ with the spectrum of $\\Delta $ , viewed as an unbounded operator on $L^2({\\mathcal {M}})$ , consists of a finite set of eigenvalues $\\lbrace \\lambda _i=s_i(d-s_i)\\rbrace _{0\\le i\\le \\ell }$ , satisfying $0<\\lambda _0=\\delta (d-\\delta )<\\lambda _1 \\le \\ldots \\le \\lambda _\\ell < \\frac{d^2}{4}.$"
],
[
"The quasi-regular representation $L^2(\\Gamma \\backslash G)$ .",
"We denote $L^2(\\Gamma \\backslash G)$ the space of square-integrable functions on $\\Gamma \\backslash G$ with respect to $dg$ .",
"The $G$ -invariance of $dg$ gives rise to a unitary representation $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )$ of $G$ , where $\\rho $ is given by the right translation: $[\\rho (g)f](x)=f(xg)$ for all $f\\in L^2(\\Gamma \\backslash G),\\, g\\in G,\\, x\\in \\Gamma \\backslash G.$ As a number of inner products on different vector spaces will show up throughout the article, we reserve now $\\langle \\cdot ,\\cdot \\rangle $ to mean the $\\rho (G)$ -invariant inner product on $L^2(\\Gamma \\backslash G)$ .",
"All other inner products will have some additional notation to distinguish them.",
"The subspace of $\\rho (K)$ -invariant vectors in $L^2(\\Gamma \\backslash G)$ is denoted $L^2(\\Gamma \\backslash G)^K$ .",
"Similarly, $L^2(\\Gamma \\backslash G)^M$ denotes the subspace of $\\rho (M)$ -invariant vectors.",
"We use $L^2(\\Gamma \\backslash G)^K$ to construct subrepresentations of $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )$ as follows: define $L^2(\\Gamma \\backslash G)_{\\mathrm {sph}}:=\\text{the closure of }{\\lbrace \\rho (g)f\\,:\\, f\\in L^2(\\Gamma \\backslash G)^K,\\,g\\in G\\rbrace }.$ Then $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )= \\big (\\rho ,L^2(\\Gamma \\backslash G)_{\\mathrm {sph}}\\big )\\bigoplus \\big (\\rho ,L^2(\\Gamma \\backslash G)_{\\mathrm {sph}}^{\\perp }\\big );$ note that both $L^2(\\Gamma \\backslash G)_{\\mathrm {sph}}$ and $L^2(\\Gamma \\backslash G)_{\\mathrm {sph}}^{\\perp }$ are $\\rho (G)$ -invariant closed subspaces.",
"Viewing the base eigenfunction $\\phi _0$ as an element of $L^2(\\Gamma \\backslash G)^K$ , we define in a similar manner ${\\mathcal {B}}_{\\delta }:=\\text{the closure of the span of }\\lbrace \\rho (g)\\phi _0\\,:\\, g\\in G\\rbrace \\subset L^2(\\Gamma \\backslash G)_{\\mathrm {sph}},$ and $\\big (\\rho ,L^2(\\Gamma \\backslash G)_{\\mathrm {sph}}\\big )=(\\rho ,{\\mathcal {B}}_{\\delta })\\oplus (\\rho ,{\\mathcal {W}}),$ ${\\mathcal {W}}$ being the orthogonal complement of ${\\mathcal {B}}_{\\delta }$ in $L^2(\\Gamma \\backslash G)_{\\mathrm {sph}}$ .",
"It will be of importance later that $(\\rho ,{\\mathcal {B}}_{\\delta })$ is an (irreducible) complementary series representation and that Theorem REF gives a complete understanding of the complementary series representations contained in $(\\rho ,{\\mathcal {W}})$ .",
"A useful fact (cf.",
"[23]) that we make of in Section is that for all $f\\in L^2(\\Gamma \\backslash G)^K$ , ${m^{\\mathrm {BR}}}(f)={m^{\\mathrm {BR}_*}}(f)= \\langle f,\\phi _0\\rangle .$ Finally, the direct integral decomposition of $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )$ reads $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )\\cong \\int _{\\mathsf {Z}}^{\\oplus } \\big (\\mathcal {\\pi }_{\\zeta }, \\mathcal {H}_{\\zeta }\\big )\\,d\\mu _{\\mathsf {Z}}(\\zeta ),$ where $(\\pi _{\\zeta }, \\mathcal {H}_{\\zeta })$ is an irreducible unitary representation of $G$ for $\\mu _{\\mathsf {Z}}$ -a.e.",
"$\\zeta $ (cf.",
"[45])."
],
[
"Sobolev norms on unitary representations of $K$",
"Given a unitary representation $(\\pi ,V)$ of $K$ with invariant inner product $(\\cdot ,\\cdot )_V$ , a basis $\\lbrace X_j\\rbrace $ of $\\mathfrak {k}$ and $m\\in {\\mathbb {N}}$ , we define a Sobolev norm $\\Vert \\cdot \\Vert _{{\\mathcal {S}}^m(V)}$ on $V$ by $\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^m(V)}^2:=\\Vert {\\text{$v$}}\\Vert ^2_V+\\sum _{U} \\Vert d\\pi (U){\\text{$v$}}\\Vert _V^2\\qquad \\text{ for any } {\\text{$v$}}\\in V,$ where $\\Vert \\cdot \\Vert _V$ denotes the corresponding norm and the sum runs over all monomials in $\\lbrace X_j\\rbrace $ of order up to $m$ .",
"We set ${\\mathcal {S}}^m(V):=\\lbrace {\\text{$v$}}\\in V:\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^m(V)}<\\infty \\rbrace $ .",
"Observe that different choices of the basis $\\lbrace X_j\\rbrace $ give rise to equivalent norms, and that in the case when $V$ is finite-dimensional, we have ${\\mathcal {S}}^m(V)=V$ for any $m\\ge 0$ .",
"Viewing $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )$ as a unitary representation of $K$ , given a function $f\\in L^2(\\Gamma \\backslash G)$ , we let either $\\Vert f\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}$ or simply ${\\mathcal {S}}^m(f)$ denote the norm $\\Vert f\\Vert _{{\\mathcal {S}}^m(L^2(\\Gamma \\backslash G))}$ as defined by (REF ) above.",
"We denote by ${\\mathcal {S}}^m(\\Gamma \\backslash G)$ the space of all functions $f\\in L^2(\\Gamma \\backslash G)$ with ${\\mathcal {S}}^m(f)<\\infty $ .",
"For $f\\in L^2(\\Gamma \\backslash G)$ , $m^{\\operatorname{BR}}(f)$ may be infinite in general when $\\delta <d$ .",
"However we have the following lemma: Lemma 2.2 If $m>\\frac{(d+1)d}{4}$ , then for any $f\\in {\\mathcal {S}}^m(\\Gamma \\backslash G)$ , $m^{\\operatorname{BR}}(f),\\; m^{\\operatorname{BR}_*}(f) \\; \\ll \\; {\\mathcal {S}}^{m}(f).$ Given $f\\in C(\\Gamma \\backslash G)\\cap L^2(\\Gamma \\backslash G)$ , define $f_K( x):= \\max _{k\\in K} |f(xk)| \\qquad \\text{ for } x\\in \\Gamma \\backslash G.$ By construction, $f_K$ is $\\rho (K)$ -invariant and $ |f(x)|\\le f_K(x)$ for all $ x\\in \\Gamma \\backslash G$ , hence $|m^{\\operatorname{BR}}(f)|\\le m^{\\operatorname{BR}}(f_K)$ .",
"In view of (REF ), the Sobolev embedding theorem on the compact manifold $K$ gives $|m^{\\operatorname{BR}}( f )| \\le m^{\\operatorname{BR}}(f_K)=\\langle f_K,\\phi _0\\rangle \\le \\Vert f_K\\Vert \\ll \\Vert f\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}$ for all $m>\\frac{\\operatorname{dim}(K)}{2}$ (cf.",
"[1]).",
"Since $\\operatorname{dim}K=\\frac{d(d+1)}{2}$ , the chain of inequalities above holds for any $m>\\frac{(d+1)d}{4}$ .",
"The claim for $m^{\\operatorname{BR}_*}$ follows similarly."
],
[
"Complementary series representations and Eisenstein integrals",
"In this section we recall the definition and $K$ -type structure of the complementary series representations of $G=\\operatorname{SO}^\\circ (d+1, 1)$ , as well as the Eisenstein integral representations of matrix coefficients for these representations.",
"We start by reviewing the representation theory of the special orthogonal groups $\\operatorname{SO}(n)$ ."
],
[
"Irreducible representations of $K$ and {{formula:9f724e12-d780-4036-8c25-64af3a19b2d8}}",
"The primary reference for the first facts listed below is [4].",
"Let $m\\ge 1$ .",
"The irreducible representations of $\\operatorname{SO}(2m)$ are parameterized by $m$ -tuples $\\tau =(\\tau _1,\\tau _2,\\ldots ,\\tau _m)\\in {\\mathbb {Z}}^{m}$ such that $\\tau _1\\ge \\tau _2\\ge \\ldots \\ge \\tau _{m-1}\\ge \\tau _{m-1}\\ge |\\tau _m|.$ Similarly, the irreducible representations of $\\operatorname{SO}(2m+1)$ are parameterized by $m$ -tuples $\\tau =(\\tau _1,\\tau _2,\\ldots ,\\tau _m)\\in {\\mathbb {Z}}^{m}$ such that $\\tau _1\\ge \\tau _2\\ge \\ldots \\ge \\tau _{m-1}\\ge \\tau _{m}\\ge 0.$ We recall that all irreducible representations of $\\operatorname{SO}(2m+1)$ and $\\operatorname{SO}(4m)$ are self-dual, and that the dual $\\tau ^*$ of an irreducible representation $\\tau =(\\tau _1,\\tau _2,\\ldots ,\\tau _{2m+1})$ of $\\operatorname{SO}(4m+2)$ is given by $\\tau ^*=(\\tau _1,\\tau _2,\\ldots ,-\\tau _{2m+1})$ ; the dual $\\tau ^*$ is defined by $\\tau ^*(k)=\\tau (k^{-1})$ , acting on the dual (or complex conjugate) of the underlying vector space.",
"A key result that we will make repeated use of is the branching law for restrictions of representations of $K$ to $M$ which we recall (cf.",
"[21] or [48]).",
"We henceforth call irreducible representations of $K$ or $M$ a $K$ or $M$ -type, respectively.",
"Let $\\tau $ be a $K$ -type.",
"Then the decomposition of $\\tau |_M$ reads $\\tau |_M=\\bigoplus _{\\sigma \\in {\\hat{M}}} m_{\\sigma ,\\tau }\\,\\cdot \\, \\sigma ,$ where $m_{\\sigma ,\\tau }=1$ if $\\sigma =(\\sigma _1,\\ldots ,\\sigma _{\\lfloor \\frac{d}{2}\\rfloor })$ satisfies the interlacing property $\\tau _1\\ge \\sigma _1\\ge \\tau _2\\ge \\sigma _2\\ge \\ldots \\ge \\sigma _{m-1}\\ge |\\tau _m| \\qquad \\mathrm {for\\;} d=2m-1,$ and $\\tau _1\\ge \\sigma _1\\ge \\tau _2\\ge \\sigma _2\\ge \\tau _m\\ge |\\sigma _m| \\qquad \\mathrm {for\\;} d=2m,$ and $m_{\\sigma ,\\tau }=0$ otherwise.",
"If $m_{\\sigma ,\\tau }=1$ , we say that $\\tau $ contains $\\sigma $ , and write $\\sigma \\subset \\tau $ .",
"The fact that each $M$ -type occurs at most once in $\\tau $ will allow us to make repeated use of Schur's lemma in the following manner: if a linear operator on $V_{\\tau }$ ($V_{\\tau }$ being a vector space on which $\\tau $ is realized) commutes with $\\tau (M)$ , then it acts as a scalar on each $\\sigma \\subset \\tau $ .",
"From now on we write $\\operatorname{dim}(\\tau )$ and $\\operatorname{dim}(\\sigma )$ for $\\operatorname{dim}(V_{\\tau })$ and $\\operatorname{dim}(V_{\\sigma })$ , respectively.",
"We connect the dimensions of $K$ -types with the Sobolev norms introduced in Section REF .",
"Firstly, let $(\\pi ,{\\mathcal {H}})$ be a unitary representation of $G$ .",
"For each $\\tau \\in {\\hat{K}}$ , we define $\\chi _{\\tau }(k)=\\operatorname{dim}(V_{\\tau })\\,\\cdot \\, \\mathrm {tr}\\big (\\tau (k)\\big )$ (the trace being defined with respect to any invariant inner product on any realization of $\\tau $ ) and an operator $\\mathsf {P}_{\\tau }$ by $\\mathsf {P}_{\\tau }=\\int _K \\overline{\\chi _{\\tau }(k)}\\pi (k)\\,dk.$ Note that $\\mathsf {P}_{\\tau }$ is the orthogonal projection onto the space ${\\mathcal {H}}_{\\tau }$ , where ${\\mathcal {H}}_{\\tau }:=\\lbrace {\\text{$v$}}\\in {\\mathcal {H}}\\,:\\, \\mathsf {P}_{\\tau }{\\text{$v$}}={\\text{$v$}}\\rbrace .$ This gives rise to a decomposition of ${\\mathcal {H}}$ as the orthogonal direct sum ${\\mathcal {H}}=\\bigoplus _{\\tau \\in {\\hat{K}}} {\\mathcal {H}}_{\\tau }.$ If $(\\pi ,{\\mathcal {H}})$ is irreducible then each ${\\mathcal {H}}_{\\tau }$ is finite-dimensional.",
"Each ${\\mathcal {H}}_{\\tau }$ has the property that $\\pi (k)|{{\\mathcal {H}}_{\\tau }}\\cong \\tau (k) $ for all $k\\in K$ .",
"If ${\\mathcal {H}}_{\\tau }\\ne 0$ , we say that $\\pi $ contains $\\tau $ , and write $\\tau \\subset \\pi $ .",
"There is a similar decomposition of ${\\mathcal {H}}$ with respect to $M$ -types, and projection operators $\\mathsf {P}_{\\sigma }=\\int _M \\overline{\\chi _{\\sigma }(m)}\\pi (m)\\,dm$ for $\\sigma \\in \\hat{M}$ .",
"As usual, $\\lceil \\beta \\rceil $ denotes the smallest integer greater than or equal to $\\beta $ .",
"Lemma 3.1 There exists $m_0\\in {\\mathbb {N}}$ depending only on $K$ such that for any unitary representation $(\\pi ,V)$ of $K$ , $\\alpha >0$ , $m\\in {\\mathbb {N}}$ , and for all ${\\text{$v$}}\\in V$ , $\\sum _{\\tau \\subset V} \\operatorname{dim}(\\tau )^{\\alpha }\\Vert \\mathsf {P}_{\\tau }{\\text{$v$}}\\Vert _{{\\mathcal {S}}^{m}(V)}\\ll \\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^{m+m_0\\lceil \\frac{\\alpha +1}{2}\\rceil }(V)},$ where the implied constant depends only on $K$ .",
"Though the argument is standard (cf.",
"[46] or [20]), we briefly recount it: using the Harish-Chandra isomorphism and the highest weight theory, we find an element $\\omega _K$ in the center of the universal enveloping algebra of $\\mathfrak {k}$ such that for each $\\tau \\in {\\hat{K}}$ , $d\\tau (\\omega _K)$ acts as a scalar $c_{\\tau }$ on $\\tau $ , where $|c_{\\tau }|\\ge \\operatorname{dim}(\\tau )^2$ .",
"Letting $m_0$ be the order of $\\omega _K$ , for any $\\ell \\in {\\mathbb {N}}$ , we have (using the Cauchy-Schwarz inequality), $\\sum _{\\tau \\subset V} \\operatorname{dim}(\\tau )^{\\alpha }\\Vert \\mathsf {P}_{\\tau }{\\text{$v$}}\\Vert _{{\\mathcal {S}}^{m}(V)}&=\\sum _{\\tau \\subset V} \\frac{\\operatorname{dim}(\\tau )^{\\alpha }}{|c_{\\tau }|^\\ell }\\Vert \\mathsf {P}_{\\tau }d\\pi (\\Omega _K^\\ell ){\\text{$v$}}\\Vert _{{\\mathcal {S}}^{m}(V)}\\\\ & \\le \\left( \\sum _{\\tau \\subset V} \\operatorname{dim}(\\tau )^{2\\alpha -4\\ell }\\sum _{\\tau \\subset V} \\Vert \\mathsf {P}_{\\tau }d\\pi (\\Omega _K^\\ell ){\\text{$v$}}\\Vert ^2_{{\\mathcal {S}}^{m}(V)}\\right)^{1/2}\\\\&\\ll \\left(\\sum _{\\tau \\subset V} \\operatorname{dim}(\\tau )^{2\\alpha -4\\ell }\\right)^{1/2} \\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^{m+\\ell m_0}(V)}.$ The sum $\\sum _{\\tau } {\\operatorname{dim}(\\tau )^{-2}}$ is finite by [20], so choosing $\\ell =\\lceil \\frac{ \\alpha +1}{2}\\rceil $ gives $\\sum _{\\tau \\subset V} \\operatorname{dim}(\\tau )^{2\\alpha -4\\ell } <\\infty $ ."
],
[
"Complementary series representations",
"We will now recall Hirai's classification [16] of the non-tempered unitary dual of $G$ and construct the models of the complementary series that we will work with, see [46] (cf.",
"also [27]).",
"Given $s\\in {\\mathbb {C}}$ , we define the standard representation $U^s$ of $G$ on $L^2(K)$ by $[U^s(g){\\text{$v$}}](k)=e^{-sH(g^{-1}k)}{\\text{$v$}}\\big (\\kappa (g^{-1}k)\\big )$ for all ${\\text{$v$}}\\in L^2(K),\\; g\\in G,$ and $ k\\in K$ .",
"We let $\\lambda $ and $\\rho $ denote the left- and right-regular representations of $K$ on $L^2(K)$ respectively.",
"Observe that $U^s|_K=\\lambda $ ; for this reason, it is practical to always view $L^2(K)$ as the unitary representation $(\\lambda , L^2(K))$ of $K$ .",
"The decomposition of $L^2(K)$ into $K$ -types reads $L^2(K)=\\bigoplus _{\\tau \\in {\\hat{K}}} L^2(K)_{\\tau },$ where each $L^2(K)_{\\tau }$ is isomorphic to $\\operatorname{dim}(\\tau )$ copies of $\\tau $ .",
"Given $\\upsilon \\in {\\hat{M}}$ , we define $L^2(K:\\upsilon ):=\\left\\lbrace {\\text{$v$}}\\in L^2(K)\\,:\\, \\int _M\\overline{\\chi _{\\upsilon }(m)}\\rho (m){\\text{$v$}}\\,dm={\\text{$v$}}\\right\\rbrace .$ The fact that $U^s(g)$ and $\\rho (m)$ commute for all $g\\in G$ and $m\\in M$ shows that $\\big (U^s,L^2(K:\\upsilon ) \\big )$ is a representation of $G$ ; in fact $\\big (U^s,L^2(K:\\upsilon ) \\big )$ is isomorphic to $\\operatorname{dim}(\\upsilon )$ copies of the representation $\\mathrm {ind}_{MAN}^G ( \\upsilon \\otimes s\\otimes 1)$ (cf.",
"[20]).",
"Note that $L^2(K:\\upsilon )_{\\tau }\\subset L^2(K)_{\\tau }$ for all $\\tau \\in {\\hat{K}}$ .",
"From Hirai's classification of the unitary dual of $G$ [16], for each $\\upsilon \\in {\\hat{M}}$ , there exists an interval ${\\mathcal {I}}_{\\upsilon }\\subset (\\frac{d}{2},d)$ such that if $s\\in {\\mathcal {I}}_{\\upsilon }$ , then there exists an irreducible, unitarizable subrepresentation ${\\mathcal {U}}(\\upsilon ,s)$ of $\\big (U^s,L^2(K:\\upsilon ) \\big )$ .",
"Furthermore, every non-tempered representation of $G$ may be realized as ${\\mathcal {U}}(\\upsilon ,s)$ for some $\\upsilon \\in \\hat{M}$ and $s\\in {\\mathcal {I}}_{\\upsilon }$ in this way, and each $\\tau \\in \\hat{K}$ that contains $\\upsilon $ occurs exactly once in ${\\mathcal {U}}(\\upsilon ,s)$ (cf.",
"[46] and [20]).",
"The inner product that makes ${\\mathcal {U}}(\\upsilon ,s)$ a unitary representation is denoted by $\\langle \\cdot ,\\cdot \\rangle _{{\\mathcal {U}}(\\upsilon ,s)}$ .",
"Note that for $g\\in G$ , and ${\\text{$v$}}, {\\text{$u$}}\\in {\\mathcal {U}}(\\upsilon ,s)$ , $\\langle U^s(g) {\\text{$v$}}, {\\text{$u$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}=\\langle {\\mathcal {U}}(\\upsilon ,s)(g) {\\text{$v$}}, {\\text{$u$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)} .$ We conclude this section by recalling two general facts regarding complementary series representations that let us classify the representation ${\\mathcal {B}}_{\\delta }$ (cf.",
"Section REF ).",
"Firstly, the spherical complementary series representation are the ${\\mathcal {U}}(1,s)$ ; i.e.",
"precisely those where the representation $\\upsilon $ is trivial.",
"Secondly, the Casimir operator ${\\mathcal {C}}$ of $G$ acts as the scalar $-s(d-s)$ on the smooth vectors of ${\\mathcal {U}}(\\upsilon ,s)$ .",
"Combining these facts with the observation that the restriction of ${\\mathcal {C}}$ to right $K$ -invariant functions on $G$ is the Laplace-Beltrami operator lets one conclude that the representation ${\\mathcal {B}}_{\\delta }$ is isomorphic to ${\\mathcal {U}}(1,\\delta )$ ."
],
[
"Eisenstein integrals",
"Here we develop the Eisenstein integrals needed to obtain the desired asymptotic expansion of matrix coefficients (cf.",
"[27], [47]).",
"Before starting, we make the following elementary but important observation: Lemma 3.2 If $\\sigma \\in {\\hat{M}}\\setminus \\lbrace \\upsilon ^*\\rbrace $ , then for any ${\\text{$v$}}\\in L^2(K:\\upsilon )_{\\sigma }$ , we have ${\\text{$v$}}(m)=0\\quad \\text{ for all $m\\in M$.",
"}$ Since ${\\text{$v$}}\\in L^2(K:\\upsilon )$ , we have ${\\text{$v$}}=\\int _M \\rho (m){\\text{$v$}}\\overline{\\chi _{\\upsilon }(m)}\\,dm$ .",
"Hence for any $m\\in M$ , ${\\text{$v$}}(m)&=\\int _M {\\text{$v$}}(mm_1)\\overline{\\chi _{\\upsilon }(m_1)}\\,dm_1\\\\&=\\int _M {\\text{$v$}}(m_1m)\\overline{\\chi _{\\upsilon }(m^{-1}m_1m)}\\,dm_1\\\\&=\\int _M {\\text{$v$}}(m_1^{-1}m)\\overline{\\chi _{\\upsilon }(m_1^{-1})}\\,dm_1\\\\&=\\int _M {\\text{$v$}}(m_1^{-1}m)\\overline{\\chi _{\\upsilon ^*}(m_1)}\\,dm_1\\\\&=[\\mathsf {P}_{\\upsilon ^*}{\\text{$v$}}](m).$ Since ${\\text{$v$}}\\in L^2(K:\\upsilon )_{\\sigma }$ for $\\sigma \\in {\\hat{M}}\\setminus \\lbrace \\upsilon ^*\\rbrace $ , the orthogonality of the $M$ -types of $L^2(K)$ gives $\\mathsf {P}_{\\upsilon ^*}{\\text{$v$}}=0$ , yielding the claim.",
"Before stating the main result of this section, we introduce some more notation.",
"Firstly, we let $\\langle \\cdot ,\\cdot \\rangle _K$ denote the usual inner product on $L^2(K)$ .",
"The corresponding norm on $L^2(K)$ is denoted $\\Vert \\cdot \\Vert _{K}$ , and similarly for the operator norm defined with respect to it.",
"In the rest of this section, we fix a complementary series representation ${\\mathcal {U}}(\\upsilon ,s)$ , $s\\in {\\mathcal {I}}_{\\upsilon }$ .",
"For each $K$ -type $\\tau $ of ${\\mathcal {U}}(\\upsilon ,s)$ we define a vector ${\\text{$\\chi $}}_{\\tau }\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau }$ by ${\\text{$\\chi $}}_{\\tau }=\\sum _{i=1}^{\\operatorname{dim}(\\tau )} \\overline{{\\text{$v$}}_i(e)}\\,{\\text{$v$}}_i,$ where $\\lbrace {\\text{$v$}}_i\\rbrace _{i=1}^{\\operatorname{dim}(\\tau )}$ is an orthornormal basis of ${\\mathcal {U}}(\\upsilon ,s)_{\\tau }$ (recall that $\\tau $ has multiplicity one in ${\\mathcal {U}}(\\upsilon ,s)$ ).",
"Note that ${\\text{$\\chi $}}_{\\tau }$ is independent of the choice of basis, and for all $ {\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau }$ and $ k\\in K$ , we have ${\\text{$v$}}(k)=\\langle {\\text{$v$}}, U^s(k){\\text{$\\chi $}}_{\\tau }\\rangle _K.$ Lemma 3.3 Let $\\tau $ be a $K$ -type of $ {\\mathcal {U}}(\\upsilon ,s)$ .",
"For all ${\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau }\\cap {\\mathcal {U}}(\\upsilon ,s)_{\\upsilon ^*}$ , we have $\\int _M |{\\text{$v$}}(m)|^2\\,dm=\\frac{\\operatorname{dim}(\\tau )\\Vert {\\text{$v$}}\\Vert _{K}^2}{\\operatorname{dim}(\\upsilon )}.$ $\\int _M |{\\text{$\\chi $}}_{\\tau }(m)|^2\\,dm=\\frac{\\operatorname{dim}(\\tau )^2}{\\operatorname{dim}(\\upsilon )}$ .",
"Let $\\lbrace {\\text{$v$}}_j\\rbrace $ be an orthonormal basis of ${\\mathcal {U}}(\\upsilon ,s)_{\\tau }\\cap {\\mathcal {U}}(\\upsilon ,s)_{\\upsilon ^*}$ with respect to $\\langle \\cdot ,\\cdot \\rangle _K$ .",
"Using (REF ) and the Schur orthogonality relations, we get $\\int _M |{\\text{$v$}}(m)|^2\\,dm &=\\int _M |\\langle {\\text{$v$}},U^s(m){\\text{$\\chi $}}_{\\tau } \\rangle _K|^2\\,dm\\\\&=\\sum _{i,j} \\overline{{\\text{$v$}}_i(e)}{\\text{$v$}}_j(e)\\int _M \\langle U^s(m){\\text{$v$}},{\\text{$v$}}_i \\rangle _K\\overline{\\langle U^s(m){\\text{$v$}},{\\text{$v$}}_j \\rangle _K}\\,dm\\\\& =\\frac{\\Vert {\\text{$v$}}\\Vert _{K}^2\\sum _i |{\\text{$v$}}_i(e)|^2}{\\operatorname{dim}(\\upsilon )}.$ Complete $\\lbrace {\\text{$v$}}_j\\rbrace $ into an orthonormal basis $\\lbrace {\\text{$v$}}_j\\rbrace \\cup \\lbrace {\\text{$w$}}_i\\rbrace $ for all of ${\\mathcal {U}}(\\upsilon ,s)_{\\tau }$ .",
"Note that ${\\text{$w$}}_i(e)=0$ for all $i$ by Lemma REF .",
"Therefore we have $\\sum _j |{\\text{$v$}}_j(e)|^2=\\sum _j |{\\text{$v$}}_j(e)|^2+\\sum _i |{\\text{$w$}}_i(e)|^2=\\operatorname{dim}(\\tau ).$ Hence the claim (1) follows.",
"Claim (2) follows from (1).",
"For $K$ -types $\\tau _1, \\tau _2$ of ${\\mathcal {U}}(\\upsilon ,s)$ , we define an operator $ T_{\\tau _1}^{\\tau _2}: {\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}\\rightarrow {\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}$ by $T_{\\tau _1}^{\\tau _2}{\\text{$v$}}=\\int _M {\\text{$v$}}(m) U^s(m){\\text{$\\chi $}}_{\\tau _2}\\,dm\\qquad \\text{for all } {\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}.$ We have the following interpretation of the Eisenstein integral (a similar formula appears in [27]): Theorem 3.4 For any $K$ -types $\\tau _1$ , $\\tau _2$ of ${\\mathcal {U}}(\\upsilon ,s)$ and $g\\in G$ , we have $\\mathsf {P}_{\\tau _2}U^s(g)\\mathsf {P}_{\\tau _1}=\\int _K e^{(s-d)H(gk)} U^s\\big (\\kappa (gk)\\big ) T_{\\tau _1}^{\\tau _2} U^s(k^{-1})\\,dk,$ where $\\mathsf {P}_{\\tau _2}U^s(g)\\mathsf {P}_{\\tau _1}$ is viewed as an operator from ${\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}$ to ${\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}$ .",
"Following [47], the key fact is that for any $g\\in G$ , $dk=e^{d\\, H(gk)}d \\big (\\kappa (gk)\\big ).$ Let ${\\text{$v$}}\\in \\mathcal {U}(\\upsilon ,s)_{\\tau _1}$ and ${\\text{$w$}}\\in \\mathcal {U}(\\upsilon ,s)_{\\tau _2}$ .",
"Then $\\langle U^s(g){\\text{$v$}},{\\text{$w$}}\\rangle _K &=\\int _K e^{-sH(g^{-1}k)}{\\text{$v$}}\\big (\\kappa (g^{-1}k)\\big )\\overline{{\\text{$w$}}(k)}\\,dk\\\\ &=\\int _K e^{-sH(g^{-1}k)}{\\text{$v$}}\\big (\\kappa (g^{-1}k)\\big )\\overline{{\\text{$w$}}(k)}\\,e^{d\\,H(g^{-1}k)}\\,d\\big (\\kappa (g^{-1}k)\\big ).$ We now carry out the change of variables $\\tilde{k}=\\kappa (g^{-1}k)$ ; this gives $k=\\kappa (g\\tilde{k})$ , allowing us to rewrite the above integral as follows $&=\\int _K e^{(d-s)H\\big (g^{-1}\\kappa (gk)\\big )}{\\text{$v$}}(k)\\overline{{\\text{$w$}}\\big (\\kappa (gk)\\big )}\\,\\,dk\\\\&=\\int _K e^{(s-d)H(gk)}{\\text{$v$}}(k)\\overline{{\\text{$w$}}\\big (\\kappa (gk)\\big )}\\,dk,$ where we used the identity $g^{-1}\\kappa (gk)=k\\exp (-H(gk)) \\big (\\exp (H(gk)n_{kg}^{-1}\\exp (-H(gk)\\big )$ to obtain $H(g^{-1}\\kappa (gk))=-H(gk)$ .",
"By (REF ), we get ${\\text{$v$}}(k)\\overline{{\\text{$w$}}\\big (\\kappa (gk)\\big )}=&\\langle {\\text{$v$}},U^s(k){\\text{$\\chi $}}_{\\tau _1}\\rangle _K \\cdot \\overline{\\langle {\\text{$w$}},U^s\\big (\\kappa (gk)\\big ){\\text{$\\chi $}}_{\\tau _2}\\rangle _K}\\\\=&\\langle U^s(k^{-1}){\\text{$v$}},{\\text{$\\chi $}}_{\\tau _1}\\rangle _K \\cdot \\langle U^s\\big (\\kappa (gk)\\big ){\\text{$\\chi $}}_{\\tau _2},{\\text{$w$}}\\rangle _K\\\\=&\\Big \\langle \\langle U^s(k^{-1}){\\text{$v$}},{\\text{$\\chi $}}_{\\tau _1}\\rangle \\,\\cdot \\, U^s\\big (\\kappa (gk)\\big ){\\text{$\\chi $}}_{\\tau _2},{\\text{$w$}}\\Big \\rangle _K.$ Hence $\\langle U^s(g){\\text{$v$}},{\\text{$w$}}\\rangle _K=&\\int _K e^{(s-d)H(gk)}\\Big \\langle \\langle U^s(k^{-1}){\\text{$v$}},{\\text{$\\chi $}}_{\\tau _1}\\rangle \\,\\cdot \\, U^s\\big (\\kappa (gk)\\big ){\\text{$\\chi $}}_{\\tau _2},{\\text{$w$}}\\Big \\rangle _K\\,dk\\\\=&\\left\\langle \\int _K e^{(s-d)H(gk)}\\langle U^s(k^{-1}){\\text{$v$}},{\\text{$\\chi $}}_{\\tau _1}\\rangle \\,\\cdot \\, U^s\\big (\\kappa (gk)\\big ){\\text{$\\chi $}}_{\\tau _2}\\,dk,{\\text{$w$}}\\right\\rangle _K.$ We define $T_0:{\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}\\rightarrow {\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}$ by $T_0{\\text{$v$}}= \\langle {\\text{$v$}}, {\\text{$\\chi $}}_{\\tau _1}\\rangle _K \\cdot {\\text{$\\chi $}}_{\\tau _2}={\\text{$v$}}(e){\\text{$\\chi $}}_{\\tau _2}.$ Then $\\mathsf {P}_{{\\tau _2}}U^s(g)\\mathsf {P}_{{\\tau _1}}= \\int _K e^{(s-d)H(gk)} U^s\\big (\\kappa (gk)\\big ) T_0U^s(k^{-1}) \\,dk.$ Using (REF ) and (REF ), we observe that $T_{\\tau _1}^{\\tau _2}=\\int _M U^s(m) T_0 U^s(m^{-1})\\,dm.$ Writing $g=k_1 a_t k_2$ , we then have $&\\mathsf {P}_{{\\tau _2}}U^s(g)\\mathsf {P}_{{\\tau _1}} = U^s(k_1) \\mathsf {P}_{{\\tau _2}}U^s(a_t)\\mathsf {P}_{{\\tau _1}} U^s(k_2)\\\\&=\\int _M U^s(k_1) U^s(m)\\mathsf {P}_{{\\tau _2}}U^s(a_t)\\mathsf {P}_{{\\tau _1}}U^s(m^{-1}) U^s(k_2)\\,dm,$ where we used the facts that $M=Z_K(A)$ and $M\\subset K$ to commute $U^s(m)$ past $U^s(a_t)$ and $\\mathsf {P}_{\\tau _i}$ .",
"Now using (REF ) gives $&= U^s(k_1) \\left(\\int _M \\int _K e^{(s-d)H(a_tk)} U^s\\big (m\\kappa (a_tk)\\big )T_0U^s(k^{-1}m^{-1})\\,dk\\,dm\\, \\right)U^s(k_2)\\\\&= \\int _K e^{(s-d)H(a_tk)} U^s\\big (k_1\\kappa (a_tk)\\big )\\left( \\int _M U^s(m)T_0 U^s({m^{-1}})\\,dm\\right) U^s(k^{-1}k_2)\\,dk.$ Finally, using the identities $k_1\\kappa (a_tk)=\\kappa (k_1a_tk)$ and $H(a_tk)=H(k_1a_tk)$ together with the change of variables $k=k_2\\tilde{k}$ gives $\\mathsf {P}_{{\\tau _2}}U^s(g)\\mathsf {P}_{{\\tau _1}}= \\int _K e^{(s-d)H(g\\tilde{k})} U^s\\big (\\kappa (g\\tilde{k})\\big )T_{\\tau _1}^{\\tau _2} U^s(\\tilde{k}^{-1})\\,d\\tilde{k},$ proving the first identity claimed in the theorem.",
"The second identity follows from the fact that $\\langle U^s(m^{-1}){\\text{$v$}},{\\text{$\\chi $}}_{\\tau _1}\\rangle _K={\\text{$v$}}(m)$ for all ${\\text{$v$}}\\in \\mathcal {U}(\\upsilon ,s)_{\\tau _1}$ and $m\\in M$ ."
],
[
"On the operator $T_{\\tau _1}^{\\tau _2}$ .",
"Using Lemma REF , we deduce: Proposition 3.5 Let $\\tau _1$ , $\\tau _2$ be $K$ -types of ${\\mathcal {U}}(\\upsilon ,s)$ .",
"If ${\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}$ is orthogonal to ${\\mathcal {U}}(\\upsilon ,s)_{\\upsilon ^*}$ , then $T_{\\tau _1}^{\\tau _2}{\\text{$v$}}=0.$ Consequently, $T_{\\tau _1}^{\\tau _2}\\, {\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}\\subset {\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}\\cap {\\mathcal {U}}(\\upsilon ,s)_{\\upsilon ^*}$ .",
"We have $T_{\\tau _1}^{\\tau _2}{\\text{$v$}}=\\int _M {\\text{$v$}}(m) U^s(m){\\text{$\\chi $}}_{\\tau _2}\\,dm$ .",
"By Lemma REF , if ${\\text{$v$}}$ is orthogonal to ${\\mathcal {U}}(\\upsilon ,s)_{\\upsilon ^*}$ , then ${\\text{$v$}}(m)=0$ for all $m\\in M$ and thus $T_{\\tau _1}^{\\tau _2}{\\text{$v$}}=0$ .",
"A direct computation using (REF ) shows that $T_{\\tau _1}^{\\tau _2}$ commutes with $U^s(m)$ for all $m\\in M$ , so Schur's lemma then gives that if $T_{\\tau _1}^{\\tau _2}{\\text{$v$}}$ is non-zero, it must be contained in ${\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}\\cap {\\mathcal {U}}(\\upsilon ,s)_{\\upsilon ^*}$ .",
"If ${\\text{$v$}}$ is $M$ -invariant and $\\upsilon \\in \\hat{M}$ is non-trivial, then $\\upsilon ^*$ is non-trivial, and hence ${\\text{$v$}}$ is orthogonal to all ${\\mathcal {U}}(\\upsilon ,s)_{\\upsilon ^*}$ .",
"Therefore we deduce the following corollary: Corollary 3.6 If $\\upsilon \\in \\hat{M}$ is non-trivial, then for any $M$ -invariant ${\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}$ , $T_{\\tau _1}^{\\tau _2}{\\text{$v$}}=0.$ Lemma 3.7 For any $K$ -types $\\tau _1, \\tau _2$ of ${\\mathcal {U}}(\\upsilon ,s)$ , we have $(T_{\\tau _1}^{\\tau _2})^*=T_{\\tau _2}^{\\tau _1},$ where the adjoint is defined with respect to $\\langle \\cdot ,\\cdot \\rangle _K$ .",
"For any ${\\text{$u$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}$ and ${\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}$ , we have $\\langle T_{\\tau _1}^{\\tau _2}{\\text{$u$}},{\\text{$v$}}\\rangle _K&=\\int _M {\\text{$u$}}(m)\\langle U^s(m){\\text{$\\chi $}}_{\\tau _2},{\\text{$v$}}\\rangle _K\\,dm\\\\&=\\int _M \\langle {\\text{$u$}}, U^s(m){\\text{$\\chi $}}_{\\tau _1}\\rangle _K\\langle U^s(m){\\text{$\\chi $}}_{\\tau _2},{\\text{$v$}}\\rangle _K\\,dm\\\\&=\\left\\langle {\\text{$u$}},\\int _M\\langle U^s(m^{-1}){\\text{$v$}},{\\text{$\\chi $}}_{\\tau _2}\\rangle _K\\,\\cdot \\, U^s(m){\\text{$\\chi $}}_{\\tau _1}\\,dm \\right\\rangle \\\\&=\\langle {\\text{$u$}},T_{\\tau _2}^{\\tau _1}{\\text{$v$}}\\rangle _K.$ Corollary 3.8 For any $K$ -types $\\tau _1, \\tau _2$ of ${\\mathcal {U}}(\\upsilon ,s)$ , $\\Vert T_{\\tau _1}^{\\tau _2}\\Vert _{K}\\le {\\frac{\\sqrt{\\operatorname{dim}(\\tau _1)\\operatorname{dim}(\\tau _2)}}{\\operatorname{dim}(\\upsilon )}}.$ For any unit vector ${\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}\\cap {\\mathcal {U}}(\\upsilon ,s)_{\\upsilon ^*}$ , we have $&\\langle T_{\\tau _1}^{\\tau _2}{\\text{$v$}},T_{\\tau _1}^{\\tau _2}{\\text{$v$}}\\rangle _K=\\int _M \\overline{{\\text{$v$}}(m)}\\langle T_{\\tau _1}^{\\tau _2}{\\text{$v$}},U^s(m){\\text{$\\chi $}}_{\\tau _2}\\rangle _K\\,dm\\\\&=\\int _M\\int _M \\overline{{\\text{$v$}}(m_1)}{\\text{$v$}}(m_2)\\langle U^s(m_2){\\text{$\\chi $}}_{\\tau _2},U^s(m_1){\\text{$\\chi $}}_{\\tau _2}\\rangle _K\\,dm_2\\,dm_1\\\\&=\\int _M\\int _M \\overline{\\langle {\\text{$v$}},U^s(m_1){\\text{$\\chi $}}_{\\tau _1}\\rangle _K}\\langle {\\text{$v$}},U^s(m_2){\\text{$\\chi $}}_{\\tau _1}\\rangle _K{\\text{$\\chi $}}_{\\tau _2}(m_2^{-1}m_1)\\,dm_2\\,dm_1\\\\&=\\int _M\\left(\\int _M \\overline{\\langle {\\text{$v$}},U^s(m_2m_3){\\text{$\\chi $}}_{\\tau _1}\\rangle _K}\\langle {\\text{$v$}},U^s(m_2){\\text{$\\chi $}}_{\\tau _1}\\rangle _K\\,dm_2\\right){\\text{$\\chi $}}_{\\tau _2}(m_3)\\,dm_3.$ Since ${\\text{$v$}}$ and ${\\text{$\\chi $}}_{\\tau _1}$ are both in ${\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}\\cap {\\mathcal {U}}(\\upsilon ,s)_{\\upsilon ^*}$ , the Schur orthogonality relations for $M$ give $\\int _M& \\overline{\\langle {\\text{$v$}},U^s(m_2m_3){\\text{$\\chi $}}_{\\tau _1}\\rangle _K}\\langle {\\text{$v$}},U^s(m_2){\\text{$\\chi $}}_{\\tau _1}\\rangle _K\\,dm_2\\\\&=\\frac{\\Vert {\\text{$v$}}\\Vert ^2}{\\operatorname{dim}(\\upsilon )}\\overline{\\langle {\\text{$\\chi $}}_{\\tau _1},U^s(m_3){\\text{$\\chi $}}_{\\tau _1}\\rangle }=\\frac{\\overline{{\\text{$\\chi $}}_{\\tau _1}(m_3)}}{\\operatorname{dim}(\\upsilon )}.$ We thus have $\\langle T_{\\tau _1}^{\\tau _2}{\\text{$v$}},T_{\\tau _1}^{\\tau _2}{\\text{$v$}}\\rangle _K&=\\frac{1}{\\operatorname{dim}(\\upsilon )}\\int _M\\overline{{\\text{$\\chi $}}_{\\tau _1}(m)}{\\text{$\\chi $}}_{\\tau _2}(m)\\,dm\\\\ &\\le \\frac{1}{\\operatorname{dim}(\\upsilon )}\\sqrt{\\int _M|{\\text{$\\chi $}}_{\\tau _1}(m)|^2\\,dm\\int _M|{\\text{$\\chi $}}_{\\tau _2}(m)|^2\\,dm}\\\\ &= \\frac{\\operatorname{dim}(\\tau _1)\\operatorname{dim}(\\tau _2)}{\\operatorname{dim}(\\upsilon )^2}$ by Lemma REF (2).",
"Combining this estimate with Proposition REF , the claim follows.",
"Corollary 3.9 For any $K$ -type $\\tau $ of ${\\mathcal {U}}(\\upsilon ,s)$ , we have $T_{\\tau }^{\\tau }=\\frac{\\operatorname{dim}(\\tau )}{\\operatorname{dim}(\\upsilon )} \\mathsf {P}_{\\upsilon ^*}.$ Following the proof of Corollary REF , we obtain $\\Vert T_{\\tau }^{\\tau }\\Vert _K^2=\\frac{1}{\\operatorname{dim}(\\upsilon )}\\int _M|{\\text{$\\chi $}}_{\\tau }(m)|^2\\,dm=\\frac{\\operatorname{dim}(\\tau )^2}{\\operatorname{dim}(\\upsilon )^2}.$ So by Proposition REF , we have $T_{\\tau }^{\\tau }=c\\cdot \\mathsf {P}_{\\upsilon ^*},$ where $c$ is one of $\\pm \\frac{\\operatorname{dim}(\\tau )}{\\operatorname{dim}(\\upsilon )}$ .",
"Since $c=\\langle T_{\\tau }^{\\tau }{\\text{$v$}},{\\text{$v$}}\\rangle _K=\\int _M|{\\text{$v$}}(m)|^2\\,dm\\ge 0$ for any unit vector ${\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau }\\cap {\\mathcal {U}}(\\upsilon ,s)_{\\upsilon ^*}$ , we get $c=\\frac{\\operatorname{dim}(\\tau )}{\\operatorname{dim}(\\upsilon )}$ ."
],
[
"Asymptotic expansions of matrix coefficients",
"We fix a complementary series representation ${\\mathcal {U}}(\\upsilon ,s)$ of $G$ for some $\\upsilon \\in \\hat{M}$ and $s\\in {\\mathcal {I}}_{\\upsilon }$ .",
"The main goal of this section is to obtain effective expansions of matrix coefficients for ${\\mathcal {U}}(\\upsilon ,s)$ .",
"More precisely, we will write a matrix coefficient $\\langle U^s(a_t) {\\text{$v$}}, {\\text{$u$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}$ as a main term that decays like $e^{(s-d)t}$ as $t\\rightarrow \\infty $ and an error term that decays exponentially faster depending only on $2s-d>0$ .",
"To do this, we first work out the asymptotics of matrix coefficients with respect to $\\langle \\cdot ,\\cdot \\rangle _K$ and then use explicit formulas for intertwining operators to convert our results into statements for $\\langle \\cdot ,\\cdot \\rangle _{{\\mathcal {U}}(\\upsilon ,s)}$ ."
],
[
"Matrix coefficients with respect to $\\langle \\cdot ,\\cdot ,\\rangle _K$ .",
"We start by proving a bound on how far elements of $K$ move vectors in representations of $K$ .",
"We let $\\operatorname{dist}_K$ denote the bi-invariant Riemannian metric on $K$ induced from the negative of the Killing form on $\\mathfrak {k}$ , and let $|\\cdot |_K$ denote the corresponding norm on $\\mathfrak {k}$ .",
"Lemma 4.1 Let $(\\pi ,V)$ be a finite-dimensional unitary representation of $K$ with invariant inner product $(\\cdot ,\\cdot )_V$ .",
"Then for all ${\\text{$v$}}\\in V$ and $k\\in K$ , we have $\\Vert \\pi (k){\\text{$v$}}-{\\text{$v$}}\\Vert _{V} \\ll \\operatorname{dist}_K(k,e)\\,\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^1(V)},$ where the implied constant depends solely on the choice of basis defining $\\Vert \\cdot \\Vert _{{\\mathcal {S}}^m(V)}$ .",
"Since $\\operatorname{dist}_K$ is bi-invariant, there exists $J\\in \\mathfrak {k}$ with $|J|_K=1$ such that the $k=\\exp (\\operatorname{dist}_K(k,e) J)$ .",
"This gives $\\pi (k){\\text{$v$}}-{\\text{$v$}}&=\\int _0^{\\operatorname{dist}_K(k,e)} \\frac{d}{dt} \\pi \\big ( \\exp (tJ)\\big ){\\text{$v$}}\\,dt\\\\&=\\int _0^{\\operatorname{dist}_K(k,e)} \\pi (\\exp (tJ))d\\pi \\big ( \\mathrm {Ad}({\\exp (-tJ)})J\\big ){\\text{$v$}}\\,dt.$ Since $\\pi $ is unitary, $|J|_K=1$ , and $K$ is compact, we get $\\big \\Vert \\pi (\\exp (tJ))d\\pi \\big ( \\mathrm {Ad}({\\exp (-tJ)})J\\big ){\\text{$v$}}\\big \\Vert _V\\le \\max _{\\overset{X\\in \\mathfrak {k}}{|X|_K=1}}\\Vert d\\pi ( X){\\text{$v$}}\\Vert _V\\ll \\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^1(V)}.$ Definition 4.1 The Harish-Chandra $c$ -function $C_+(s): {\\mathcal {U}}(\\upsilon ,s)\\rightarrow {\\mathcal {U}}(\\upsilon ,s)$ is defined as follows: $C_+(s)=\\int _{\\overline{N}} U^s\\big (\\kappa (\\overline{n})^{-1}\\big ) e^{-s H(\\overline{n})}\\,d\\overline{n}.$ Since $U^s$ is unitary on ${\\mathcal {U}}(\\upsilon ,s)$ and $\\int _{\\overline{N}} e^{-s H(\\overline{n})}\\,d\\overline{n}<\\infty $ for all $s>\\frac{d}{2}$ (cf.",
"[20]), $C_+(s)$ is a well-defined bounded operator on ${\\mathcal {U}}(\\upsilon ,s)$ .",
"Since $C_+(s)$ is defined using only the restriction of $U^s$ to $K$ , it preserves the $K$ -types of ${\\mathcal {U}}(\\upsilon ,s)$ .",
"Lemma 4.2 $C_+(s)$ preserves the $M$ -types of ${\\mathcal {U}}(\\upsilon ,s)$ .",
"Note that $M$ normalizes $\\overline{N}$ , $d\\overline{n}=d(m\\overline{n}m^{-1})$ and $H(mgm^{-1})=H(g)$ for all $m\\in M$ , $\\overline{n}\\in \\overline{N}$ , and $g\\in G$ .",
"Therefore for all $m\\in M$ , $C_+(s)U^s(m)&=\\int _{\\overline{N}} U^s\\big (\\kappa (\\overline{n})^{-1}m\\big ) e^{-s H(\\overline{n})}\\,d\\overline{n}\\\\&=\\int _{\\overline{N}} U^s\\big (m\\kappa (m^{-1}\\overline{n}m)^{-1}\\big ) e^{-s H(\\overline{n})}\\,d\\overline{n}=U^s(m)C_+(s);$ hence the claim follows.",
"For $d/2<s< d$ , set $\\eta _s=\\min \\lbrace 2s-d, 1\\rbrace >0.$ We remark that the following theorem was shown in [27] for some $\\eta _s$, based on Harish-Chandra's expansion formula and the maximum modulus principle.",
"We give a more direct argument, with the explicit $\\eta _s$ given in (REF ).",
"Let ${\\mathcal {S}}^1(K)$ denote the Sobolev norm ${\\mathcal {S}}^1(L^2(K))$ defined in Section REF .",
"Theorem 4.3 Let $\\tau _1, \\tau _2$ be $K$ -types of ${\\mathcal {U}}(\\upsilon ,s)$ .",
"For all ${\\text{$u$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _1}$ and ${\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}$ , we have for any $t\\ge 0$ , $\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{K}=e^{(s-d)t}&\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}},{\\text{$v$}}\\rangle _{K}\\\\&\\quad + O_s\\left(e^{(s-d-\\eta _s)t}\\Vert T_{\\tau _1}^{\\tau _2}\\Vert _K\\Vert {\\text{$u$}}\\Vert _{K} \\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^1(K)}\\right),$ where the implied constant is uniformly bounded over $s$ in compact subsets of $(\\frac{d}{2},d)$ .",
"Applying Theorem REF , we have $\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{K}=\\int _K e^{(s-d)H(a_tk)} \\langle U^s\\big (\\kappa (a_tk)\\big ) T_{\\tau _1}^{\\tau _2}U^s(k^{-1}){\\text{$u$}},{\\text{$v$}}\\big \\rangle _{K}\\,dk.$ Since the function $k\\mapsto e^{(s-d)H(a_tk)} \\langle U^s\\big (\\kappa (a_tk)\\big ) T_{\\tau _1}^{\\tau _2}U^s(k^{-1}){\\text{$u$}},{\\text{$v$}}\\big \\rangle _{{\\mathcal {U}}(\\upsilon ,s)}$ is right $M$ -invariant, we may use the integration formula [20] to obtain $\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{K}\\\\=\\int _{\\overline{N}}e^{(s-d)H(a_t\\kappa (\\overline{n}))} \\big \\langle U^s\\big (\\kappa (a_t\\kappa (\\overline{n}))\\big ) T_{\\tau _1}^{\\tau _2}U^s(\\kappa (\\overline{n})^{-1}){\\text{$u$}},{\\text{$v$}}\\big \\rangle _{K}e^{-d\\,H(\\overline{n})}\\,d\\overline{n}.$ The identities $H\\big (a_t\\kappa (\\overline{n})\\big )&=H(a_t\\overline{n}a_{-t})+H(a_t)-H(\\overline{n})\\quad \\text{ and }\\\\\\kappa \\big (a_t\\kappa (\\overline{n})\\big )&=\\kappa (a_t\\overline{n}a_{-t})$ then give that the previous integral is equal to $e^{(s-d)t}\\int _{\\overline{N}}\\big \\langle T_{\\tau _1}^{\\tau _2}U^s(\\kappa (\\overline{n})^{-1}){\\text{$u$}},U^s\\big (\\kappa (a_t\\overline{n}a_{-t})^{-1}\\big ){\\text{$v$}}\\big \\rangle _{K}e^{(s-d)H(a_t\\overline{n}a_{-t})-s\\,H(\\overline{n})}\\,d\\overline{n}.$ We now use the identification of $\\overline{N}$ with ${\\mathbb {R}}^d$ to again rewrite: $e^{(s-d)t}\\int _{{\\mathbb {R}}^d}\\big \\langle T_{\\tau _1}^{\\tau _2}U^s(\\kappa (\\overline{n}_{{\\text{$x$}}})^{-1}){\\text{$u$}},U^s\\big (\\kappa (\\overline{n}_{e^{-t}{\\text{$x$}}})^{-1}\\big ){\\text{$v$}}\\big \\rangle _{K}e^{(s-d)H(\\overline{n}_{e^{-t}{\\text{$x$}}})-s\\,H(\\overline{n}_{{\\text{$x$}}})}\\,d{\\text{$x$}};$ note that the integral is absolutely convergent due to $s>\\frac{d}{2}$ .",
"Using the fact that $e^{H(\\overline{n}_{{\\text{$x$}}})}= 1+\\Vert {\\text{$x$}}\\Vert ^2$ (cf.",
"[21]) gives $\\langle & U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{K}=e^{(s-d)t}\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}},{\\text{$v$}}\\rangle _{K}+\\\\&e^{(s-d)t}\\int _{{\\mathbb {R}}^d} \\big \\langle T_{\\tau _1}^{\\tau _2}U^s(\\kappa (\\overline{n}_{{\\text{$x$}}})^{-1}){\\text{$u$}},(1+\\Vert e^{-t}{\\text{$x$}}\\Vert ^2)^{s-d}U^s\\big (\\kappa (\\overline{n}_{e^{-t}{\\text{$x$}}})^{-1}\\big ){\\text{$v$}}-{\\text{$v$}}\\big \\rangle _{K} {\\textstyle \\frac{d{\\text{$x$}}}{(1+\\Vert {\\text{$x$}}\\Vert ^2)^{s}}}.$ Changing to spherical coordinates, we let ${\\text{$x$}}=(r, {\\text{$\\theta $}})$ , $r\\ge 0$ , ${\\text{$\\theta $}}\\in \\mathbb {S}^{d-1}$ and set ${\\text{$w$}}(r,{\\text{$\\theta $}})=(1+r^2)^{s-d}U^s\\big (\\kappa (\\overline{n}_{(r,{\\text{$\\theta $}})})^{-1}\\big ){\\text{$v$}}-{\\text{$v$}}.$ Using this, we deduce $\\langle & U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{K}=e^{(s-d)t}\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}},{\\text{$v$}}\\rangle _{K}+\\\\&O\\left(e^{(s-d)t}\\int _0^{\\infty } \\int _{\\mathbb {S}^{d-1}} \\big \\langle T_{\\tau _1}^{\\tau _2}U^s(\\kappa (\\overline{n}_{(r,{\\text{$\\theta $}})})^{-1}){\\text{$u$}},{\\text{$w$}}(e^{-t}r,{\\text{$\\theta $}})\\big \\rangle _{K} \\,dm({\\text{$\\theta $}})\\, \\tfrac{r^{d-1}}{ (1+r^2)^{s}}\\,dr\\right),$ where $dm({\\text{$\\theta $}})$ denotes the spherical measure.",
"Since the map $\\overline{n}\\mapsto \\kappa (\\overline{n})$ is smooth, we get $d_K\\big (\\kappa (\\overline{n}_{(r,{\\text{$\\theta $}})}),e\\big )\\ll r\\qquad \\text{for all } r>0, {\\text{$\\theta $}}\\in \\mathbb {S}^{d-1}.$ Using $(1+r^2)^{s-d}= 1 -O_s(r)$ (with the implied constant depending continuously on $s$ ) and Lemma REF , we have $\\Vert {\\text{$w$}}(r,\\theta )\\Vert _K\\ll _{s} \\min \\lbrace 1, r\\rbrace \\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^1(K)}.$ This gives $&\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{K}=e^{(s-d)t}\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}},{\\text{$v$}}\\rangle _{K}\\\\&\\qquad + O_s\\left(\\Vert T_{\\tau _1}^{\\tau _2}\\Vert _{K}\\Vert {\\text{$u$}}\\Vert _K \\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^1(K)} e^{(s-d)t}\\int _0^{\\infty } \\min \\lbrace 1, e^{-t}r\\rbrace (1+r^2)^{-s} r^{d-1}\\,dr\\right).$ The proof is completed by writing the integral $\\int _0^{\\infty }$ as $\\int _0^1+\\int _1^{e^t}+\\int _{e^t}^{\\infty }$ to obtain $&\\int _0^{\\infty } \\min \\lbrace 1, e^{-t}r\\rbrace (1+r^2)^{-s} r^{d-1}\\,dr\\\\&\\le e^{-t}+e^{-t}\\int _1^{e^t} r\\,\\cdot \\,r^{-2s}\\,\\cdot \\,r^{d-1}\\,dr+\\int _{e^t}^{\\infty } r^{-2s}\\,\\cdot r^{d-1}\\,dr\\\\&\\ll _s\\, e^{-t} +e^{(d-2s)t}.$"
],
[
"The invariant inner product on ${\\mathcal {U}}(\\upsilon ,s)$",
"The intertwining operator ${\\mathcal {A}}(\\upsilon ,s)$ on ${\\mathcal {U}}(\\upsilon ,s)$ is defined so that $\\langle {\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}=\\langle {\\text{$u$}}, {\\mathcal {A}}(\\upsilon ,s){\\text{$v$}}\\rangle _K$ for all $K$ -finite vectors ${\\text{$u$}},{\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)$ .",
"The key intertwining property of ${\\mathcal {A}}(\\upsilon ,s)$ reads (cf.",
"[22]) ${\\mathcal {A}}(\\upsilon ,s)U^s(g)=U^{d-s}(g){\\mathcal {A}}(\\upsilon ,s)\\qquad \\text{for all $g\\in G.$}$ In particular, ${\\mathcal {A}}(\\upsilon ,s)$ commutes with $U^s(k)$ for all $k\\in K$ .",
"Since each $K$ -type occurs at most once in ${\\mathcal {U}}(\\upsilon ,s)$ , by Schur's lemma, ${\\mathcal {A}}(\\upsilon ,s)$ acts as a scalar $a(\\upsilon ,s,\\tau )$ on each $K$ -type $\\tau $ of ${\\mathcal {U}}(\\upsilon ,s)$ : ${\\mathcal {A}}(\\upsilon ,s)=\\sum _{\\tau \\supset \\upsilon } a(\\upsilon ,s,\\tau )\\mathsf {P}_{\\tau }.$ The positive definiteness of the inner product $\\langle \\cdot ,\\cdot \\rangle _{{\\mathcal {U}}(\\upsilon ,s)}$ implies that for all $\\tau \\in \\hat{K}$ contained in ${\\mathcal {U}}(\\upsilon ,s)$ , we have $a(\\upsilon ,s,\\tau )>0.$ Recalling the parameterization of $K$ and $M$ types given in Section REF , we now assume that ${\\mathcal {U}}(\\upsilon ,s)$ has a non-trivial $M$ -invariant vector.",
"There is then a $K$ -type $\\sigma =(\\sigma _1,\\sigma _2,\\ldots )$ of ${\\mathcal {U}}(\\upsilon ,s)$ that contains the trivial representation of $M$ .",
"Thus $(\\sigma _1,\\sigma _2,\\ldots )$ satisfies the interlacing relation with the trivial representation $(0,0,0,\\ldots ,0)$ of $M$ : $ \\sigma _1 \\ge 0 \\ge \\sigma _2\\ge 0 \\ge \\sigma _3\\ge \\ldots .",
"$ From this, we conclude that $\\sigma =(\\sigma _1,0,0,\\ldots ,0)$ .",
"Now writing $\\upsilon =(\\upsilon _1,\\upsilon _2,\\ldots )$ , the classification of $K$ -types of ${\\mathcal {U}}(\\upsilon ,s)$ ensures that $\\upsilon $ is a subrepresentation of $\\sigma $ , and so $(\\upsilon _1,\\upsilon _2,\\ldots )$ must satisfy the interlacing relation with $(\\sigma _1,0,0,0,\\ldots )$ .",
"We therefore see that $\\upsilon =(\\upsilon _1,0,0,0,\\ldots )$ .",
"For notational convenience we will simply write $\\upsilon =(\\upsilon ,0,0,0,\\ldots )\\in {\\mathbb {Z}}^{\\lfloor \\frac{d}{2}\\rfloor }$ , where $\\upsilon \\ge 0$ .",
"Note that if $d=1$ or 2, then by the classification of the unitary dual of $\\operatorname{SO}(2,1)$ and $\\operatorname{SO}(3,1)$ , $\\upsilon =0$ [16].",
"Combining the fact that each $K$ -type of ${\\mathcal {U}}(\\upsilon ,s)$ must have $\\upsilon $ as a subrepresentation with the interlacing relation gives that all $K$ -types $\\tau $ of ${\\mathcal {U}}(\\upsilon ,s)$ may be parameterized as vectors $\\tau =(t_1,t_2,0,\\ldots ,0)\\in {\\mathbb {Z}}^{\\lfloor \\frac{d+1}{2}\\rfloor }$ with $t_1\\ge \\upsilon \\ge t_2\\ge 0$ if $d>3$ , $t_1\\ge \\upsilon \\ge |t_2|$ if $d=3$ , $t_1\\ge 0$ if $d=2$ , and just $t_1\\in {\\mathbb {Z}}$ if $d=1$ .",
"We will thus write $\\tau =(t_1,t_2)$ and $a(\\upsilon ,s,\\tau )=a(\\upsilon ,s,t_1,t_2)$ , with $\\upsilon =t_2=0$ for $d=1,2$ .",
"Again using the classification of the unitary dual of $G$ , observe that if $s\\ge d-1$ , then $\\upsilon =t_2=0$ .",
"Finally, we let $\\Omega _K\\in Z(\\mathfrak {k}_{{\\mathbb {C}}})$ denote the Casimir operator of $K$ .",
"The following lemma is the main technical result needed to establish the bounds on the quotients $\\frac{a(\\upsilon ,s,\\tau _1)}{a(\\upsilon ,s,\\tau _2)}$ given in Propositions REF and REF .",
"Lemma 4.4 Assume that ${\\mathcal {U}}(\\upsilon ,s)$ has a non-trivial $M$ -invariant vector.",
"Then for any $K$ -types $\\tau _1=(t_1,t_2)$ and $\\tau _2=(t_3,t_4)$ of ${\\mathcal {U}}(\\upsilon ,s)$ , if $s<d-1$ , $&\\frac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)}=\\frac{\\Gamma (d-s+t_3)}{\\Gamma (s+t_3)}\\,\\cdot \\, \\frac{\\Gamma (d-s+t_4-1)}{\\Gamma (s+t_4-1)}\\,\\cdot \\,\\frac{\\Gamma (s+t_1)}{\\Gamma (d-s+t_1)}\\,\\cdot \\, \\frac{\\Gamma (s+t_2-1)}{\\Gamma (d-s+t_2-1)},$ and if $s\\ge d-1$ , $&\\frac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)}=\\frac{\\Gamma (d-s+t_3)}{\\Gamma (s+t_3)}\\,\\cdot \\, \\frac{\\Gamma (s+t_1)}{\\Gamma (d-s+t_1)}.$ The claimed formula follows from a recursion formula similar to that for the Harish-Chandra $c$ -function given in [10].",
"We start by letting $H\\in \\mathfrak {a}$ be such that $a_t=\\exp (t H)$ , and defining an inner product $\\langle \\cdot ,\\cdot \\rangle _{\\fg }$ on $\\fg _{{\\mathbb {C}}}$ by $\\langle X,Y\\rangle _{\\fg }:=c\\cdot \\left( -B(X,\\theta Y) \\right),$ where $B(\\cdot ,\\cdot )$ denotes the Killing form on $\\fg $ , $\\theta $ is the Cartan involution defining $K$ , and $c\\in {\\mathbb {R}}_{>0}$ is chosen so that $ \\langle H,H\\rangle _{\\fg }=1$ .",
"Denote the $-1$ -eigenspace of $\\theta $ by $\\mathfrak {p}$ .",
"Then for all $X\\in \\mathfrak {p}_{{\\mathbb {C}}}$ , ${\\text{$v$}}\\in C^{\\infty }(K)$ , and $k\\in K$ , we have $&[dU^s(X){\\text{$v$}}](k)= (s-{\\textstyle \\frac{d}{2}})\\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\fg } {\\text{$v$}}(k)\\\\&\\quad \\quad \\quad +{\\textstyle \\frac{1}{2}}\\Big ( d\\rho (\\Omega _K)\\big \\lbrace \\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\mathfrak {g}} {\\text{$v$}}(k) \\big \\rbrace -\\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\mathfrak {g}} [d\\rho (\\Omega _K){\\text{$v$}}](k) \\Big ),$ where $\\rho $ denotes right-translation (cf.",
"[42] and [10]).",
"Let $\\tau =(r_1,r_2)$ be an arbitrary $K$ -type of ${\\mathcal {U}}(\\upsilon ,s)$ .",
"By e.g.",
"[42] (cf.",
"also [21]), for any vector ${\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau }$ , $d\\rho (\\Omega _K){\\text{$v$}}=dU^s(\\Omega _K){\\text{$v$}}=(r_1^2+r_2^2+(d-1)r_1+(d-3)r_2){\\text{$v$}}.$ We now denote the orthogonal projection onto $\\tau $ by $\\mathsf {P}_{(r_1,r_2)}$ .",
"Combining the above expression for $d\\rho (\\Omega _K){\\text{$v$}}$ with (REF ) gives $&[\\mathsf {P}_{(r_1+1,r_2)}dU^s(X){\\text{$v$}}](k)\\\\&= \\left( s-{\\textstyle \\frac{d}{2}} +{\\textstyle \\frac{(r_1+1)^2+r_2^2+(d-1)(r_1+1)+(d-3)r_2}{2}}-{\\textstyle \\frac{r_1^2+r_2^2+(d-1)r_1+(d-3)r_2}{2}} \\right)\\\\&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\times \\mathsf {P}_{(r_1+1,r_2)}\\left\\lbrace \\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\fg } {\\text{$v$}}(k) \\right\\rbrace \\\\&= \\left( s +r_1\\right)\\mathsf {P}_{(r_1+1,r_2)}\\left\\lbrace \\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\fg } {\\text{$v$}}(k) \\right\\rbrace , $ and similarly $[\\mathsf {P}_{(r_1,r_2+1)}dU^s(X){\\text{$v$}}](k)= \\left( s +r_2-1\\right)\\mathsf {P}_{(r_1,r_2+1)}\\left\\lbrace \\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\fg } {\\text{$v$}}(k) \\right\\rbrace .",
"$ From (REF ), we obtain $[\\mathsf {P}_{(r_1,r_2)}dU^s(X){\\mathcal {A}}(\\upsilon ,s){\\text{$v$}}](k)=[\\mathsf {P}_{(r_1,r_2)}{\\mathcal {A}}(\\upsilon ,s)dU^{d-s}(X){\\text{$v$}}](k).$ Combining this with (REF ) and (REF ) and (REF ), respectively, gives $&a(\\upsilon ,s,r_1,r_2) (s+r_1)\\mathsf {P}_{(r_1+1,r_2)}\\left\\lbrace \\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\fg } {\\text{$v$}}(k) \\right\\rbrace \\\\&\\qquad =a(\\upsilon ,s,r_1+1,r_2) (d-s+r_1)\\mathsf {P}_{(r_1+1,r_2)}\\left\\lbrace \\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\fg } {\\text{$v$}}(k) \\right\\rbrace ,$ and $&a(\\upsilon ,s,r_1,r_2) (s+r_2-1)\\mathsf {P}_{(r_1,r_2+1)}\\left\\lbrace \\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\fg } {\\text{$v$}}(k) \\right\\rbrace \\\\&\\qquad =a(\\upsilon ,s,r_1,r_2+1) (d-s+r_2-1)\\mathsf {P}_{(r_1,r_2+1)}\\left\\lbrace \\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\fg } {\\text{$v$}}(k) \\right\\rbrace .$ The decomposition of $(\\mathrm {Ad},\\mathfrak {p}_{{\\mathbb {C}}})\\otimes (U^s,{\\mathcal {U}}(\\upsilon ,s)_{\\tau })$ into irreducible representations of $K$ ensures the existence of $X,Y\\in \\mathfrak {p}_{{\\mathbb {C}}}$ and ${\\text{$v$}}_1,{\\text{$v$}}_2\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau }$ such that $\\mathsf {P}_{(r_1+1,r_2)}\\left\\lbrace \\langle \\mathrm {Ad}_{k^{-1}}X,H\\rangle _{\\fg } {\\text{$v$}}_1(k)\\right\\rbrace \\lnot \\equiv 0$ , and $\\mathsf {P}_{(r_1,r_2+1)}\\left\\lbrace \\langle \\mathrm {Ad}_{k^{-1}}Y,H\\rangle _{\\fg } {\\text{$v$}}_2(k)\\right\\rbrace \\lnot \\equiv 0$ if $(r_1+1,r_2)$ , and $(r_1,r_2+1)$ are $K$ -types of ${\\mathcal {U}}(\\upsilon ,s)$ , cf., e.g., [42] and [44].",
"This gives $(s+r_1)\\cdot a(\\upsilon ,s,r_1+1,r_2)=(d-s+r_1)\\cdot a(\\upsilon ,s,r_1,r_2)$ and $(s+r_2-1)\\cdot a(\\upsilon ,s,r_1,r_2+1)=(d-s+r_2-1)\\cdot a(\\upsilon ,s,r_1,r_2)$ (cf.",
"[10]).",
"Note that if one of the four factors that are multiplied with $a(\\upsilon ,s,\\ldots )$ 's is zero, then the representation given by that choice of $\\upsilon $ and $s$ is not in the unitary dual of $G$ .",
"In particular, if $s\\ge d-1$ , then ${\\mathcal {U}}(\\upsilon ,s)$ is spherical, hence $\\upsilon =r_2=0$ .",
"These two recursion formulas imply that if $\\upsilon \\ne 0$ , $a(\\upsilon ,s,r_1,r_2)=& \\tfrac{\\Gamma (d-s+r_1)}{\\Gamma (s+r_1)}\\,\\cdot \\, \\tfrac{\\Gamma (d-s+r_2-1)}{\\Gamma (s+r_2-1)}\\,\\cdot \\, \\tfrac{\\Gamma (s+\\upsilon )}{\\Gamma (d-s+\\upsilon )}\\cdot \\tfrac{\\Gamma (s+\\upsilon -1)}{\\Gamma (d-s+\\upsilon -1)}\\,\\cdot \\, a(\\upsilon ,s,\\upsilon ,\\upsilon ),$ and if $\\upsilon =0$ , $a(0,s,r_1,0)=& \\tfrac{\\Gamma (d-s+r_1)}{\\Gamma (s+r_1)}\\,\\cdot \\, \\tfrac{\\Gamma (s)}{\\Gamma (d-s)}\\,\\cdot \\, a(0,s,0,0).$ The formulas claimed in the proposition then follow.",
"By Schur's lemma, $\\Omega _K$ acts on any realization of a $K$ -type $\\tau $ by the same scalar, which we denote $\\tau (\\Omega _K)$ .",
"Proposition 4.5 Assume that ${\\mathcal {U}}(\\upsilon ,s)$ has a non-trivial $M$ -invariant vector.",
"Then for any $K$ -types $\\tau _1,\\tau _2$ of ${\\mathcal {U}}(\\upsilon ,s)$ , $\\frac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)} \\ll _s \\big (1+\\tau _1(\\Omega _K)^{d}\\big )\\big (1+\\tau _2(\\Omega _K)^{d}\\big ),$ and the implied constant is uniformly bounded over $s$ in compact subsets of ${\\mathcal {I}}_{\\upsilon }$ .",
"The key result needed in the proof is a consequence of [17]: for all $t\\ge 0$ , $\\tfrac{\\Gamma (s+t)}{\\Gamma (d-s+t)}\\asymp _s 1+t^{2s-d},$ where the implied constant is uniformly bounded over $s$ in compact subsets of $(\\frac{d}{2},d)$ .",
"We now write $\\tau _1=(t_1,t_2)$ , $\\tau _2=(t_3,t_4)$ .",
"Assuming first that $d>3$ and $\\upsilon >0$ , we then have $t_i\\ge 0$ for all $i$ .",
"Since $\\upsilon >0$ , $s<d-1$ , and so rewriting Lemma REF gives $&\\tfrac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)}=\\tfrac{\\Gamma (d-s+t_3)}{\\Gamma (s+t_3)}\\,\\cdot \\, \\tfrac{\\Gamma (d-s+t_4)}{\\Gamma (s+t_4)}\\,\\cdot \\,\\tfrac{\\Gamma (s+t_1)}{\\Gamma (d-s+t_1)}\\,\\cdot \\, \\tfrac{\\Gamma (s+t_2)}{\\Gamma (d-s+t_2)}\\,\\cdot \\, \\tfrac{s+t_4-1}{d-s+t_4-1}\\,\\cdot \\,\\tfrac{d-s+t_2-1}{s+t_2-1},$ and so (after recalling the formula (REF ) for $\\tau _1(\\Omega _K)$ and $\\tau _2(\\Omega _K)$ ), $\\tfrac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)} \\asymp _s &(1+t_1^{2s-d})(1+t_2^{2s-d})(1+t_3^{2s-d})(1+t_3^{2s-d})\\\\&\\ll \\big (1+\\tau _1(\\Omega _K)^{d}\\big )\\big (1+\\tau _2(\\Omega _K)^{d}\\big ),$ with the implied constant uniformly bounded over $(\\frac{d}{2},d-1)={\\mathcal {I}}_{\\upsilon }$ .",
"In the case $d\\ge 2$ and $\\upsilon =0$ (and so $t_2=t_4=0$ ), Lemma REF gives $&\\tfrac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)}=\\tfrac{\\Gamma (d-s+t_3)}{\\Gamma (s+t_3)}\\,\\cdot \\, \\tfrac{\\Gamma (s+t_1)}{\\Gamma (d-s+t_1)},$ so by a direct application of (REF ) and (REF ), $\\tfrac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)}\\ll _s (1+ t_1^{2s-d})(1+ t_3^{2s-d}) \\ll \\big (1+\\tau _1(\\Omega _K)^{\\frac{d}{2}}\\big )\\big (1+\\tau _2(\\Omega _K)^{\\frac{d}{2}}\\big ),$ with the implied constant uniformly bounded over $s$ in compact subsets of $(\\frac{d}{2},d)$ .",
"For the remaining cases, $d=1$ and $d=3$ with $\\upsilon >0$ , negative values of the $t_i$ can appear in the formulas given in Lemma REF .",
"It thus remains to bound quotients of the form $ \\frac{\\Gamma (s+t)}{\\Gamma (d-s+t)}$ , where $t$ is a negative integer and $s\\in (\\frac{1}{2},1)$ if $d=1$ , and $s\\in (\\frac{3}{2},2)$ if $d=3$ .",
"By the reflection formula, in both cases, $\\tfrac{\\Gamma (s+t)}{\\Gamma (d-s+t)}=\\tfrac{\\Gamma (s-|t|)}{\\Gamma (d-s-|t|)}=\\tfrac{\\Gamma (|t|+1+s-d)}{\\Gamma (|t|+1-s)}.$ If $d=1$ , we then have $\\tfrac{\\Gamma (s+t)}{\\Gamma (1-s+t)}=\\tfrac{\\Gamma (|t|+s)}{\\Gamma (|t|+1-s)},$ and so (REF ) gives $\\tfrac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)}&=\\tfrac{\\Gamma (1-s+|t_3|)}{\\Gamma (s+|t_3|)}\\,\\cdot \\, \\tfrac{\\Gamma (s+|t_1|)}{\\Gamma (1-s+|t_1|)} \\\\&\\ll _s(1+ |t_1|^{2s-1})(1+ |t_2|^{2s-1}) \\ll \\big (1+\\tau _1(\\Omega _K)^{\\frac{1}{2}}\\big )\\big (1+\\tau _2(\\Omega _K)^{\\frac{1}{2}}\\big ).$ If $d=3$ , using (REF ), we have $\\tfrac{\\Gamma (s+t)}{\\Gamma (3-s+t)}=\\tfrac{\\Gamma (|t|+s-2)}{\\Gamma (|t|+2-s)}=\\tfrac{(|t|+1-s)(|t|+2-s)}{(s+|t|-1)(s+|t|-2)}\\,\\cdot \\,\\tfrac{\\Gamma (s+|t|)}{\\Gamma (3-s+|t|)}.$ Now using $\\frac{(|t|+1-s)(|t|+2-s)}{(s+|t|-1)(s+|t|-2)}\\asymp _s 1$ , where the implied constants are uniformly bounded over $s$ in compact subsets of $(\\frac{3}{2},2)$ , together with (REF ) as previously completes the proof.",
"For $K$ -types that contain $M$ -invariant vectors, we have the following strengthening of the bound in Proposition REF : Proposition 4.6 Let $\\tau _1,\\tau _2$ be $K$ -types of ${\\mathcal {U}}(\\upsilon ,s)$ containing non-trivial $M$ -invariant vectors.",
"Then $\\frac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)} \\ll _s \\big (1+\\tau _1(\\Omega _K)^{\\frac{d}{2}}\\big )\\big (1+\\tau _2(\\Omega _K)^{\\frac{d}{2}}\\big ),$ and the implied constant is uniformly bounded over $s$ in compact subsets of $(\\frac{d}{2},d)$ .",
"For $d\\ge 3$ , we observe that if a $K$ -type $\\tau =(r_1,r_2)$ contains an $M$ -invariant vector, the interlacing relation implies $r_2=0$ , hence for $t_1,t_2$ as in the statement of the lemma, we have $\\tau _1=(t_1,0)$ and $\\tau _2=(t_2,0)$ .",
"Lemma REF in this case reads $&\\tfrac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)}=\\tfrac{\\Gamma (d-s+t_2)}{\\Gamma (s+t_2)}\\,\\cdot \\, \\tfrac{\\Gamma (s+t_1)}{\\Gamma (d-s+t_1)}.$ The lemma then follows from (REF ) for $d\\ge 2$ and (REF ) for $d=1$ .",
"Remark 4.9 The main point of Proposition REF is that the implied constant remains bounded even as $s$ approaches $d-1$ in the case $\\upsilon >0$ .",
"This allows us to use the bound uniformly over complementary series representations appearing in the direct integral decomposition of $L^2(\\Gamma \\backslash G)$ .",
"The bounds from Propositions REF and REF allow us to restate Theorem REF in terms of $\\langle \\cdot ,\\cdot \\rangle _{{\\mathcal {U}}(\\upsilon ,s)}$ .",
"This can then be applied to the matrix coefficients of irreducible unitary representations weakly contained in $L^2(\\Gamma \\backslash G)$ .",
"Retaining the notation of Theorem REF , we have: Proposition 4.7 There exists $m\\in {\\mathbb {N}}$ such that for any ${\\mathcal {U}}(\\upsilon ,s)$ with a non-trivial $M$ -invariant vector, for all ${\\text{$u$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _1},{\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}$ , and $t\\ge 0$ , $\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}=&e^{(s-d)t}\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}\\\\&\\qquad \\qquad +O_s\\big (e^{(s-d-\\eta _s)t}\\Vert {\\text{$u$}}\\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)} \\big ),$ where the implied constant is uniformly bounded over $s$ in compact subsets of ${\\mathcal {I}}_{\\upsilon }$ .",
"Furthermore, if the $K$ -types $\\tau _1$ and $\\tau _2$ both contain non-trivial $M$ -invariant vectors, the the implied constant is uniformly bounded over $s$ in compact subsets of $(\\frac{d}{2},d)$ .",
"Since ${\\text{$v$}}\\in {\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}$ , using the expression for $\\langle \\cdot ,\\cdot \\rangle _{{\\mathcal {U}}(\\upsilon ,s)}|_{{\\mathcal {U}}(\\upsilon ,s)_{\\tau _2}}$ , Theorem REF gives $&\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}=a(\\upsilon ,s,\\tau _2)\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{K}\\\\&=\\begin{multlined}[t]e^{(s-d)t}a(\\upsilon ,s,\\tau _2)\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}},{\\text{$v$}}\\rangle _{K}\\\\+ O_s\\left(e^{(s-d-\\eta _s)t} a(\\upsilon ,s,\\tau _2)\\Vert T_{\\tau _1}^{\\tau _2}\\Vert _K\\Vert {\\text{$u$}}\\Vert _{K} \\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^1(K)}\\right)\\end{multlined} \\\\&=\\begin{multlined}[t] e^{(s-d)t}\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}\\\\ +O_s\\left(e^{(s-d-\\eta _s)t}\\sqrt{{\\textstyle \\frac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)}}}\\Vert T_{\\tau _1}^{\\tau _2}\\Vert _K\\Vert {\\text{$u$}}\\Vert _{{\\mathcal {U}}(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^1(\\upsilon ,s)}\\right).\\end{multlined}$ By Proposition REF , or Proposition REF if $\\tau _1$ and $\\tau _2$ both have $M$ -invariant vectors, $\\sqrt{{\\textstyle \\frac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)}}}&\\Vert {\\text{$u$}}\\Vert _{{\\mathcal {U}}(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^1(\\upsilon ,s)}\\\\\\ll _s& \\sqrt{(1+\\tau _1(\\Omega _K)^{d})(1+\\tau _2(\\Omega _K)^{d})}\\Vert {\\text{$u$}}\\Vert _{{\\mathcal {U}}(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^1(\\upsilon ,s)}\\\\\\ll & \\Vert (1+dU^s(\\Omega _K^{\\lceil d/2 \\rceil })){\\text{$u$}}\\Vert _{{\\mathcal {U}}(\\upsilon ,s)}\\Vert (1+dU^s(\\Omega _K^{\\lceil d/2 \\rceil })){\\text{$v$}}\\Vert _{{\\mathcal {S}}^1(\\upsilon ,s)}\\\\&\\ll \\Vert {\\text{$u$}}\\Vert _{{\\mathcal {S}}^{d+1}(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^{d+2}(\\upsilon ,s)},$ with the implied constant uniformly bounded over $s$ in compact subsets of ${\\mathcal {I}}_{\\upsilon }$ , or $(\\frac{d}{2},d)$ , respectively.",
"Now Corollary REF and Lemma REF give $\\Vert T_{\\tau _1}^{\\tau _2}\\Vert _K\\Vert {\\text{$u$}}\\Vert _{{\\mathcal {S}}^{d+1}(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^{d+2}(\\upsilon ,s)}\\le &\\sqrt{\\operatorname{dim}(\\tau _1)\\operatorname{dim}(\\tau _2)}\\Vert {\\text{$u$}}\\Vert _{{\\mathcal {S}}^{d+1}(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^{d+2}(\\upsilon ,s)}\\\\&\\ll \\Vert {\\text{$u$}}\\Vert _{{\\mathcal {S}}^{m}(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^{m}(\\upsilon ,s)}$ for some $m\\in {\\mathbb {N}}$ , completing the proof.",
"Remark 4.10 Both Proposition REF and Proposition REF are expected to hold for all complementary series ${\\mathcal {U}}(\\upsilon ,s)$ .",
"We have proved them only for those representations with $M$ -invariant vectors since in this case slight simplifications occur in the proof of Lemma REF .",
"Note that if Proposition REF were proved for all complementary series representations, Proposition REF would also follow automatically for all complementary series.",
"Theorem 4.8 There exists $m\\in {\\mathbb {N}}$ such that for any complementary series representation ${\\mathcal {U}}(\\upsilon ,s)$ containing a non-trivial $M$ -invariant vector, for all ${\\text{$u$}},{\\text{$v$}}\\in {\\mathcal {S}}^m(\\upsilon ,s)$ and $t\\ge 0$ , $\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)} =&e^{(s-d)t}\\left(\\sum _{\\tau _1,\\tau _2\\in {\\hat{K}}}\\langle T_{\\tau _1}^{\\tau _2}C_+(s)\\mathsf {P}_{\\tau _1}{\\text{$u$}},\\mathsf {P}_{\\tau _2}{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}\\right)\\\\&\\qquad \\qquad +O_s\\big (e^{(s-d-\\eta _s)t}\\Vert {\\text{$u$}}\\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)} \\big ),$ and the sum $\\sum _{\\tau _1,\\tau _2\\in {\\hat{K}}}\\langle T_{\\tau _1}^{\\tau _2}C_+(s)\\mathsf {P}_{\\tau _1}{\\text{$u$}},\\mathsf {P}_{\\tau _2}{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}$ converges absolutely.",
"Since smooth vectors are dense in ${\\mathcal {S}}^m(\\upsilon ,s)$ for all $m\\in {\\mathbb {N}}$ , and both sides of the inequality are continuous with respect to $\\Vert \\cdot \\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)}$ , we start by assuming that ${\\text{$u$}}$ and ${\\text{$v$}}$ are smooth, and decompose them according to the $K$ -types of ${\\mathcal {U}}(\\upsilon ,s)$ : ${\\text{$u$}}=\\sum _{\\tau _1\\subset {\\mathcal {U}}(\\upsilon ,s)} {\\text{$u$}}_{\\tau _1},\\qquad {\\text{$v$}}=\\sum _{\\tau _2\\subset {\\mathcal {U}}(\\upsilon ,s)} {\\text{$v$}}_{\\tau _2},$ where ${\\text{$u$}}_{\\tau _1}=\\mathsf {P}_{\\tau _1}{\\text{$u$}}$ and ${\\text{$u$}}_{\\tau _2}=\\mathsf {P}_{\\tau _2}{\\text{$v$}}$ .",
"By [46], $\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}=\\sum _{\\tau _1,\\tau _2} \\langle U^s(a_t){\\text{$u$}}_{\\tau _1},{\\text{$v$}}_{\\tau _2}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)},$ with the sum converging absolutely.",
"Applying Proposition REF gives $\\langle U^s(a_t){\\text{$u$}}_{\\tau _1},{\\text{$v$}}_{\\tau _2}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}=e^{(s-d)t}&\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}}_{\\tau _1},{\\text{$v$}}_{\\tau _2}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}\\\\&+O_s\\big (e^{(s-d-\\eta _s)t}\\Vert {\\text{$u$}}_{\\tau _1}\\Vert _{{\\mathcal {S}}^{m^{\\prime }}(\\upsilon ,s)}\\Vert {\\text{$v$}}_{\\tau _2}\\Vert _{{\\mathcal {S}}^{m^{\\prime }}(\\upsilon ,s)} \\big )$ for some $m^{\\prime }\\in {\\mathbb {N}}$ .",
"Hence $\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)} =e^{(s-d)t}\\left(\\sum _{\\tau _1,\\tau _2}\\langle T_{\\tau _1}^{\\tau _2}C_+(s)\\mathsf {P}_{\\tau _1}{\\text{$u$}},\\mathsf {P}_{\\tau _2}{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}\\right)\\\\+O_s\\left( e^{(s-d-\\eta _s)t}\\left(\\sum _{\\tau }\\Vert {\\text{$u$}}_{\\tau }\\Vert _{{\\mathcal {S}}^{m^{\\prime }}(\\upsilon ,s)} \\right) \\left(\\sum _{\\tau }\\Vert {\\text{$v$}}_{\\tau }\\Vert _{{\\mathcal {S}}^{m^{\\prime }}(\\upsilon ,s)} \\right)\\right).$ By Lemma REF , there exists $m\\ge m^{\\prime }$ (depending only on $K$ ) large enough such that $\\sum _{\\tau \\in {\\mathcal {U}}(\\upsilon , s)}\\Vert {\\text{$u$}}_{\\tau }\\Vert _{{\\mathcal {S}}^{m^{\\prime }}(\\upsilon ,s)} \\ll \\Vert {\\text{$u$}}\\Vert _{{\\mathcal {S}}^{m}(\\upsilon ,s)}\\quad \\text{ and }\\quad \\sum _{\\tau \\in {\\mathcal {U}}(\\upsilon , s)}\\Vert {\\text{$v$}}_{\\tau }\\Vert _{{\\mathcal {S}}^{m^{\\prime }}(\\upsilon ,s)} \\ll \\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^{m}(\\upsilon ,s)}$ where the implied constants depend only on $K$ .",
"It remains to prove that the sum $\\sum _{\\tau _1,\\tau _2}\\langle T_{\\tau _1}^{\\tau _2}C_+(s)\\mathsf {P}_{\\tau _1}{\\text{$u$}},\\mathsf {P}_{\\tau _2}{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}$ converges absolutely.",
"Looking at an individual summand, we have $|\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}}_{\\tau _1},{\\text{$v$}}_{\\tau _2}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}| &=a(\\upsilon ,s,\\tau _2)|\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}}_{\\tau _1},{\\text{$v$}}_{\\tau _2}\\rangle _{K}|\\\\&\\le a(\\upsilon ,s,\\tau _2)\\Vert T_{\\tau _1}^{\\tau _2}\\Vert _{K} \\cdot \\Vert C_+(s){\\text{$u$}}_{\\tau _1}\\Vert _K \\cdot \\Vert {\\text{$v$}}_{\\tau _2}\\Vert _K$ Since $U^s|_K$ is unitary on $L^2(K)$ , from the definition of $C_+(s)$ (see Theorem REF ), we get $\\Vert C_+(s){\\text{$u$}}_{\\tau _1}\\Vert _K\\ll _s \\Vert {\\text{$u$}}_{\\tau _1}\\Vert _K,$ giving $|\\langle T_{\\tau _1}^{\\tau _2}C_+(s){\\text{$u$}}_{\\tau _1},{\\text{$v$}}_{\\tau _2}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}|&\\ll _s a(\\upsilon ,s,\\tau _2)\\Vert T_{\\tau _1}^{\\tau _2}\\Vert _{K} \\cdot \\Vert {\\text{$u$}}_{\\tau _1}\\Vert _K \\cdot \\Vert {\\text{$v$}}_{\\tau _2}\\Vert _K\\\\&\\le \\sqrt{{\\textstyle \\frac{a(\\upsilon ,s,\\tau _2)}{a(\\upsilon ,s,\\tau _1)}}}\\Vert T_{\\tau _1}^{\\tau _2}\\Vert _{K} \\cdot \\Vert {\\text{$u$}}_{\\tau _1}\\Vert _{{\\mathcal {U}}(\\upsilon ,s)} \\cdot \\Vert {\\text{$v$}}_{\\tau _2}\\Vert _{{\\mathcal {U}}(\\upsilon ,s)}.$ This expression is now bounded using Proposition REF , Corollary REF , and Lemma REF as in the proof of Proposition REF .",
"Lemma REF then gives the desired convergence of the sum.",
"Theorem 4.9 There exists $m\\in {\\mathbb {N}}$ such that for any non-spherical complementary series representation ${\\mathcal {U}}(\\upsilon ,s)$ , for all $M$ -invariant vectors ${\\text{$u$}}$ , ${\\text{$v$}}\\in {\\mathcal {S}}^m(\\upsilon ,s)$ , and $t\\ge 0$ , we have $|\\langle U^s(a_t){\\text{$u$}},{\\text{$v$}}\\rangle _{{\\mathcal {U}}(\\upsilon ,s)}|\\ll _s e^{(s-d-\\eta _s)t}\\Vert {\\text{$u$}}\\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)}\\Vert {\\text{$v$}}\\Vert _{{\\mathcal {S}}^m(\\upsilon ,s)},$ where the implied constant is uniformly bounded over $s$ in compact subsets of $(\\frac{d}{2},d)$ .",
"Note that since ${\\text{$u$}}$ , ${\\text{$v$}}$ are both $M$ -invariant and $M\\subset K$ , $\\mathsf {P}_{\\tau }{\\text{$u$}}$ and $\\mathsf {P}_{\\tau }{\\text{$v$}}$ are as well for any $K$ -type $\\tau $ of ${\\mathcal {U}}(\\upsilon ,s)$ .",
"Proposition REF then gives that the implied constant in Theorem REF is uniformly bounded over $s$ in compact subsets of $(\\frac{d}{2},d)$ .",
"Thus, in order to prove the theorem, it suffices to show that for any $K$ -types $\\tau _1, \\tau _2$ of ${\\mathcal {U}}(\\upsilon ,s)$ and an arbitrary $M$ -invariant vector ${\\text{$w$}}$ of ${\\mathcal {U}}(\\upsilon ,s)$ , $ T_{\\tau _1}^{\\tau _2}C_+(s)\\mathsf {P}_{\\tau _1}{\\text{$w$}}=0.$ Since $C_+(s)$ preserves the $M$ -types of each $K$ -type by Lemma REF and ${\\text{$w$}}$ is $M$ -invariant (hence $\\mathsf {P}_{\\tau _1}{\\text{$w$}}$ is as well), $C_+(s)\\mathsf {P}_{\\tau _1}{\\text{$w$}}$ is $M$ -invariant.",
"Therefore (REF ) follows from Corollary REF ."
],
[
"Leading term for Sobolev functions",
"In this section, we will extend Roblin's result on mixing of the geodesic flow for continuous functions with compact support to general functions in a Sobolev space of sufficiently high order.",
"Recall the following theorem of Roblin [33]: Theorem 5.1 For all $M$ -invariant $f_1,f_2\\in C_c(\\Gamma \\backslash G)$ , $\\lim _{t\\rightarrow +\\infty } e^{(d-\\delta )t}\\langle \\rho (a_t)f_1,f_2\\rangle =m^{\\mathrm {BR}}(f_1)\\,m^{\\mathrm {BR_*}}(f_2).$ We note that $m^{\\operatorname{BR}}(f)$ may be infinite for a general function $f\\in L^2(\\Gamma \\backslash G)$ , and hence Theorem REF does not generalize to arbitrary $L^2$ -functions.",
"In order to extend this theorem to Sobolev functions (which are not necessarily compactly supported), we first prove the following bound on matrix coefficients of $L^2(\\Gamma \\backslash G)$ : Lemma 5.2 There exists $m\\in {\\mathbb {N}}$ such that for all $f_1,f_2\\in {\\mathcal {S}}^m(\\Gamma \\backslash G)$ and $t\\ge 0$ , $|\\langle \\rho (a_t)f_1,f_2\\rangle |\\ll _{\\Gamma } e^{(\\delta -d)t}{\\mathcal {S}}^m(f_1){\\mathcal {S}}^m(f_2).$ We start by assuming that $f_1$ and $f_2$ are $\\rho (K)$ -invariant.",
"Using the notation of Section REF , the decomposition of the functions according to $L^2(\\Gamma \\backslash G)_{\\mathrm {sph}}={\\mathcal {B}}_{\\delta }\\oplus {\\mathcal {W}}$ reads $f_i = \\langle f_i,\\phi _0\\rangle \\phi _0+f_i^{\\prime },\\qquad i=1,2,$ where $f_i^{\\prime }\\in {\\mathcal {W}}$ are $K$ -invariant, and $\\phi _0\\in {\\mathcal {B}}_{\\delta } $ is the base-eigenfunction in $L^2(\\Gamma \\backslash \\mathbb {H}^{d+1})=L^2(\\Gamma \\backslash G)^K$ of unit norm.",
"Since ${\\mathcal {B}}_\\delta $ is a complementary series representation ${\\mathcal {U}}(1, \\delta )$ (cf.",
"Sections REF and REF ), it follows from Theorem REF that for all $t\\ge 0$ , $|\\langle \\rho (a_t)\\phi _0,\\phi _0\\rangle |\\ll e^{(\\delta -d)t}.$ As a consequence of Theorem REF , $f_1^{\\prime }$ and $f_2^{\\prime }$ are orthogonal to any complementary series representation ${\\mathcal {U}}(\\upsilon ,s)$ with $s_1<s\\le \\delta $ .",
"It follows that for any $\\varepsilon >0$ , for any $K$ -invariant $f_1,f_2\\in L^2(\\Gamma \\backslash G)$ and $t>0$ , $|\\langle \\rho (a_t)f_1^{\\prime },f_2^{\\prime }\\rangle |\\ll _{\\varepsilon } e^{(s_1-d+\\varepsilon )t}\\Vert f_1^{\\prime }\\Vert \\Vert f_2^{\\prime }\\Vert $ (see [37], [23]).",
"Therefore, choosing $0<\\varepsilon < \\delta -s_1$ gives $|\\langle \\rho (a_t) f_1,f_2\\rangle | \\ll e^{(\\delta -d)t}\\big ( |\\langle f_1,\\phi _0\\rangle ||\\langle f_2,\\phi _0\\rangle |+\\Vert f_1^{\\prime }\\Vert \\Vert f_2^{\\prime }\\Vert \\big )\\le e^{(\\delta -d)t} \\Vert f_1\\Vert \\Vert f_2\\Vert .$ In view of [37], this bound extends for all $K$ -finite functions $f_1, f_2$ in $L^2(\\Gamma \\backslash G)$ : $| \\langle \\rho (a_t)f_1,f_2\\rangle |\\ll e^{(\\delta -d)t}\\Vert f_1\\Vert \\Vert f_2\\Vert \\sqrt{\\operatorname{dim}(\\rho (K)f_1)\\,\\operatorname{dim}(\\rho (K)f_2)}.$ To pass to Sobolev functions, we first observe that $\\operatorname{dim}(\\rho (K)\\mathsf {P}_{\\tau }f)\\le \\operatorname{dim}(\\tau )^2$ for all $f\\in L^2(\\Gamma \\backslash G)$ (cf.",
"[20]).",
"Then for all $f_1,f_2\\in {\\mathcal {S}}^m(\\Gamma \\backslash G)$ , $|\\langle \\rho (a_t)f_1,f_2\\rangle | &\\le e^{(\\delta -d)t}\\sum _{\\tau _1,\\tau _2\\in {\\hat{K}}} \\operatorname{dim}(\\tau _1)\\operatorname{dim}(\\tau _2)\\Vert \\mathsf {P}_{\\tau _1}f_1\\Vert \\Vert \\mathsf {P}_{\\tau _2}f_2\\Vert \\\\&\\ll e^{(\\delta -d)t} \\Vert f_1\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)} \\Vert f_2\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}$ for $m\\in {\\mathbb {N}}$ large enough, by Lemma REF .",
"Proposition 5.3 There exists $m\\in {\\mathbb {N}}$ such that for all $M$ -invariant $f_1,f_2\\in {\\mathcal {S}}^m(\\Gamma \\backslash G)$ , $\\lim _{t\\rightarrow +\\infty } e^{(d-\\delta )t}\\langle \\rho (a_t)f_1,f_2\\rangle =m^{\\mathrm {BR}}(f_1)\\,m^{\\mathrm {BR_*}}(f_2).$ Let $m>d(d+1)/4$ satisfy the conclusion of Lemma REF .",
"For simplicity, we write $\\Vert f\\Vert _{{\\mathcal {S}}^m}=\\Vert f\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}$ .",
"Using the density of $C^{\\infty }_c(\\Gamma \\backslash G)^M$ in ${\\mathcal {S}}^m(\\Gamma \\backslash G)^M$ , given $\\varepsilon >0$ , there exist $f_1^{\\varepsilon },f_2^{\\varepsilon }\\in C_c^{\\infty }(\\Gamma \\backslash G)$ such that $\\Vert f_i-f_i^{\\varepsilon }\\Vert _{{\\mathcal {S}}^m}\\le \\varepsilon \\qquad i=1,2.$ We then write, using Lemma REF , $&e^{(d-\\delta )t}\\langle \\rho (a_t)f_1,f_2\\rangle \\\\ &=e^{(d-\\delta )t}\\Big (\\langle \\rho (a_t)f_1^{\\varepsilon },f_2^{\\varepsilon }\\rangle +\\langle \\rho (a_t)(f_1-f_1^{\\varepsilon }),f_2\\rangle +\\langle \\rho (a_t)f_1^{\\varepsilon },f_2-f^{\\varepsilon }_2\\rangle \\Big )\\\\ &= e^{(d-\\delta )t}\\langle \\rho (a_t)f_1^{\\varepsilon },f^{\\varepsilon }_2\\rangle + O\\big (\\varepsilon (\\Vert f_1\\Vert _{{\\mathcal {S}}^m}+\\Vert f_2\\Vert _{{\\mathcal {S}}^m})\\big ).$ By Theorem REF , we have $\\lim _{t\\rightarrow \\infty } e^{(d-\\delta )t}\\langle \\rho (a_t)f_1^{\\varepsilon },f_2^{\\varepsilon }\\rangle ={m^{\\mathrm {BR}}}(f_1^{\\varepsilon })\\,{m^{\\mathrm {BR}_*}}(f_2^{\\varepsilon }).$ So $\\limsup _{t\\rightarrow \\infty } e^{(d-\\delta )t}\\langle \\rho (a_t)f_1,f_2\\rangle ={m^{\\mathrm {BR}}}(f_1^{\\varepsilon })\\,{m^{\\mathrm {BR}_*}}(f_2^{\\varepsilon })+ O\\big (\\varepsilon (\\Vert f_1\\Vert _{{\\mathcal {S}}^m}+\\Vert f_2\\Vert _{{\\mathcal {S}}^m})\\big ).$ Since $m> d(d+1)/4$ , we may apply Lemma REF to get ${m^{\\mathrm {BR}}}(f_1^{\\varepsilon })\\,{m^{\\mathrm {BR}_*}}(f_2^{\\varepsilon })={m^{\\mathrm {BR}}}(f_1)\\,{m^{\\mathrm {BR}_*}}(f_2)+ O\\big (\\varepsilon (\\Vert f_1\\Vert _{{\\mathcal {S}}^m}+\\Vert f_2\\Vert _{{\\mathcal {S}}^m})\\big ).$ Therefore $\\limsup _{t\\rightarrow \\infty } e^{(d-\\delta )t}\\langle \\rho (a_t)f_1,f_2\\rangle ={m^{\\mathrm {BR}}}(f_1)\\,{m^{\\mathrm {BR}_*}}(f_2)+ O\\big (\\varepsilon (\\Vert f_1\\Vert _{{\\mathcal {S}}^m}+\\Vert f_2\\Vert _{{\\mathcal {S}}^m})\\big ).$ Since $\\varepsilon >0$ was arbitrary, we in fact have $\\limsup _{t\\rightarrow \\infty } e^{(d-\\delta )t}\\langle \\rho (a_t)f_1,f_2\\rangle ={m^{\\mathrm {BR}}}(f_1)\\,{m^{\\mathrm {BR}_*}}(f_2).$ An analogous calculation with $\\liminf $ in place of $\\limsup $ proves the proposition."
],
[
"Proof of Theorem ",
"In this final section, we prove Theorem REF .",
"Recall the isomorphism $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )\\cong \\int _{\\mathsf {Z}}^{\\oplus } (\\pi _{\\zeta },{\\mathcal {H}}_{\\zeta })\\,d\\mu _{\\mathsf {Z}}(\\zeta ).$ Fix arbitrary $0< r< \\delta -s_1$ .",
"We partition $\\mathsf {Z}$ as $\\mathsf {Z}=\\mathsf {Z}_{r}^-\\cup \\mathsf {Z}_{r}^+,$ where $\\mathsf {Z}_{r}^-=\\left\\lbrace \\zeta \\in \\mathsf {Z}\\,:\\, (\\pi _{\\zeta },{\\mathcal {H}}_{\\zeta })\\cong {\\mathcal {U}}(\\upsilon ,s),\\;\\mathrm {where}\\;\\upsilon \\in {\\hat{M}}\\;\\mathrm {and}\\; s\\in [s_1+r,\\delta ]\\right\\rbrace $ and $\\mathsf {Z}_{r}^+=\\mathsf {Z}\\setminus \\mathsf {Z}_{r}^-$ (cf.",
"(REF )).",
"This partition is then used to decompose $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )$ as $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )= \\big (\\rho ,L^2(\\Gamma \\backslash G)^-\\big )\\oplus \\big (\\rho ,L^2(\\Gamma \\backslash G)^+\\big ),$ where $\\big (\\rho ,L^2(\\Gamma \\backslash G)^{\\pm }\\big )\\cong \\int _{\\mathsf {Z}_{r}^{\\pm }}^{\\oplus } (\\pi _{\\zeta },{\\mathcal {H}}_{\\zeta })\\,d\\mu _{\\mathsf {Z}}(\\zeta )$ with the dependency on $r$ being slightly suppressed.",
"Recall that ${\\mathcal {B}}_{\\delta }$ occurs as a subrepresentation of $\\big ( \\rho , L^2(\\Gamma \\backslash G)^-\\big ) $ and ${\\mathcal {B}}_{\\delta }\\cong {\\mathcal {U}}(1,\\delta )$ (cf.",
"Sections REF and REF ).",
"We may further decompose $\\big (\\rho ,L^2(\\Gamma \\backslash G)^-\\big )$ accordingly: let $L^2(\\Gamma \\backslash G)_0^-$ be the orthogonal complement of ${\\mathcal {B}}_{\\delta }$ in $L^2(\\Gamma \\backslash G)^-$ .",
"We thus have $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )=(\\rho ,{\\mathcal {B}}_{\\delta })\\oplus \\big (\\rho ,L^2(\\Gamma \\backslash G)_0^{-}\\big )\\oplus \\big (\\rho ,L^2(\\Gamma \\backslash G)^{+}\\big ).$ Note that Theorem REF and the duality between eigenfunctions of the Laplacian on ${\\mathcal {M}}=\\Gamma \\backslash G/K$ and spherical representations in the decomposition of $L^2(\\Gamma \\backslash G)$ imply that no spherical representation is weakly contained in $\\big (\\rho ,L^2(\\Gamma \\backslash G)_0^-\\big )$ .",
"At the level of functions, we write $f_i=f_i^0+f_i^-+f_i^+,\\qquad i=1,2$ where $ f_i^0\\in {\\mathcal {B}}_{\\delta }$ , $f_i^-\\in L^2(\\Gamma \\backslash G)_0^-$ , and $f_i^+\\in L^2(\\Gamma \\backslash G)^+$ .",
"The matrix coefficients we are interested in now decompose as $\\langle \\rho (a_t)f_1,f_2\\rangle =\\langle \\rho (a_t)f_1^0,f_2^0\\rangle +\\langle \\rho (a_t)f_1^-,f_2^-\\rangle +\\langle \\rho (a_t)f_1^+,f_2^+\\rangle .$ We deal with the three summands in turn: by construction, $\\big (\\rho ,L^2(\\Gamma \\backslash G)^+\\big )$ does not weakly contain any ${\\mathcal {U}}(\\upsilon ,s)$ with $s>s_1+r$ (this also uses the fact $\\big (\\rho ,L^2(\\Gamma \\backslash G)\\big )$ does not weakly contain any ${\\mathcal {U}}(\\upsilon ,s)$ with $s>\\delta $ ; cf.",
"[27]).",
"We assume that $m$ is large enough so that all the results of the previous sections hold for $f_1, f_2\\in {\\mathcal {S}}^m(\\Gamma \\backslash G)$ .",
"By [23] (cf.",
"also [27] or [37] combined with the argument from the proof of Lemma REF ), we have that for all $\\xi >0$ , $|\\langle \\rho (a_t)f_1^+,f_2^+\\rangle | &\\ll _{\\xi } e^{(s_1-d +r+\\xi )t} \\Vert f_1^+\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\Vert f_2^+\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\\\&\\le e^{(s_1-d+r+\\xi )t} \\Vert f_1\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\Vert f_2\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}.$ We will now use Theorem REF to bound $\\langle \\rho (a_t)f_1^-,f_2^-\\rangle $ .",
"Let $\\widetilde{\\mathsf {Z}}_{r}^{-}\\subset \\mathsf {Z}_{r}^-$ be such that $\\big (\\rho ,L^2(\\Gamma \\backslash G)_0^{-}\\big )\\cong \\int _{\\widetilde{\\mathsf {Z}}_{r}^{-}}^{\\oplus } (\\pi _{\\zeta },{\\mathcal {H}}_{\\zeta })\\,d\\mu _{\\mathsf {Z}}(\\zeta )$ .",
"The corresponding decomposition of the functions $f_1^-$ , $f_2^-$ reads $f_i^-=\\int _{\\widetilde{\\mathsf {Z}}_{\\varepsilon }^{-}} (f_i^-)_{\\zeta }\\,d\\mu _{\\mathsf {Z}}(\\zeta )$ ($i=1,2$ ).",
"Since $f_i$ is $M$ -invariant, so is $f_i^-$ and $\\mu _{\\mathsf {Z}}$ -a. e. $(f_i^-)_{\\zeta }$ .",
"The matrix coefficient $\\langle \\rho (a_t)f_1^-,f_2^-\\rangle $ is now written as $\\langle \\rho (a_t)f_1^-,f_2^-\\rangle =\\int _{\\widetilde{\\mathsf {Z}}_{\\varepsilon }^{-}} \\langle \\pi _{\\zeta }(a_t)(f_1^-)_{\\zeta },(f_2^-)_{\\zeta }\\rangle _{{\\mathcal {H}}_{\\zeta }}\\,d\\mu _{\\mathsf {Z}}(\\zeta ).$ Each $(\\pi _{\\zeta },{\\mathcal {H}}_{\\zeta })$ is isomorphic to some ${\\mathcal {U}}(\\upsilon ,s)$ with $\\upsilon $ non-trivial and $s\\in [s_1+r,\\delta ]$ .",
"Setting $\\lambda (\\delta ,r)=\\max _{s\\in [s_1+r,\\delta ]} s-d-\\eta _s$ where $\\eta _s=\\min (2s-d,1)$ , we apply Theorem REF to each $\\langle \\pi _{\\zeta }(a_t)(f_1^-)_{\\zeta },(f_2^-)_{\\zeta }\\rangle _{{\\mathcal {H}}_{\\zeta }}$ and obtain $| &\\langle \\rho (a_t)f_1^-,f_2^-\\rangle |\\ll _{r,\\delta } e^{\\lambda (\\delta ,r)t}\\int _{\\widetilde{\\mathsf {Z}}_{r}^{-}} \\Vert (f_1^-)_{\\zeta }\\Vert _{{\\mathcal {S}}^m({\\mathcal {H}}_{\\zeta })}\\Vert (f_2^-)_{\\zeta }\\Vert _{{\\mathcal {S}}^m({\\mathcal {H}}_{\\zeta })} \\,d\\mu _{\\mathsf {Z}}(\\zeta )\\\\\\le & e^{\\lambda (\\delta ,r)t}\\sqrt{\\int _{\\widetilde{\\mathsf {Z}}_{r}^{-}} \\Vert (f_1^-)_{\\zeta }\\Vert _{{\\mathcal {S}}^m({\\mathcal {H}}_{\\zeta })}^2\\,d\\mu _{\\mathsf {Z}}(\\zeta )}\\sqrt{\\int _{\\widetilde{\\mathsf {Z}}_{r}^{-}} \\Vert (f_2^-)_{\\zeta }\\Vert _{{\\mathcal {S}}^m({\\mathcal {H}}_{\\zeta })}^2\\,d\\mu _{\\mathsf {Z}}(\\zeta )}\\\\=& e^{\\lambda (\\delta ,r)t} \\Vert f_1^-\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\Vert f_2^-\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)} \\\\ \\le &e^{\\lambda (\\delta ,r)t} \\Vert f_1\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\Vert f_2\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)},$ where the implied constant remains uniformly bounded as $r\\rightarrow 0$ .",
"The remaining term in the left-hand side is $\\langle \\rho (a_t)f_1^0,f_2^0\\rangle $ .",
"Using the fact that $(\\rho ,{\\mathcal {B}}_{\\delta })$ is isomorphic to ${\\mathcal {U}}(1,\\delta )$ , applying Proposition REF gives $\\langle \\rho (a_t)f_1^0,f_2^0\\rangle &= \\Phi (f_1^0,f_2^0)\\,e^{(\\delta -d)t}+O_{\\delta }\\left( e^{(\\delta -d-\\eta _{\\delta })t}\\Vert f_1^0\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\Vert f_2^0\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\right)\\\\ =& \\Phi (f_1^0,f_2^0)\\,e^{(\\delta -d)t}+O_{\\delta }\\left( e^{(\\delta -d-\\eta _{\\delta })t}\\Vert f_1\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\Vert f_2\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\right),$ where $\\Phi (f_1^0,f_2^0)$ is given by the sum (REF ) under the aforementioned isomorphism.",
"Note that $\\beta &:= \\min \\big \\lbrace \\eta _{\\delta },\\delta -s_1-r-\\xi ,\\delta -d-\\lambda (\\delta ,r)\\big \\rbrace \\\\&=\\min \\left\\lbrace 1 ,2\\delta -d,\\delta -s_1-r-\\xi , \\min _{s\\in [s_1+r,\\delta ]}\\big ( \\delta -s+\\min \\lbrace 1, 2s-d\\rbrace \\big )\\right\\rbrace \\\\&\\ge \\min \\left\\lbrace 1 ,2\\delta -d,\\delta -s_1-r, \\min _{s\\in [s_1+r,\\delta ]}\\big ( \\delta -s+\\min \\lbrace 1, 2s-d\\rbrace \\big )\\right\\rbrace -\\xi \\\\&= \\min \\left\\lbrace 1 ,2\\delta -d,\\delta -s_1-r,\\delta +s_1+r-d\\right\\rbrace -\\xi \\\\&= \\min \\left\\lbrace 1,\\delta -s_1-r,\\delta +s_1+r-d\\right\\rbrace -\\xi \\\\&= \\min \\left\\lbrace 1,\\delta -s_1-r\\right\\rbrace -\\xi \\\\&\\ge \\min \\left\\lbrace 1,\\delta -s_1\\right\\rbrace -(\\xi +r).$ Entering (REF ), (REF ), and (REF ) into (REF ) gives $e^{(d-\\delta )t}\\langle \\rho (a_t)f_1,f_2\\rangle =\\Phi (f_1^0,f_2^0)+O_{r,\\xi }\\big (e^{ - \\beta t} \\Vert f_1\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\Vert f_2\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\big ).$ Writing $\\eta =\\min (1, \\delta -s_1)$ and $\\varepsilon =\\xi +r$ , we thus have $e^{(d-\\delta )t}\\langle \\rho (a_t)f_1,f_2\\rangle =\\Phi (f_1^0,f_2^0)+O_{\\varepsilon }\\big (e^{ - (\\eta -\\varepsilon ) t} \\Vert f_1\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\Vert f_2\\Vert _{{\\mathcal {S}}^m(\\Gamma \\backslash G)}\\big ).$ Choosing $0<\\varepsilon <\\eta $ gives $\\lim _{t\\rightarrow \\infty } e^{(d-\\delta )t}\\langle \\rho (a_t)f_1,f_2\\rangle =\\Phi (f_1^0,f_2^0),$ so $\\Phi (f_1^0,f_2^0)=m^{\\mathrm {BR}}(f_1)\\,m^{\\mathrm {BR_*}}(f_2)$ by Proposition REF , completing the proof of Theorem REF .",
"Remark 6.0 In the case when $\\Gamma $ is a lattice, i.e., when $\\delta =d$ , the above proof leads to the claim in Remark REF (1) as follows.",
"Firstly, we have $\\langle \\rho (a_t)f_1^0,f_2^0\\rangle =\\int f_1\\, dx \\cdot \\int f_2\\, dx $ for any $t\\in {\\mathbb {R}}$ .",
"As before, we use (REF ) to bound $\\langle \\rho (a_t)f_1^+,f_2^+\\rangle $ .",
"Now, since complementary series representations ${\\mathcal {U}}(\\upsilon ,s)$ with non-trivial $\\upsilon $ exist only for $ s\\le d-1$ , the constant $\\lambda (d,r)$ appearing in the bound (REF ) for $|\\langle \\rho (a_t)f_1^-,f_2^-\\rangle |$ is at most $ \\alpha := \\max \\lbrace s-d-\\eta _s : {s\\in [s_1+r, d-1]}\\rbrace $ .",
"We note that $\\alpha <-2+r$ , and if, in addition, $\\lbrace s>s_1+1: {\\mathcal {U}}(\\upsilon ,s) \\textrm { is weakly contained in } L^2(\\Gamma \\backslash G)\\rbrace =\\emptyset $ , then one can take $\\lambda (d,r) < s_1-d+r$ .",
"Hence combining these bounds, we obtain Remark REF (1)."
]
] | 2001.03377 |
[
[
"Auction-based Charging Scheduling with Deep Learning Framework for\n Multi-Drone Networks"
],
[
"Abstract State-of-the-art drone technologies have severe flight time limitations due to weight constraints, which inevitably lead to a relatively small amount of available energy.",
"Therefore, frequent battery replacement or recharging is necessary in applications such as delivery, exploration, or support to the wireless infrastructure.",
"Mobile charging stations (i.e., mobile stations with charging equipment) for outdoor ad-hoc battery charging is one of the feasible solutions to address this issue.",
"However, the ability of these platforms to charge the drones is limited in terms of the number and charging time.",
"This paper designs an auction-based mechanism to control the charging schedule in multi-drone setting.",
"In this paper, charging time slots are auctioned, and their assignment is determined by a bidding process.",
"The main challenge in developing this framework is the lack of prior knowledge on the distribution of the number of drones participating in the auction.",
"Based on optimal second-price-auction, the proposed formulation, then, relies on deep learning algorithms to learn such distribution online.",
"Numerical results from extensive simulations show that the proposed deep learning-based approach provides effective battery charging control in multi-drone scenarios."
],
[
"Introduction",
"The possibility to use commercial drones in a broad range of applications is being extensively studied by the research community, and they are expected to manned operations in remote locations [1].",
"In general, commercial drones have inherent limitations in the amount of energy available to support their operations.",
"This is due to the energy/weight ratio of current energy storage technologies, where increasing the capacity of the battery beyond a certain point degrades flight time due to excessive weight.",
"As a consequence, effective battery management is one of the main enablers of practical deployments of drone-based technologies and applications.",
"Importantly, in applications requiring extensive flight time, the energy constraint problem can not be solved by only optimizing power consumption.",
"Thus, charging during the completion of long-term tasks has been proposed to extend the operational range of the drones [1], [2].",
"There are several ways to powering the drones which have been proposed in the literature.",
"We can divide them into two main classes: (i) harvesting energy directly from the surrounding environment, and (ii) taking energy from an electrical source such as a charging station [2].",
"Within the latter class of approaches, the charging stations can be either stationary or mobile.",
"However, solutions based on stationary charging stations may constrain the geographical area of operations around specific locations.",
"In order to deal with this issue, mobile charging stations can be used although they face other challenges [2].",
"As they are mobile, the size of these charging stations needs to be comparably smaller to that of fixed stations.",
"As a consequence, the capacity of the system has limitations, leading to relatively low chargin speeds and a relatively smaller number of drones that can be charged simultaneously [2], [3].",
"Motivated by this compelling problem, we consider a scenario where multiple drones compete to access the services provided by a mobile charging station (see Fig.",
"REF ).",
"The framework proposed in this paper controls the charging process of the drones, where the charging station takes the role of leader in the distributed drone-charging system, and coordination within the system is supported by Internet-of-Vehicle (IoV) networking functions [4], [5], [6].",
"Figure: Multi-drone network model for mobile charging stations.We take an econometric approach, where the problem of controlling the scheduling is formulated as an auction, whose objective is to maximize the utility of the drones (i.e., the difference between payment and bid during auction computation) as well as the station's revenue (i.e., payment received by the drones through charging scheduling).",
"In general auction problems, buyers (the drones in this system) bid to access services periodically auctioned by a seller (the mobile charging station in the considered setting).",
"The value of the bid is individually, and privately, estimated by each drone based on the urgency of its charging needs.",
"The auction approach is especially useful when there is no accurate estimation of the buyer's true valuation, and buyers are not aware of the private true values of other buyers.",
"In the drone network model considered herein, the drones are assumed to be non-cooperative, that is, they operate independently and distributely.",
"Furthermore, the mobile charging station is not assumed to know the exact true values associated with each drone, which became available only when the actual values are submitted.",
"The auction approach we take is especially suitable to solve the problem of assigning time slots to drones in this information-limited system.",
"Among the various auction formulations available (e.g., ascending auction, descending auction, first price auction, second price auction), we choose a second price auction formulation, where the highest bidder wins but the price paid is set to the second highest bid.",
"One of the main benefits of the second price auction is that it results in a truthful auction process.",
"In the considered system model, the mobile station is the auctioneer and owner/seller of the resource (that is, the charging time slot) and each drone is considered as a buyer.",
"The drones are in competition for scheduling battery charging with price bidding via its own private valuation for auction.",
"As auctioneer, the mobile station (i) receives all bids from the drones, (ii) calculates the charging time allocation probabilities and payments, (iii) assigns the charging time to the drone (i.e., the winner in auction) who bids the highest value, corresponding to the largest allocation probability, (iv) announces the value which should be paid by the winner drone, and (v) receives the payment.",
"During the auction, drones strategically submit bids to increase their profits, i.e., utility.",
"Similarly, the resource-owned auctioneer is not a sacrificial seller, thus it is required to consider the revenue in auctioneer, i.e., profitable.",
"Therefore, revenue-optimal auctions have been considered as one of major objectives in auction design.",
"Although there are many variants already available in the literature auction theory, the problem of simultaneously optimizing auctioneer's revenue and buyers utility is still open [7], [8], [9], [10].",
"Among various auction algorithms, Myerson auction is one of the most efficient revenue-optimal single-item auctions [11].",
"The auction transforms the bid value, and then winner and payment is determined based on the transformed bid.",
"At that point, if the transformation function is monotonic, the revenue-optimal auction is configured.",
"Therefore, the proposed auction designs the revenue-optimal auction based on the concept of the Myerson auction.",
"However, it is difficult to apply the existing auction as it is in the distributed drone network environment considered in this paper.",
"The charging scheduling system of the drones is still in the early stages of research; and key properties of the system such as drones location distribution and residual energy distribution have not been fully characterized in the literature.",
"Therefore, a system that can extract the desired data (i.e., distribution of drones), from the actual system without prior knowledge or assumptions is desirable.",
"Therefore, this paper takes advantage of deep learning to learn important features on-the-fly from the operating environment.",
"Recently, frameworks combining game theory and deep learning have been active subject of research [12], [13].",
"Results illustrate applications of such approach in various domains [14], [15], [16].",
"The key is that deep learning can automatically extract and learn important features from data, and it has been widely demonstrated that neural network structures can approximate complex non-linear functions [17], [18], [19].",
"In this paper, we use this feature to approximate some key – monotonic – functions governing the behavior of the system using relatively simple neural networks [16].",
"Specifically, we use deep learning to learn the features necessary for the virtual transformation step of Myerson auctions.",
"Then, the proposed auction is configured by replacing the trained deep learning network with a virtual transformation function.",
"We remark that the functions to be learn by the deep learning layer is non-decreasing monotonic [11].",
"The proposed deep learning network uses the ReLU (activation function) and softmax (classification function) which are widely used in optimization procedures.",
"In addition, due to the fact that the operations mostly amount to linear multiplications, the proposed approach has low complexity, and its execution takes a limited amount of time.",
"Contributions.",
"Our proposed auction-based charging scheduling algorithm makes the following contributions.",
"First, the revenue of auctioneer is considered even if the drones submit false/fake bids, i.e., thus the proposed algorithm is self-configurable and truthful.",
"The proposed auction automatically learns environmental features.",
"In distributed drone scenarios, various time varying features exist that make self-configurable nature essential to adapt to different scenarios and environments.",
"The proposed deep learning based auction structure is simple to be implemented and imposes a small computation burden.",
"Organization.",
"The rest of this paper is organized as follows.",
"Sec.",
"discusses related work, and then Sec.",
"describes the auction-based mobile charging model.",
"In Sec.",
", the deep learning based approach is presented.",
"In Sec.",
", performance evaluation results are presented.",
"Sec.",
"concludes the paper."
],
[
"Related Work",
"There have been several research results to solve limited-battery and limited resource scheduling problems through auctions [1], [2], [20], [21].",
"The method in [1] aims at optimizing battery assignment and drone scheduling, assuming that the battery can be quickly replaced.",
"The joint assignment and scheduling problem is formulated as a two-stage problem, where the assignment problem is solved by a heuristic and the scheduling problem is formulated as an integer-linear programming (ILP) problem.",
"This paper proposes the scheduling algorithm based on auction.",
"The proposed method uses information provided by drones capable of communicating with mobile charging stations to overcome the inability of a central service provider to acquire perfect state information in a distributed drone network.",
"However, a solution based on battery assignment necessarily maps to a stationary service station.",
"This imposes some limitations [2], which are mitigated when using mobile charging stations.",
"In [2], a systems of mobile robots executing a transportation task supported by a charging station is considered.",
"The location of the charging station is a major factor in determining the operations and performance of the robots, and the paper assumes that the mobile charging station is itself an autonomous robot that attempts to incrementally improve its location.",
"Although this work considers a mobile charging station, the problem of charging scheduling is not considered.",
"In a more general scenario, the resources of the charging station are limited and the number of robots to be charged may be larger than the actual charging capacity of the station.",
"Therefore, the charging system will need to implement forms of prioritization to optimize the charging process.",
"The method proposed herein incorporates a notion of priority using an auction formulation based on the valuation of the drones.",
"In [20], an auction mechanism is proposed to solve a resource allocation problem in a distributed computing system.",
"The inherently distributed nature of the system makes the resolution of the problem much harder.",
"The paper proposes an auction-based solution to address such challenge.",
"The proposed mechanism is configured as a two auction mechanism, used to compute optimal solutions at the single unit within the distributed scheduling problem in a computationally efficient manner.",
"However, [20] assumes prior knowledge of the environment where the auction mechanism is executed, which may limit its application in real-world distributed scenarios.",
"The method we propose herein uses an auction-based solution to solve the resource allocation problem, and employs deep learning to extract the required features automatically from the environment, so that prior knowledge is not necessary.",
"The method in [21] addresses a distributed train scheduling problem using an auction method.",
"The determination of the winner is formulated as a mixed-integer problem.",
"The bidding strategy of the buyers is solved via dynamic programming.",
"In the proposed method, the auctioneer computes the set of bids that maximizes revenue.",
"Both the method proposed in [21] and the one proposed in this paper are based on an auction formulation to effectively solve resource scheduling in distributed environments and maximize the revenue of the auctioneer.",
"However, the method in [21] differs from the proposed deep learning based auction in terms of the required prior information to conduct the auction.",
"The deep learning-based auction proposed in this paper only requires limited information since as it can learn in real-time environmental characteristics and parameters."
],
[
"Charging Scheduling Mechanism Design",
"Drone Network Model.",
"The system is composed of the mobile charging station $S$ and $U$ dronesThe notation used in this paper is summarized in Table REF ..",
"The mobile station is governed by the charging service controller; and the service controller collects revenue by providing charging services.",
"The revenue of the charging service controller is recorded and will be requested later to be paid to drone operators.",
"This paper assumes that the mobile station can provide charging service to only one drone in each time slot.",
"Thus, drones competes to obtain charging opportunities.",
"Note that we consider a short-range Internet of Vehicles (IoV) multi-drone network supporting short-distance communications among drones based on IEEE 802.11-based wireless local area network (WLAN) technologies.",
"Therefore, the size of the network composed of one single mobile charging station and multiple drones is relatively small, and we assume that the flight time from drones' current positions to the mobile charging station is negligible.",
"Thus, unexpected operational problems due to the delay induced by long flight time toward the mobile charging station are not considered in this paper.",
"Furthermore, we note that the specific design, system capabilities and state of the drones participating in the auction can vary in terms of battery capacity, residual battery, charging rates and so forth.",
"Formally, each drone $u_i$ is characterized by the battery capacity $c_i$ , average amperage draw $e_i$ , and battery residual charge $h_i$ , which determines the mission lifetime.",
"The average amperage draw $e_i$ denotes the amount of amperage required to the drone to operate on-board systems such as motors, embedded computers, sensors, etc.",
"Each drone continuously monitors its own state and requests the scheduling of a charging slot to the mobile station if needed.",
"The requests from multiple drones to the mobile station for charging services can be interpreted as a distributed competition for a limited resource, which here is modeled and solved using an auction-based approach.",
"In the considered setting, the auctioneer is the mobile station, which is also the owner and provider of the resource, and the drones are the buyers.",
"The mobile station and drones exchange information, i.e., bids and other auction variables, over wireless links.",
"The mobile station announces the start of the auction to the drones when the charging system is ready to serve (i.e., idle).",
"Upon reception of the announcement, each drone makes its own private, and independent, valuation for the use of the charging system.",
"The private valuation $v_i$ of drone $u_i$ is used to compete for the charging service.",
"Note that the charging resource is assigned at the granularity of individual time slots as illustrated in Fig REF .",
"The mobile station $\\mathcal {S}$ sells the charging service and obtains revenue $p_i$ paid by the winner drone $u_i$ via auction.",
"Figure: Auction procedure.Drone Scheduling Auction Design.",
"We use second price auction (SPA) as a baseline to design the auction in the considered setting.",
"In SPA, all buyers submit their bids privately.",
"The auctioneer receives the sealed bids and selects as winner the buyer who made the highest bid.",
"The amount paid by the winner is set to be equal to the second highest bid value.",
"Herein, the problem of assigning slots to drones is formulated as a single item auction based on SPA.",
"Therefore, drones compete for one item, i.e., the charging service.",
"Since the proposed approach is based on SPA, it is guaranteed that the charging service will be assigned to the drone with the highest valuation to the service [22], [23], [24].",
"Myerson presents provable analytical results for single item auctions optimizing the auctioneer revenue where each buyer has its own private valuation of the resource [11], [18].",
"When the auction-based mechanism is designed, it is important to let the participants act truthfully to ensure system stability [11], [18], [19], [25], [26], [27].",
"Previous studies attempted to achieve this objective by enforcing truthfulness to individual participants.",
"The concepts such as incentive compatibility (IC) and individual rationality (IR) are the characteristics of auctions inducing the truthful action of participants.",
"Based on this approach, we use a Myerson auction where the following characteristic is used as the baseline mechanism: The Myerson auction guarantees dominant strategy incentive compatibility (DSIC) and IR.",
"Definition 1 (Incentive Compatibility [28]) Incentive compatibility is defined by the following property: if for every bidder $j$ , every valuation $v_j$ , all declarations of the other bidders $v_{-j}$ , and all possible \"false declarations\" $v^{\\prime }_j$ , we have that bidder $j$ 's utility with bidding $v^{\\prime }_{j}$ is no more than his utility with bidding the truth $v_j$ .",
"Formally, let $\\lambda _j$ and $P_j$ be the mechanism 's output with input $(v_j, v_{-j})$ and $\\lambda ^{\\prime }_j$ and $P^{\\prime }_j$ be the mechanism's output with input $(v^{\\prime }_j,v_{-j})$ , then $v_j(\\lambda _j) - P_j > v_j(\\lambda ^{\\prime }_j) - P^{\\prime }_j$ .",
"Thus, this weaker degree of DSIC guarantees IC, where IC means that the utility a participant can obtain by acting truthfully is greater than that by fake acting according to Definition REF .",
"Definition 2 (Dominant Strategy Incentive Compatibility) Dominant strategy incentive compatibility is defined as the following property.",
"For each bidder $i$ , and for every possible report of the other bidders bid $b_{-i}$ , bidder $i$ weakly maximizes utility by reporting $b_i = v_i$ .",
"That is, for all possible reports $b^*_i$ , $u_i(v_i, b_{-i}) \\ge u_i(b^*_i, b_{-i})$ .",
"Thus, the DISC is a stronger degree of IC, meaning that a truthful action is a weakly dominant strategy, that is, the action is guaranteed to be the best, regardless of the actions of others, as shown in Definition REF .",
"Definition 3 (Individual Rationality) Individual rationality (IR) is defined by the following property: for every bidder $i$ and for every $v_i$ , we have $v_i \\ge p_i$ , that is, no bidder is ever asked to pay more than its bid valuation.",
"In a DSIC and IR auction, it is in the best interest of each bidder to report truthfully.",
"Therefore, these characteristics make the overall auction truthful.",
"The Myerson auction guarantees DSIC and IR, thus encouraging the bidders to report truthfully [11], [18].",
"Furthermore, the Myerson auction also guarantees auctioneer's revenue optimality.",
"In the considered drone network model, we remark that the charging service controller obtains revenue by providing charging services.",
"The following subsections describe in detail the components of the Myerson auction mechanism, i.e., private valuation, allocation rule, payment rule, reserve price, utility, and auction design.",
"Private Valuation.",
"In the proposed auction, each drone $u_i$ has its own individual private valuation $v_i$ .",
"Each drone $u_i$ has an maximum battery capacity (denoted by $c_i$ ) and a current remaining battery (denoted by $r_i$ ).",
"If the drone is assigned the mobile charging service time slot, the charged energy will be added to the residual energy in its own battery $r_i$ .",
"The amount of energy charged by the mobile charging station $S$ can be expressed as $q_i = \\min (f\\cdot t_i, (c_i - r_i))$ where $f$ is the charging rate per unit of time in the mobile charging station.",
"The scheduled charging time to $u_i$ via auction is denoted by $t_i$ .",
"$t$ denotes the item being sold by auction, i.e., charging time.",
"The higher $f\\cdot t$ , the higher the valuation of $t$ by drone $u_i$ , and the drone is willing to pay a higher amount for the charging service.",
"The expected drone flight time with current battery status is denoted as $l_i$ , calculated as $l_i = \\frac{r_i\\cdot h_i}{e_i}$ .",
"If $l_i$ is larger, the drone will give a smaller valuation to $t$ .",
"Let $v_i$ denote the private valuation of drone $u_i$ .",
"Then, $v_i$ can be expressed as $v_i = \\frac{f\\cdot t}{l_i}$ .",
"Allocation Rule.",
"The allocation rule $g$ is used to determine the winner drone $u_i$ based on the valuation, i.e., to find which drone $u_i$ should be scheduled for charging.",
"In the Myerson auction, the allocation rule that awards the item to the highest bidder is monotone.",
"Therefore, in the proposed auction model, the allocation rule $g$ used to award the charging service to the highest bidder is monotone.",
"Therefore, the allocation rule can be expressed as follows: $u^* \\in \\arg \\max _{u_i\\in u} g(v_i).$ Payment Rule.",
"The payment rule $p$ is used to determine the payment by the winner drone $u_i$ based on the valuation.",
"In the proposed auction, the payment rule $p$ chooses a payment which is not higher than the private valuation, and it can be expressed as follows: $p_i(v_i) \\in [0, x_i v_i]$ where $x_i \\in \\left\\lbrace 0,1 \\right\\rbrace $ , $\\forall i = \\left\\lbrace 1,\\dots ,U\\right\\rbrace $ stands for the variable to represent the winning valuation in the auction, and $u^*$ is the winner drone in the auction.",
"Reserve Price.",
"The proposed auction sets a specific price called a reserve price.",
"The reserve price is the minimum reward the seller accepts [11].",
"In this paper, the reserve price is set to 0.",
"In the auction, the auctioneer solicits the private bids from the bidders and computes the allocation rule $g = (g_1, \\dots , g_U)$ and payment rule $p = (p_1,\\dots , p_U)$ .",
"Utility.",
"The proposed auction guarantees DSIC and IR; and thus each bidder reports truthfully to maximize its own utility.",
"Note that $x_i \\in \\left\\lbrace 0,1 \\right\\rbrace , \\forall i = \\left\\lbrace 1,\\dots ,U\\right\\rbrace $ stands for the variable corresponding to the winning in the auction.",
"If the drone wins in auction, $x$ is set to 1 whereas the $x$ is set to 0 otherwise.",
"Thus, the utility of drone $u_i$ can be calculated as $utility(u_i) = g(v_i) - x_i \\cdot p_i(v_i), \\forall i = \\lbrace 1, \\dots , U\\rbrace $ .",
"Revenue Optimal Auction Design.",
"We define the virtual valuation and virtual surplus as in Myerson [11].",
"The virtual valuation $\\phi _i(v_i)$ of a buyer $u_i$ in the auction is a function used to calculate the expected revenue of the auctioneer from that buyer $u_i$ .",
"The virtual surplus is the expected revenue excluding the computing cost defined below.",
"In Myerson auctions, each bidder $i$ has its own individual private valuation $v_i$ which is drawn from the strictly increasing cumulative density function $F_i(v_i)$ where the probability density function of $v_i$ is denoted as $f_i(v_i)$ [29], [30].",
"The virtual valuation of bidder $i$ with private valuation $v_i$ can be expressed as follows: $\\phi _i(v_i) = v_i - \\frac{1-F_i(v_i)}{f_i(v_i)},$ There is a cost in computing the outcome $c(g)$ which must be payed by the auction [31].",
"Given valuation $v_i$ , virtual valuation $\\phi _i(v_i)$ , and allocation rule $g$ , the virtual surplus can be calculated as follows: $\\sum _{\\forall i}\\nolimits \\phi _i(v_i)x_i - c(g).$ In Myerson auction, the expected payment is proportional to the expected virtual surplus; and it can be computed as follows [11], [31]: $\\mathbb {E}_{b_i}[p_i(b_i)] = \\mathbb {E}_{b_i}[\\phi _i(b_i)x_i(b_i)].$ Therefore, if the virtual valuations $\\phi (b)$ are non-decreasing in valuations $b$ , the virtual surplus $\\mathbb {E}_b$ is non-decreasing in valuations $b$ .",
"The bid $b$ is drawn from the distribution $F(b)$ with probability density function $f(b)$ .",
"Then, the expected payment can be computed as follows: $\\mathbb {E}_{b}[p(b)] &=&\\int _{b=0}^h bg(b)f(b)\\texttt {d}b-\\int _{b=0}^h g(b)[1-F(b)]\\texttt {d}b.\\\\&=&\\int _{b=0}^h \\left[b-\\frac{1-F(b)}{f(b)}\\right]g(b)f(b)\\texttt {d}b.\\\\&=&\\mathbb {E}_b[\\phi (b)g(b)].$ As a result, the proposed auction approach, which consists of a variant of the Myerson auction, is DSIC, IR, and revenue optimal.",
"However, Myerson auctions require full knowledge of the distributions $F_1, F_2, \\dots , F_U$ according to Eq.",
"(REF ).",
"In the considered scenario, it is hard to obtain such information a priori, and we propose to use deep learning to estimate the distributions.",
"In previous research results, it has been shown that the deep learning with limited structure can approximate specific functions [17], [15], [16].",
"Specifically, herein, we use neural networks and unsupervised learning [18], [19] to approximate the virtual valuation function $\\phi (v)$ .",
"The strength of deep learning is that the approximated function can be continually updated as inputs are acquired.",
"The use of unsupervised learning makes the learning process possible, as it does not require the true values as input.",
"The resulting auction is not only easily applicable to the distributed multi-drone network problem, but is also capable to adapt to continuously changing environments.",
"Figure: The proposed deep learning framework (revenue network) for revenue-optimal auction computation."
],
[
"Deep Learning based Auction Design",
"In this section, a deep learning based method for single item auctions is introduced.",
"The method defines allocation rule $g$ , payment rule $p$ , and virtual valuation function $\\phi $ for maximizing the revenue of the mobile charging station via deep learning.",
"The deep learning model constitutes the auction that guarantees DSIC and IR as well as enables the revenue optimal computation for auctioneer [18], [19].",
"The revenue optimal auction can be configured through a relatively simple deep learning structure, i.e., composed of max/min operations and a loss function shaping the training process.",
"Theorem 1 (Myerson [11]).",
"There exist a collection of monotonically increasing functions $\\phi _i$ : $V_i \\rightarrow R$ , referred to as the virtual valuation functions, for selling a single item in the DSIC mechanism, which assigns the item to the buyer $i$ with the highest virtual value $\\phi _i(v_i)$ assuming this quantity is positive and charges the winning bidder the smallest bid that ensures that the bidder is winning.",
"As mentioned earlier, the proposed deep learning based auction is a variant of Myerson auctions; and thus the bid set $b$ is transformed to the virtual valuation $\\overline{b_i}$ via virtual valuation transformation.",
"Specifically, as expressed in Theorem REF , the bid set of $b_i, \\forall i = \\lbrace 1, \\dots , U\\rbrace $ are converted to $\\overline{b_i} = \\phi ^{mononet}(b_i), i = \\lbrace 1, \\dots , U\\rbrace $ , where $\\overline{b_i}$ denotes that the transformed bid of $u_i$ .",
"In this procedure, the trained deep network $\\phi ^{mononet}$ (a monotonic network) is utilized to replace the virtual valuation function $\\phi $ .",
"The $\\phi ^{mononet}$ consist of two layers, and is composed of linear computation units and min/max operation units.",
"Based on the transformed bid $\\overline{b_i}$ , the SPA with reserve price 0 (SPA-0) is performed.",
"The SPA-0 calculates the allocation probability and the payment of the winner drone based on the rules (i.e., payment rule $p$ and allocation rule $g$ ) as follow: Theorem 2 (Myerson [11]).",
"For any set of strictly monotonically increasing functions $\\phi _1 , \\dots , \\phi _U : R_{\\ge 0} \\rightarrow R_{\\ge 0}$ , an auction defined by the allocation rule $g_i = \\texttt {softmax}(\\overline{b_i}) $ and payment rule $p_i = \\phi ^{-1}_i( \\underset{j\\ne i}{ \\max } (\\phi (b_i)) ) $ is DSIC and IR.",
"The $\\phi ^{mononet}$ should have non-decreasing monotone feature when converting $b$ into transformed bid $\\overline{b}$ .",
"Therefore, the proposed deep learning network has a parameter constraint and a specific structure so that the deep learning network can be approximated to monotonic function via training process.",
"The used parameters for deep learning, i.e., weights and biases, are positive.",
"The structure of the network, shown in Fig REF , is rather simple.",
"The two layers network $\\phi ^{mononet}$ is represented as $\\phi ^{shared}(\\phi _i(b_i)), \\forall i = \\lbrace 1, \\dots , U\\rbrace $ [16], [18].",
"The assignment rule consist of the softmax operation which has been used in deep learning based multimodal classification.",
"The payment rules is composed of max operation and ReLU.",
"The ReLU makes the transformed bid $\\overline{b_i}$ which is less than the reserve price of SPA-0 to be 0.",
"The max unit is used to make $\\overline{p_i}$ be the highest transformed bid except $p_i$ .",
"The results of ReLU and max unit are denoted by $\\overline{p}$ .",
"$\\overline{p_i}$ is the value, before conversion to $p_i$ , which should be paid by the winner drone $u_i$ .",
"Note that $\\overline{p_i}$ can be larger than $b_i$ .",
"Therefore, in a IR auction, $\\overline{p_i}$ can not be the payment.",
"Thus $\\overline{p_i}$ is converted to $p_i$ via $\\phi ^{-1}_{mononet}$ .",
"This process makes the result of deep learning based auction to be IR when revenue optimal auction is designed as shown in Fig REF .",
"The $\\phi ^{-1}_{mononet}$ can be expressed as $\\phi ^{-1}_{shared}(\\phi ^{-1}_i(\\overline{p_i})), \\forall i=\\lbrace 1,\\dots ,U\\rbrace $ .",
"The computations of $\\phi ^{-1}_{shared}$ and $\\phi ^{-1}_i$ is described as follows.",
"$p_i^\\prime &=& \\max _{1 \\le g \\le \\mathcal {G}} \\left\\lbrace \\min _{1 \\le n \\le N}(w^{shared}_{g,n})^{-1}(\\overline{p_i} - \\beta ^{shared}_{g,n}) \\right\\rbrace \\\\p_i &=& \\max _{1 \\le g \\le \\mathcal {G}} \\left\\lbrace \\min _{1 \\le n \\le N}(w^{i}_{g,n})^{-1}(p_i^\\prime - \\beta ^{i}_{g,n}) \\right\\rbrace $ The two layers network $\\phi ^{mononet}$ constitutes the virtual valuation function $\\phi $ of Myerson auction, as shown in Fig REF .",
"In the $\\phi ^{-1}_{mononet}$ , it is important to reuse the weights from the $\\phi ^{mononet}$ network as presented in () and (REF ).",
"This forces $b_i$ to be equal to $\\phi ^{-1}_{mononet}(\\phi ^{mononet}(b_i))$ as in the case in which the Myerson virtual valuation function is based on full knowledge of the distribution $F$ .",
"The result of $\\phi ^{-1}_{mononet}$ is the payment which should be paid by winner drone $u_i$ .",
"Therefore, the result $p_i$ is greater than the second highest bid of $b$ and smaller than the winning bid $b_i$ [11].",
"Additional networks are required to implement the rules in overall auction processes.",
"The deep learning networks used in the proposed auction consist of three modular networks as follows: (i) a network that can replace the virtual valuation function $\\phi _i$ of the Myerson auction, (ii) a network for the allocation rule $g_i$ , and (iii) a network for the payment rule $p_i$ .",
"The above networks are optimized according to a loss function via a training process.",
"The loss function is essential to enable the deep learning computation of the same structure to have different characteristics [32], [33], and plays an important role in deep learning.",
"Loss Function.",
"In this paper, the negative expected virtual surplus is used as a loss function where the virtual surplus is equivalent to the revenue of the mobile charging station, that is, the auctioneer and seller.",
"The loss function is used to train deep neural network parameters (weights and biases).",
"The deep neural network that configures a revenue-optimal auction is composed of weights (denoted as $w$ ) and biases (denoted as $\\beta $ ) which replaces the virtual valuation function of Myerson.",
"Hhere, the deep neural network model automatically learns the distribution, and fits its parameters to the actual distribution of the data during the training process.",
"The trained neural networks are approximated by a virtual valuation function which is based on fully distributed knowledge.",
"The parameters $w$ and $\\beta $ of the deep neural network are trained through unsupervised learning without ground truth information, i.e., the winner (which drone will be scheduled for charging) and payment (how many the winner drone will pay).",
"Therefore, the results of the allocation and payment rules are used for training parameters (i.e., $\\vec{w}$ and $\\vec{\\beta }$ ) can be explained as follows: $\\mathcal {R}\\left(\\vec{w},\\vec{\\beta }\\right) = - \\sum ^{U}_{i=1}\\nolimits g_i(b_i) \\ast p_i(b_i)$ where the loss function (REF ) stands for the expected negative revenue of auctioneer, i.e., the maximization of the expected revenue of the auctioneer since the loss function should be minimized eventually during the training procedure.",
"Based on the loss function, the benefit of the deep learning based auction is seen in the training process.",
"The proposed networks which replace the virtual valuation function as well as auction rules are optimized to DSIC, IR and the revenue optimal auction.",
"Deep Learning Training.",
"The detailed training process of the three networks are summarized in Algorithm REF , where, based on the bid $b$ , the payment and allocation probabilities are calculated (line $[4-9]$ ).",
"$\\mathcal {R}(w,\\beta )$ is the loss function to guide the deep learning network training.",
"The negative expected revenue is used as the loss function.",
"The loss function can be calculated by allocation probability $g$ and payment $p$ (line $[10]$ ).",
"The $L_2(w,\\beta )$ is regularization factor which are used to regularize the deep learning parameters (weights and bias) (line $[11]$ ).",
"The $L_2(w,\\beta )$ regularization prevents parameters from becoming excessively large.",
"The training process is based on unsupervised learning; and thus the allocation probability and payment are the only required information.",
"This means that the environmental information such as distribution of private valuation is not required.",
"As a result, the proposed deep learning network can be easily applied to mobile charging stations.",
"The parameters are determined by means of empirical experiments as the payments of winners are updated sensitively due to the weight range (line $[15-16]$ )."
],
[
"Deep Learning Networks",
"In this paper, the virtual valuation function is replaced by the two layers network $\\phi ^{mononet}$ , composed of the monotonic networks.",
"Monotonic Network.",
"As shown in Fig REF , the monotonic network is a three-layer deep neural network.",
"The input layer is configured with multiple groups composed of sets of linear units.",
"The maximum value of each group is calculated in the second layer.",
"The last layer selects the minimum value of the given output of the second layer.",
"As the name suggests, the monotonic network is monotonic, and this characteristic is preserved regardless of the number of groups, units, and the order of min/max operations.",
"[t!]",
"0.5em0.8em 1.0 InputInput OutputOutput InitializeInitialize Optimized weights $w$ and $\\beta $ The network weights $w$ and $\\beta $ using Xavier initialization epoch r: $1 \\rightarrow \\mathcal {R}$ $t: 1 \\rightarrow \\mathcal {T}$ Forward: $\\triangleright $ $b_i^\\prime = \\phi _i{(b_i)} = \\underset{1 \\le g \\le \\mathcal {G}}{\\min }\\left\\lbrace \\underset{1 \\le n \\le \\mathcal {N}}{\\max }(w^{i}_{g,n}b_{i}+\\beta ^{i}_{g,n}) \\right\\rbrace $ $\\triangleright $ $\\overline{b_i} = \\phi _{shared}(b_i^\\prime )$ $\\triangleright $ $g_i = \\texttt {softmax}\\left(\\overline{b_1}, \\dots , \\overline{b_U};k\\right) = \\frac{e^{k\\overline{b_i}}}{\\sum _{j=1}^{U} e^{k\\overline{b_j}}}$ $\\triangleright $ $\\overline{p_i} = \\texttt {ReLU}\\left\\lbrace \\max _{j\\ne i} ( \\overline{b_i}) \\right\\rbrace $ $\\triangleright $ $p_i^\\prime $ = $\\phi _{shared}^{-1}(\\overline{p_i})$ $\\triangleright $ $p_i = \\phi ^{-1}_i(p_i^\\prime ) = \\underset{1 \\le g \\le \\mathcal {G}}{\\max } \\left\\lbrace \\underset{1 \\le n \\le \\mathcal {N}}{\\min }(w^{i}_{g,n})^{-1}(p_i^\\prime - \\beta ^{i}_{g,n}) \\right\\rbrace $ $\\triangleright $ Compute the expected negative revenue $\\mathcal {R}(w,\\beta ) = - \\sum ^{\\mathcal {U}}_{i=1} g_i(b_i) * p_i(b_i)$ $\\triangleright $ Compute $L_2$ weight loss $L_2(w, \\beta ) = \\sum ^{U}_{i=1}\\sum ^{\\mathcal {G}}_{g=1} \\sum ^{\\mathcal {N}}_{n=1}\\left\\lbrace (w^i_{g,n})^2 + (\\beta ^i_{g,n})^2\\right\\rbrace $ $\\triangleright $ Compute $\\texttt {Loss}(w, \\beta )$ = $\\mathcal {R}(w,\\beta ) + L_2(w, \\beta )$ Optimize: $\\triangleright $ Update $\\vec{w}$ and $\\vec{\\beta }$ for minimizing $\\texttt {Cost}(w, \\beta )$ $\\triangleright $ Clip $\\vec{w}$ $(\\min B, \\max \\infty )$ $\\triangleright $ Clip $\\vec{\\beta }$ $(\\min 0, \\max \\infty )$ Deep Learning Training Virtual Valuation Network.",
"The virtual valuation function in the Myerson auction is replaced with the monotonic network.",
"The computation of $\\phi ^{shared}$ and $\\phi _i$ is implemented as follows.",
"$\\small \\overline{b_i} = \\phi _i^{shared}{(b_i^\\prime )} = \\underset{1 \\le g \\le \\mathcal {G}}{\\min } \\left\\lbrace \\underset{1 \\le n \\le \\mathcal {N}}{\\max }(w^{shared}_{g,n}b_{i}^\\prime + \\beta ^{shared}_{g,n}) \\right\\rbrace $ $b_i^\\prime = \\phi _i{(b_i)} = \\underset{1 \\le g \\le \\mathcal {G}}{\\min } \\left\\lbrace \\underset{1 \\le n \\le \\mathcal {N}}{\\max }(w^{i}_{g,n}b_{i} + \\beta ^{i}_{g,n}) \\right\\rbrace $ The bid $b_i$ of the drone $u_i$ is transformed to $\\overline{b_i}$ via the virtual valuation network $\\phi ^{mononet}$ .",
"In the $\\phi ^{mononet}$ , all outcomes of $\\phi ^{shared}$ are calculated on the same weights, whereas the $\\phi _i$ calculates the outcome using different weights for each bid.",
"The inverse computation of $\\phi ^{mononet}$ is denoted by $\\phi ^{-1}_{mononet}$ .",
"The $\\phi ^{-1}_{mononet}$ that determines the payment of the winner drone $u_i$ which is composed of two networks.",
"In the computation of $\\phi ^{-1}_{mononet}$ , the weights of $\\phi ^{mononet}$ are used.",
"Thus the computations of two layers can be expressed as follows: $\\begin{medsize}p_i^\\prime = \\phi ^{-1}_{shared}(\\overline{p_i}) = \\underset{1 \\le g \\le \\mathcal {G}}{\\max } \\left\\lbrace \\underset{1 \\le n \\le \\mathcal {N}}{\\min }(w^{shared}_{g,n})^{-1}(\\overline{p_i} - \\beta ^{shared}_{g,n}) \\right\\rbrace \\end{medsize}$ $p_i = \\phi ^{-1}_i(p_i^\\prime ) = \\underset{1 \\le g \\le \\mathcal {G}}{\\max } \\left\\lbrace \\underset{1 \\le n \\le \\mathcal {N}}{\\min }(w^{i}_{g,n})^{-1}(p_i^\\prime - \\beta ^{i}_{g,n}) \\right\\rbrace $ The payment $\\overline{p_i}$ of the drone $u_i$ is transformed to $p_i$ via the $\\phi ^{-1}_{mononet}$ .",
"The $\\phi ^{-1}_{mononet}$ consists of $\\phi ^{-1}_i$ and $\\phi ^{-1}_{shared}$ .",
"The same weights are used to calculate all outcomes of $\\phi ^{-1}_{shared}$ .",
"The outcome of $\\phi ^{-1}_i$ is calculated based on different weights for each bid, as shown in (REF ).",
"The monotonic network is responsible for the transformation of the virtual bid $\\overline{b_i}$ in auction.",
"As mentioned above, the optimal revenue is equivalent to the optimal virtual surplus.",
"Thus, the monotonic network is major component of the auction.",
"However, in order to configure the revenue-optimal auction, the additional network by allocation and payment rules is required.",
"In this paper, the payment and allocation rules are configured with ReLU and softmax which have been mainly used in deep learning as an activation function.",
"This makes backpropagation easy during the training process.",
"Allocation Rule Network ($g_i$ ).",
"This section describes in more details the structure of the allocation rules ($g_i$ ).",
"In this paper, since this allocation rule is implemented using a deep neural network, the probability is calculated using softmax which converts the input vector into a probability vector.",
"The allocation rule $g$ awards the charging service to the highest bidder drone; and thus the highest probability is assigned to the highest bidder.",
"The continuous function (REF ) traditional auction is approximated using the deep network, which converts the input vector $\\overline{b}$ to the probability vector.",
"In the SPA auction with reserve price 0 (SPA-0), the allocation rule assigns the highest winning probability to the highest bidder whose transformed bid $\\overline{b_i}$ is greater than 0, $\\overline{b_i}>0$ .",
"The softmax based assignment can be calculated as follows: $g_i = \\texttt {softmax}\\left(\\overline{b_1}, \\dots , \\overline{b_U};k\\right) = \\frac{e^{k\\overline{b_i}}}{\\sum _{j=1}^{U} e^{k\\overline{b_j}}}.$ The parameter $k$ is a constant value and it determines the quality of the approximation.",
"As the $k$ increases, the quality of the approximation increases, whereas the smoothness in the allocation network decreases.",
"For simplicity, this means that the higher $k$ makes a large difference between the allocation probabilities of users [18].",
"When the networks are trained to minimize (REF ), the value of $g_i(b_i)$ increases.",
"As a result, since the profit of the auctioneer is related to the second highest $g_i(b_i)$ , it is also a function of the parameter $k$ .",
"Results in Sec. )",
"show how larger values of $k$ lead to higher profits.",
"Payment Rule Network ($\\overline{p_i}$ ).",
"This section describes the structure of the allocation rule ($\\overline{p_i}$ ).",
"The ReLU is widely used in deep learning computation as an activation function.",
"In the proposed auction, the payment $p_i$ of drone $u_i$ is calculated from the transformed bid $\\overline{b_i}$ .",
"Before the computation of $\\phi ^{-1}_{mononet}$ , the deep network excludes the bid below the reserve price 0 via ReLU$(\\overline{b_i})$ $\\triangleq \\max (\\overline{b_i}, 0)$ .",
"The input $\\overline{b_i}$ is the second highest transformed bid which is the output of $\\max _{j\\ne i}\\left( \\overline{b_i}\\right)$ .",
"The payment rule network can be, then, calculated as: $\\overline{p_i} = \\texttt {ReLU} \\left\\lbrace \\max _{j\\ne i}\\left( \\overline{b_i}\\right) \\right\\rbrace $ and the result $\\overline{p_i}$ is used as an input of (REF ), i.e., the actual payment of winner drone.",
"0.5em0.8em [t] 1.00 InputInput OutputOutput allocation probability set $g_i \\triangleq (g_1, \\dots , g_U)$ , payment set $p_i \\triangleq (p_1,\\dots ,p_U)$ Mobile charging system is $idle$ $\\triangleright $ Drones: charging scheduling valuation $v_i$ $\\triangleright $ Drones: submit bid $b_i$ $\\triangleright $ $b_i^\\prime = \\phi _i{(b_i)} = \\underset{1 \\le g \\le \\mathcal {G}}{\\min }\\left\\lbrace \\underset{1 \\le n \\le \\mathcal {N}}{\\max }(w^{i}_{g,n}b_{i}+\\beta ^{i}_{g,n}) \\right\\rbrace $ $\\triangleright $ $\\overline{b_i} = \\phi _{shared}(b_i^\\prime )$ $\\triangleright $ $g_i = \\texttt {softmax}\\left(\\overline{b_1}, \\dots , \\overline{b_U};k\\right) = \\frac{e^{k\\overline{b_i}}}{\\sum _{j=1}^{U} e^{k\\overline{b_j}}}$ $\\triangleright $ $\\overline{p_i} = \\texttt {ReLU}\\left\\lbrace \\max _{j\\ne i} ( \\overline{b_i}) \\right\\rbrace $ $\\triangleright $ $p_i^\\prime $ = $\\phi _{shared}^{-1}(\\overline{p_i})$ $\\triangleright $ $p_i = \\phi ^{-1}_i(p_i^\\prime ) = \\underset{1 \\le g \\le \\mathcal {G}}{\\max } \\left\\lbrace \\underset{1 \\le n \\le \\mathcal {N}}{\\min }(w^{i}_{g,n})^{-1}(p_i^\\prime - \\beta ^{i}_{g,n}) \\right\\rbrace $ $\\triangleright $ Calculate winner and payment $(g_k, p_k)$ $\\triangleright $ Winner Drone: Pay payment $\\triangleright $ Allocate charging system to the winner Deep Learning-Based Algorithm for the Auction Controlling the Charging Scheduling"
],
[
"Overall Auction Mechanism",
"The overall deep learning-based auction mechanism is summarized in Algorithm REF .",
"If the mobile charging system becomes idle, the auction is initiated (line $[1]$ ).",
"The valuation $v_i$ for the charging time is computed by each drone $u_i$ based on its own private criteria.",
"Then, based on the individual private valuation, each drone submits its bid $b_i$ (line $[2-3]$ ).",
"The mobile charging station runs the auction using the pre-trained networks.",
"If $\\overline{p} = 0$ , then all the drones assign a low valuation to the charging time and the mobile charging system does not allocate the charging time to users.",
"If there exist bids which are larger than reserve price 0, the corresponding allocation and payment probabilities are calculated using the proposed deep learning networks, i.e., virtual valuation network, allocation network, and payment network (line $[4-10]$ ).",
"Because the proposed deep learning auction is the variant of SPA-0, any bid below the reserve price 0 is converted to 0 (line $[7]$ ).",
"As shown in line $[11]$ , the mobile charging station assigns the payment of $p_i$ to the drone $u_i$ with the highest $g_i$ .",
"Finally, the mobile charging station allocates the charging time to the winner drone $u_i$ (line $[12]$ ).",
"The drone, then, reaches the charging station and occupy it for the duration of the slot.",
"After the winner drone leaves the charging station, next iteration starts if the mobile station is idle.",
"Figure: Revenue changes by kk and UU.Table: Revenue changes by kk, UU (in Fig.-)Table: Parameters"
],
[
"Performance Evaluation",
"Software Prototype.",
"First, we describe the software developed to test the auction mechanism.",
"The Xavier initializer was used for weight value initialization, where the biases were initialized as 0.",
"As mentioned earlier, $L_2$ regularization is used to prevent excessive parameter growth during training and reduce overfitting.",
"The regularization factor was set to $0.001$ .",
"During the training phase, the Adam [34] optimizer was us to iteratively.",
"This choice is motivated by the need to keep separated the learning rates for each weight.",
"An exponentially decaying average of previous gradients was used for iteration-based optimization.",
"In the experiments, different uniform distributions were used for data generation, as shown in Table REF .",
"Data-intensive evaluation was conducted with $100,000$ generated data sets.",
"Among the data sets, 70% of sets were used for training; and the remaining 30% were used for testing.",
"The proposed deep learning-based auction mechanism was implemented in Python/TensorFlow [35] and Keras [36].",
"A multi-GPU platform (equipped with 2 NVIDIA Titan XP GPUs using 1405 MHz main clock and 12 GB memory) was used for training and testing.",
"Experimental Setting.",
"The test environment includes 5, 10, or 15 drones.",
"During performance evaluation, the parameter $k$ is determined to control the quality of approximation.",
"First, we compare the proposed model with SPA-0 with a priori knowledge to demonstrate revenue-optimality.",
"Results show the ability of the proposed deep learning-based approach to adapt to different scenarios.",
"The valuation results of drones are generated based on various distributions as defined in Table REF .",
"Table REF summarizes the used parameters.",
"Figure: Revenue analysis.Revenue Analysis - Parameter $k$ .",
"The proposed framework is based on the Myerson optimal auction, which produces an increased revenue to the mobile charging station compared to SPA-0 auctions.",
"The experiments shown in Fig.",
"REF confirm this effect, and illustrate the effect of the parameter $k$ .",
"Fig.",
"(REF )-(REF ) show a comparison between the revenue of the mobile charging station – the auctioneer – as a function of the parameter $k$ defined in (REF ).",
"In the experiments, the bid set is uniformly generated in the range of $0-10$ .",
"The bid is calculated based on the private valuation as discussed in Sec. .",
"The value of system parameters, such as battery consumption rate, and weight, is also assumed to be uniformly distributed.",
"The results in Fig.",
"(REF )-(REF ) show that the revenue increases as the $k$ increases.",
"The numerical results are presented in the Table REF .",
"The revenue gap between the SPA-0 and the proposed auction when the number of drone is 5 is near $2.25$ when $k=1$ , near $2.26$ when $k=3$ and about $2.35$ when $k=5$ .",
"The mobile charging station can take the highest revenue, i.e., the case where $k=5$ .",
"This result shows that the revenue of mobile charging station increases in the order of $SPA-0$ , $k=1$ , $k=3$ and $k=5$ .",
"The parameter $k$ determines not only the approximation quality of softmax function but also the revenue of charging station.",
"In Fig.",
"REF , the number of drones which participate in the proposed charging scheduling auction is updated.",
"In general, more drones participate in auction, the higher the bid can be submitted to the auction with high probability; and thus the second highest bid value of the auction can be increased while the number of drones increases.",
"Note that the revenue of auctioneer is equivalent to the payment of user.",
"Therefore, the payment of winner drone increases.",
"As a result, the revenue of mobile charging station becomes larger.",
"In the SPA-0, the revenue is increased from $4.7532$ to $5.8493$ when the number of drones increases from 5 to 10.",
"Similarly, the revenue of proposed charging scheduling auction increases to $7.0345$ when $k=1$ , $7.1408$ when $k=3$ and $7.2235$ when $k=5$ .",
"This tendency is maintained when the number of drones increases from 10 to 15, as shown in (Fig.REF -REF ).",
"Fig.",
"REF and Table.",
"REF shows the revenue of $k=3$ model when the number of drones increases from 5 to 15.",
"In this evaluation, the training of deep learning network uses the pre-trained weights.",
"Based on this experiment result, we can confirm that the proposed deep learning auction provides higher revenue to mobile charging system when the number of drones increases.",
"In Fig.REF , the horizontal axis of the experiment means the iteration of the proposed deep learning network training.",
"The convergence of the deep learning networks during small number of iteration shows high adaptability to specific applications.",
"Fig.",
"(REF -REF ) show that the proposed deep learning-based auction can achieve stability in approximately 300 iterations.",
"Fig.",
"REF shows that the stability can be achieved much faster when the pre-trained network is used.",
"This results mean that the proposed deep learning based auction has high adaptability; and thus it can be applied to the various environment with partial knowledge valuation distribution as presented in Sec. .",
"In Fig.REF , the results show that the proposed auction guarantees the increased revenue of mobile charging station over SPA-0 and has a highly adaptive algorithm under partial knowledge distribution (a.k.a., not fully distributed knowledge).",
"Table: Revenue statistics (in Fig.",
")Table: Revenue of mobile charging station (in Fig.",
")Figure: Payment comparison among drones.Statistical Analysis (Parameter $k$ ).",
"In this section, we show the case where the penalty given to participant who bids a false bid.",
"We confirm that the proposed method imposes penalty on the false bidder.",
"In addition, the experimental results shows that how the penalty varies depending on $k$ values.",
"In Fig.",
"REF , the effect of parameter $k$ can be observed while the number of drones varies.",
"Fig.",
"REF shows that the statistics analysis of revenue values for difference $k$ values in (REF ) and SPA-0 when the number of drones does not vary.",
"The experiment results compare the average revenue, maximum revenue, minimum revenue, top 25 percentile, and top 75 percentile.",
"The evaluation uses the deep learning networks when the $k$ values are 1 and 3.",
"As $k$ increases, the gap between the average revenue of model and the average revenue of SPA-0 get larger as shown in Fig.",
"REF and Table.",
"REF .",
"When the proposed model is $k=1$ , the revenue average is $8.0548$ , similar to the revenue average of SPA-0.",
"However, when the $k$ value of model is 3, the revenue average is $8.6032$ ; and thus the model gets near 7% higher revenue average than SPA-0.",
"When $k=1$ , the gap between the proposed model and SPA-0 is near $0.008$ in terms of top 25 percentile whereas the gap is about $0.57$ , when $k=3$ .",
"In addition, in terms of top 75 percentile, the revenue of $k=3$ model is about $0.3$ larger than $k=1$ model and SPA-0.",
"This result shows that the proposed model with large $k$ takes higher revenue.",
"Therefore, we can confirm that the revenue of mobile charging station declines in the order of $k=3$ , $k=1$ , and SPA-0.",
"In Fig.",
"REF , the graph shows the 10 results of validation experiments, i.e., 10 cases are considered in the validation experiments.",
"The number on the $X$ -axis in Fig.",
"REF represents the indices of individual cases.",
"The result stands for the revenue of mobile charging station via deep learning auction.",
"We can confirm that the revenue with $k=1$ is always smaller than the one with $k=3$ .",
"The gap between the $k=1$ and $k=3$ models is about $0.2$ in Case 1; and the Case 1 is the minimum, whereas the maximum gap is about 5 in Case 7 as shown in Table REF .",
"The revenue with $k=1$ is larger than the SPA-0, but similar to SPA-0.",
"The gap between the $k=1$ and SPA-0 is $0.13$ in Case 4; and the Case is the minimum.",
"The maximum gap is about $0.18$ in Case 1.",
"This experiments also show that the gap between the results by the two models with $k=1$ /$k=3$ and the results of SPA-0 are not always equivalent.",
"For example, the gap between $k=1$ and $k=3$ models is near $0.2$ in Case 1m however near $1.7$ in Case 2.",
"This is due to the fact that the transformation depends on the weight of $\\phi _i$ ; and thus the transformation via $\\phi ^{mononet}$ is not applied equally to the same bids.",
"This means that if $b_{1}=2$ and also $b_{3}=2$ , these two bids can be transformed differently.",
"Therefore, the payment is not always equivalent.",
"This means that the proposed deep learning auction adapts to the bid distribution at the time of the auction procedure, giving the mobile charging station high revenue.",
"It can be seen that higher revenue is guaranteed by increasing the value of parameter $k$ .",
"Table: Payment of drone (in Fig.",
")The proposed deep learning based auction algorithm has a strength in terms of giving penalty to false bidder.",
"In Fig.",
"REF , experiment results present the payment of drone when the drone submits bid falsely (i.e., fake bid).",
"This experiment conducts with the models of $k=1$ and $k=3$ .",
"The experiment assumes that the number of drones which participate in auction is 5.",
"The truth valuation of drone which submits the bid falsely is set to $15.9835$ .",
"The bid values of the other drones are generated by uniform distribution.",
"This experiment uses the scenario where 5 drones exist and one is with fake bid and the other four are with truthful bids.",
"In Fig.",
"REF , the second highest bid is set to $8.6177$ as shown in Table REF .",
"This result shows that a drone cannot win the auction when it bids up to $0.2-0.8$ times larger than the true valuation $15.9835$ .",
"As a result, the drone is defeated in the auction due to false bid.",
"On the other hand, when a drone submits bid as $1.2-2$ times larger than true valuation, the fake bid leads to win in auction.",
"However, the fake bid increases the payment in the fake-bid drone.",
"For example, if the drone submits near 40 bid falsely in the SPA-based auction, the drone can only pay about $8.4$ .",
"However, the payment is $32.4915$ in the proposed auction with $k=1$ .",
"This means that the bidding of drone which falsely submit the bid for getting charging increases the payment.",
"In addition, the payment increases when the $k$ value of models increases.",
"Table REF also shows the payment increment while $k$ increases.",
"The increased payment of the proposed model is at least 30% greater than that of the auction using SPA-0 as shown in Table REF .",
"When the drone submits true valuation, the payment is just about 50% higher than the SPA-0 auction.",
"However, if the bid is $1.2$ times larger than the true valuation, the payment is about 200% larger than the SPA-0 auction.",
"This experiment shows that if drone submits false bid for winning the auction, the drone gets a loss in terms of the payment; and thus the loss let the drone avoid fake bidding.",
"Figure: Performance improvements under various bid distributions.Fig.",
"(REF ) shows the proposed deep learning based auction can be trained through transfer learning when the distribution of bid values varies.",
"The dotted lines are revenue when the SPA-0 is executed.",
"The red and blue lines stand for the revenue when the proposed algorithm is used.",
"The revenue is higher than SPA-0 as shown in previous experiments.",
"The proposed model is stabilized with 400 training iterations if training starts from the initialized model.",
"If the training starts from the trained model (i.e., transfer learning), the proposed model is stabilized with approximately 100 training iterations when the bid distribution varies.",
"This experiment shows that the proposed model adapts to the change of the bid distribution and can provide reasonable results at various distribution.",
"In Fig.",
"(REF ), we consider the flight energy consumption of drones in this experiment as follows [37], [38].",
"$E(t) = (\\beta + \\alpha z)\\cdot t + P_{\\max }\\left(\\frac{z}{s}\\right)$ where $\\alpha $ is a motor speed multiplier, $\\beta $ is the minimum power needed to hover just over the ground (when altitude is almost zero), $z$ means the height at time step $t$ , and $s$ is the speed of drone and $P_{\\max }$ is the maximum power of motor to flight.",
"Therefore, the term $P_{\\max }(\\frac{z}{s})$ refers to the power consumption needed to lift to height $h$ with speed $v$ [37], [38].",
"In this experiment, we set the values of $\\alpha $ to $5.5$ , $\\beta $ to 15, $z$ to $1m$ , and $P_{\\max }$ to 45.",
"Fig.",
"(REF ) shows that the number of drones discharged from battery.",
"In this experiment, we assumed that the charging service fully charges the battery of the drones which wins in the auction.",
"The charging service is only for 1 time slot.",
"Therefore, the drones consume $65.5mAh$ per 1 time step (1 hour) and recharge $1000mAh$ when recharged.",
"In this experiment, we assume that 15 drones exist and they want to constantly join the charging service scheduling.",
"In Fig.",
"(REF ), when the proposed deep learning auction is used to schedule charging services, 11 of the 15 drones can be charged without being discharged.",
"This experiment shows that the proposed method can increase the drone flight time in multi-drone networks.",
"Figure: φ i (b i )\\phi _i(b_i) changes according to drone's valuation.Figure: φ shared (b i ' )\\phi ^{shared}(b_i^\\prime ) changes according to drone's valuation.Deep Learning Model Characteristics.",
"These experimental results explain why monotonic network is considered as the baseline structure of the proposed deep learning auction architecture.",
"In the proposed auction, when bids are transformed to virtual values through the monotonic network, the order of bids must be maintained.",
"For example, if the bid $b_1$ of user $u_1$ is larger than the bid $b_2$ of user $u_2$ , converted $\\overline{b_1}$ must be larger than $\\overline{b_2}$ .",
"The corresponding experimental results show that the monotonic network performs the transformation that maintains the order.",
"Fig.",
"REF shows the transformed value which is the result of virtual valuation function $\\phi _i$ when the bid of the winner drone and the second highest bid increase from 1 to 10.",
"The fixed weights of the $\\phi _i$ were used when the winning bid and the second highest bid were transformed because the weights are continuously updated during deep learning training process.",
"For this experiment, the networks are trained when the winning bid is $7.48292044$ , the second highest bid is $4.75319804$ , and other bids are generated along the uniform distribution.",
"In Fig.",
"REF , the transformed value $p_i^\\prime $ of the second highest bid is higher than the one of winning bid.",
"This tendency can be shown in Fig.",
"REF and Fig.",
"REF .",
"By comparing the Fig.",
"REF and Fig.",
"REF , it can be observed that the transformed bid decreases as the number of drones increases.",
"However, through Fig.",
"REF and Fig.",
"REF , we can confirm that the transformed value $b_i^\\prime $ is independent to the number of drones.",
"Instead of the number of drones, the weights are affected by the allocation probability and payment of other bids because the loss function is configured based on the allocation rule and payment rule.",
"Fig.",
"REF also shows the $\\phi _i$ network conducts non-decreasing monotonic transformation.",
"Therefore, the $\\phi _i$ network is able to replace $\\phi $ function in Myerson auction because the $\\phi _i$ network performs a monotonic transformation.",
"In Fig.",
"REF , the changes of $\\phi ^{shared}(\\phi _i(b_i))$ is presented when the bid of the winner drone and the second highest bid increase from 1 to 10.",
"The fixed weights of the $\\phi ^{shared}$ were also used.",
"The deep learning network is trained when the winning bid is $7.48292044$ , the second highest bid is $4.75319804$ , and other bids are generated along the uniform distribution similar to the evaluation for Fig.",
"REF .",
"It can be seen that the result of $\\phi ^{shared}$ is larger than the one of the $\\phi _i$ shown in Fig.",
"REF because the computation of $\\phi ^{shared}$ is conducted based on the result of the $\\phi _i$ as well as the $\\phi ^{shared}$ does non-decreasing monotonic transformation.",
"Through comparison of Fig.",
"REF and Fig.",
"REF , it can be seen that the result of $\\phi ^{shared}$ decreases when the number of drones increases, as shown in Fig.",
"REF .",
"However, in Fig.",
"REF , it can be observed that the result of $\\phi ^{shared}$ with 15 drones becomes larger than the one of $\\phi ^{shared}$ with 10 drones.",
"Therefore, this experiment shows that the weights are affected by the allocation probability and payment and they are independent to the number of drones.",
"The transformed bid $\\overline{p_i}$ of the second highest bid is larger than the one of the winning bid.",
"The tendency which is shown in Fig.",
"REF is maintained.",
"The $\\phi _i$ network has independent weights per input data.",
"That is, if there are 5 inputs, there are 5 $\\phi _i$ networks, and all learning networks have different weights.",
"However, $\\phi ^{shared}$ network is just one network regardless of the number of inputs.",
"If there are 5 inputs, only one $\\phi ^{shared}$ network exists.",
"Through Fig.",
"REF and Fig.",
"REF , it can be observed that the non-decreasing monotonic feature of $\\phi _i$ networks and $\\phi ^{shared}$ are trained by the limited network structure and the loss function regardless of whether the weights are shared or not."
],
[
"Concluding Remarks and Future Work",
"The proposed deep learning based auction is revenue optimal for mobile charging scheduling in distributed multi-drone networks.",
"In this paper, the mobile charging scheduling problem is interpreted as auction problem where each drone bids its own valuation and then the charging station schedules drones based on it in terms of revenue-optimality.",
"Through the proposed deep-learning based solution approach, the charging auction enables efficient scheduling by automatically learning the required knowledge (i.e., bids distribution), which is required in conventional auction mechanisms.",
"Therefore, environmental information is not required anymore in auction computation.",
"This makes effective troubleshooting possible in distributed multi-drone networks.",
"The proposed algorithm only requires payment and allocation probabilities by the multi-drones.",
"The loss function in deep learning computation is an important factor that allows the proposed auction to be constructed based on environment independent information.",
"As verified via software prototype based performance evaluation, following facts are observed: (i) guaranteeing optimal revenue in terms of individual rationality and dominant strategy incentive compatibility, (ii) limiting the false bids of drones by increasing the payment to the false-bid drones, and (iii) enabling a revenue optimal auction to be constructed without complex prior knowledge, i.e., bids distribution.",
"As future research directions, advanced auction mechanism designs with multiple mobile charging stations are worthy to consider.",
"In this case, the problem can be formulated with multi-item auction and then the corresponding mathematical formulation, verification, and analysis are desired.",
"Furthermore, the proposed deep learning-based auction mechanisms can be advantageous in various applications.",
"For example, visual attention is considerable because it can be reformulated as resource allocation [39], [40], [41], [42].",
"[Figure: NO_CAPTION [Figure: NO_CAPTION He is a senior member of the IEEE.",
"He was a recipient of the Annenberg Graduate Fellowship with his Ph.D. admission from USC (2009) and the Haedong Young Scholar Award (2018) which is for recognizing a young Korean researcher under the age of 40 who has made outstanding scholarly contributions to communications and information sciences research.",
"[Figure: NO_CAPTION"
]
] | 2001.03611 |
[
[
"A geometric vof method for interface resolved phase change and\n conservative thermal energy advection"
],
[
"Abstract We present a novel numerical method to solve the incompressible Navier-Stokes equations for two-phase flows with phase change, using a one-fluid approach.",
"Separate phases are tracked using a geometric Volume-Of-Fluid (VOF) method with piecewise linear interface construction (PLIC).",
"Thermal energy advection is treated in conservative form and the geometric calculation of VOF fluxes at computational cell boundaries is used consistently to calculate the fluxes of heat capacity.",
"The phase boundary is treated as sharp (infinitely thin), which leads to a discontinuity in the velocity field across the interface in the presence of phase change.",
"The numerical difficulty of this jump is accommodated with the introduction of a novel two-step VOF advection scheme.",
"The method has been implemented in the open source code PARIS and is validated using well-known test cases.",
"These include an evaporating circular droplet in microgravity (2D), the Stefan problem and a 3D bubble in superheated liquid.",
"The accuracy shown in the results were encouraging.",
"The 2D evaporating droplet showed excellent prediction of the droplet volume evolution as well as preservation of its circular shape.",
"A relative error of less than 1% was achieved for the Stefan problem case, using water properties at atmospheric conditions.",
"For the final radius of the bubble in superheated liquid at a Jacob number of 0.5, a relative error of less than 6% was obtained on the coarsest grid, with less than 1% on the finest."
],
[
"Introduction",
"A popular approach used in the wider problem space of the numerical solution of multiphase-flows is the so-called one-fluid formulation, in which the governing equations (incompressible Navier-Stokes equations for two-phase flow in our case) are applied to both phases [1].",
"Some phase tracking technique is then employed to calculate the phase-dependent fluid properties.",
"For an overview of different interface tracking techniques and associated methods to account for phase change, the reader can refer to the work of Kharangate & Mudawar [2] and Tryggvason et al.",
"[1].",
"In this work, the focus will be on the Volume-of-Fluid (VOF) method [3].",
"Even though VOF methods are relatively common for incompressible, isothermal two-phase flows, very few include temperature driven phase change.",
"Some of the pertinent VOF works dealing with interface resolved phase change problems will be mentioned here as a reference.",
"Welch and Wilson [4] developed a VOF method for phase change on staggered (MAC) grids.",
"The interface is reconstructed using piecewise linear segments, based on the method of Youngs [5].",
"A time-split geometric advection technique, also by Youngs, is used to advect the interface.",
"The energy conservation equation is treated non-conservatively.",
"Welch and Rachidi [6] expanded this method to study film boiling with conjugate heat transfer.",
"Agarwal et al.",
"[7] used the Welch method [4] with the addition of temperature dependence on fluid properties.",
"Schlottke and Weigand used a geometric based method to study evaporating droplets [8].",
"They developed an intricate, iterative method to deal with the calculation of a volume source due to mass transfer, which they apply only in interface cells (sharply).",
"However, they neglect the velocity disparity between the interface and the liquid phase.",
"Kunkelmann [9] used an algebraic VOF advection technique in the OpenFOAM framework and implemented a microlayer and contact angle model to simulate nucleate boiling on unstructured meshes.",
"The mass transfer between phases is accounted for using a smearing technique: source terms are redistributed in some finite region around the interface, which smoothes out the sharp discontinuity in velocity at the interface.",
"Guedon [10] also used the OpenFOAM framework for arbitrary meshes, but attempted to apply a sharp interface approach instead of the smearing technique used by Kunkelmann.",
"A ghost-fluid approach was used at the interface to apply the jump conditions.",
"The author, however, reported difficulties from spurious currents due to unbalanced surface tension models.",
"Another difference compared to the method of Kunkelmann, is the non-conservative treatment of energy conservation.",
"In summary, there remains several non-trivial challenges to the numerical treatment of the discontinuities at the interface in VOF phase change modelling schemes.",
"More specifically, the authors are unaware of a method that combines conservative thermal energy advection with a sharp interface treatment of the velocity discontinuities that arise from phase change.",
"This is particularly pressing due to the large heat capacity ratio (per unit volume) between liquid and gas phases of many substances of practical interest.",
"In this work we aim to address the above challenges.",
"A novel method is developed which simulates the incompressible flow of pure substances undergoing phase change.",
"A VOF method with piecewise linear interface calculation (PLIC) is employed.",
"The interface is treated as sharp (infinitely thin), separating immiscible fluids.",
"A novel geometric VOF advection method is developed to deal with the discontinuity in the velocity that arises from phase change.",
"It is applied consistently to the calculation of the thermal energy advection term, which is treated in conservative form.",
"The method is designed such that it can be easily incorporated into an existing VOF framework using geometric advection algorithms, such as that of Weymouth and Yue [11]."
],
[
"Mathematical Formulation",
"This work entails modelling the incompressible flow of pure substances undergoing phase change and in this section we describe the integral form of the governing equations as applied in a generic control volume, $V$ .",
"For a detailed derivation, refer to [12].",
"Fig.",
"REF shows volume $V$ , which is fixed in space and bounded by surface $S$ of arbitrary shape with outward pointing normal $\\mathbf {n}$ .",
"Surface $\\Gamma $ is assumed to have zero thickness, intersects $V$ and represents the phase boundary (interface) between two pure phases.",
"Note that we will not consider molecular diffusion between phases in this work and will name the two phases liquid ($\\ell $ ) and gas ($g$ ) throughout.",
"The surface $\\Gamma $ has a normal $\\mathbf {n}_\\Gamma $ which points from the liquid into the gas phase.",
"We assume constant material properties (density, viscosity and heat conductivity) in the respective liquid and gas phases.",
"Figure: The control volume under consideration, VV, contains a sharp interface Γ\\Gamma (red shading) and is bounded in space by fixed surfaces S ℓ S_\\ell and S g S_g.",
"The interface has velocity 𝐮 Γ \\mathbf {u}_\\Gamma , which may differ from the velocity of the adjacent fluid.For phase change at the interface, $\\Gamma $ , we define $\\dot{m}^{\\prime \\prime } $ as the rate at which liquid is converted to gas (or vapour).",
"It has units of mass per unit area per unit time, so that $\\dot{m}^{\\prime \\prime } = \\rho _\\ell \\left(\\mathbf {u}_\\ell - \\mathbf {u}_{\\Gamma } \\right) \\cdot \\mathbf {n}_{\\Gamma } = \\rho _g \\left(\\mathbf {u}_g - \\mathbf {u}_{\\Gamma } \\right) \\cdot \\mathbf {n}_{\\Gamma } \\,.$ Here $\\mathbf {u_\\ell }$ and $\\mathbf {u_g}$ are the respective liquid and gas phase velocities adjacent to the interface.",
"The velocity of $\\Gamma $ is $\\mathbf {u}_\\Gamma $ .",
"We use the Heaviside function, $H$ , as an indicator for the liquid phase, so that: $H(\\mathbf {x},t) ={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } \\, \\mathbf {x} \\, \\text{ is inside } \\, V_\\ell \\\\0, & \\text{if } \\, \\mathbf {x} \\, \\text{ is inside } \\, V_g \\,.\\end{array}\\right.",
"}$ The definition of $H$ can be constructed by using the integral of consecutive products of one dimensional Dirac delta functions, which is extended to three dimensions for $V$ from the derivation in [13].",
"The rate of change of $H$ in $V$ is given by $ \\int _V \\dfrac{\\partial H}{\\partial t} \\, dV + \\oint _S H \\mathbf {u} \\cdot \\mathbf {n} \\, dA + \\int _\\Gamma \\left(\\mathbf {u}_\\ell - \\mathbf {u}_{\\Gamma } \\right) \\cdot \\mathbf {n}_{\\Gamma }dA = 0\\,,$ where the last term on the left hand side accounts for phase change at the interface along with the previously defined convention that $H=1$ in the liquid phase.",
"The VOF represents the volumetric fraction of liquid in $V$ .",
"We will denote it $c$ , from the popular usage of colour function $ c = \\dfrac{1}{V} \\int _{V} H \\left(\\mathbf {x},t \\right) \\, dV \\,.$ From the equation above, it can be deduced that $0 \\le c \\le 1$ .",
"For $V$ completely filled with liquid $c=1$ and $c=0$ when only gas is present.",
"Note that $c$ is a scalar function of which the definition is always relative to the volume in which it is defined.",
"Using (REF ), (REF ) and (REF ), we can write for $c$ that $ \\dfrac{\\partial c}{\\partial t} + \\nabla \\cdot \\left( \\mathbf {u} c \\right) + \\dfrac{\\dot{m}^{\\prime \\prime } }{\\rho _{\\ell }} \\delta _\\Gamma = 0 \\,,$ where $\\delta _\\Gamma $ is a Dirac delta function defined on $\\Gamma $ .",
"Equation (REF ) is intended in a weak sense, since $c$ is not a continuous function and is solved by integrating over a volume.",
"Let $\\overline{\\phi }$ be some volume averaged material property.",
"We then have that $ \\overline{\\phi } = \\phi _\\ell c + (1-c)\\phi _g$ with the subscripts indicating the phase.",
"We can now write a single set of the governing equations, also referred to as the “one-fluid” approach.",
"The fluid density ($\\rho $ ), kinematic viscosity ($\\mu $ ) and heat capacity ($C_p=\\rho c_p$ ) is calculated using (REF ), with the product of density and specific heat capacity creating a heat capacity per unit volume.",
"From conservation of mass in $V$ we have that $ \\int _{V} \\mathbf {\\nabla } \\cdot \\mathbf {u} \\, d V = \\left( \\frac{1}{\\rho _g} - \\frac{1}{\\rho _\\ell } \\right) \\int _{V} \\dot{m}^{\\prime \\prime } \\delta _\\Gamma \\, dV \\,.$ Here $\\mathbf {u}$ is the fluid velocity.",
"The conservation of momentum is written $ \\int _{V} \\dfrac{\\partial \\left( \\rho \\mathbf {u} \\right)}{\\partial t} \\, dV + \\oint _{S} \\rho \\mathbf {u} \\left( \\mathbf {u} \\cdot \\mathbf {n} \\right) \\, dA = \\oint _{S} \\mathbf {\\tau } \\cdot \\mathbf {n} \\ dA + \\int _{V} \\rho \\mathbf {g} \\ d V + \\int _{V} \\sigma \\kappa \\delta _\\Gamma \\mathbf {n}_\\Gamma \\, dV \\,,$ with $\\mathbf {\\tau }$ the stress tensor, which for our problem is $\\mathbf {\\tau } = -p\\mathbf {\\mathrm {I}} + \\mu \\left( \\nabla \\mathbf {u} + \\left(\\nabla \\mathbf {u}\\right)^{T}\\right) \\,.$ Here $p$ is the static pressure, $\\mathbf {\\mathrm {I}}$ the unit tensor and $\\mu $ the kinematic viscosity.",
"The last integral in (REF ) represents the surface tension force on $\\Gamma $ with $\\sigma $ the surface tension coefficient and $\\kappa $ the curvature of $\\Gamma $ .",
"Conservation of thermal energy is given by $ \\int _{V} \\frac{\\partial \\left( C_p T \\right) }{\\partial t} \\, dV + \\oint _S C_p T \\mathbf {u} \\cdot \\mathbf {n} \\, dA = \\oint _{S} k \\mathbf {\\nabla } T \\cdot \\mathbf {n} \\, dA - \\int _V \\dot{q}_\\Gamma \\delta _\\Gamma \\, dV \\,,$ with the thermal energy flux term in the second integral calculated conservatively in this work, contrary to many approaches where that integral is written as $\\overline{C_p} \\oint _S T \\mathbf {u} \\cdot \\mathbf {n} \\, dA$ .",
"$T$ is the temperature and $k$ the thermal conductivity coefficient.",
"The last term in (REF ) is a thermal energy source located on the interface due to phase change latent heat transfer $ \\dot{q}_\\Gamma = \\dot{m}^{\\prime \\prime } h_{fg} = k_g \\nabla _{n_\\Gamma } T_g - k_\\ell \\nabla _{n_\\Gamma } T_\\ell \\,,$ with $h_{fg}$ the latent heat of evaporation and $\\nabla _{n_\\Gamma } T$ the normal temperature gradient on $\\Gamma $ in the respective phase.",
"The above equations are discretized on a regular Cartesian grid prior to an iterative numerical solution process.",
"This is described next."
],
[
"Numerical Method",
"The system of equations described in the previous section is solved numerically on a marker-and-cell (MAC) type computational mesh (also known as a staggered grid), created by Harlow and Welch [14].",
"Discrete velocity components are located on cell faces, while other scalar variables – like pressure, VOF and temperature – are located at cell centres.",
"The time integration procedure will first be presented to provide an overview of the scheme, after which novel numerical techniques will be detailed.",
"These include a PLIC based mass transfer rate calculation, a novel method of dealing with interface velocity jump conditions while maintaining a sharp interface definition and the discretisation of the thermal energy advection in conservative form using consistent VOF flux calculations for thermal heat capacity."
],
[
"Solution overview",
"We next detail the time integration procedure, assuming that all variables are known at some time step $n$ for all computational cells and initial values provided when $n=0$ .",
"Time integration is done with an explicit, first order (forward Euler) scheme, using a split pressure projection method [15].",
"The forward Euler time integration of the thermal energy conservation equation is first applied to each cell and reads $ C_pT\\vert ^{n+1}=C_pT\\vert ^n + \\Delta t \\left[ \\mathbf {\\nabla }^h\\cdot k \\mathbf {\\nabla }^hT\\vert ^{n+1} - \\mathbf {\\nabla }^h\\cdot \\left( C_pT\\mathbf {u} \\right)\\vert ^n - \\dot{q}\\delta _\\Gamma \\vert ^n \\right] \\,,$ where $\\Delta t$ is the discrete time step.",
"The $h$ superscript for the gradient operator indicates that it is defined in a discrete sense.",
"The diffusion of thermal energy is calculated implicitly and will be detailed in Section REF .",
"Calculation of the thermal energy advection term will be discussed in more detail in Section REF .",
"The last term deals with latent energy transfer due to phase change at the interface and is calculated from (REF ), with the mass transfer rate at the interface, $\\dot{m}^{\\prime \\prime } $ , calculated from the temperature field, $T^n$ $\\dot{m}^{\\prime \\prime } = \\dfrac{1}{h_{fg}} \\left( k_g\\nabla _{n_\\Gamma }^h T\\vert ^n_g - k_\\ell \\nabla _{n_\\Gamma }^h T\\vert ^n_\\ell \\right) \\,.$ This is done for all cells where $\\epsilon <c<1-\\epsilon $ , with $\\epsilon =10^{-12}$ in this work.",
"The method used to calculate this term is described in Section REF .",
"The VOF equation is integrated in time as $ \\dfrac{c^{n+1} - c^n}{\\Delta t} + \\mathbf {\\nabla }^h \\cdot \\left( \\mathbf {u} c \\right)\\vert ^n + \\dfrac{\\dot{m}^{\\prime \\prime } }{\\rho _{\\ell }} \\delta _\\Gamma \\vert ^n = 0 \\,.$ This equation is solved numerically using a geometric (PLIC) method.",
"The procedure is detailed in Section REF .",
"All fluid properties that depend on $c$ can now be calculated at time $n+1$ ($\\rho ^{n+1}$ , $\\mu ^{n+1}$ , $C_p^{n+1}$ , $\\kappa ^{n+1}$ , $\\delta _\\Gamma ^{n+1}$ , $\\mathbf {n}^{n+1}_\\Gamma $ ).",
"The predicted velocity $\\mathbf {u}^{\\ast }$ in the momentum equation is calculated from $\\dfrac{\\mathbf {u}^{\\ast }-\\mathbf {u}^{n}}{\\Delta t}= - \\mathbf {u}^{n} \\cdot \\mathbf {\\nabla }^{h} \\mathbf {u}^n + \\dfrac{1}{\\rho ^{n+1}} \\left( \\mathbf {\\nabla }^{h} \\cdot 2 \\mu \\mathbf {\\mathrm {S}} + \\sigma \\kappa \\delta _\\Gamma \\mathbf {n}_\\Gamma \\vert ^{n+1} \\right) + \\mathbf {g} \\, .$ Here $\\mathbf {\\mathrm {S}}$ is the fluid strain rate tensor, given by $\\mathbf {\\mathrm {S}}={}^{1}\\!/_{2} \\left( \\nabla \\mathbf {u} + \\left(\\nabla \\mathbf {u}\\right)^{T}\\right)$ .",
"Note that the viscous term can be calculated explicitly ($\\mathbf {\\nabla }^{h} \\cdot 2 \\mu \\mathbf {\\mathrm {S}}\\vert ^{n}$ ) or implicitly ($\\mathbf {\\nabla }^{h} \\cdot 2 \\mu \\mathbf {\\mathrm {S}}\\vert ^{n+1}$ ).",
"For more detail on the discretization of this equation, refer to [16].",
"A pressure Poisson equation is then solved for the pressure at $n+1$ $\\mathbf {\\nabla }^{h} \\cdot \\left[\\frac{\\Delta t}{\\rho ^{n+1}} \\mathbf {\\nabla }^{h} p^{n+1} \\right] = \\mathbf {\\nabla }^{h} \\cdot \\mathbf {u}^{*} - \\dot{m}^{\\prime \\prime } \\left( \\dfrac{1}{\\rho _g} - \\dfrac{1}{\\rho _\\ell }\\right) \\delta _\\Gamma ^n$ The one fluid velocity $\\mathbf {u}^{n+1}$ is obtained with a correction step using $p^{n+1}$ $\\dfrac{\\mathbf {u}^{n+1}-\\mathbf {u}^{*}}{\\Delta t} = -\\dfrac{1}{\\rho ^{n+1}}\\mathbf {\\nabla }^{h} p^{n+1}$ All variables have now been solved at time step $n+1$ and the cycle can repeat for the next time step.",
"The numerical method presented here has been implemented in PARIS with full parallel processing capability for three dimensional flow."
],
[
"Calculating the mass transfer rate",
"This section describes the calculation of the mass transfer rate, $\\dot{m}^{\\prime \\prime } $ , as given by (REF ).",
"The numerical procedure calculates a value for $\\dot{m}^{\\prime \\prime } $ in all cells containing a portion of the interface, known as mixed cells.",
"We repeat (REF ) here for some mixed cell $i,j,k$ , $ \\dot{m}^{\\prime \\prime } _{i,j,k} = \\dfrac{1}{h_{fg}} \\left( k_g \\nabla _{n_\\Gamma }^h T\\vert ^g_{i,j,k} - k_\\ell \\nabla _{n_\\Gamma }^h T\\vert ^\\ell _{i,j,k} \\right) \\,,$ where $\\nabla _{n_\\Gamma }^h$ indicates a discrete gradient either side of the interface.",
"To calculate the normal temperature gradients, we disregard the temperature in mixed cells, since these temperatures essentially represents some volume average of the temperatures in both phases.",
"A method to calculate the interface normal temperature gradients was proposed by Kunkelmann [9] and also applied by Guedon [10].",
"Temperature gradients are calculated in pure, single-phase cells which lie adjacent to mixed cells.",
"Kunkelmann's method uses interpolation to obtain the $c=0.5$ iso-surface, the location of which is used to calculate a finite difference temperature gradient in these pure cells.",
"The mass transfer rate in mixed cells are then obtained using an average from pure cell neighbours.",
"We employ a similar approach to Kunkelmann, but instead of using an interpolation technique for the interface location, we use the PLIC reconstruction.",
"The process is illustrated in Fig.",
"REF .",
"Pure cells are indicated with filled dots, while mixed cells are indicated with open dots and contain a plane (PLIC) reconstruction of the interface.",
"The temperature gradient normal to the interface is calculated in pure cells which are first (darker shade) or second (lighter shade) neighbours of mixed cells.",
"Figure: Mass transfer rate calculation for mixed cells.For a pure cell, the heat flux is simply estimated using a finite difference $k_p \\nabla _{n_p}^h T = k_p \\dfrac{T - T_\\Gamma }{d_\\Gamma } \\, ,$ with $d_\\Gamma $ the normal distance from the cell node to the interface and $p = \\ell , g$ a phase indicator.",
"It is assumed that $T_\\Gamma =T_{sat}(p_{sys})$ .",
"The interface is defined in a mixed cell as a plane given by the implicit equation $\\alpha = \\mathbf {m} \\cdot \\mathbf {x} \\, ,$ with $\\alpha $ a constant and $\\mathbf {m} = \\langle m_1,m_2,m_3\\rangle $ the interface vector.",
"A local coordinate system is defined in each cell, with its origin in the corner closest to the larger domain origin.",
"The interface vector is scaled such that $\\vert m_1 \\vert + \\vert m_2 \\vert + \\vert m_3 \\vert = 1 \\,,$ and is calculated using the VOF field with the Mixed Youngs-centered method [17].",
"The plane constant $\\alpha $ is obtained by using geometrical volume calculations [18].",
"In the local cell coordinate system of a mixed cell, the normal distance to the interface from the cell origin is therefore simply found by normalizing the interface vector (REF ) $d_{i,j,k}= \\dfrac{\\alpha }{\\vert \\vert \\mathbf {m} \\vert \\vert } = \\mathbf {n}_\\Gamma ^h \\cdot \\mathbf {x} \\,,$ with $\\mathbf {n}_\\Gamma ^h$ the planar interface normal vector.",
"To obtain the normal distance to the interface from any neighbouring pure cell, we apply a coordinate transformation.",
"Let $ \\Delta \\mathbf {x}$ be the translation vector from the mixed cell origin to the pure cell node.",
"The normal distance from this pure cell node is then $d_\\Gamma = d_{i,j,k} - \\mathbf {n}_\\Gamma ^h \\cdot \\Delta \\mathbf {x} \\,.$ Inside a pure cell, one can imagine that through the transformation in (REF ), the distance can be obtained to the interface representation in various neighbouring mixed cells.",
"The interface distance is determined using the neighbouring mixed cell of which the collinearity of the plane normal in that cell and the translation vector is greatest, with the collinearity given by $\\xi = \\vert \\mathbf {n}_\\Gamma ^h \\cdot \\Delta \\mathbf {x} \\vert \\,.$ The interface heat flux can now be calculated in the pure cell using (REF ), since its temperature is known and the interface normal distance is calculated, Fig.",
"REF .",
"This is done for all pure cells that are first or second neighbours to mixed cells.",
"Finally, the fluxes required to compute phase change in (REF ) are computed by using a weighted average of pure cell heat fluxes.",
"This is done by considering all the first and second, pure cell neighbours in the desired phase within a $5 \\times 5$ stencil, as shown in Fig.",
"REF .",
"The example of liquid is given here, but the same applies for the gas phase: $ \\nabla _{n_\\ell }^h T\\vert ^\\ell _{i,j,k} = \\sum \\limits _{q=1}^n \\chi _q \\dfrac{T^{\\ell }_q - T_\\Gamma }{d_{\\Gamma ,q}} \\,,$ where $q$ is an index for all $n$ neighbours in the stencil.",
"The normalized weighting factor $\\chi _q$ is given by $\\chi _q = \\dfrac{\\gamma _q}{\\sum \\limits _{c=1}^n \\gamma _c} \\,,$ with $\\gamma $ determined by the collinearity of the neighbour and the square of its distance from the mixed cell $\\gamma _q = \\dfrac{\\xi _q}{\\vert \\vert \\Delta \\mathbf {x}_q \\vert \\vert ^2}$ The temperature gradients calculated in (REF ) on either side of the interface are then used in (REF ) to obtain $\\dot{m}^{\\prime \\prime } $ in all mixed cells."
],
[
"Geometric VOF advection for phase change",
"Several VOF advection schemes exist for incompressible, divergence free flow fields.",
"For more details on geometric methods, refer to [19],[13].",
"The schemes available in PARIS include the method of Li [20] (sometimes referred to as CIAM or Lagrangian Explicit) and the conservative method of Weymouth and Yue [11].",
"However, these methods cannot be applied directly to (REF ), due to the existence of the phase change source term.",
"In this work we present a novel method to solve (REF ) using existing geometric advection methods designed for divergence-free velocity fields, while retaining the sharp interface approach.",
"The method splits (REF ) into two steps: $\\dfrac{\\partial c}{\\partial t} \\approx \\dfrac{c^{n+1} - c^n}{\\Delta t} = \\dfrac{1}{V} \\int _V \\dfrac{\\partial H_1}{\\partial t} + \\dfrac{\\partial H_2}{\\partial t} \\, dV \\,,$ where the numeric subscripts indicate the step numbers.",
"The first step can be seen as a material advection of the liquid, in which the Heaviside function (REF ) gets advected with the liquid velocity field.",
"The second step accounts for the phase change term, or the relative velocity between the interface and liquid velocities.",
"These steps are detailed next and are shown schematically in Fig.",
"REF ."
],
[
"Step 1: Advecting with an extended, divergence free liquid velocity",
"The first step advects the Heaviside function with the liquid velocity $\\mathbf {u}_\\ell $ , to produce an intermediate VOF field, $\\tilde{c}$ : $\\dfrac{\\tilde{c} - c^n}{\\Delta t} = \\dfrac{1}{V} \\int _V \\dfrac{\\partial H_1}{\\partial t} \\, dV = - \\dfrac{1}{V} \\oint _S H^n \\mathbf {u}_\\ell \\cdot \\mathbf {n} \\, dA\\,= - \\nabla \\cdot \\left( \\mathbf {u}_\\ell c \\right) .$ This step effectively assumes that the VOF field is advected at the velocity of the liquid phase, which is indeed the case everywhere inside the liquid.",
"In order to use the existing geometric advection schemes, the velocity in the liquid is extended across the interface in a manner that ensures $\\mathbf {\\nabla } \\cdot \\mathbf {u}_\\ell = 0 \\,.$ For this purpose we propose a simple yet effective technique to obtain $\\mathbf {u}_\\ell $ , which is readily implementable into existing incompressible flow solvers.",
"After solution of equations (REF ) and (REF ), the one–fluid velocity field ($\\mathbf {u}^{n+1}$ ) is obtained, which will contain a discontinuity at the interface when phase change occurs (for phases with different densities).",
"We now construct a sub-domain around the interface, of which a 2D example is shown in Fig.",
"REF .",
"The sub-domain is delimited on the liquid side of the interface by defining all the cell faces between mixed and liquid cells as domain boundaries on which a no-slip boundary condition is applied (red line).",
"On the other side of the interface, all the cell faces between second and third pure gas cell neighbours (blue dotted line) are considered as free-flow faces (a fixed pressure is applied with a zero velocity gradient normal to the boundary).",
"Figure: Schematic of split geometric VOF advection method for an evaporating droplet.A Poisson problem is set up on this newly created sub-domain in a manner similar to (REF ) $\\mathbf {\\nabla }^{h} \\cdot \\left[\\frac{\\Delta t}{\\rho ^{n+1}} \\mathbf {\\nabla }^{h} \\tilde{p} \\right] = \\dot{m}^{\\prime \\prime } \\left( \\frac{1}{ \\rho _g} - \\frac{1}{\\rho _\\ell } \\right) \\delta _\\Gamma \\,,$ with the same phase change source term, but of opposite sign.",
"This equation is solved with the same convergence tolerance as for the previous problem.",
"A phase change related velocity correction is then obtained on the cell faces of the newly defined sub-domain, similar to (REF ) $\\tilde{\\mathbf {u}} = -\\dfrac{\\Delta t}{\\rho ^{n+1}}\\mathbf {\\nabla }^{h} \\tilde{p}^{n+1} \\,.$ with $\\tilde{\\mathbf {u}}$ the newly obtained velocity field.",
"It is defined everywhere on the cell faces inside the light blue sub-domain in Fig.",
"REF , elsewhere it is zero.",
"The divergence free liquid velocity field can now be found using $\\mathbf {u}_\\ell ^{n+1} = \\mathbf {u}^{n+1} + \\tilde{\\mathbf {u}} \\,.$ This step extends the velocity of the liquid phase across the interface for all faces up to the third neighbours in the gas phase.",
"Velocities that are further into the gas phase are not of concern, as they do not affect the VOF field.",
"The $\\mathbf {u}_\\ell $ field is divergence free inside the liquid and up to the extension into the gas phase by construction $\\mathbf {\\nabla } \\cdot \\mathbf {u}_{\\ell } = \\mathbf {\\nabla } \\cdot \\mathbf {u}^{n+1} + \\mathbf {\\nabla } \\cdot \\tilde{\\mathbf {u}} = \\dot{m}^{\\prime \\prime } \\left( \\frac{1}{ \\rho _g} - \\frac{1}{\\rho _\\ell } \\right) \\delta _\\Gamma - \\dot{m}^{\\prime \\prime } \\left( \\frac{1}{ \\rho _g} - \\frac{1}{\\rho _\\ell } \\right) \\delta _\\Gamma = 0 \\,.$ We now use existing geometric advection schemes to calculate $\\tilde{c}$ , shown in Fig.",
"REF ."
],
[
"Step 2: Shifting the interface to account for phase change",
"In the second step of the split VOF equation, the interface shift due to phase change is accounted for $ \\dfrac{c^{n+1} - \\tilde{c}}{\\Delta t} = \\dfrac{1}{V} \\int _V \\dfrac{\\partial H_2}{\\partial t} \\, dV = - \\dfrac{1}{V} \\int _\\Gamma \\left(\\mathbf {u}_\\ell - \\mathbf {u}_{\\Gamma } \\right) \\cdot \\mathbf {n}_{\\Gamma }dA = - \\dfrac{\\dot{m}^{\\prime \\prime } }{\\rho _{\\ell }} \\, .$ This is illustrated in Fig.",
"REF .",
"This step accounts for the relative movement between the interface and neighbouring liquid molecules (bearing in mind that the liquid is the tracked phase).",
"The relative velocity $(\\mathbf {u}_\\Gamma - \\mathbf {u}_\\ell ) \\cdot \\mathbf {n}_\\Gamma $ is a consequence of mass transfer, (REF ).",
"To satisfy (REF ), we need to shift the interface in the normal direction.",
"We define the shift in the interface $\\Delta d$ $\\Delta d = (\\mathbf {u}_\\Gamma - \\mathbf {u}_\\ell ) \\cdot \\mathbf {n}_\\Gamma = - \\dfrac{\\dot{m}^{\\prime \\prime } }{\\rho _{\\ell }} \\dfrac{\\Delta t}{h} \\,,$ with $h$ the cell length used to re-scale the interface shift to the local cell coordinate system (see Fig.",
"REF ), wherein the PLIC plane constant is defined (REF ).",
"We once again use the PLIC reconstruction and existing geometric VOF library in PARIS [18] to calculate $c^{n+1}$ $\\alpha ^{n+1} &=\\tilde{\\alpha } + \\vert \\vert \\mathbf {m} \\vert \\vert \\Delta d \\ldots 0 \\le \\alpha ^{n+1} \\le 1 \\\\c^{n+1} &= \\ f(\\alpha ^{n+1},\\mathbf {m}) \\,.$ The movement is capped to avoid local over- and undershoots and respect $0 \\le c^{n+1} \\le 1$ .",
"When they do occur, the clipped amount is accounted for in the neighbour that is located in the direction of interface movement in order to still respect mass conservation.",
"When this step is completed, the VOF equation (REF ) is satisfied.",
"The advection of thermal energy is solved such that fluxes in heat capacity are calculated in a consistent manner with the VOF fluxes, as detailed next."
],
[
"Energy conservation",
"A thermal energy conservation equation (REF ) was added to PARIS to facilitate the simulation of phase change flows."
],
[
"Calculating the energy advection term",
"The advection term is calculated as $C_p^\\ast T^\\ast - C_p^{n} T^{n} = - \\Delta t \\sum \\limits _{f} \\, C_p^{n}\\vert _f \\, T^n_f u_f^n A_f \\,,$ where $C_p^\\ast T^\\ast $ indicates the thermal energy per unit volume after advection is considered.",
"The right-hand-side represents the sum of thermal energy fluxes at all $f$ cell faces for time step $n+1$ .",
"Each face has an area $A_f=h^2$ (in 3D) and face velocity $u_f = \\mathbf {u} \\cdot \\mathbf {n}_f$ , which is calculated explicitly on the staggered (MAC) grid.",
"Care needs to be taken when calculating $C_p^{n}\\vert _f$ as well as $T^n_f$ , respectively the fluxed heat capacity and face temperature.",
"One reason is that the volumetric heat capacity ratio becomes significant in many problems of interest.",
"Consider, as an example, the properties of water at atmospheric conditions in Table REF .",
"In this case ${}^{C_{p,\\ell }}\\!/_{C_{p,g}} \\approx 3200$ , which motivates why a conservative formulation is preferable, as opposed to the non-conservative approximation $C_p^\\ast T^\\ast - C_p^n T^n \\vert _{i,j,k} = - \\Delta t \\, C_p^n \\vert _{i,j,k} \\, \\sum \\limits _{f} \\, T^n_f u_f^n A_f \\,.$ Table: Properties for saturated water at atmospheric conditionsA flux-consistent, geometric advection technique is employed.",
"The term flux-consistent is used, since the flux terms in (REF ) are calculated using the same geometric advection procedure as for the VOF function.",
"Note that the flux-consistent implementation here is inspired by a similar method already present in PARIS, but applied to the momentum term [21].",
"In the same manner as the directionally split VOF advection methods ([20], [11]) three separate, consecutive sweeps are performed, one for each coordinate direction (in 3D).",
"Let $s$ be the sweep counter, so that $s=1$ corresponds to time step $n$ , $s=1,4$ and $s=4$ corresponds to time step $n+1$ .",
"Before the sweeps start, a thermal energy $e_t$ is calculated for every cell $e_t^{(s=1)} = e_t^n = C_p^n T^n \\,,$ where the cell temperature is known from the solution at time step $n$ and the volumetric heat capacity in a cell is obtained by using a volume average of the phase heat capacities $C_p^{(s=1)} = C_p^n = C_{p,\\ell } \\, c + C_{p,g} \\left( 1 - c \\right) \\,.$ For each direction sweep, the following process is followed: The face temperatures $T_f$ are obtained by using a fifth order WENO reconstruction scheme [22].",
"The fluxed volumes and their respective VOF values are known from the geometric VOF flux calculations at each of the two faces in the specific sweep direction.",
"This allows us to calculate a volume weighted volumetric heat capacity $C_p\\vert _f$ and the fluxed thermal energy at each face from the temperature in the WENO scheme $C_p\\vert _f\\,T_f$ .",
"We now have the intermediate thermal energy in the cell (after the sweep) and can calculate the intermediate temperature $T^{(s)} = \\dfrac{e_t^{(s)}}{C_p^{(s)}} \\,,$ The thermal energy and VOF sweeps are done in tandem for every sweep direction.",
"Before the next direction is swept, the interface is reconstructed from the newly calculated intermediate VOF $c^{(s+1)}$ , along with the thermal energy at the same sweep $e_t^{(s+1)}$ .",
"This ensures that the heat capacity that is used at each fluxing step is consistent with the VOF function.",
"The face temperatures are calculated by using the updated temperature field after each sweep.",
"After the three coordinate directions have been swept using $\\mathbf {u}_\\ell $ , (just like the VOF process) the energy change due to phase change is calculated by $\\Delta e_t = \\Delta c \\left( C_{p,\\ell } - C_{p,g}\\right) T_\\Gamma $"
],
[
"Calculating the energy diffusion term",
"The heat diffusion term needs to be evaluated to complete the integration of (REF ) in time.",
"This term is calculated implicitly using a similar approach to Sato and Ničeno [23].",
"The saturation temperature at the interface is applied directly by solving the diffusion term separately for each phase.",
"The interface is treated as a boundary, where a Dirichlet boundary condition is specified for the temperature.",
"This is equivalent to solving two heat diffusion equations with constant fluid properties, with the interface an arbitrary boundary separating the two phases in which each respective problem is solved.",
"First, the phase of a specific node is determined by simply evaluating the VOF function: A cell node is determined to be inside the liquid whenever $c \\ge 0.5$ , otherwise it is in the gas.",
"Fig.",
"REF shows two 2D stencils used to compute the diffusion term for a liquid (left hand side) and gas (right hand side) node.",
"Figure: Discretisation for temperature neighbouring the interface.",
"On the left the computational stencil for a liquid cell is shown, with the stencil for its neighbour in the gas on the right.Consider the liquid node on the left as an example.",
"The finite volume discretiation of the diffusion term for the 2D cell at position $i,j$ is approximated using the irregular stencil as $\\dfrac{1}{V}\\int _{V_{i,j}} \\mathbf {\\nabla } \\cdot \\left( k \\mathbf {\\nabla } T \\right) dV \\approx k_{\\ell } \\left( \\dfrac{\\nabla ^{h}_y \\,T_{i,j+{}^{1}\\!/_{2}}- \\nabla ^{h}_y \\,T_{i,j-{}^{1}\\!/_{2}}}{{}^{1}\\!/_{2} \\left( h + \\theta ^{\\ell }_{i,j-{}^{1}\\!/_{2}} \\right) } + \\dfrac{\\nabla ^{h}_x \\,T_{i+{}^{1}\\!/_{2},j} - \\nabla ^{h}_x \\,T_{i-{}^{1}\\!/_{2},j}}{ {}^{1}\\!/_{2} \\left( \\theta ^{\\ell }_{i+{}^{1}\\!/_{2},j} + h \\right) } \\right) \\,,$ noting that $\\Delta x = \\Delta y = h$ and $k=k_{\\ell }$ since we are only considering the liquid.",
"The temperature gradient is approximated using a finite difference.",
"Consider again the liquid node: $ \\mathbf {\\nabla }^{h}_x \\,T_{i+{}^{1}\\!/_{2},j} = \\dfrac{T_\\Gamma -T_{i,j}}{\\theta ^{\\ell }_{i+{}^{1}\\!/_{2},j}} \\,\\,;\\quad \\mathbf {\\nabla }^{h}_x \\,T_{i-{}^{1}\\!/_{2},j} = \\dfrac{T_{i,j} -T_{i-1,j}}{h} \\, .$ For the temperature gradient to the right of the node in question, the interface distance $\\theta ^{\\ell }$ is used to calculate the finite difference.",
"A standard finite difference is used when the temperature gradient is approximated between two nodes in the same phase.",
"The distance to the interface from a node is simply taken as the height function in that direction.",
"When the interface configuration is such that a height cannot be obtained in the required direction, the distance is approximated by using a plane reconstruction of the interface in the staggered volume.",
"The distance is capped so that $\\epsilon h \\le \\theta \\le h$ , with a value of $\\epsilon = 10^{-3}$ typically used."
],
[
"A note on Momentum Conservation",
"Before proceeding to present results, a word is due on the numerical treatment of the momentum conservation equation.",
"For the purpose of this work a non-conservative momentum advection formulation was used.",
"The two-step advection process was not applied to the existing momentum conserving scheme [21] in PARIS due to additional numerical routines required to do so.",
"One challenge is the fact that the momentum control volume is staggered to the pressure and thermal energy control volumes on the staggered (MAC) grid.",
"The error introduced with the use of the non-conservative discretization of the momentum equation manifests in the force balance on the interface.",
"Without a conservative discretisation, an error is introduced for the vapour recoil pressure.",
"This error was not found to be detrimental for the purpose of this work, but a conservative, consistent implementation for the momentum is envisaged in future work."
],
[
"Simulation Cases",
"In this section we validate the novel numerical scheme with analytical benchmark test-cases.",
"The numerical method was implemented in PARIS with full parallel processing capability in three spatial dimensions using the MPI libraries [24]."
],
[
"Interface distance calculation",
"This test case measures the accuracy of the PLIC–based interface distance calculation, as explained in Section REF .",
"It is clear from (REF ) that the interface distance calculation is important when the normal temperature gradients are calculated for the mass transfer rate.",
"As a first validation test, the VOF field was initialized as a plane in 3D.",
"The theoretical interface distances were compared to the calculated distances in PARIS for all first and second pure cell neighbours.",
"The calculated distances were found to correspond to the theoretical ones to machine precision (as expected).",
"Next, a 3D droplet test was performed.",
"A droplet of radius $R_0=0.25+\\delta _R$ is initialized at $x_0+\\delta _x, \\, y_0+\\delta _y, \\, z_0+\\delta _z$ of a unit cube, with $x_0=y_0=z_0=0.5$ .",
"The $\\delta $ -values are each a randomly generated number such that $-h \\le \\delta \\le h$ , with $h={}^{1}\\!/_{N}$ the cell size for a mesh with $N^3$ grid points.",
"Five different resolutions were tested: $N=\\left\\lbrace 16,\\,32,\\,64,\\,128,\\,256\\right\\rbrace $ .",
"At each mesh resolution, 100 different droplets were generated for a total of 500 tests.",
"For each droplet, the calculated interface normal distances $d^h$ were calculated for all first and second pure cell neighbours (in each phase).",
"The relative error for a pure cell is calculated as $e_{i,j,k} = \\dfrac{\\vert d_{i,j,k}^h - d \\vert }{R_0} \\,.$ For each droplet, the max error is recorded as well as an $L_2$ -norm of all $n$ first- and second pure cell neighbours to mixed cells, with $L_2 = \\sqrt{\\dfrac{\\sum _q^n e_q^2}{n}} \\,.$ The results are presented in Fig.",
"REF .",
"The convergence rate for the ($L_2$ ) average distance error is around second order at lower mesh resolutions.",
"For higher mesh resolutions it is between first and second order.",
"Figure: Convergence of the PLIC interface distance calculation."
],
[
"Two-dimensional droplet evaporating test",
"One crucial aspect of the novel numerical method is the proposed two-step VOF advection procedure.",
"This includes the correct calculation of a divergence free liquid velocity and subsequent phase change adjustment to ensure mass conservation.",
"A two-dimensional evaporating droplet is modelled to validate this part of the implementation.",
"The discretization errors that may be present from the mass transfer rate calculations are removed by specifying a constant mass transfer rate of $\\dot{m}^{\\prime \\prime } =0.05$ , ensuring that this test specifically evaluates the two-step VOF-advection procedure.",
"A two-dimensional, circular liquid droplet with initial radius $R=0.23$ is initialized in the center of a square domain of unit length.",
"The units in this problem are irrelevant and we can consider the problem dimensionless.",
"The boundaries are given an outflow boundary condition (fixed pressure and zero normal gradient for the velocity).",
"The constant mass transfer rate causes the droplet to evaporate.",
"The radius evolution over time is simply $R \\left( t \\right) = R_0 - \\dot{m}^{\\prime \\prime } t \\,.$ Three grid resolutions were used: $N_x=32,64$ and 128 with respective time step sizes of $t=0.002\\,s$ , $t=0.001\\,s$ and $t=0.0005\\,s$ .",
"The time evolution of the droplet volume is shown in Fig.",
"REF and compared to the analytical solution.",
"As shown, an accurate solution is achieved on all meshes.",
"Fig.",
"REF shows a time sequence of the droplet evolution for the finest resolution.",
"The velocity magnitude is shown, revealing resulting radial flow patterns.",
"It is encouraging to note that the circular shape is maintained throughout and the velocity field around the droplet remains symmetric.",
"Figure: The volume evolution an evaporating cylinder in 2D with constant rate of evaporation.Figure: A time series showing the velocity magnitude for the finest resolution simulation (N x =128N_x=128), taken at 0.010.01, 2.012.01 and 4.014.01s.",
"A zoom showing the mesh is on the right."
],
[
"Stefan problem",
"The Stefan problem was used by Welch and Wilson [4] to validate their VOF-based method and has since often been used to validate incompressible phase change methods [10], [23].",
"In this problem, a liquid at saturation temperature is initially at rest.",
"A thin vapour layer is present between the liquid and a wall, which is at a fixed, elevated temperature of $T_{sup}$ degrees above the saturation temperature, $T_{wall} = T_{sat} + T_{sup}$ .",
"Gravity is neglected and an outflow boundary condition is used opposite the heated wall.",
"The temperature gradient between the wall and the interface, which is at saturated temperature, causes phase change and the vapour layer to grow in time.",
"The liquid is pushed out of the domain by the growing vapour layer.",
"Let the thickness of the vapour layer be $\\delta $ .",
"An analytical solution is available for the evolution of the interface thickness and temperature profile [4].",
"The interface thickness is given by $\\delta (t) = 2 \\lambda \\sqrt{\\alpha t} \\,,$ with $t$ the time and $\\alpha ={}^{k_g}\\!/_{\\rho _g c_{p,g}}$ the thermal diffusivity of the vapour phase.",
"$\\lambda $ is found by solving the transcendental equation $\\lambda \\exp ( \\lambda ^2)erf(\\lambda ) = \\dfrac{c_{p,g}T_{sup}}{h_{fg}\\sqrt{\\pi }} \\,.$ A two-dimensional test case was defined with a domain size of $10 \\, mm$ , a $T_{sup} = 10 \\,K$ superheat and fluid properties for water at atmospheric conditions.",
"These properties are given in Table REF .",
"Table: Properties for saturated water at atmospheric conditionsThe initial temperature field was $T_{sat}$ in the liquid and a linear profile from the interface to the wall temperature.",
"Three grid spacings were used on the $10 \\, mm$ domain: $N_x=64,128,256$ with respective time step sizes of $t=0.002\\,s$ , $t=0.001\\,s$ and $t=0.0005\\,s$ .",
"The initial vapour layer thickness was taken as $322.5 \\, \\mu m$ , which corresponds to a time $t=0.282435\\,s$ , if the vapour layer would have grown from zero thickness at time $t=0\\,s$ .",
"The simulated time was $10s$ .",
"The theoretical solution of the evolution of the interface thickness was obtained by solving (REF ) numerically with a relative convergence error of $10^{-6}$ for $\\lambda $ .",
"The results from PARIS for the three grid resolutions are plotted in Fig.",
"REF .",
"An excellent agreement was obtained.",
"The relative errors of the interface position at $t=10\\,s$ for the three grid resolutions are presented in Table REF .",
"The theoretical solution of (REF ) for $t=10\\,s$ is $\\delta (10)=1.919 \\times 10^{-3} \\,m$ .",
"Table: Relative errors for Stefan problem interface location at t=10st=10\\,s.Figure: The interface time evolution results for the Stefan problem.The results show an error of less that 1% for the coarsest grid.",
"Since the errors obtained here are small, the spatial convergence rate will be determined with a more demanding test case."
],
[
"Bubble in superheated liquid",
"A more demanding test case is that of a three-dimensional gas bubble at saturated temperature in a superheated liquid.",
"The superheated temperature is denoted $T_\\infty $ and no gravity force is applied.",
"There is an analytical solution to the problem assuming spherical symmetry, obtained by Scriven [25].",
"The solution gives the bubble radius $R$ as Table: Properties for the bubble in superheated liquid case.$R = 2 \\, \\beta _g \\sqrt{\\dfrac{k_\\ell }{c_{p,\\ell } \\rho _\\ell } t}$ with $t$ the time and the fluid properties as defined before.",
"The value of $\\beta _g$ , sometimes referred to as the “growth constant” is obtained by solving $&\\dfrac{\\rho _{\\ell } c_{p,\\ell } \\left(T_\\infty - T_{sat}\\right)}{\\rho _g \\left(h_{fg}+\\left(c_{p,\\ell }-c_{p,g}\\right) \\left(T_\\infty - T_{sat}\\right)\\right)} = \\\\&2 \\, \\beta _g^2 \\int _0^1 exp \\left( -\\beta _g^2 \\left( \\left(1-\\zeta \\right)^{-2} - 2 \\left(1 - \\dfrac{\\rho _g}{\\rho _\\ell } \\right) \\zeta -1 \\right) \\right) \\, d\\zeta $ The solution is obtained numerically using the native numerical tools in the GNU package Octave .",
"The temperature field is then given by $T(r<R) &= T_{sat} \\\\T(r>R) &= T_\\infty - 2 \\, \\beta _g^2 \\left( \\dfrac{\\rho _g \\left(h_{fg}+\\left(c_{p,\\ell }-c_{p,g}\\right) \\left(T_\\infty - T_{sat}\\right)\\right)}{\\rho _{\\ell } c_{p,\\ell }} \\right) \\\\&\\int _{1-{}^{R}\\!/_{r}}^1 exp \\left( -\\beta _g^2 \\left( \\left(1-\\zeta \\right)^{-2} - 2 \\left(1 - \\dfrac{\\rho _g}{\\rho _\\ell } \\right) \\zeta -1 \\right) \\right) \\, d\\zeta $ The problem can be characterized by the Jacob number, given by $Ja = \\dfrac{\\rho _\\ell c_{p,\\ell } \\left(T_\\infty - T_{sat}\\right)}{\\rho _g h_{fg}} \\,.$ A test case is defined with fluid properties given in Table REF .",
"Note that these properties do not correspond to a specific fluid, but was chosen for the purposes of a test with ${}^{\\rho _\\ell }\\!/_{\\rho _g} = {}^{\\mu _\\ell }\\!/_{\\mu _g} = {}^{k_\\ell }\\!/_{k_g} = 10$ .",
"The heat capacity ratio is ${}^{c_{p,\\ell }}\\!/_{c_{p,g}}=2.5$ .",
"The saturated temperature is $T_{sat}=1\\,K$ with the temperature at infinity in the liquid $T_\\infty =3\\,K$ .",
"This results in $Ja=0.5$ .",
"A bubble of initial radius $R(t_0)=0.12$ is placed in the center of a unit cube.",
"The initial temperature is taken from the analytical solution and the simulation is run for a time $t_f \\approx 4 \\times t_0$ , with $t_f$ the final time.",
"Table: Relative errors for the bubble radius RR at t=4t 0 t=4 \\, t_0.Figure: The time evolution of the bubble radius for a bubble at saturated temperature in a superheated liquid.Three test cases were run: $N=64^3,128^3,256^3$ with respective time step sizes of $t=0.01\\,s$ , $t=0.005\\,s$ and $t=0.0025\\,s$ .",
"The time evolution of the bubble radius in the simulations is compared to the theoretical value in Fig.",
"REF .",
"The theoretical value of the bubble radius at the final time was determined by the analytical solution: $R(t_f=2.2156\\,s)=0.2413$ and compared to simulation.",
"The results are shown in Table REF , along with the relative error compared to the theoretical value.",
"The largest error was 5.56% on the coarsest grid.",
"One important source of the error is the resolution of the thermal boundary layer in the superheated liquid.",
"Lower resolution results in an inaccurate calculation of the temperature gradient in the liquid, which leads to an inaccurate mass transfer rate, (REF ).",
"A time sequence of the simulation is shown in Fig.",
"REF , indicating the bubble growth and superheated temperature around the interface.",
"It is encouraging to notice that the thermal boundary layer as well as the bubble shape remains spherically symmetric along with a relatively accurate prediction of the bubble radius evolution.",
"Figure: A time sequence presentation of a vapor bubble in superheated liquid, taken at 0.40.4s intervals starting at t=0t=0s at the top left.",
"The blue surface indicates the c=0.5c=0.5 iso-surface, which is a representation of the interface.",
"The colour scale shows temperature at a mid-plane through the bubble."
],
[
"Conclusion",
"The Direct Numerical Simulation of flows with phase change was studied via a geometric PLIC VOF scheme.",
"The discontinuity in velocity across the interface due to phase change poses a particular challenge to conventional VOF schemes that rely on a smooth, divergence–free velocity field.",
"In this work a novel two-step VOF advection method was proposed.",
"In the first step, the interface is advected with a divergence free liquid velocity, which is obtained from the solution of a Poisson problem that decomposes the one-fluid velocity into a liquid velocity and a phase-change component.",
"In the second step, the phase change component is accounted for with an explicit interface shift in the local normal direction.",
"The geometric treatment of VOF advection is applied consistently to the thermal energy advection term.",
"The thermal energy diffusion term is solved implicitly, using a similar technique to Sato and Ničeno [23], where an asymmetric stencil is used to apply the interface temperature directly.",
"The method was implemented in PARIS (with full parallel computation capability) and was tested using three benchmark test cases.",
"The two-step VOF advection method was tested on a two-dimensional evaporating droplet.",
"The results on three different grids were indistinguishable and showed excellent agreement to the ${}^{1}\\!/_{t^2}$ evolution of the volume.",
"The one-dimensional Stefan problem was solved to excellent accuracy.",
"A three dimensional bubble at saturated temperature in a superheated liquid with $Ja=0.5$ was simulated at three different grid resolutions.",
"It was found that the spherical shape was preserved and the radius evolution was predicted with good accuracy, especially in the high resolution case where the temperature gradient in the thermal boundary layer could be captured with greater accuracy than in the cases with lower resolution."
],
[
"Acknowledgements",
"This work is based on research supported by the National Research Foundation of South Africa (Grant Numbers: 89916).",
"The opinions, findings and conclusions or recommendations expressed is that of the authors alone, and the NRF accepts no liability whatsoever in this regard.",
"The authors thank Roy Horwitz for the creation of a numerical solver for the theoretical bubble radius and temperature profile in the flashing saturated bubble test case that was used to validate the numerical results.",
"Screen shots presented in this work are from VisIt [26]."
]
] | 2001.03477 |
[
[
"QoE-driven Coupled Uplink and Downlink Rate Adaptation for 360-degree\n Video Live Streaming"
],
[
"Abstract 360-degree video provides an immersive 360-degree viewing experience and has been widely used in many areas.",
"The 360-degree video live streaming systems involve capturing, compression, uplink (camera to video server) and downlink (video server to user) transmissions.",
"However, few studies have jointly investigated such complex systems, especially the rate adaptation for the coupled uplink and downlink in the 360-degree video streaming under limited bandwidth constraints.",
"In this letter, we propose a quality of experience (QoE)-driven 360-degree video live streaming system, in which a video server performs rate adaptation based on the uplink and downlink bandwidths and information concerning each user's real-time field-of-view (FOV).",
"We formulate it as a nonlinear integer programming problem and propose an algorithm, which combines the Karush-Kuhn-Tucker (KKT) condition and branch and bound method, to solve it.",
"The numerical results show that the proposed optimization model can improve users' QoE significantly in comparison with other baseline schemes."
],
[
"Introduction",
"360-degree video technology has recently become more and more popular with the increasing demands on interactive applications.",
"By wearing a head-mounted display (HMD), 360-degree video users can freely move their heads to change the viewing directions, which provides an immersive viewing experience.",
"To improve the quality of experience (QoE) [1], most 360-degree videos have 6K or even higher resolution.",
"Streaming such high-resolution videos is non-trivial because of the limited bandwidth of wireless communication channels [2].",
"In addition, there has been a recent trend toward high-quality 360-degree video content creation using 3D panoramic VR cameras.",
"Compared to the traditional live streaming, 360-degree video live streaming is considerably more challenging due to its panoramic nature [3], and it has more stringent QoE requirements to prevent motion sickness.",
"Many published works have investigated 360-degree video streaming [4], [5], [6], [7], [8].",
"For example, the authors of [5] designed a rate adaptation algorithm that can maximize the defined QoE metrics for 360-degree video streaming.",
"Given the FOV and bandwidth estimation, Xie et al.",
"[6] proposed a probabilistic tile-based adaptive 360-degree video streaming system, named 360ProbDASH, which combined viewport adaptation and rate adaptation to solve the QoE-driven optimization problem.",
"In [7], the authors presented Vortex, a live VR video streaming system that works in a computationally and bandwidth-efficient manner.",
"In 360-degree video live streaming applications, a capturing device is used to record scene and transmit the captured video to the server in real time.",
"Then, multiple users can access and watch the 360-degree video with their HMDs by downloading the coresponding video content from the server.",
"Obviously, under a limited bandwidth constraint, the uplink (VR cameras to video server) and downlink rate (video server to end users) should be carefully selected to improve the viewing experiences.",
"In this paper, we formulate this transmission problem as a nonlinear integer programming problem.",
"To solve this, we propose an optimal algorithm that combines the KKT condition and the branch and bound method.",
"Extensive simulations are conducted based on the real-world LTE network traces, and the simulation results show that the proposed solution can significantly improve users' QoE in comparison with other baseline schemes.",
"To the best of our knowledge, this is the first research jointly considering rate adaptation for coupled uplink and downlink in 360-degree video streaming system."
],
[
"System Overview",
"In this section, we present our 360-degree video live streaming system, which jointly allocates uplink and downlink wireless resources to maximize the overall QoE."
],
[
"System Model",
"We consider a scenario where multiple users are watching 360-degree video live streaming, as illustrated in Fig.",
"REF .",
"$C$ cameras are installed in the 360-degree video capturing device, and each camera can record high-quality 2D video.",
"Constrained by the limited wireless uplink bandwidth, the recorded $C$ videos may not be transmitted to video server with the highest quality.",
"Assuming that each camera is capable of encoding the original video into several versions with different bitrates, the video with the most appropriate bitrate should be selected and uploaded to the video server.",
"Video is partitioned into tiles at the server and the corresponding tiles for user's FoV are transmitted to the user from the server.",
"The downlink transmission is also limited by the bandwidth.",
"Then, one representation is selected for every tile and transmitted to the user, according to the corresponding FOV and channel status under the assumption that such information is transmitted to the video server in real time.",
"Obviously, for each tile, the best quality received by a user cannot exceed the quality of the video uploaded by the corresponding camera, where the original video is captured.",
"Thus, this scenario can be modeled as a typical coupled video transmission problem.",
"In this paper, we consider this coupled uplink and downlink transmission system to maximize the QoE of all users through rate adaptation.",
"The architecture is shown in Fig.",
"REF .",
"The system consists of four parts: a 360-degree video capturing device with $C$ cameras, processing modules on both the server side and the client, and the end users wearing HMDs.",
"Each camera can encode the captured video into different video qualities, expressed as different constant bitrates, but only one video bitrate can be uploaded to the video server.",
"Figure: QoE driven coupled uplink and downlink 360-degree video transmission system.The server side includes two modules: the QoE-driven uplink processing module and the downlink processing module.",
"The uplink processing module is responsible for obtaining a raw 360-degree video by stitching the $C$ uploaded videos.",
"The downlink processing module selects the appropriate video quality for each tile based on the FOVs and channel information from all the users.",
"The system workflow is as follows: the uplink processing module selects a bitrate level for each camera.",
"Then, the $C$ videos with different qualities are stitched and projected into a panoramic video.",
"Next, the downlink processing module divides the raw panoramic video into tiles and encodes each tile into different quality representations.",
"With the help of feedback information from the client side, the downlink processing module can determine the range of tiles to be transmitted and select the appropriate quality level for each tile.",
"Finally, all the selected tiles are transmitted through the downlink channel to maximize the users' QoE.",
"At the client side, as the tiles are received from the server, they are decoded, projected, rendered and displayed in the HMD.",
"In addition, the client side sends the user's FOV and channel information in real time to the video server through each user's uplink feedback channel."
],
[
"Problem Formulation",
"Suppose the system includes $N$ users (indexed by $n$ ).",
"The evenly deployed $C$ cameras form a 360-degree video capturing system.",
"Each camera $c$ can record and generate videos at up to ${D}^{^{\\prime }}$ bitrate levels (indexed by ${d}^{^{\\prime }}$ ).",
"The bitrate of camera $c$ uploaded with bitrate level ${d}^{^{\\prime }}$ is denoted by $\\lambda _{c,d^{^{\\prime }}}^{UL}$ .",
"The total bandwidth of the uplink channel is $B^{UL}$ .",
"When the videos generated by the $C$ cameras are uploaded to the server, the server processes them to produce a 360-degree video.",
"Prior to downlink transmission, the 360-degree video is divided into $T$ tiles (indexed by $t$ ), and each tile is encoded into $D$ different representations (indexed by $d$ ) at different quality levels, which is the same as how the HTTP DASH processes the video.",
"We denote the bitrates of tile $t$ with representation $d$ in GOP $k$ by $\\lambda _{t,d,k}^{DL}$ .",
"As aforementioned, the quality of each video $d$ in the downlink cannot exceed the quality of the corresponding video ${d}^{^{\\prime }}$ in the uplink transmission.",
"Assume that the bandwidth of the k-th GOP in the downlink channel is $B_{k}^{D L}$ .",
"For user $n$ , $T_{FoV}^{n}$ denotes the tiles covered by his FOV.",
"Then user $n$ 's expected QoE can be defined as follows: $\\begin{small}\\begin{array}{l}QoE_{n} = \\sum \\limits _{\\forall t \\in T_{FOV}^n} {\\sum \\limits _{\\forall d \\in D} {\\sum \\limits _{\\forall k \\in K} {q\\left( {\\lambda _{t,d,k}^{DL} \\cdot \\chi _{t,d,k}^{DL}} \\right)} } } - \\alpha \\times \\\\\\sum \\limits _{\\forall t \\in T_{FOV}^n} {\\sum \\limits _{\\forall d \\in D} {\\sum \\limits _{\\forall k \\in K} {\\mathbb {I} \\left( {\\lambda _{t,d,k}^{DL} \\cdot \\chi _{t,d,k}^{DL} > B_k^{DL}} \\right)} } } \\cdot {T_k} - \\beta \\times \\\\\\sum \\limits _{\\forall t \\in T_{FOV}^n} {\\sum \\limits _{\\forall d \\in D} {\\sum \\limits _{\\forall k \\in K} \\left(q\\left(\\lambda _{t, d, k+1}^{D L} \\cdot \\chi _{t, d, k+1}^{D L}\\right)-q\\left(\\lambda _{t, d, k}^{D L} \\cdot \\chi _{t, d, k}^{D L}\\right)\\right)^{2} }},\\end{array}\\end{small}$ where $\\lambda _{t,d,k}^{DL}$ denotes the bitrate of tile $t$ with bitrate level $d$ in the $k$ -th GOP, and $DL$ means “downlink\" channel.",
"$\\chi _{t,d,k}^{DL}$ is a binary variable, which equals 1 if tile $t$ is transmitted with bitrate level $d$ in the $k$ -th GOP, and 0 otherwise.",
"Function $q$ is a mapping function, which maps the bitrate of tile $t$ to the quality perceived by the user.",
"The form of function $q$ is the logarithmic of the received bitrate.",
"The second term of this equation is used to show the impact of stalls.",
"We assume stall will occur when the bandwidth for the $k$ -th GOP is less than the video bitrate, and stall time is approximately equal to the duration of the $k$ -th GOP In a practical system, due to the existence of the playback buffer, stalls are related to the transmission rate, the playback speed and buffer status.",
"Once the buffer drains, a stall will occur.",
"In this manuscript, to simplify the formulation, we assume when the bandwidth value in a GOP is less than the video bitrate, stall will occur.",
"How to more previously model this is left as the future work.. $\\mathbb {I}\\left(\\lambda _{t, d, k}^{D L} \\cdot \\chi _{t, d, k}^{D L}>B_{k}^{D L}\\right)$ is an indicator function, and its value is 1 only when $\\lambda _{t, d, k}^{D L} \\cdot \\chi _{t, d, k}^{D L}>B_{k}^{D L}$ , otherwise 0.",
"This means that in one GOP, when the video bitrate is greater than the bandwidth, the stalls will occur.",
"$T_{k}$ is the duration of the $k$ -th GOP.",
"$B_{k}^{D L}$ is the bandwidth when transmitting the $k$ -th GOP.",
"Thus we can use the average of the bandwidth during $T_{k}$ as the value of $B_{k}^{D L}$ .",
"In addition, $\\left(q\\left(\\lambda _{t, d, k+1}^{D L} \\cdot \\chi _{t, d, k+1}^{D L}\\right)-q\\left(\\lambda _{t, d, k}^{D L} \\cdot \\chi _{t, d, k}^{D L}\\right)\\right)^{2}$ considers the quality switches between the consequent GOPs.",
"Similarly, $\\lambda _{t, d, k+1}^{D L}$ denotes the bitrate of tile $t$ with bitrate level $d$ in the $k+1$ -th GOP.",
"$\\chi _{t, d, k+1}^{D L}$ is also a binary variable, and equals 1 if tile $t$ is transmitted with bitrate level $d$ in the $k+1$ -th GOP, and 0 otherwise.",
"Finally, constants $\\alpha $ and $\\beta $ are the non-negative weight parameters to balance the three factors.",
"With this QoE model, our uplink and downlink optimization problem is defined as follows: problem 1: $\\begin{small}\\max \\sum _{\\forall n \\in N} QoE_{n}\\end{small}$ s.t.",
"$\\begin{small}\\sum _{\\forall d^{\\prime } \\in D^{\\prime }} \\chi _{c, d^{\\prime }}^{U L}=1, \\forall c \\in C\\end{small}$ $\\begin{small}\\sum _{\\forall d^{\\prime } \\in D^{\\prime }} \\sum _{\\forall c \\in C} \\lambda _{c, d^{\\prime }}^{U L} \\cdot \\chi _{c, d^{\\prime }}^{U L} \\le B^{U L}\\end{small}$ $\\begin{small}\\sum _{\\forall d \\in D} \\chi _{t, d, k}^{D L}=1, \\forall t \\in T_{F O V}^{n}, \\forall k \\in K\\end{small}$ $\\begin{small}\\sum _{\\forall t \\in T_{F O V}^{n}} \\sum _{\\forall d \\in D} \\sum _{\\forall k \\in K} \\lambda _{t, d, k}^{D L} \\cdot \\chi _{t, d, k}^{D L} \\le \\sum _{\\forall k \\in K} B_{k}^{D L}\\end{small}$ $\\begin{split}\\sum _{\\forall d \\in D} \\sum _{\\forall k \\in K} \\lambda _{t, d, k}^{D L} \\cdot \\chi _{t, d, k}^{D L} \\le \\frac{1}{T} \\sum _{\\forall d^{\\prime } \\in D^{\\prime }} \\lambda _{c, d^{\\prime }}^{U I} \\cdot \\chi _{c, d^{\\prime }}^{U L},\\\\\\forall c \\in C, \\forall t \\in T_{F O V}^{n}\\end{split}$ where $\\chi _{c, d^{\\prime }}^{U L}$ and $\\chi _{t,d,k}^{DL}$ are the optimization variables.",
"$UL$ stands for \"uplink\".",
"$\\chi _{c, d^{\\prime }}^{U L}$ is a binary variable, which equals 1 when the video from camera $c$ is transmitted with bitrate level $d^{\\prime }$ and 0 otherwise.",
"$\\chi _{t,d,k}^{DL}$ is also binary variable (same as in equation (1)), which equals 1 if tile $t$ is transmitted with representation $d$ in $k$ -th GOP and 0 otherwise.",
"$\\lambda _{c, d^{\\prime }}^{U L}$ denotes the bitrate of camera $c$ uploaded at bitrate level $d^{\\prime }$ .",
"Constraints (3)-(4) apply to the uplink.",
"Constraint (3) indicates that the video of camera $c$ can be uploaded with only one quality level.",
"The total bitrate of all the uploaded videos cannot exceed the total bandwidth of the uplink as specified in Constraint (4).",
"Constraints (5)-(6) are the downlink constraints.",
"Constraint (5) ensures that only one representation can be selected for tile $t$ in k-th GOP, which is transmitted to the client side.",
"Constraint (6) ensures that the sum of the tile bitrates cannot exceed the total bandwidth of the downlink channel.",
"Constraint (7) discusses the coupled uplink and downlink and ensures that the quality levels of tiles in the downlink cannot exceed the quality level of the videos generated by the corresponding camera and transmitted in the uplink."
],
[
"QoE driven Optimal Rate Adaptation Algorithm",
"In this section, we introduce the optimal solution of the above problem, which is a nonlinear integer programming problem and can be proven to be NP-hard.",
"We firstly approximate the indicator function in the QoE model with a logarithmic function, so that the QoE function becomes a continuous function.",
"Then, because the constraint in problem 1 satisfies the linear constraint qualification, we can use the KKT condition to solve the relaxation problem of problem 1.",
"By using the logarithmic function to approximate the indicator function in the QoE model and relaxing the integer variables $\\chi _{c,d^{^{\\prime }}}^{UL}$ and $\\chi _{t,d,m}^{DL}$ to continuous variables, the relaxed problem can be solved by applying the KKT conditions and the Lagrangian function.",
"Then, we can obtain the optimal value of the original problem."
],
[
"KKT condition for the relaxed continuous problem",
"First, we approximate the indicator function in the QoE model with a logarithmic function and relax $\\chi _{c,d^{^{\\prime }}}^{UL}$ and $\\chi _{t,d,k}^{DL}$ to continuous variables.",
"Then, the original problem 1 can be solved by KKT conditions.",
"The Lagrangian function of the problem is as follows Eq.",
"(REF ): $ \\begin{split}L(\\chi _{c,{d^{^{\\prime }}}}^{UL},\\chi _{t,d,k}^{DL},\\lambda ,\\mu ) = - \\sum _{\\forall n \\in N} {Qo{E_n}} + {\\lambda _1}h(\\chi _{c,{d^{^{\\prime }}}}^{DL})\\\\+ {\\lambda _2}{h}(\\chi _{t,d,k}^{DL}) + {\\mu _1}g(\\chi _{c,{d^{^{\\prime }}}}^{DL})\\\\+ {\\mu _2}g(\\chi _{t,d,k}^{DL}) + {\\mu _3}g(\\chi _{c,{d^{^{\\prime }}}}^{DL},\\chi _{t,d,k}^{DL}),\\end{split}$ where $\\begin{split}h(\\chi _{c,{d^{^{\\prime }}}}^{DL}){\\text{ = }}\\sum _{\\forall d^{\\prime } \\in D^{\\prime }} {\\chi _{c,{d^{^{\\prime }}}}^{UL} - 1} \\ \\end{split}$ $\\begin{split}{h}(\\chi _{t,d,k}^{DL}){\\text{ = }}\\sum _{\\forall d \\in D} {\\chi _{t,d,k}^{DL} - 1}\\end{split}$ $\\begin{split}g(\\chi _{c,{d^{^{\\prime }}}}^{DL}){\\text{ = }}\\sum _{\\forall d^{\\prime } \\in D^{\\prime }} \\sum _{\\forall c \\in C} {\\chi _{c,{d^{^{\\prime }}}}^{UL} \\cdot \\lambda _{c,{d^{^{\\prime }}}}^{UL} - B{UL}}\\end{split}$ $\\begin{split}g(\\chi _{t,d,k}^{DL}) = \\sum _{\\forall t \\in T_{F O V}^{n}} \\sum _{\\forall d \\in D} \\sum _{\\forall k \\in K} \\lambda _{t, d, k}^{D L} \\cdot \\chi _{t, d, k}^{D L} - \\sum _{\\forall k \\in K} B_{k}^{D L}\\end{split}$ $\\begin{split}g(\\chi _{c,{d^{^{\\prime }}}}^{DL},\\chi _{t,d,k}^{DL})=\\sum _{\\forall d \\in D} \\sum _{\\forall k \\in K} \\lambda _{t, d, k}^{D L} \\cdot \\chi _{t, d, k}^{D L} \\\\- \\frac{1}{T} \\sum _{\\forall d^{\\prime } \\in D^{\\prime }} \\lambda _{c, d^{\\prime }}^{U I} \\cdot \\chi _{c, d^{\\prime }}^{U L}\\end{split}$ Thus, we can obtain the relevant KKT conditions: $\\begin{split}\\frac{{\\partial L(\\chi _{c,{d^{^{\\prime }}}}^{UL},\\chi _{t,d,k}^{DL},\\lambda ,\\mu )}}{{\\partial \\chi _{c,{d^{^{\\prime }}}}^{UL}}}{\\text{ = }}{\\lambda _1}\\frac{{\\partial h(\\chi _{c,{d^{^{\\prime }}}}^{DL})}}{{\\partial \\chi _{c,{d^{^{\\prime }}}}^{UL}}}{\\text{ + }}{\\mu _1}\\\\{\\text{ + }}{\\mu _3}\\frac{{\\partial g(\\chi _{c,{d^{^{\\prime }}}}^{DL},\\chi _{t,d,k}^{DL})}}{{\\partial \\chi _{c,{d^{^{\\prime }}}}^{UL}}}{\\text{ = }}0\\end{split}$ $\\begin{split}\\frac{{\\partial L(\\chi _{c,{d^{^{\\prime }}}}^{UL},\\chi _{t,d,k}^{DL},\\lambda ,\\mu )}}{{\\partial \\chi _{t,d,k}^{DL}}}{\\rm { = }} - \\sum _{\\forall n \\in N} {\\frac{{\\partial Qo{E_n}}}{{\\partial \\chi _{t,d,k}^{DL}}}} {\\rm { + }}{\\lambda _2}\\frac{{\\partial {h_1}(\\chi _{t,d,k}^{DL})}}{{\\partial \\chi _{t,d,k}^{DL}}}\\\\{\\rm { + }}{\\mu _2}\\frac{{\\partial g(\\chi _{t,d,k}^{DL})}}{{\\partial \\chi _{t,d,k}^{DL}}}{\\rm { + }}{\\mu _3}\\frac{{\\partial g(\\chi _{c,{d^{^{\\prime }}}}^{DL},\\chi _{t,d,k}^{DL})}}{{\\partial \\chi _{t,d,k}^{DL}}}{\\rm { = }}0\\end{split}$ $\\begin{split}g(\\chi _{c,{d^{^{\\prime }}}}^{DL}) \\leqslant 0,g(\\chi _{t,d,k}^{DL}) \\leqslant 0,g(\\chi _{c,{d^{^{\\prime }}}}^{DL},\\chi _{t,d,k}^{DL}) \\leqslant 0\\end{split}$ $h(\\chi _{c,{d^{^{\\prime }}}}^{DL}){\\text{ = 0,}}{\\hspace{1.0pt}} {h}(\\chi _{t,d,k}^{DL}){\\text{ = }}0$ $\\begin{small}{\\lambda _1},{\\lambda _2} \\ne 0,{\\mu _1},{\\mu _2},{\\mu _3} \\geqslant 0\\end{small}$ $\\begin{small}{\\mu _1}g(\\chi _{c,{d^{^{\\prime }}}}^{DL}){\\text{ = }}0,{\\mu _2}g(\\varphi _{t,d,k}^{DL}) = 0,{\\mu _3}g(\\chi _{c,{d^{^{\\prime }}}}^{DL},\\chi _{t,d,k}^{DL}) = 0.\\end{small}$ By solving equations (14)-(19), which are associated with the KKT condition, we can derive the optimal solution of the relaxed nonlinear problem.",
"Next, we use the branch and bound method to find the solution of the original binary programming problem."
],
[
"Branch and bound Method",
"The branch and bound method designed to solve problem 1, as shown in Table REF .",
"The initial inputs are $\\chi _{relax}$ and $Z_{relax}$ , where $\\chi _{relax}$ is the solution to the corresponding relaxation problem solved by the KKT condition, and $Z_{relax}$ indicates the corresponding optimal objective function value.",
"The outputs are the 0–1 variable solution $\\chi _{0-1}$ and the corresponding optimal objective function value $Z_{0-1}$ .",
"Table: Branch and Bound Algorithm for Solving Problem 1."
],
[
"Performance Evaluation",
"We conduct experiments to verify the performance of the proposed QoE-driven 360-degree video live streaming system."
],
[
"Simulation Setup",
"During the simulation, we select 6 videos captured by an Insta 360 Pro2 panoramic camera as the original video at the uplink.",
"The highest resolution of each video is $1920 \\times 1080$ .",
"The video duration is 35s, and the frame rate is 30 fps.",
"Figure.",
"REF (a) shows a snapshot of the six original videos.",
"Figure: Illustration of the video sourceThen, we use the high efficiency video coding (HEVC) method and constant bitrate (CBR) as a bitrate control technique to compress each original video, with the bitrate representations of each video being {1.5, 2, 2.5, 3} Mbps.",
"We use AVP software (Kolor Autopano Video Pro) to complete the synthesis and stitch the VR videos together.",
"Subsequently, the panoramic VR video is divided into 16 tiles, each tile covers $90 \\times 45$ degrees.",
"We assume that the video server can provide 4 different bitrate representations for each tile at constant bitrates {0.2, 0.6, 1, 1.4} Mbps.",
"Figure.",
"REF (b) shows the panoramic VR video synthesized from the 6 original videos, which is then divided into 16 tiles.",
"At the client side, we assume that there are 100 users wearing HTC Vive as the HDM device.",
"The users are located at random distances from the base station, and they can request and watch 360-degree videos.",
"The channel information can be calculated based on the fixed transmission power of the base station.",
"During the simulation, we assume that each users' FOV ($120 \\times 90$ degrees) is randomly distributed in the 360-degree video and their FOV information is transmitted to the video server in real time."
],
[
"Simulation Results",
"To better verify the performance of the proposed scheme and make the simulation more realistic, we have conducted simulations using the real-world network traces [9] with/without the perfect knowledge of future network conditions (i.e.",
"bandwidth prediction is 100% correct or partially correct.).",
"We use two baseline schemes to show the advantages of the proposed scheme.",
"In the first method (denoted as Algorithm 2), the uplink bandwidth resources are evenly distributed among the different cameras, and the downlink uses the same adaptive allocation algorithm as our method.",
"In the second method (denoted as Algorithm 3), uplink bandwidth resources are evenly distributed among the different cameras, and downlink bandwidth resources are equally distributed to different tiles.",
"We first consider the case with perfect knowledge of future network condition and use the bandwidth value of each GOP as input for simulation.",
"Then we consider the case with imperfect knowledge of future network conditions.",
"According to the state-of-the-art bandwidth prediction scheme [10], we add a Gaussian random noise, which has mean of 0 and variance of 1, to the LTE trace (with multiplying 30% ) to simulate the prediction error.",
"We then perform simulation experiments using the predicted bandwidth (which differs from the real bandwidth) as the input for the optimization, and real the correct bandwidth to calculate the value of the objective function.",
"The simulation results of both cases are shown in Table REF .",
"From the simulation results, we can observe that the proposed algorithm can achieve a higher QoE value with all the three LTE traces.",
"This is because we not only achieve the integration of uplink and downlink resource allocation but also efficiently allocate the resources.",
"We can also observe that the larger bandwidth results in larger QoE value by comparing the performances with different traces.",
"The QoE value drops due to the prediction error.",
"This is because if the predicted bandwidth value after adding Gaussian noise is larger than the real bandwidth value, the real bandwidth may not be able to satisfy the bitrate requirement, which will cause the suspension of stalling and quality switching.",
"If the predicted bandwidth value is less than the real bandwidth value, some bandwidth may be wasted, which will degrade the received video quality.",
"However, the proposed algorithm still works better than the baseline scheme due to the joint consideration of uplink and downlink rate allocation.",
"Table: Performance comparisons."
],
[
"CONCLUSIONS",
"In this paper, we proposed a multi-user QoE-driven 360-degree video live streaming system, which jointly considers the uplink and downlink transmissions.",
"In our system, the server selects the optimal bitrate settings for both the uplink and downlink channels based on the bandwidth information and the users' real-time FOV to maximize the QoE value of all users.",
"To achieve this, we proposed an algorithm that combined the KKT condition and branch and bound method to solve the defined rate adaptation problem.",
"Finally, the simulation results based on the real-world network traces demonstrated that our proposed algorithm outperformed other baseline schemes."
]
] | 2001.03536 |
[
[
"Counterexample: scale-free networked graphs with invariable diameter and\n density feature"
],
[
"Abstract Here, we propose a class of scale-free networks $G(t;m)$ with some intriguing properties, which can not be simultaneously held by all the theoretical models with power-law degree distribution in the existing literature, including (i) average degrees $\\langle k\\rangle$ of all the generated networks are no longer a constant in the limit of large graph size, implying that they are not sparse but dense, (ii) power-law parameters $\\gamma$ of these networks are precisely calculated equal to $2$, as well (iii) their diameters $D$ are all an invariant in the growth process of models.",
"While our models have deterministic structure with clustering coefficients equivalent to zero, we might be able to obtain various candidates with nonzero clustering coefficient based on original networks using some reasonable approaches, for instance, randomly adding some new edges under the premise of keeping the three important properties above unchanged.",
"In addition, we study trapping problem on networks $G(t;m)$ and then obtain closed-form solution to mean hitting time $\\langle \\mathcal{H}\\rangle_{t}$.",
"As opposed to other previous models, our results show an unexpected phenomenon that the analytic value for $\\langle \\mathcal{H}\\rangle_{t}$ is approximately close to the logarithm of vertex number of networks $G(t;m)$.",
"From the theoretical point of view, these networked models considered here can be thought of as counterexamples for most of the published models obeying power-law distribution in current study."
],
[
"INTRODUCTION",
"Complex systems, such as, friendship networks, metabolic networks, protein-protein interaction networks, predator-prey networks, can be naturally interpreted as complex network, a newborn yet useful tool that has been widely adopted in a large variety of disciplines, particularly, in statistic physics and computer science [1].",
"So, in the past two decades, complex networks have attracted considerable attention and helped us to understand some topological properties and structural dynamics on the complex systems mentioned above.",
"Two significant findings of which are the small-word property [2] and scale-free feature [3].",
"There are in general two directions in current complex network study.",
"The one is to generate complex networked models, also called synthetic networks, in order to mimic some characters prevalent in real-world networks, such as power-law degree distribution, small-world phenomena, hierarchical structure [2]-[5].",
"The other aims at determining the influence from topological structure of networks on dynamics taking place on networks themselves, for instance, the mean hitting time for trapping problem, synchronization in networks, epidemic spread [6]-[8].",
"In this paper, we not only propose a family of networked models $G(t;m)$ and discuss some commonly used topological measures for understanding models $G(t;m)$ in much detail, but also consider the trapping problem on the proposed models $G(t;m)$ and final derive the closed-form solution to mean hitting time.",
"The main concern in the previous research of theoretical networked models is to focus on constructing models which have scale-free and small-world characters as described above.",
"However, an overwhelming number of models are sparse, implying that the average degree of networks will tend to a constant in the limit of large graph size.",
"This is because a great deal of real-life networks are found to display sparsity feature.",
"By contrast, current studies in some areas turn out the existence of dense networks [9].",
"To describe these such networks, some available networked models have been proposed [10] and analytically investigated in some principled manners, including mean-field theory and master equation.",
"Yet, most of them are stochastic.",
"To our knowledge, almost no deterministic models with both density structure and scale-free feature are built in the past.",
"Although there exist some disadvantages inherited by the latter in comparison with stochastic models, the deterministic structure of model allows us to precisely derive the solutions to some quantities of great interest, such as clustering coefficient, degree distribution, average path length.",
"To some extent, determining some invariants on networked models with deterministic structure has the theoretical flavor.",
"Motivated by this, we present a class of novel networked models with hierarchical structure that are precisely proved to not only show scale-free and small-world characters but also be dense.",
"Throughout this paper, all graphs (models) addressed are simple and the terms graph and network are used indistinctly.",
"The rest of this paper can be organized by the following several sections.",
"Section II aims at introducing networked models and discussing some widely studied structural parameters, including average degree $\\langle k\\rangle $ , diameter $D$ .",
"Among of them, while our models are analytically proved to show scale-free feature, the power-law exponents $\\gamma $ is equal to an unexpected constant 2.",
"This suggests that the density feature can be found on our models.",
"Surprisingly, all the networked models have an identical diameter that is always invariable in the growth process ($t\\ge 1$ ).",
"These properties above are rarely reported in current complex network studies.",
"In addition, we take some effective measures, for instance, adding edges with some probability $p$ , to switch the original graphs with deterministic structure into stochastic ones.",
"Last but not least, we take account into a type of random walks, called trapping problem, and then analytically determine the mean hitting time $\\langle \\mathcal {H}\\rangle _{t}$ .",
"The result shows an interesting phenomenon that the closed-form solution to quantity $\\langle \\mathcal {H}\\rangle _{t}$ is asymptotically close to the logarithm of vertex number of our models under consideration, which is not covered by almost all theoretical models characterizing complex networks with scale-free feature.",
"Final, we close this paper with a concise conclusion and some future directions in Section III."
],
[
"Networked graphs ",
"The goal of this section is to build up our networked models, denoted by graphs $G(t;m)$ , with hierarchical structure and study some topological structural parameters on these proposed models both analytically and experimentally.",
"In addition, we also consider the trapping problem on the proposed models $G(t;m)$ where the trap is allocated on the largest degree vertex."
],
[
"Construction",
"Here we will introduce the graphs $G(t;m)$ ($m\\ge 2$ ).",
"First, the seed, denoted by $G(0;m)$ , is a star with $m$ leaves as shown in the top-left panel of Fig.1.",
"The next graph $G(1;m)$ can be obtained from $G(0;m)$ in the following manner, (1) generating $m$ duplications of seed $G(0;m)$ labelled as $G^{i}(0;m)$ , (2) taking an active vertex, (3) connecting that active vertex to each leaf in star $G^{i}(0;m)$ .",
"Obviously, the resulting graph $G(1;m)$ has hierarchical structure as plotted in the bottom-left of Fig.1.",
"For convenience, we divide all vertices of $G(1;m)$ into three classes, i.e., that active vertex allocated at the level 0, denoted by $L=0$ , all the central vertices of stars $G^{i}(0;m)$ at the level $L=1$ and the remaining vertices of graph $G(1;m)$ at the level $L=2$ .",
"Henceforth, for time step $t\\ge 2$ , the young graph $G(t;m)$ can be built based on $m$ duplications of the preceding graph $G(t-1;m)$ by connecting an active vertex to all the vertices at the level $L=t$ of graphs $G^{i}(t-1;m)$ .",
"As an illustrative example, the graph $G(2;2)$ is shown in the rightmost panel of Fig.1.",
"In view of the growth manner of graph $G(t;m)$ , it is not hard to obtain a couple of equations satisfied by vertex number $|V(t;m)|$ and edge number $|E(t;m)|$ as follows $\\left\\lbrace \\begin{aligned}&|V(t;m)|=m|V(t-1;m)|+1\\\\&|E(t;m)|=m|E(t-1;m)|+m^{t+1}\\end{aligned}\\right..$ With the initial conditions $|V(0;m)|=m+1$ and $|E(0;m)|=m$ , we can solve for $|V(t;m)|$ and $|E(t;m)|$ from Eq.",
"(REF ) to obtain $|V(t;m)|=\\frac{m^{t+2}-1}{m-1},\\quad |E(t;m)|=(t+1)m^{t+1}.$ So far, we already accomplish the construction of our networked models exhibiting hierarchy phenomena.",
"Note that the related work to construction of hierarchical networks has been widely reported in the literature, such as Refs.",
"[11] and [12].",
"For instance, Ravasz et al have generated a class of hierarchical networked models in [12] and then discussed some structural parameters on them.",
"Whereas, it is worthy noticing that the generative method adopted in this paper is slightly different from that used in [12].",
"While the both are manipulated in an iterative manner, our method is based on an idea that the next networked model $G(t;m)$ is created by connecting a portion of vertices of each duplication $G^{i}(t-1;m)$ of the proceeding model $G(t-1;m)$ to a new external vertex instead of a designated vertex in model $G(t-1;m)$ .",
"In addition, as will become clear in the coming sections, some properties of our models are not found on those models in [11] and [12].",
"The next tasks are to in-detail discuss several common topological properties on graphs $G(t;m)$ , for instance, average degree $\\langle k\\rangle $ and clustering coefficient $\\langle C\\rangle $ , as tried in the existing literature [13],[14]."
],
[
"Structural properties",
"As described above, the deterministic nature of the proposed graphs allows us to evaluate some well-studied topological parameters associated with the underlying structure.",
"This section aims to calculate the closed-form solutions for several structural indices, including average degree $\\langle k\\rangle $ ."
],
[
"Average degree ",
"As the simplest yet most important structural parameter, average degree $\\langle k\\rangle $ , defined as the ratio of 2 times edge number and vertex number, can be adopted to determine whether a given network is sparse or not.",
"In general, almost all published networked models with scale-free feature (discussed later), both stochastic and deterministic, are by definition sparse, that is, the value for $\\langle k\\rangle $ being finite in the large graph size limit.",
"By contrast, our networked models $G(t;m)$ turn out to be of density due to $\\langle k(t;m)\\rangle =\\frac{2|E(t;m)|}{|V(t;m)|}\\sim 2t=O(\\ln |V(t;m)|).$ From Eq.",
"(REF ), it is clear to see that average degree $\\langle k(t;m)\\rangle $ is linearly correlated with time step $t$ and no longer a constant compared with those models in [11]-[13], such as, Apollonian networks [15].",
"Meantime, as recently reported in [9], many real-world network examples have been proven to have no sparsity topological structure.",
"Therefore, Our networked models $G(t;m)$ may be able to be selected as potential models to unveil some unseen properties behind those dense networks in real life."
],
[
"Degree distribution ",
"In the past years, there are two significant findings in complex network study.",
"One of which is scale-free feature due to Barabasi and Albert [3] using statistical method for depicting vertex degree distribution of many real-world networks.",
"Such types of networks show a fact that a small fraction of vertices possess a great number of connections and however the rest of vertices have a small number of connections.",
"Since then, given a complex networked model, one always estimate using degree distribution whether or not it is scale-free.",
"Taking into account deterministic structure of graphs $G(t;m)$ , we make use of the cumulative degree distribution in discrete form $P_{cum}(k\\ge k_{i})=\\frac{\\sum _{k\\ge k_{i}}N_{k}}{|V|},$ where symbol $N_{k}$ represents the total number of vertices with degree exactly equal to $k$ in graph $G(V,E)$ .",
"As said in Eq.",
"(REF ), we need to classify all vertices of graphs $G(t;m)$ according to vertex degree in order to determine whether the presented models show scale-free feature.",
"For a graph $G(t;m)$ with $t+1$ levels, it is evident to see that the greatest degree vertex is that active vertex added at time step $t$ and has degree $m^{t+1}$ , the second greatest degree vertices are all at the level $L=1$ , and so on.",
"With such a classification, there is in fact an unexpected case where all vertices at level $L=t+1$ have degree $t+1$ .",
"This degree value must be in some range between $m^{t_{i}}$ and $m^{t_{i+1}}$ .",
"Therefore, we will adjust the initial ranks of vertices with respect to vertex degree alone for simplicity.",
"The end list is as follows Table: NO_CAPTIONIn practice, the above list is usually called degree sequence of graph $G(t;m)$ in the jargon of graph theory.",
"Based on such a list and Eq.",
"(REF ), we can calculate the degree distribution of networked model $G(t;m)$ in the following $P_{cum}(k\\ge k_{i})=\\left\\lbrace \\begin{array}{ll}\\quad k_{i}^{-\\gamma _{\\alpha }}\\; , \\qquad & \\quad k_{i}> t+1\\\\k_{i}^{-\\gamma _{\\alpha }}+\\frac{1}{2}\\; , & \\quad k_{i}\\le t+1 \\\\\\end{array}\\right.$ where power-law parameter $\\gamma _{\\alpha }=1$ .",
"Performing the derivative of both sides in Eq.",
"(REF ) with respect to $k$ yields $P(k)\\sim k^{-\\gamma }, \\quad \\gamma =\\gamma _{\\alpha }+1=2.$ This disapproves a statement in [16] that no network with an unbounded power-law degree distribution with $0\\le \\gamma \\le 2$ can exist in the limit of large graph size.",
"Technically, it is an obvious corollary from Eq.",
"(REF ) that our networked models $G(t;m)$ are all dense as in Eq.",
"(REF ) because of the feature of Riemann $\\zeta (1)$ function $\\zeta (1)=\\sum _{i=1}i^{-1}.$ In the limit of large $i$ , the right-hand side of Eq.",
"(REF ) is divergence.",
"Thus, average degree $\\langle k(t;m)\\rangle $ of networked models $G(t;m)$ must be infinite due to $\\langle k(t;m)\\rangle =\\int _{k_{min}}^{k_{max}}kP(k)dk$ .",
"As showed in [17], Caldarelli et al have pointed out the widespread occurrence of the inverse square distribution in social sciences and taxonomy and also provided some detailed discussions about mechanisms causing such a phenomenon.",
"Meanwhile, they showed that the treelike classification method is competent to lead to this behavior mentioned above.",
"At the same time, several stochastic scale-free graphs with degree parameter $\\gamma =2$ have been constructed in [18].",
"Nonetheless, there are no deterministic models following power-law degree distribution with $\\gamma =2$ found in the current theoretical models study.",
"Roughly speaking, our models, graphs $G(t;m)$ , can be regarded as the first attempt in context of constructing deterministic models.",
"In addition, graphs $G(t;m)$ have many other interesting topological structural properties as we will show shortly.",
"Some of which are not displayed by those stochastic models.",
"In particular, the finding of ultrasmall diameter is a surprising result in this sense."
],
[
"Diameter ",
"Following the previous subsection, the other intriguing finding in the complex network studies is small-world property attributed to Watts and Strogatz [2] by empirically capturing the diameter of some real-world and synthetic network models.",
"Mathematically, diameter of a graph, denoted by $D$ , is the maximum over distances of all possible vertex pairs.",
"For a pair of vertices $u$ and $v$ , distance between them, denoted by $d_{uv}$ , is the edge number of any shorted path joining vertex $u$ and $v$ .",
"Most generally, diameter can be viewed as a coarse-granularity index for measuring the information delay on a network in question.",
"With the help of concrete construction of networked models $G(t;m)$ , it is easy to find the diameters $D(t;m)$ to obey $D(t;m)=4=O(1).$ This is because (1) all vertices at level $L=t+1$ are connected to that active vertex at the highest level, namely, $L=0$ ; (2) each vertex at the intermediate levels, $L=1,...,L=t$ , always connects to a vertex at the most bottom level $L=t+1$ .",
"In fact, diameters $D(t;m)$ are equal to the distance between that two vertices that are both at the intermediate levels and in different branches of graphs $G(t;m)$ .",
"Taking into account a trivial character in a connected graph $G(V,E)$ that average path length, defined as $\\langle d\\rangle =\\sum _{u,v\\in E, u\\ne v}d_{uv}/\\frac{1}{2}[|V|(|V|-1)]$ , is no larger than the diameter, we omit the analytical solutions for average path length of networked models $G(t;m)$ .",
"Using the exact value of diameter in Eq.",
"(REF ), one can be convinced that our graphs have captured an ingredient of small-world property.",
"The other will be discussed in the subsequent section.",
"It is worthy noting that in [19], the authors have reported some results about diameter of scale-free graphs $G(V,E)$ , for instance, those with $2<\\gamma <3$ having a much smaller diameter in the limit of large graph size, which behaves as $D\\sim \\ln \\ln |V|$ .",
"However, there are few discussions about diameter of the scale-free graphs with $\\gamma =2$ in published papers.",
"Perhaps one of important reasons for this is that the previous researches focus mainly on sparse models with scale-free feature.",
"Here, our networked models $G(t;m)$ are proved to have an invariable diameter in the evolution process.",
"As a result, graphs $G(t;m)$ can serve as stronger evidences for illustrating that our ability to understand the fundamental structural properties of graphs including all scale-free ones is always limited to some specific models and hence some early demonstrations may be not complete.",
"Compared to numerous pre-existing networked models with scale-free feature, such as those in [19], the power-law parameter $\\gamma $ of our graphs $G(t;m)$ is not in the range $2<\\gamma <3$ .",
"Hence, one has mostly likely to conjecture whether the density feature of networked models $G(t;m)$ leads to such an ultrasmall diameter directly.",
"In principle, the dense graphs should show smaller diameters in comparison with those sparse ones.",
"In practice, we would like to note that density feature planted on graphs $G(t;m)$ indeed shrinks the distance between any pair of vertices and thus has vastly effect on emergence of smaller diameter, but it is not a sufficient condition.",
"As described in our recent work [20], the diameter of dense graphs $G(V,E)$ obeying power-law distribution may also be quite large compared with the widely-used value $\\ln |V|$ .",
"Again, this strongly means that there are a great number of structural properties of scale-free graphs incompletely uncovered until now and so more efforts should be paid to better understand this kinds of fascinating graphs in the future."
],
[
"Clustering coefficient ",
"The other ingredient of small-world property is clustering coefficient that plays an important role in evaluating the level of clusters in a networked model under consideration.",
"For instance, it in essence describes a phenomenon that in a friendship network two arbitrary friends of one person will have a higher likely to be friends with each other, usually called the triadic in social analysis.",
"For the theoretical point of view, such a connection trend among neighbors of one vertex $v$ with degree $k$ can be abstractly depicted in the following form $c_{v}=n_{v}/\\frac{1}{2}[k(k-1)]$ where $n_{v}$ represents the actually existing edges between neighbors of vertex $v$ .",
"For the whole graph $G(V,E)$ , the clustering coefficient $\\langle C\\rangle $ can be defined as the averaged value over clustering coefficients of all vertices, as follows $\\langle C\\rangle =\\frac{\\sum _{v\\in V}n_{v}}{|V|}.$ Figure: The diagram of clustering coefficients 〈C 1 〉\\langle C_{1}\\rangle of graphs G 1 (t;m)G_{1}(t;m).Figure: The diagram of clustering coefficients 〈C 2 〉\\langle C_{2}\\rangle of graphs G 2 (t;m)G_{2}(t;m) where parameters mm are equal to 2,4,62,4,6 and 8 from left to right, respectively.By definition, we can without difficulty obtain that the clustering coefficients $\\langle C(t;m)\\rangle $ of networked models $G(t;m)$ are equivalent to 0 because there are no triangles contained in the evolution of graphs $G(t;m)$ .",
"This suggests that small-world property can not be found in graphs $G(t;m)$ .",
"Nevertheless, there are still plenty of other potential characters behind networked models $G(t;m)$ .",
"To be more concrete, all the generated graphs $G(t;m)$ are planar, meaning that for an arbitrary parameter $m$ , graph $G(t;m)$ can be embedded in the plane so that its edges intersect only at their ends [21].",
"To make our models $G(t;m)$ small-world, i.e., achieving the transformation from zero clustering coefficient to nonzero, we have to take some effective measures.",
"There are in fact a great deal of methods for addressing problems of this kind in present research.",
"The simplest one is to add new edges between unconnected vertex pairs for generating enough triangles.",
"Here, our goal is not only to obtain nonzero clustering coefficient but also to remain the properties discussed above unchanged.",
"To this end, we make use of a simple replacement of the seed from star $G(0;m)$ to wheel $G_{1}(0;m)$ [22].",
"Besides that, all the growth mechanisms keep the same as previously.",
"This leads to a new graph $G_{1}(t;m)$ .",
"By definition, it is straightforward to get $\\langle C_{1}\\rangle =\\frac{\\sum _{i=0}^{t}\\frac{2m^{t-i}}{m^{i+1}-1}+\\frac{2(t+1)m^{t+1}}{(t+3)(t+2)}}{|V_{1}(t;m)|},$ where the vertex number $|V_{1}(t;m)|$ of graph $G_{1}(t;m)$ is equal to $|V(t;m)|$ .",
"In the large graph size limit, the clustering coefficient $\\langle C_{1}\\rangle $ of Eq.",
"(REF ) tends to a nonzero constant for small parameters $m$ as shown in Fig.2, suggesting that deterministic graph $G_{1}(t;m)$ will have scale-free feature and small-world property simultaneously.",
"To make further progress, we may delete with probability $p$ each edge between vertices at the level $L=t+1$ in graph $G_{1}(t;m)$ , leading to another graph $G_{2}(t;m)$ .",
"The introduction of randomly deleting edges will switch deterministic graphs into the opposite case, namely, stochastic ones.",
"As before, the clustering coefficient $\\langle C_{2}\\rangle $ of stochastic graphs $G_{2}(t;m)$ can be calculated as follows $\\langle C_{2}\\rangle =\\frac{(1-p)\\sum _{i=0}^{t}\\frac{2m^{t-i}}{m^{i+1}-1}+\\left[(1-p)^{2}\\frac{2(t+1)}{(t+3)(t+2)}+\\frac{2p(1-p)}{t+2}\\right]m^{t+1}}{|V_{2}(t;m)|},$ here $|V_{2}(t;m)|$ is the vertex number of graphs $G_{2}(t;m)$ .",
"To determine the tendency of $\\langle C_{2}\\rangle $ in the limit of large size, we feed graphs $G_{2}(t;m)$ into computer.",
"As plotted in Fig.3, for distinct parameters $m$ , the values $\\langle C_{2}\\rangle $ are different from each other initially but all show similar tendency in the large-$t$ limit.",
"In a word, the three types of networked models, graphs $G(t;m)$ , $G_{1}(t;m)$ and $G_{2}(t;m)$ , can be selected as counterexamples for disproving some previous statements about scale-free graphs, such as the scale-free graphs with small-world property may have invariable diameter.",
"Meantime, the lights shed by them may be helpful to construct new networked models in the future."
],
[
"Assortative structure ",
"Many real-world networks [23] have been observed to show assortative mixing on their degrees, that is, a preference for high-degree vertices to attach to other vertices like them, while others show disassortative mixing, i.e., high-degree vertices attach to ones unlike them.",
"Particularly, this is a popular phenomenon in social networks.",
"For instance, it is mostly willing of people to establish friendships with those at the same level as them rather than to get in touch with others.",
"For the purposes of quantifying this feature of networks $G(V,E)$ , Newman defined the following measure $r$ , usually called assortativity coefficient, $r=\\frac{|E|^{-1}\\sum \\limits _{e_{ij}\\in E} k_{i}k_{j}-\\left[|E|^{-1}\\sum \\limits _{e_{ij}\\in E} \\frac{1}{2}(k_{i}+k_{j})\\right]^{2}}{|E|^{-1}\\sum \\limits _{e_{ij}\\in E} \\frac{1}{2}(k^{2}_{i}+k^{2}_{j})-\\left[|E|^{-1}\\sum \\limits _{e_{ij}\\in E} \\frac{1}{2}(k_{i}+k_{j})\\right]^{2}},$ in which $k_{i}$ is the degree of vertex $v$ and $e_{ij}$ denotes an edge connecting vertex $i$ to $j$ .",
"With such an index, most social networks turn out to have significant assortative mixing, while technological and biological networks seem to be disassortatively constructed.",
"As explained above, the scalar measure $r$ in fact figures the degree of similarity between two endpoints of any edge on an observed network by means of vertex degree.",
"Empirically, our networked models $G(t;m)$ should be disassorative.",
"To show this, we can write based on Eq.",
"(REF ) Figure: The diagram of assortativity coefficient r(t;m)r(t;m) of graphs G(t;m)G(t;m).$r(t;m)=\\frac{\\frac{m^{t+2}-m}{m-1}-\\left[\\frac{t+1}{2}+\\frac{m^{t+2}-m}{(2m-2)(t+1)}\\right]^{2}}{\\frac{(t+1)^{2}}{2}+\\frac{m^{2t+4}-m^{2}}{(2m^{2}-2)(t+1)}-\\left[\\frac{t+1}{2}+\\frac{m^{t+2}-m}{(2m-2)(t+1)}\\right]^{2}}.$ As $t\\rightarrow \\infty $ , for a given parameter $m$ , $r(t;m)$ tends to zero, seeing Fig.4 for a lot.",
"By analogy with Eq.",
"(REF ), we can evaluate assortativity coefficient $r_{2}(t;m)$ of stochastic graphs $G_{2}(t;m)$ by plugging the following equations into Eq.",
"(REF ) $|E|=|E_{2}(t;m)|=m^{t+1}(t+2-p),$ $\\begin{aligned}\\sum \\limits _{e_{ij}\\in E} k_{i}k_{j}&=m^{t+1}\\sum _{i=0}^{t}[p^{2}(t+1)+2p(1-p)(t+2)]\\\\&+m^{t+1}\\sum _{i=0}^{t}(1-p)^{2}(t+3)+(1-p)^{4}(t+3)^{2}m^{t+1}\\\\&+4p(1-p)^{2}[p(t+2)^{2}+(1-p)(t+2)(t+3)]m^{t+1}\\end{aligned},$ $\\begin{aligned}\\sum \\limits _{e_{ij}\\in E}(k_{i}+k_{j})&=m^{t+1}(t+1)+\\sum _{i=0}^{t}(1-p)^{2}(t+3)m^{i} \\\\&+\\sum _{i=0}^{t}m^{i}[p^{2}(t+1)+2p(1-p)(t+2)]\\\\&+4p(1-p)^{2}[p(2t+4)+(1-p)(2t+5)]m^{t+1}\\\\&+(1-p)^{4}(2t+6)m^{t+1}\\end{aligned},$ $\\begin{aligned}\\sum \\limits _{e_{ij}\\in E}(k_{i}^{2}+k_{j}^{2})&=m^{t+1}\\sum _{i=0}^{t}m^{i+1}+\\sum _{i=0}^{t}(1-p)^{2}(t+3)^{2}m^{i} \\\\&+\\sum _{i=0}^{t}m^{i}[p^{2}(t+1)^{2}+2p(1-p)(t+2)^{2}]\\\\&+2(1-p)^{2}[4p^{2}(t+2)^{2}+(1-p)^{2}(t+3)^{2}]m^{t+1}\\\\&+4p(1-p)^{3}[(t+2)^{2}+(t+3)^{2}]m^{t+1}\\end{aligned}.$ In order to evaluate whether the random deletion of edges has influence on assortativity coefficient $r_{2}(t;m)$ , we conduct extensive simulations in terms of Eq.",
"(REF ) and Eqs.",
"(REF )-(REF ) and experimental results are shown in Fig.5.",
"Interestingly, from the panels in Fig.5, it can be easy to see a phenomenon that all theoretical values for assortativity coefficients $r_{2}(t;m)$ are bounded from above the critical condition 0 while approaching to zero in the large graph size limit.",
"This is sharply different from those previously reported results in [23] where all assortativity coefficients associated with most of studied networked models are negative while also tending to zero.",
"Figure: The diagram of assortativity coefficients r 2 (t;m)r_{2}(t;m) of graphs G 2 (t;m)G_{2}(t;m) where parameters mm are equal to 2,4,62,4,6 and 8 from left to right, respectively."
],
[
"Mean hitting time",
"In this section, we formulate the trapping problem on our graphs $G(t;m)$ .",
"In practice, this is a simple unbiased Markovian random walk with a tap, say a perfect absorber, allocated on a designated vertex on a graph in question.",
"As shown above, that active vertex at the level $L=0$ of graphs $G(t;m)$ has the largest degree and hence is called the hub vertex, denoted by $h_{t}$ .",
"In order to further probe its importance under trapping problem, we put an absorber on the hub vertex.",
"And then, a particle located on vertex $v$ but for the hub will hop to one of its neighbor $N_{v}$ with the transition probability $1/d_{v}(t;m)$ before arriving at that absorber where $d_{v}(t;m)$ is the degree of vertex $v$ in graphs $G(t;m)$ .",
"Consider that a particle starts from vertex $v$ at initial time, the jumping probability $P_{vu}$ of starting out from $v$ to $u$ satisfies the following master equation $P_{vu}(l+1)=\\sum _{i\\in V(t;m)}\\frac{a_{iu}}{d_{i}(t;m)}P_{vi}(l),$ where $a_{iu}$ is the element of adjacency matrix of graph $G(t;m)$ , $a_{iu}=1$ if this pair of vertices $i$ and $u$ are connected by an edge and $a_{iu}=0$ otherwise.",
"According to the rule above, we are particularly interested in the quantity, called hitting time $\\mathcal {H}$ , for measuring the expected time for a particle, which starts from an arbitrary vertex, to first visit at the trap in the trapping problem.",
"For graph $G(t;m)$ as a whole, we denote the hitting time for a particle placed on vertex $v$ by $\\mathcal {H}_{v}$ and then let $P(\\mathcal {H}_{v}=l)$ be the probability for that particle to first hit the trap, i.e., hub vertex $h_{t}$ , after $l$ steps.",
"By analogy with Eq.",
"(REF ), we can obtain $P(\\mathcal {H}_{v}=l)=\\sum _{i\\in V(t;m),i\\ne h_{t}}\\frac{a_{iu}}{d_{i}(t;m)}P(\\mathcal {H}_{v}=l-1).$ A commonly used approach for the preceding equation is generating function.",
"Without loss of generality, we may define the corresponding generating function of quantity $P(\\mathcal {H}_{v}=l)$ in the following form $\\mathcal {P}_{v}(x)=\\sum _{t=0}^{\\infty }P(\\mathcal {H}_{v}=l)x^{l}.$ As we will show later, a trial yet useful fact related to $\\mathcal {P}_{v}(x)$ , that is, the expected time $\\mathcal {H}_{v}$ is exactly equal to the value $\\mathcal {P}^{\\prime }_{v}(1)$ , helps us to consolidate all the results in the subsequent section.",
"Before beginning to derive our calculations, for convenience, we need to introduce two notations $P_{t}(s)$ and $Q_{L}(s)$ .",
"The former represents the probability for a particle on an arbitrary vertex at the level $L=t+1$ of graphs $G(t;m)$ to first arrive at the hub $h_{t}$ after $s$ steps, and the latter is defined as the probability that a particle originating from an arbitrary vertex $w$ at the level $L$ ($L=1,\\dots ,t$ ) hits one at random chosen vertex at the level $L=t+1$ of graphs $G(t;m)$ , which connects to vertex $w$ , after $s$ jumps.",
"Based on the structure of graphs $G(t;m)$ and the statements above, we can write the following equation $P_{t}(s)=\\frac{\\delta _{s,1}}{k_{L=t+1}}+\\frac{1}{k_{L=t+1}}\\sum _{L=1}^{t}\\sum _{i=1}^{s-1}Q_{L}(i)P_{t}(s-1-i),$ here $\\delta _{s,1}$ is the Kronecker delta function in which $\\delta _{s,1}=1$ as $s=1$ and $\\delta _{s,1}=0$ otherwise, $k_{L=t+1}$ is the degree of vertex at the level $L=t+1$ and equals $t+1$ as above.",
"Using the lights shed by Eq.",
"(REF ), the generating function $\\mathcal {P}_{t}(x)$ corresponding to quantity $P_{t}(s)$ can be expressed as follows $\\mathcal {P}_{t}(x)=\\frac{x}{k_{L=t+1}}+\\frac{tx^{2}}{k_{L=t+1}}\\mathcal {P}_{t}(x),$ in which we make use of an evident result $Q_{L}(i)=1$ only for both $L=1,\\dots ,t$ and $i=1$ , as well $Q_{L}(i)=0$ otherwise.",
"At the same time, we let $\\mathcal {H}_{t+1}^{t}$ stand for the hitting time for a particle initially set on any vertex at the level $L=t+1$ which is by definition written $\\mathcal {H}_{t+1}^{t}=\\left.\\dfrac{d}{dx}\\mathcal {P}_{t}(x)\\right|_{x=1}.$ Taking into consideration Eq.",
"(REF ), performing the derivative of both sides of Eq.",
"(REF ) produces the exact solution of $\\mathcal {H}_{t+1}^{t}$ as follows $\\mathcal {H}_{t+1}^{t}=2t+1.$ For each vertex at the level $L=1,\\dots ,t$ , combining the definition of $Q_{L}(s)$ and the hierarchical structure of graphs $G(t;m)$ , the hitting time $\\mathcal {H}_{L}^{t}$ for a particle originally allocated on any vertex at the level $L$ may be obtained in terms of $\\mathcal {H}_{t+1}^{t}$ $\\mathcal {H}_{L}^{t}=\\mathcal {H}_{t+1}^{t}+1.$ By far, the hitting times $\\mathcal {H}_{L}^{t}$ for a particle at the level $L=1,\\dots ,t+1$ are all precisely calculated in a rigorous manner.",
"The next task is to derive the mean hitting time $\\langle \\mathcal {H}\\rangle _{t}$ , which characterizes the trapping process on average, in the following fashion $\\langle \\mathcal {H}\\rangle _{t}=\\frac{1}{|V(t;m)|-1}\\sum _{L=1}^{t+1}\\mathcal {H}_{L}^{t}|N_{t}(L)|,$ here we again utilize the hierarchy of graphs $G(t;m)$ and $|N_{t}(L)|$ denotes the total number of vertices at the level $L$ ($L=1,\\dots ,t+1$ ).",
"As pointed out before, $|N_{t}(L)|$ is in essence equal to $N_{L,t;m}$ .",
"Substituting Eqs.",
"(REF )-(REF ) and the value of $|V(t;m)|$ in Eq.",
"(REF ) into Eq.",
"(REF ) yields $\\langle \\mathcal {H}\\rangle _{t}=O(2t+\\frac{1}{m}),$ where we take useful advantage of some simple arithmetics.",
"To make process further, we consider the logarithm of vertex number of graphs $G(t;m)$ , namely, $\\ln |V(t;m)|\\sim t\\ln m$ .",
"It is clear to see that for the whole graphs $G(t;m)$ , the mean hitting time $\\langle \\mathcal {H}\\rangle _{t}$ has a close relationship with the vertex number of graphs $G(t;m)$ as shown below $\\langle \\mathcal {H}\\rangle _{t}\\sim \\ln |V(t;m)|.$ This is completely different from some previous results in the existing literature, such as, the complete graph on $N$ vertices having the mean hitting time exactly equal to $N-1$ , which is quite approximately close to its vertex number, the hierarchical models considered in [24] with the mean hitting time also being the same order of magnitude as their own vertex number.",
"Compared to our models, the former, i.e., complete graph, has no both scale-free feature and hierarchical structure while with the smallest diameter.",
"On the other hand, the models in [24] have a larger enough diameter than our graphs while showing scale-free feature and the hierarchy of structure.",
"Following the discussions above, it has been shown that when the trap is allocated on the greatest degree vertex, those hierarchical networks presented in [11] and [12] have mean hitting time approximately close to the power of number of vertices for some exponent $\\alpha $ in form, a demonstration that is distinct with the consequence obtained in this paper.",
"On average, this suggests that the proposed networked models $G(t;m)$ outperform those models [11] and [12] with respect to mean hitting time in the trapping problem considered here.",
"As a consequence, our models can be capable of serving as counterexamples for in-depth understanding many other fundamental properties on theoretical models, in particular, with respect to scale-free graphs.",
"One of the most important reasons is that scale-free feature is ubiquitously observed in a large amount of complex networks, both synthetic and real-world."
],
[
"Conclusion",
"In summary, we present a family of scale-free networked models of significant interest.",
"Based on both theoretical arguments and experimental simulations, we derive some striking results unseen in pre-existing theoretical models.",
"They shows that (1) our graphs $G(t;m)$ follow power-law degree distribution with exponent 2 and thus are dense, (2) an invariable diameter can be found on our graphs $G(t;m)$ compared to almost all previously proposed scale-free models, (3) using random method leads the end graphs $G_{2}(t;m)$ to display a nonnegative assortativity coefficient, and (4) when the trap is allocated on the hub in graphs $G(t;m)$ , the mean trapping time is approximately related to the logarithm of vertex number of graphs.",
"To the best of our knowledge, this work seems the first to probe novel scale-free models particularly because in the last, a significant amount of attention have been paid to discuss sparse graphs with scale-free feature.",
"From the respect of theoretical research, our models can be used as counterexamples to disprove some previous demonstrations corresponding to the scale-free graph family in current study and so enable researchers to well understand the fundamental structure properties planted on scale-free models."
],
[
"Acknowledgments",
"The research was supported by the National Key Research and Development Plan under grant 2017YFB1200704 and the National Natural Science Foundation of China under grant No.",
"61662066."
]
] | 2001.03525 |
[
[
"Quasilinear systems of Jordan block type and the mKP hierarchy"
],
[
"Abstract We demonstrate that commuting quasilinear systems of Jordan block type are parametrised by solutions of the modified KP hierarchy.",
"Systems of this form naturally occur as hydrodynamic reductions of multi-dimensional linearly degenerate dispersionless integrable PDEs."
],
[
"Introduction",
"Quasilinear systems of the form $u_t=v(u)u_x$ have been thoroughly investigated in the literature.",
"Here $u=(u^1, \\dots , u^n)^T$ is a column vector of the dependent variables and $v$ is a $n\\times n$ matrix.",
"The main emphasis has always been on the strictly hyperbolic case where the matrix $v$ has real distinct eigenvalues.",
"Under the additional condition that the Haantjes tensor of matrix $v$ vanishes, any such system can be reduced to the diagonal form $R^i_t=\\lambda ^i(R) R^i_x,$ $i=1, \\dots , n$ , in specially adapted coordinates $R^1, \\dots , R^n$ known as Riemann invariants.",
"Systems of type (REF ) govern a wide range of problems in pure and applied mathematics, see e.g.",
"[15], [1], [13].",
"It was shown by Tsarev [15] that under the so-called semi-Hamiltonian constraint, $\\left(\\frac{\\lambda ^i_j}{\\lambda ^j-\\lambda ^i}\\right)_k=\\left(\\frac{\\lambda ^i_k}{\\lambda ^k-\\lambda ^i}\\right)_j,$ system (REF ) possesses infinitely many conservation laws and commuting flows, and can be solved by the generalised hodograph method (here $i\\ne j\\ne k$ and low indices indicate differentiation by the variables $R^j$ ).",
"In this paper we study quasilinear systems (REF ) of Jordan block type.",
"More precisely, we assume the existence of special coordinates (which we will also denote $R^1, \\dots , R^n$ ) where the equations reduce to upper-triangular Toeplitz form $R_t=(\\lambda ^0E+\\sum _{i=1}^{n-1}\\lambda ^iP^i)R_x;$ here $R=(R^1, \\dots , R^n)^T$ , $E$ is the $n\\times n$ identity matrix, $P$ is the $n\\times n$ Jordan block with zero eigenvalue (note that $P^n=0$ ), and $\\lambda ^0, \\lambda ^i$ are functions of $R$ .",
"Explicitly, a three-component version of system (REF ) is as follows: $\\left(\\begin{array}{c}R^1\\\\ R^2\\\\R^3\\end{array}\\right)_t=\\left(\\begin{array}{ccc}\\lambda ^0 & \\lambda ^1 & \\lambda ^2\\\\0&\\lambda ^0&\\lambda ^1\\\\0&0&\\lambda ^0\\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\\\ R^3\\end{array}\\right)_x.$ The main properties of systems (REF ) can be summarised as follows: (a) the corresponding matrix $v$ is pointwise of Jordan block type; (b) the Haantjes tensor of matrix $v$ vanishes.",
"The vanishing of the Haantjes tensor makes systems (REF ) natural parabolic analogues of hydrodynamic type systems (REF ) in Riemann invariants.",
"Note that upper-triangular Toeplitz matrices form a commutative family (cyclic Haantjes algebra in the terminology of [14]).",
"Systems of type (REF ) appear as degenerations of hydrodynamic type systems associated with multi-dimensional hypergeometric functions [5], in the context of parabolic regularisation of the Riemann equation [6], and as reductions of hydrodynamic chains and linearly degenerate dispersionless PDEs in 3D [11].",
"The most well-studied case of system (REF ) corresponds to the choice $\\lambda ^0=R^1, \\ \\lambda ^1=1, \\ \\lambda ^i=0,\\ i\\ge 2$ .",
"In section we classify commuting systems of type (REF ).",
"Our main observation is that integrable hierarchies of Jordan block type are governed by the modified Kadomtsev-Petviashvili (mKP) hierarchy.",
"Here is a brief summary of our results in this direction.",
"Any two-component hierarchy of type (REF ) can be parametrised in the form $\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_t=\\left(\\begin{array}{cc}\\psi & \\psi _1 \\\\0&\\psi \\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_x$ where $\\psi $ satisfies the Lax equation of the mKP hierarchy, $\\psi _2=\\psi _{11}+\\rho \\psi _1;$ here low indices indicate differentiation by $R^1, R^2$ .",
"Fixing the potential $\\rho $ and varying $\\psi $ we obtain commuting flows of the corresponding hierarchy.",
"Similarly, any three-component hierarchy of type (REF ) can be parametrised in the form $\\left(\\begin{array}{c}R^1\\\\ R^2\\\\R^3\\end{array}\\right)_t=\\left(\\begin{array}{ccc}\\psi & \\psi _1 & \\psi _{11}+w_1\\psi _1\\\\0&\\psi &\\psi _1\\\\0&0&\\psi \\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\\\ R^3\\end{array}\\right)_x,$ where $w$ solves the mKP equation $4 w_{13}+6 w_1^2 w_{11}-w_{1111}-3w_{22}-6 w_{2}w_{11}=0,$ and $\\psi $ satisfies the corresponding Lax equations $\\psi _{2}={\\psi _{11}}+2w_1\\psi _{1},\\qquad \\psi _{3}={\\psi _{111}}+{3w_1}\\psi _{11}+\\frac{3}{2}(w_{2}+w_{11}+w_1^2)\\psi _{1}.$ Fixing $w$ and varying $\\psi $ we obtain commuting flows of the hierarchy.",
"We show that the corresponding conserved densities are governed by the adjoint Lax equations.",
"In section we demonstrate that systems of Jordan block type naturally occur as hydrodynamic reductions of multi-dimensional linearly degenerate PDEs: the 3D Mikhalev system [10] is used as an illustrating example, see also [11]."
],
[
"Form-invariance",
"The class of two-component Toeplitz systems (REF ), $\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_t=\\left(\\begin{array}{cc}\\lambda ^0 & \\lambda ^1 \\\\0&\\lambda ^0\\\\\\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_x,$ is form-invariant under triangular changes of variables $(R^1, R^2)\\leftrightarrow (r^1, r^2)$ of the form $R^1=F(r^1, r^2), \\qquad R^2=G(r^2),$ where $F$ and $G$ are arbitrary functions of the indicated arguments.",
"Similarly, the class of three-component Toeplitz systems (REF ), $\\left(\\begin{array}{c}R^1\\\\ R^2\\\\R^3\\end{array}\\right)_t=\\left(\\begin{array}{ccc}\\lambda ^0 & \\lambda ^1 & \\lambda ^2\\\\0&\\lambda ^0&\\lambda ^1\\\\0&0&\\lambda ^0\\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\\\ R^3\\end{array}\\right)_x,$ is form-invariant under triangular changes of variables $(R^1, R^2, R^3)\\leftrightarrow (r^1, r^2, r^3)$ of the following form: $R^1=r^1\\frac{(\\partial _{r^2}G)^2}{\\partial _{r^3}H}+F(r^2, r^3), \\qquad R^2=G(r^2, r^3), \\qquad R^3=H(r^3),$ where $F, G$ and $H$ are arbitrary functions of the indicated arguments.",
"This transformation freedom will be utilised to simplify the classification results.",
"Note that the group preserving the class of diagonal systems (REF ) is far more narrow, generated by transformations of the form $R^i=F^i(r^i)$ , functions of one variable only."
],
[
"Commuting flows",
"Let us recall that commuting flows of the diagonal system (REF ) are governed by the equations $w^i_j=a^{ij}(w^j-w^i)$ where $a^{ij}(R)={\\lambda ^i_j}/{(\\lambda ^j-\\lambda ^i)}$ are fixed and solutions $w^i$ to linear system (REF ) vary [15].",
"Note that the consistency conditions of system (REF ), $a^{ij}_k=a^{ij}a^{jk}+a^{ik}a^{kj}-a^{ik}a^{ij},$ are equivalent to the integrable $2+1$ -dimensional $n$ -wave system.",
"Thus, commuting flows of diagonal form (REF ) are governed by the $n$ -wave hierarchy.",
"Below we demonstrate that commuting flows of Toeplitz type (REF ) are governed by the mKP hierarchy.",
"Furthermore, the corresponding conserved densities of hydrodynamic type satisfy the adjoint Lax equations."
],
[
"Two-component case",
"Direct calculation shows that the compatibility (commutativity) of two-component systems of the form $\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_t=\\left(\\begin{array}{cc}\\lambda ^0 & \\lambda ^1 \\\\0&\\lambda ^0\\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_x, \\qquad \\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_y=\\left(\\begin{array}{cc}\\mu ^0 & \\mu ^1 \\\\0&\\mu ^0\\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_x$ is equivalent to the following two conditions: $\\frac{\\lambda ^0_1}{\\lambda ^1}=\\frac{\\mu ^0_1}{\\mu ^1},$ $\\frac{\\lambda ^0_2-\\lambda ^1_1}{\\lambda ^1}=\\frac{\\mu ^0_2-\\mu ^1_1}{\\mu ^1}.$ Let us introduce the notation $\\frac{\\lambda ^0_1}{\\lambda ^1}=m, \\ \\frac{\\lambda ^0_2-\\lambda ^1_1}{\\lambda ^1}=\\rho $ ; note that the quantities $m, \\rho $ are shared by all commuting flows.",
"If $m\\ne 0$ , using symmetry (REF ) one can set $m=1$ .",
"Denoting $\\lambda ^0=\\psi $ we obtain $\\lambda ^1=\\psi _1, \\ \\psi _2=\\psi _{11}+\\rho \\psi _1$ .",
"Thus, members of the commuting hierarchy can be parametrised in the form $\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_t=\\left(\\begin{array}{cc}\\psi & \\psi _1 \\\\0&\\psi \\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_x,$ where $\\psi $ solves the Lax equation of the mKP hierarchy, $\\psi _2=\\psi _{11}+\\rho \\psi _1.$ Here the `potential' $\\rho $ is fixed and the solution $\\psi $ of the Lax equation varies.",
"The choice $\\rho =0,\\ \\psi =R^1$ gives rise to the system $\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_t=\\left(\\begin{array}{cc}R^1 & 1 \\\\0&R^1\\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_x$ whose commuting flows are parametrised by solutions to the heat equation [5], [11].",
"Remark 1.",
"Conservation laws of system (REF ) are relations of the form $\\phi _t=g_x$ which hold identically modulo (REF ).",
"This gives $g_1=\\psi \\phi _1, \\quad g_2=\\psi \\phi _2+\\psi _1 \\phi _1,$ and the elimination of $g$ results in the adjoint Lax equation for the conserved density $\\phi $ : $\\phi _2=-\\phi _{11}+\\rho \\phi _1.$"
],
[
"Three-component case",
"Direct calculation shows that the compatibility of three-component systems of the form $\\left(\\begin{array}{c}R^1\\\\ R^2\\\\R^3\\end{array}\\right)_t=\\left(\\begin{array}{ccc}\\lambda ^0 & \\lambda ^1 & \\lambda ^2\\\\0&\\lambda ^0&\\lambda ^1\\\\0&0&\\lambda ^0\\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\\\ R^3\\end{array}\\right)_x, \\qquad \\left(\\begin{array}{c}R^1\\\\ R^2\\\\R^3\\end{array}\\right)_y=\\left(\\begin{array}{ccc}\\mu ^0 & \\mu ^1 & \\mu ^2\\\\0&\\mu ^0&\\mu ^1\\\\0&0&\\mu ^0\\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\\\ R^3\\end{array}\\right)_x$ is equivalent to the following six conditions (we assume $\\lambda ^1\\ne 0$ and $\\mu ^1\\ne 0$ ): $m\\equiv \\frac{\\lambda ^0_{1}}{\\lambda ^{1}} =\\frac{\\mu ^{0}_{1}}{\\mu ^{1}},$ $q\\equiv \\frac{\\lambda ^{1}_{1}-m\\lambda ^{2}}{\\lambda ^{1}} =\\frac{\\mu ^{1}_{1}-m\\mu ^{2}}{\\mu ^{1}},$ $p\\equiv \\frac{\\lambda ^0_{2}-m\\lambda ^{2}}{\\lambda ^{1}} =\\frac{\\mu ^{0}_{2}-m\\mu ^{2}}{\\mu ^{1}},$ $s\\equiv \\frac{\\lambda ^{1}_{2}-\\lambda ^{2}_{1}-p \\lambda ^{2}}{\\lambda ^{1}} =\\frac{\\mu ^{1}_{2}-\\mu ^{2}_{1}-p \\mu ^{2}}{\\mu ^{1}},$ $r\\equiv \\frac{\\lambda ^0_{3}-\\lambda ^{1}_{2}+q \\lambda ^{2}}{\\lambda ^{1}} =\\frac{\\mu ^{0}_{3}-\\mu ^{1}_{2}+q \\mu ^{2}}{\\mu ^{1}},$ $h\\equiv \\frac{\\lambda ^{1}_{3}-\\lambda ^{2}_{2}-(s+r) \\lambda ^{2}}{\\lambda ^{1}} =\\frac{\\mu ^{1}_{3}-\\mu ^{2}_{2}-(s+r) \\mu ^{2}}{\\mu ^{1}}.$ These conditions can be reduced to the mKP equation and its Lax pair as follows.",
"Assuming $m\\ne 0$ and denoting $\\lambda ^0=\\psi $ , from (REF ) and (REF ) one obtains $\\lambda ^{1}=\\frac{\\psi _{1}}{m},\\quad \\lambda ^{2}=\\frac{\\psi _{11}}{m^2}-\\left(\\frac{q}{m^2}+\\frac{m_1}{m^3}\\right) \\psi _{1}.$ Inserting the above into (REF ) yields $\\psi _{2}=\\frac{\\psi _{11}}{m}+\\left(\\frac{p-q}{m}-\\frac{m_1}{m^2}\\right) \\psi _{1},$ so that (REF ) gives $\\frac{m_1}{m}\\psi _{11}-\\left[sm-pq-p_1+m_2+\\frac{m_1}{m}q+\\left(\\frac{m_1}{m}\\right)^2\\right]\\psi _1=0.$ This implies $m_1=0, \\quad s=\\frac{1}{m}(q p+p_{1}-m_2).$ As $m\\ne 0$ , modulo transformations (REF ) one can set $m=1,\\ s=q p+p_{1},$ so that the formulae for $\\lambda ^0, \\lambda ^1, \\lambda ^2$ simplify to $\\lambda ^0=\\psi , \\ \\lambda ^1=\\psi _1, \\ \\lambda ^2=\\psi _{11}-q\\psi _1$ where $\\qquad \\psi _2=\\psi _{11}+(p-q)\\psi _1.$ Then (REF ) yields $\\psi _{3}={\\psi _{111}}+{(p-2q)}\\psi _{11}+\\big [q^2+r+(p-q)_1\\big ]\\psi _{1}.$ Finally, it follows from (REF ) that $(p+q)_{1}{ \\psi _{11}}-\\left(q^2p-h+rq+ q_{2}+q q_{1} +2 q p_{1}+ r_{1}\\right) \\psi _{1}=0,$ which allows us to set $q=-p,\\quad h=(p)^3-rp- p_{2}-p p_{1}+ r_{1}.$ The compatibility of (REF ) and (REF ), i.e., the condition $\\psi _{23}=\\psi _{32}$ , leads to $\\big [p_{11}-2pp_{1}+2r_{1}-3p_{2}\\big ]\\psi _{11}=\\big [2p_{1}^2+2(r-p^2)p_{1}-2p r_{1}+2pp_{2}-r_{11}+r_{2}+2 p_{12}-2p_{3}\\big ]\\psi _{1},$ from which we obtain the following two equations: $p_{11}-2pp_{1}+2r_{1}-3p_{2}=0,$ $2p_{1}^2+2(r-p^2)p_{1}-2p r_{1}+2pp_{2}-r_{11}+r_{2}+2 p_{12}-2p_{3}=0.$ In order to solve (REF ) for $r$ we introduce the potential variable $w$ such that $p= w_{1}$ .",
"Then integrating (REF ) gives $r=\\frac{1}{2}w_1^2-\\frac{1}{2} w_{11}+\\frac{3}{2}w_{2}.$ Finally from (REF ) we obtain $4 w_{13}+6 w_1^2 w_{11}-w_{1111}-3w_{22}-6 w_{2}w_{11}=0,$ which is the potential mKP equation.",
"Note that the mKP hierarchy was introduced in [7], [4].",
"The generalised Miura transformation connecting KP and mKP equations was constructed in [7], see also [8] for the first classification results of integrable equations in 2+1 dimensions.",
"Exact solutions of mKP equation were constructed in [9], [2], [3], see also references therein.",
"To summarise, members of the three-component commuting hierarchy can be parametrised in the form $\\left(\\begin{array}{c}R^1\\\\ R^2\\\\R^3\\end{array}\\right)_t=\\left(\\begin{array}{ccc}\\psi & \\psi _1 & \\psi _{11}+w_1\\psi _1\\\\0&\\psi &\\psi _1\\\\0&0&\\psi \\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\\\ R^3\\end{array}\\right)_x$ where $w$ satisfies the mKP equation (REF ) and $\\psi $ solves the corresponding Lax equations (REF ), (REF ): $\\psi _{2}={\\psi _{11}}+2w_1\\psi _{1},\\quad \\psi _{3}={\\psi _{111}}+{3w_1}\\psi _{11}+\\frac{3}{2}(w_{2}+w_{11}+w_1^2)\\psi _{1}.$ Fixing $w$ and varying $\\psi $ we obtain commuting flows of the hierarchy.",
"Example.",
"Set $w=0$ , then equations for $\\psi $ and $\\lambda ^0, \\ \\lambda ^1, \\ \\lambda ^2$ become $\\psi _{2}={\\psi _{11}},\\quad \\psi _{3}={\\psi _{111}},\\qquad \\lambda ^0=\\psi , \\quad \\lambda ^{1}={\\psi _{1}},\\quad \\lambda ^{2}={\\psi _{11}}.$ We can choose $\\psi =R^{1},\\quad \\lambda ^0=R^1, \\quad \\lambda ^{1}=1,\\quad \\lambda ^{2}=0,$ or $\\psi =e^{k R^{1}+k^2 R^{2}+k^3 R^{3}},\\quad \\lambda ^0=\\psi , \\quad \\lambda ^{1}=k \\psi ,\\quad \\lambda ^{2}=k^2 \\psi ,$ where $k$ is an arbitrary constant (the former solution was considered in [11]).",
"Remark 2.",
"Conservation laws of system (REF ) are relations of the form $\\phi _t=g_x$ which hold identically modulo (REF ).",
"This gives $g_1=\\psi \\phi _1, \\quad g_2=\\psi \\phi _2+\\psi _1 \\phi _1,\\quad g_3=\\psi \\phi _3+\\psi _1 \\phi _2+(\\psi _{11}+w_1\\psi _1)\\phi _{1}.$ The elimination of $g$ results in the adjoint Lax equations for the conserved density $\\phi $ : $\\phi _{2}=-{\\phi _{11}}+2w_1\\phi _{1},\\quad \\phi _{3}={\\phi _{111}}-{3w_1}\\phi _{11}+\\frac{3}{2}(w_{2}-w_{11}+w_1^2)\\phi _{1}.$"
],
[
"Four-component case",
"Omitting details of calculations we present the final result: four-component commuting flows of Toeplitz type (REF ) can be parametrised in the form ${\\small \\left(\\begin{array}{c}R^1\\\\ R^2\\\\R^3\\\\R^4\\end{array}\\right)_t=\\left(\\begin{array}{cccc}\\psi & \\psi _1 & \\psi _{11}+w_1\\psi _1 & \\psi _{111}+ 2w_{1}\\psi _{11}+ \\frac{1}{2}(w_{2}+3w_{11}+w_{1}^2)\\psi _1\\\\0&\\psi &\\psi _1&\\psi _{11}+w_1\\psi _1\\\\0&0&\\psi &\\psi _1\\\\0&0&0&\\psi \\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\\\ R^3\\\\R^4\\end{array}\\right)_x}$ where $w$ solves the first three equations of the mKP hierarchy, $4 w_{13}+6 w_1^2 w_{11}-w_{1111}-3w_{22}-6 w_{2}w_{11}=0,$ $2 w_{23} = 3 w_{14}-w_{1112}+(3 w_{1}^2-3 w_{2}) w_{12}+6 w_{1} w_{11} w_{2}+(w_{111}-4 w_{3}-2 w_{1}^3) w_{11},$ $w_{24} = \\frac{8}{9} w_{33}-\\frac{1}{9} w_{111111}+\\frac{2}{9} w_{1113}-2 w_{1} w_{2} w_{12}+4 w_{111} w_{1} w_{11}+(w_{1}^2-w_{2}) w_{1111}$ $\\quad \\quad +\\frac{2}{3}( w_{1}^3+2 w_{3}-2 w_{111}) w_{12}+\\frac{4}{3} w_{11}^3+(4 w_{1}^2 w_{2}-2 w_{1}^4-2 w_{2}^2-w_{112}) w_{11},$ and $\\psi $ satisfies the corresponding Lax equations: $\\begin{array}{c}\\psi _{2}={\\psi _{11}}+2w_1\\psi _{1},\\quad \\psi _{3}={\\psi _{111}}+{3w_1}\\psi _{11}+\\frac{3}{2}(w_{2}+w_{11}+w_1^2)\\psi _{1},\\\\\\ \\\\\\psi _{4}={\\psi _{1111}}+{4w_1}\\psi _{111}+(2w_2+4w_{11}+4w_1^2)\\psi _{11}+\\Delta \\psi _{1};\\end{array}$ here $\\Delta =\\frac{4}{3} w_{3}+\\frac{5}{3}w_{111}+\\frac{2}{3}w_{1}^3+4w_{1}w_{11}+w_{12}+2 w_{1} w_{2}.$ Fixing $w$ and varying $\\psi $ we obtain commuting flows of the hierarchy.",
"Remark 3.",
"For the matrix elements of (REF ), $\\lambda ^0=\\psi , \\quad \\lambda ^1=\\psi _1, \\quad \\lambda ^2=\\psi _{11}+w_1\\psi _1, \\quad \\lambda ^3=\\psi _{111}+ 2w_{1}\\psi _{11}+ \\frac{1}{2}(w_{1}^2+w_{2}+3w_{11})\\psi _1,$ we have simple recursive formulae: $\\lambda ^1=\\psi _1, \\quad \\lambda ^2+w_1\\lambda ^1=\\psi _2, \\quad \\lambda ^3+w_1\\lambda ^2+w_2\\lambda ^1=\\psi _3.$ This recurrence generalises to the general $n$ -component case: $\\lambda ^k+w_1\\lambda ^{k-1}+w_2\\lambda ^{k-2}+\\dots +w_{k-1}\\lambda ^1=\\psi _k, \\quad 1\\le k\\le n-1.$"
],
[
"Generalised hodograph method",
"Solutions to system (REF ) can be obtained by the following recipe which is analogous to the generalised hodograph method of Tsarev [15].",
"Let $R_y=(\\mu ^0E+\\sum _{i=1}^{n-1}\\mu ^iP^i)R_x$ be a commuting flow of system (REF ).",
"Then the matrix equation $\\mu ^0E+\\sum _{i=1}^{n-1}\\mu ^iP^i=Ex+(\\lambda ^0E+\\sum _{i=1}^{n-1}\\lambda ^iP^i)t$ defines an implicit solution of (REF ).",
"In components, this is equivalent to $n$ implicit relations $\\mu ^0=x+\\lambda ^0t, \\quad \\mu ^1=\\lambda ^1t, \\quad \\dots ,\\quad \\mu ^n=\\lambda ^n t.$"
],
[
"Jordan type reductions of linearly degenerate PDEs",
"It is remarkable that, although for `strongly nonlinear' PDEs such as the dispersionless KP/Toda equations, the Jordan type reductions do not occur, they naturally arise in the context of multi-dimensional linearly degenerate PDEs (such as linearly degenerate systems of hydrodynamic type, Monge-Ampére equations, etc).",
"Below we illustrate this phenomenon for the 3D Mikhalev system [10]."
],
[
"3D Mikhalev system",
"Here we consider the system $u_t=v_y+uv_x-vu_x, \\qquad v_x=u_y.$ Two-component hydrodynamic reductions (of Jordan block type) of system (REF ) are exact solutions of the form $u=u(R^1, R^2), \\quad v=v(R^1, R^2)$ where the variables $R^1, R^2$ satisfy a pair of commuting $2\\times 2$ systems (REF ): $\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_t=\\left(\\begin{array}{cc}\\psi & \\psi _{1} \\\\0&\\psi \\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_x, \\qquad \\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_y=\\left(\\begin{array}{cc}\\varphi & \\varphi _{1} \\\\0&\\varphi \\end{array}\\right)\\left(\\begin{array}{c}R^1\\\\ R^2\\end{array}\\right)_x,$ with $\\psi _2=\\psi _{11}+\\rho \\psi _1,\\quad \\varphi _2=\\varphi _{11}+\\rho \\varphi _1.$ All such reductions can be described explicitly.",
"Direct calculation shows that $u$ and $v$ must be polynomial in $R^1$ of degree 2 and 4, respectively: $u=-\\frac{1}{4}a^{\\prime } (R^{1})^2+ bR^1+c, \\quad v=-\\frac{1}{2}u^2+au +d,$ where the coefficients $a, b, c, d$ are functions of $R^2$ satisfying a single relation $d^{\\prime }+b^2+ca^{\\prime }=0$ (prime denotes differentiation by $R^2$ ).",
"The functions $\\varphi , \\psi $ and $\\rho $ are expressed in terms of $u$ by the formulae $\\varphi =-u+a,\\quad \\psi =-v+a\\varphi , \\quad \\rho =\\frac{2 u_{2}-a^{\\prime }}{2u_{1}}.$ A particular choice $a=0, \\ b=-1,\\ c=0, \\ d=-R^2$ leads to $u=-R^1, \\quad v=-R^2-\\frac{1}{2}(R^1)^2, \\quad \\varphi =R^1, \\quad \\psi =R^2+\\frac{1}{2}(R^1)^2,\\quad \\rho =0,$ the case considered in [11].",
"Three component reductions of Jordan block type can be sought in the form $\\displaystyle {R_t=\\left(\\begin{array}{ccc}\\psi & \\psi _1 & \\psi _{11}+{w}_1\\psi _1\\\\0&\\psi &\\psi _1\\\\0&0&\\psi \\end{array}\\right)R_{x},\\qquad R_{y}=\\left(\\begin{array}{ccc}\\varphi & \\varphi _1 & \\varphi _{11}+{w}_1\\varphi _1\\\\0&\\varphi &\\varphi _1\\\\0&0&\\varphi \\end{array}\\right)R_{x}},$ where ${w}$ satisfies the mKP equation (REF ) and $\\psi $ , $\\varphi $ are two solutions of the corresponding Lax equations (REF ).",
"In this case the formulae become more complicated.",
"Direct calculation shows that $u$ and $v$ must be polynomial in $R^1, R^2$ : $u = b R^1-\\frac{4}{27}a^{\\prime \\prime }(R^2)^3+\\frac{2}{3}\\gamma ^{\\prime }(R^2)^2+\\alpha R^2+\\beta ,$ $v = -\\frac{1}{2} u^2+ au-\\frac{4}{27} a^{\\prime 2} (R^2)^3-b\\gamma R^2+\\zeta ,\\quad $ where $b=\\gamma -\\frac{2}{3} a^{\\prime } R^2$ and $a$ , $\\alpha $ , $\\beta $ , $\\gamma $ , $\\zeta $ are functions of $R^3$ with the condition $\\zeta ^{\\prime } +a^{\\prime }\\beta +\\frac{3}{2}\\alpha \\gamma =0.$ The functions $\\varphi , \\psi $ and $w$ are as follows: $\\varphi =-u+a,\\quad \\psi =-v+a\\varphi ,$ $w = -\\frac{a^{\\prime }}{ 6b}(R^1)^2-\\frac{1}{ 18b}R^1\\left(4a^{\\prime \\prime }(R^2)^2-12\\gamma ^{\\prime }R^2-9\\alpha \\right)+f,$ where $f$ is a function of $R^2$ and $R^3$ which satisfies the equation $\\frac{1}{2} (3 \\gamma -2 a^{\\prime } R^2)^2 f_{2}=x^4 (R^2)^4+x^3 (R^2)^3+x^2 (R^2)^2+x^1 (R^2)+3x^0.$ Here the coefficients $x^i$ are functions of $R^3$ defined as $x^4\\equiv \\frac{8}{ 27} a^{\\prime \\prime \\prime } a^{\\prime }- \\frac{2}{9} a^{\\prime \\prime 2},\\quad x^3\\equiv \\frac{4}{ 3} a^{\\prime \\prime }\\gamma ^{\\prime }-\\frac{4}{ 9} a^{\\prime \\prime \\prime } \\gamma -\\frac{4}{ 3}\\gamma ^{\\prime \\prime } a^{\\prime },$ $x^2\\equiv \\alpha a^{\\prime \\prime } -2 \\alpha ^{\\prime } a^{\\prime }+2 \\gamma ^{\\prime \\prime } \\gamma -2\\gamma ^{\\prime 2},\\quad x^1\\equiv 3 \\alpha ^{\\prime } \\gamma -2 \\beta ^{\\prime } a^{\\prime }-3 \\alpha \\gamma ^{\\prime }+a^{\\prime 2},\\quad x^0\\equiv \\beta ^{\\prime } \\gamma - \\frac{1}{ 2} a^{\\prime } \\gamma -\\frac{3}{8} \\alpha ^2.$ If $a^{\\prime }=0$ , then (REF ) gives $f =\\frac{4}{27}(R^2)^3 \\left(\\frac{\\gamma ^{\\prime }}{\\gamma } \\right)^{\\prime }+\\frac{1}{3} (R^2)^2\\left( \\frac{\\alpha }{\\gamma } \\right)^{\\prime }+R^2\\left(\\frac{2\\beta ^{\\prime }}{3\\gamma } -\\frac{\\alpha ^2}{4\\gamma ^2} \\right) +\\eta ,$ otherwise, $f= \\frac{(R^2)^3}{6a^{\\prime 2}} x^{4}+\\frac{(R^2)^2}{ 4a^{\\prime 3}} (a^{\\prime } x^{3}+3 \\gamma x^{4})+\\frac{R^2}{ 8 a^{\\prime 4}} (4 a^{\\prime 2} x^{2}+12 a^{\\prime } \\gamma x^{3}+27 \\gamma ^2 x^{4})$ $+\\frac{1}{ 16 a^{\\prime 5} b} (16 a^{\\prime 4} x^{0}+8 a^{\\prime 3} \\gamma x^{1}+12 a^{\\prime 2} \\gamma ^2 x^{2}+18 a^{\\prime } \\gamma ^3 x^{3}+27 \\gamma ^4 x^{4})$ $+\\frac{\\ln {b}}{ 8 a^{\\prime 5}} (4 a^{\\prime 3} x^{1}+12 a^{\\prime 2} \\gamma x^{2}+27 a^{\\prime } \\gamma ^2 x^{3}+54 \\gamma ^3 x^{4})+\\eta ;$ here $\\eta $ is an extra arbitrary function of $R^3$ ."
],
[
"Acknowledgements",
"We thank Maxim Pavlov and Vladimir Novikov for useful comments.",
"EVF also thanks Alexey Bolsinov and David Calderbank for a discussion on quasilinear systems of Jordan block type and their role in the general method of hydrodynamic reductions.",
"LX was supported by the National Natural Science Foundation of China (Grant No.",
"11501312) and the K.C.",
"Wong Magna Fund in Ningbo University.",
"LX also thanks Loughborough University for a kind hospitality."
]
] | 2001.03601 |
[
[
"Learning credit assignment"
],
[
"Abstract Deep learning has achieved impressive prediction accuracies in a variety of scientific and industrial domains.",
"However, the nested non-linear feature of deep learning makes the learning highly non-transparent, i.e., it is still unknown how the learning coordinates a huge number of parameters to achieve a decision making.",
"To explain this hierarchical credit assignment, we propose a mean-field learning model by assuming that an ensemble of sub-networks, rather than a single network, are trained for a classification task.",
"Surprisingly, our model reveals that apart from some deterministic synaptic weights connecting two neurons at neighboring layers, there exist a large number of connections that can be absent, and other connections can allow for a broad distribution of their weight values.",
"Therefore, synaptic connections can be classified into three categories: very important ones, unimportant ones, and those of variability that may partially encode nuisance factors.",
"Therefore, our model learns the credit assignment leading to the decision, and predicts an ensemble of sub-networks that can accomplish the same task, thereby providing insights toward understanding the macroscopic behavior of deep learning through the lens of distinct roles of synaptic weights."
],
[
"Learning credit assignment Chan Li Haiping Huang PMI Lab, School of Physics, Sun Yat-sen University, Guangzhou 510275, People's Republic of China Deep learning has achieved impressive prediction accuracies in a variety of scientific and industrial domains.",
"However, the nested non-linear feature of deep learning makes the learning highly non-transparent, i.e., it is still unknown how the learning coordinates a huge number of parameters to achieve a decision making.",
"To explain this hierarchical credit assignment, we propose a mean-field learning model by assuming that an ensemble of sub-networks, rather than a single network, are trained for a classification task.",
"Surprisingly, our model reveals that apart from some deterministic synaptic weights connecting two neurons at neighboring layers, there exist a large number of connections that can be absent, and other connections can allow for a broad distribution of their weight values.",
"Therefore, synaptic connections can be classified into three categories: very important ones, unimportant ones, and those of variability that may partially encode nuisance factors.",
"Therefore, our model learns the credit assignment leading to the decision, and predicts an ensemble of sub-networks that can accomplish the same task, thereby providing insights toward understanding the macroscopic behavior of deep learning through the lens of distinct roles of synaptic weights.",
"Introduction.— As deep neural networks become an increasingly important tool in diverse domains of scientific and engineering applications [1], [2], [3], [4], [5], the black box properties of the tool turn out to be a challenging obstacle puzzling researchers in the field [6].",
"In other words, the decision making behavior of a network output can not be easily understood in terms of interactions among building components of the network.",
"This shares the same spirit as another long-standing puzzle in theory of the brain—how the emergent behavior of a neuronal population hierarchy can be traced back to its elements [6], [5].",
"This challenging issue is the well-known credit assignment problem (CAP), determining how much credit a given component (either neuron or connection) should take for a particular behavior output [5].",
"To solve this problem, one needs to bridge the gap between microscopic interactions of components and macroscopic behavior.",
"Excitingly, recent works showed that there exist sub-networks of random weights that are able to produce better-than-chance accuracies [7], [8], [9].",
"This property seems to be universal across different architectures, datasets and computational tasks [10].",
"Even one can start with no connections and add complexity as needed by assuming a single shared weight parameter [11].",
"Moreover, it was recently revealed that the innate template of face-selective neurons can spontaneously emerge from sufficient statistical variations present in the random initial wirings of neural circuits, while the template may be fine-tuned during early visual experiences [12].",
"Therefore, from both artificial and biological neural networks perspectives, exploring how and why these random wirings exist will definitely provide us a powerful lens through which we can better understand and even further improve the computational capacities of deep neural networks.",
"Here, we propose a statistical model of learning credit assignment from training data of a computational task.",
"According to the model, we search for an optimal random network ensemble as an inductive bias about the hypothesis space [6].",
"The hypothesis space is composed of all candidate networks with different assignments of weight values that accomplish the computation task.",
"Consistent with previous studies [7], [8], [9], the optimal ensemble contains sub-networks of the original full network, which further allows for capturing uncertainty in the hypothesis space.",
"The model can be solved by mean-field methods, thereby providing a physics interpretation of how credit assignment occurs in a hierarchical deep neural system.",
"Figure: The schematic illustration of the model learning credit assignment.A deep neural network of four layers including two hidden layers is used to recognize a handwritten digit, say zero,with the softmax output indicating the probability of the categorization.",
"Each connection is specified by a spike and slab distribution, wherethe spike indicates the probability of the absence of this connection, and the slab is modeled by a Gaussian distribution of weight values as pictorially shown onlyon strong connections withdifferent means and variances.",
"Other weak connections indicate nearly unit spike probabilities, although they also carry a slabdistribution (not shown in the illustration for simplicity).Model.— To learn credit assignment, we search for an optimal random neural network ensemble to accomplish a classification task of handwritten digits [13].",
"More precisely, we design a deep neural network of $L$ layers, including $L-2$ hidden layers.",
"The depth of the network $L$ can be made arbitrarily large.",
"The size (width) of the $l$ -th layer is denoted by $N_l$ , therefore $N_1$ is determined by the total number of pixels in an input image, and $N_L$ is the number of classes (e.g., 10 for the handwritten digit dataset).",
"The weight value of the connection from neuron $i$ at the upstream layer $l$ to neuron $k$ at the downstream layer $l+1$ is defined by $w_{ik}^l$ , and the activation of the neuron $k$ at the $(l+1)$ -th layer $h_k^{l+1}$ is a non-linear function of the pre-activation $z_k^{l+1}=\\frac{1}{\\sqrt{N_l}}\\sum _iw_{ik}^lh_i^{l}$ , where the weight scaling factor $1/\\sqrt{N_l}$ ensures that the weighted sum is independent of the upstream layer width.",
"We use the rectified linear unit (ReLU) function [14] as the transfer function from pre-activation $z$ to activation $h$ defined as $h={\\rm max}(0,z)$ .",
"The output transfer function specifies a probability over all classes of the input image by using a softmax function $h_k=\\frac{e^{z_k}}{\\sum _ie^{z_i}}$ where $z_i$ is the pre-activation of the $i$ -th neuron at the output layer.",
"For the categorization task, we define $\\hat{h}_i$ as the target label (one-hot representation), and use the cross entropy $\\mathcal {C}=-\\sum _i\\hat{h}_i\\ln h_i$ as the objective function to be minimized.",
"Training the neural network corresponds to adjusting all connection weights to minimize the cross entropy until the network is able to classify unseen handwritten digits with a satisfied accuracy (so-called generalization ability).",
"Therefore, after the network is trained with a training data size of $T$ , the network's generalization ability is verified with a test data size of $V$ .",
"Remarkably, the state-of-the-art test accuracy is able to surpass the human performance in some complex tasks [3].",
"However, the decision making behavior of the output neurons in a deep network is still challenging to understand in terms of computational principles of single building components (either neurons or weights).",
"Recent empirical machine learning works showed that a subnetwork of random weights can produce a better-than-chance accuracy [7], [9], [8].",
"This clearly suggests that there may exist a random ensemble of neural networks that fulfill the computational task given the width and depth of the deep network.",
"This ensemble may occupy a tiny portion of the entire model space.",
"Therefore, a naive random initialization of the neural network can only yield a chance-level accuracy.",
"To incorporate all these challenging issues into a theoretical model, we propose to model the weight by a spike and slab (SaS) distribution as follows: $P(w_{ik}^l)=\\pi _{ik}^l\\delta (w_{ik}^l)+(1-\\pi _{ik})\\mathcal {N}(w_{ik}^l|m_{ik}^l,\\Xi _{ik}^l),$ where the discrete probability mass at zero defines the spike, and the slab is characterized by a Gaussian distribution with mean $m_{ik}^l$ and variance $\\Xi _{ik}^l$ over a continuous domain (see Fig.",
"REF for an illustration).",
"The SaS distribution has been widely used in statistics literatures [15], [16].",
"Here, the spike and slab have respectively their own physics interpretations in learning credit assignment of neural networks.",
"The spike is intimately related to the concept of network compression [17], [18], [10], where not all resources of connections are used in a task.",
"This parameter allows to identify very important weights, and further evaluate remaining capacities for learning new tasks [19].",
"A recent physics study has already showed that a deep neural network can be robust against connection removals [20].",
"The slab continuous support characterizes the ensemble of neural networks with random weights producing better-than-chance accuracies.",
"Among these weights, some are very important, indicated by a vanishing spike probability mass, and could thus explain the decision making of the output neurons, while the variance of the corresponding Gaussian distribution captures the uncertainty of the decision making solutions [21].",
"Therefore, the inductive bias of connections and their associated weights can be learned through the SaS model of credit assignment.",
"Note that the Gaussian slab is not used here as an additional regularization complexity term in the objective function [22].",
"Instead, the continuous slab is combined coherently with the spike probability to model the uncertainty of weights facing noisy sensory inputs (Fig.",
"REF ).",
"Next, we derive a mean-field method to learn the SaS parameters $\\mathbf {\\theta }_{ik}^l\\equiv (\\pi _{ik}^l,m_{ik}^l,\\Xi _{ik}^l)$ for all layers.",
"The first and second moments of the weight $w_{ik}^l$ are given by $\\mu _{ik}^l\\equiv \\mathbb {E}[w_{ik}^l]=m_{ik}^l(1-\\pi _{ik}^l)$ and $\\varrho _{ik}^l\\equiv \\mathbb {E}[(w_{ik}^l)^2]=(1-\\pi _{ik}^l)[\\Xi _{ik}^l+(m_{ik}^l)^2]$ , respectively.",
"Given a large width of the layer, the central-limit-theorem implies that the pre-activation follows approximately a Gaussian distribution $\\mathcal {N}(z_i^l|G_i^l,(\\Delta _i^l)^2)$ , where the mean and variance are given respectively by $G_i^l&=\\frac{1}{\\sqrt{N_{l-1}}}\\sum _{k}\\mu _{ki}^{l-1}h_k^{l-1},\\\\(\\Delta _i^l)^2&=\\frac{1}{N_{l-1}}\\sum _{k}(\\varrho _{ki}^{l-1}-(\\mu _{ki}^{l-1})^2)(h_k^{l-1})^2.$ Then, the feedforward transformation of the input signal can be re-parametrized by [23], [24] $z_i^l&=G_i^l+\\epsilon _i^l\\Delta _i^l,\\\\h_i^l&={\\rm ReLU}(z_i^l),$ for $l<L$ .",
"The last layer uses the softmax function.",
"$\\epsilon _i^l$ is a layer- and component-dependent standard Gaussian random variable with zero mean and unit variance.",
"$\\epsilon ^l$ is quenched for every single training mini-epoch and the same value is used in both forward and backward computations.",
"This reparametrization retains the statistical structure in Eq.",
"(REF ).",
"The objective function relies on $(\\mathbf {\\mu },\\mathbf {\\varrho })$ that can be preserved by a transformation of $\\mathbf {\\theta }$ .",
"Hence, we use a regularization strength to control the $\\ell _2$ -norm of $\\mathbf {m}$ and $\\mathbf {\\Xi }$ .",
"In addition, the transformation does not change the fraction of $\\pi =1$ or 0 [25], maintaining the qualitative behavior of the model.",
"Learning of the hyper-parameter $\\mathbf {\\theta }_{ik}^l$ can be achieved by a gradient descent of the objective function, i.e., $\\Delta \\mathbf {\\theta }_{ki}^l=-\\eta \\mathcal {K}_i^{l+1}\\frac{\\partial z_i^{l+1}}{\\partial \\mathbf {\\theta }_{ki}^l}$ where $\\eta $ denotes the learning rate, and $\\mathcal {K}_{i}^{l+1}\\equiv \\frac{\\partial \\mathcal {C}}{\\partial z_i^{l+1}}$ .",
"The entire dataset is divided into minibatches over which the gradients are evaluated.",
"Note that unlike the standard back-propagation (BP) [26], we adapt the SaS distribution rather than a particular weight, in accord with our motivation of learning statistical features of the hypothesis space for a particular computation task (e.g., image classification here).",
"On the top layer, $\\mathcal {K}_i^L$ can be directly estimated as $\\mathcal {K}_i^L=-\\hat{h}_i^{L}(1-h_i^L)$ by definition.",
"For lower layers, $\\mathcal {K}_i^l$ can be estimated by using the chain rule, resulting in a back-propagation equation of the error signal from the top layer: $\\mathcal {K}_i^l&=\\delta _i^lf^{\\prime }(z_i^l),\\\\\\delta _i^l&=\\sum _{k}\\mathcal {K}_k^{l+1}\\frac{\\partial z_k^{l+1}}{\\partial h_i^l},$ where $\\delta _i^l\\equiv \\frac{\\partial \\mathcal {C}}{\\partial h_i^l}$ , and $f^{\\prime }()$ denotes the derivative of the transfer function.",
"To proceed, we have to compute $\\frac{\\partial z_k^{l+1}}{\\partial h_i^l}$ and $\\frac{\\partial z_i^{l+1}}{\\partial \\mathbf {\\theta }_{ki}^l}$ .",
"The first derivative characterizes how sensitive the pre-activation is under the change of the input activity of one neuron, and this response is computed as follows: $\\frac{\\partial z_k^{l+1}}{\\partial h_i^l}=\\frac{\\mu _{ik}^l}{\\sqrt{N_l}}+\\frac{(\\varrho _{ik}^l-(\\mu _{ik}^l)^2)h_i^l}{N_l\\Delta _k^{l+1}}\\epsilon _k^{l+1}.$ The second derivative characterizes how sensitive the pre-activation is under the change of the hyper-parameters of the SaS distribution.",
"Because the mean and variance, $G_i^{l+1}$ and $\\Delta _i^{l+1}$ , is a function of the hyper-parameters, the second derivative for each hyper-parameter can be derived similarly as follows: $\\frac{\\partial z_i^{l+1}}{\\partial m_{ki}^{l}}&=\\frac{(1-\\pi _{ki}^{l})h_k^{l}}{\\sqrt{N_l}}+\\frac{\\mu _{ki}^{l}\\pi _{ki}^{l}}{N_l\\Delta _{i}^{l+1}}(h_k^{l})^2\\epsilon _i^{l+1},\\\\\\frac{\\partial z_i^{l+1}}{\\partial \\pi _{ki}^{l}}&=-\\frac{m_{ki}^{l}h_k^{l}}{\\sqrt{N_l}}-\\frac{(2\\pi _{ki}^{l}-1)(m_{ki}^{l})^2+\\Xi _{ki}^{l}}{2N_l\\Delta _{i}^{l+1}}(h_k^{l})^2\\epsilon _i^{l+1},\\\\\\frac{\\partial z_i^{l+1}}{\\partial \\Xi _{ki}^{l}}&=\\frac{1-\\pi _{ki}^{l}}{2N_l\\Delta _{i}^{l+1}}(h_k^{l})^2\\epsilon _i^{l+1}.$ Eq.",
"(REF ) and Eq.",
"(REF ) share the same form with the pre-activation $z_i^l$ [Eq.",
"(REF )], due to the reparametrization trick used to handle the uncertainty of weights in the hypothesis space.",
"Therefore, the learning process of our model naturally captures the fluctuation of the hypothesis space, highlighting the significant difference from the standard BP which computes only a point estimate of the connection weights.",
"In particular, if we enforce $\\mathbf {\\pi }=0$ and $\\mathbf {\\Xi }=0$ , $\\mathbf {m}$ becomes identical to the weight configuration, thus our learning equations will immediately recover the standard BP algorithm [26].",
"Therefore, our learning protocol can be thought of as a generalized back-propagation (gBP) at the weight distribution level, or the candidate-network ensemble level.",
"Our framework is a cheap way to compute the posterior distribution of the weights given the data, which can be alternatively realized by a Bayesian inference where a variational free energy is commonly optimized through Monte-Carlo samplings [27], [24], [28].",
"The optimization of the variational free energy for deep neural networks is computationally challenging, especially for the SaS prior whose entropy has no analytic form as well.",
"In contrast, gBP stores two times more parameters than BP, while other computational steps are exactly the same; the training time is affordable, as a typical deep network training scales linearly with the number of parameters [29].",
"We remark that during learning, the spike mass $\\pi $ should be clipped as ${\\rm max}(0,{\\rm min}(1,\\pi ))$ , and the variance $\\Xi \\leftarrow {\\rm max}(0,\\Xi )$ .",
"Learning the SaS model allows for credit assignment to each connection, considering a network ensemble realizing the same task.",
"An effective network can be constructed by sampling the learned SaS distribution.",
"Figure: Properties of gBP in testing performances on unseen data.",
"(a) Training trajectories of a network architecture of three or four layers.",
"The architecture is defined as 784-100-100-10 (four layers) or 784-100-10 (three layers), where each number indicatesthe corresponding layer width.",
"Networks are trained on the data size of T=10 4 T=10^4 images and tested on another unseen-data size of V=10 4 V=10^4 images.",
"The fluctuation is computed from five independent runs.",
"(b) Test errors of gBP versus training data size.",
"The error bar characterizes the fluctuation across ten independently-sampled network architectures from the SaS distribution.",
"The samenetwork of four layers as in (a) is used.",
"The inset shows the sparsity per connection as a function of layers, for which networks of five layers are used and the hidden layer width is still 100nodes.",
"Each marker is an average over ten independent runs.",
"k in the inset means the training data size in the unit of 10 3 10^3.The size of the hypothesis space can be approximated by the model entropy $S=-\\int _{\\mathbb {R}^{\\mathcal {D}}}P(\\mathbf {w})\\ln P(\\mathbf {w})d\\mathbf {w}$ , where $\\mathcal {D}$ is the number of weight parameters in the network.",
"Because the joint distribution of weights is assumed to be factorized across individual connections, $S=\\sum _{\\ell }S_{\\ell }$ , where the entropy contribution of each individual connection ($\\ell $ ) is expressed as [25] $S_{\\ell }=-\\pi _{\\ell }\\ln \\Biggl [\\pi _{\\ell }+(1-\\pi _{\\ell })\\mathcal {N}(0|m_{\\ell },\\Xi _{\\ell })\\Biggr ]-\\frac{(1-\\pi _{\\ell })}{\\mathcal {B}}\\sum _s\\Gamma (\\epsilon _s),$ where $\\mathcal {B}$ denotes the number of standard Gaussian random variables $\\epsilon _s$ , and $\\Gamma (\\epsilon _s)=\\ln \\Bigl [\\pi _{\\ell }\\delta (m_{\\ell }+\\sqrt{\\Xi _{\\ell }}\\epsilon _s)+\\frac{(1-\\pi _{\\ell })}{\\sqrt{\\Xi _{\\ell }}}\\mathcal {N}(\\epsilon _s|0,1)\\Bigr ]$ .",
"If $\\pi _{\\ell }=0$ , the entropy $S_{\\ell }$ can be analytically computed as $\\frac{1}{2}\\ln (2\\pi e\\Xi _{\\ell })$ .",
"If $\\Xi _{\\ell }=0$ , the Gaussian distribution reduces to a Dirac delta function, and the entropy becomes an entropy of discrete random variables.",
"Results.— Despite working at the synaptic weight distribution level, gBP can reach a similar even better test accuracy than that of BP [Fig.",
"REF (a)].",
"As expected, the test error for gBP decreases with training data size [Fig.",
"REF (b)].",
"The error bar implies that effective networks sampled from the learned SaS distribution still yield accurate predictions, confirming the network ensemble assumption.",
"Our theory also reveals that the sparsity, obtained from the statistics of $\\lbrace \\pi _{ij}^l\\rbrace $ , grows first and then decreases [the inset of Fig.",
"REF (b)], suggesting that the actual working network does not used up all synaptic connections, consistent with empirical observations of network compression [17], [30], [10] and theoretical studies of toy models [20], [31].",
"Along hierarchical stages of the deep neural network, the initial stage is responsible for encoding, therefore the sparsity must be low to ensure that there is sufficient space for encoding the important information, while at the middle stage, the encoded information is re-coded through hidden representations, suggesting that the noisy information is further distilled, which may explain why the sparsity goes up.",
"Finally, to guarantee the feature-selective information being extracted, the sparsity must drop to yield an accurate classification.",
"Therefore, our model can interpret the deep learning as an encoding-recoding-decoding process, as also argued in a biological neural hierarchy [32].",
"To inspect more carefully what distribution of hyper-parameters gBP learned, we plot Fig.",
"REF (a) showing the evolution of the distribution across the hierarchical stages.",
"First, the spike mass $\\mathbf {\\pi }$ has a U-shaped distribution.",
"One extreme is at $\\pi =0$ , suggesting that the corresponding connection carries feature-selective information and thus can not be pruned, while the other extreme is at $\\pi =1$ , suggesting that the corresponding connection can be completely pruned.",
"Apart from these two extremes, there exist a relatively small number of connections which can be present or absent with certain probabilities.",
"These connections may reflect nuisance factors in sensory inputs [33].",
"The mean of the slab distribution has a relatively broad distribution, but the peak is located around zero; an L-shaped distribution of the variance hyper-parameter is observed.",
"Note that if $\\Xi =0$ , the SaS distribution reduces to a Bernoulli distribution with two delta peaks.",
"Due to Eq.",
"(REF ), the point solution $(\\pi =0,\\Xi =0)$ is not a stable attractor for gBP [25].",
"Moreover, as the training data size increases, the variance per connection displays a non-monotonic behavior with a residual value depending on the hierarchical stage [25].",
"We can use the entropy $S_{\\ell }$ for each connection to characterize the variability of the weight value.",
"Note that the entropy can be negative for continuous random variables.",
"The peak at zero [Fig.",
"REF (b)] indicates that a large number of connections are deterministic, including two cases: (i) $\\pi =1$ (unimportant (UIP) weight); (ii) $\\pi =0$ and $\\Xi =0$ (very important (VIP) weight).",
"Fig.",
"REF (c) shows that the entropy per connection grows first, and then decreases.",
"This non-monotonic behavior is the same with that of the sparsity in Fig.",
"REF (b).",
"The large entropy at the middle stage suggests that during the recoding process, the network has more degrees of freedom to manipulate the hypothesis space of the computational task [34].",
"Furthermore, more training data reduce the uncertainty (yet not to zero) until saturation [Fig.",
"REF (c)].",
"Figure: Statistical properties of trained four-layer networks with gBP.",
"Unless stated otherwise, other training conditions are the same as in Fig. .",
"(a) Distribution of hyper-parameters (π,𝐦,Ξ)(\\pi ,\\mathbf {m},\\Xi ) for a typical trained network.",
"(b) Distribution of connection entropy S ℓ S_{\\ell } for a typical trained network.",
"(c) Entropy per connection versus layers (ℬ=100\\mathcal {B}=100).",
"Different training data sizes are considered, and the result is averaged over ten independent runs of five-layer networks.Figure: Effects of targeted-weight perturbation on test performances.",
"Training conditions are the same as in Fig.",
", and the result is averaged over ten independent runs.",
"(a) VIP weights and all weights (without weight categorization) are stochastically turned off, keeping the same number,while UIP weights are stochastically turned on and sampled from a standard Gaussian distribution.",
"The perturbation is carried out at the first hierarchical stage of a four-layer network.",
"(b) The fraction of VIP and UIP weights between neighboring layers.Finally, our mean-field framework can be used to explore effects of targeted-weight perturbation [Fig.",
"REF (a)], which previous studies of random deep models could not address [20], [31].",
"VIP weights play a significant role in determining the generalization capability, while turning on UIP weights does not impair the performance.",
"A random dilution of all weights behaves mildly in between.",
"When going deeper, the perturbation effect becomes less evident (not shown), due to the less number of VIP weights [Fig.",
"REF (b)].",
"Conclusion.— In this work, we propose a statistical model of learning credit assignment in deep neural networks, resulting in an optimal random sub-network ensemble explaining the behavior output of the hierarchical system.",
"The model can be solved by using mean-field methods, yielding a practical way to visualize consumed resources of connections and an ensemble of random weights—two key factors affecting the emergent decision making behavior of the network.",
"Our framework can be applied to a more challenging CIFAR-10 dataset [35] and obtains qualitatively similar results [25].",
"Our model thus provides deep insights towards understanding many recent interesting empirical observations of random templates in both artificial and biological neural networks [7], [9], [8], [12], [36].",
"The model also provides a principled method for network compression saving memory and computation demands [17], and could also have implications for continual learning of sequential tasks, where those important connections for old tasks are always protected to learn new tasks [37], [30].",
"As artificial neural networks become increasingly important to model the brain [5], [12], our current study of artificial neural networks may provide cues for addressing how the brain solves the credit assignment problem.",
"Other promising directions include considering correlations among weights and generalization of this framework to the temporal credit assignment, where spatio-temporal information is considered in training recurrent neural networks [38], [39], [40].",
"This research was supported by the start-up budget 74130-18831109 of the 100-talent- program of Sun Yat-sen University, and the NSFC (Grant No.",
"11805284)."
]
] | 2001.03354 |
[
[
"A formula to calculate the invariant $J$ of a quasi-homogeneous map germ"
],
[
"Abstract In this work, we consider a quasi-homogeneous, corank $1$, finitely determined map germ $f$ from $(\\mathbb{C}^2,0)$ to $(\\mathbb{C}^3,0)$.",
"We consider the invariants $m(f(D(f))$ and $J$, where $m(f(D(f))$ denotes the multiplicity of the image of the double point curve $D(f)$ of $f$ and $J$ denotes the number of tacnodes that appears in a stabilization of the transversal slice curve of $f(\\mathbb{C}^2)$.",
"We present formulas to calculate $m(f(D(f))$ and $J$ in terms of the weights and degrees of $f$."
],
[
"Introduction",
"$ \\ \\ \\ \\ $ In this work, we consider a quasi-homogeneous, corank 1, finitely determined map germ $f$ from $(\\mathbb {C}^2,0)$ to $(\\mathbb {C}^3,0)$ .",
"Local coordinates can be chosen so that these map germs can be written in the form $f(x,y)=(x, \\tilde{p}(x,y),\\tilde{q}(x,y))$ , for some quasi-homogeneous function germs $\\tilde{p}, \\tilde{q} \\in m_2^2$ , where $m_2$ is the maximal ideal of the local ring of holomorphic function germs in two variables $\\mathcal {O}_2$ (see Lemma REF ).",
"In [7], Nuño-Ballesteros and Marar studied the transversal slice of a corank 1 map germ $f:(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^3,0)$ (see [7]).",
"They show in some sense that if a set of generic conditions are satisfied, then the transverse slice curve contains information on the geometry of $f$ .",
"They also introduced the invariants $C$ , $T$ and $J$ , which are defined as the number of cusps, triple points and tacnodes that appears in a stabilization of the transversal slice of $f$ , respectively.",
"They showed in [7] that the numbers $C$ and $T$ are the same as the numbers of cross-caps and triple points that appear in a stabilization of $f$ , which are usually denoted by $C(f)$ and $T(f)$ , respectively, and were defined by Mond in [11].",
"On the other hand, the invariant $J$ is related to both, the delta invariant of the transverse slice curve of $f(\\mathbb {C}^2)$ and $N(f)$ (see [7]), where $N(f)$ is the Mond's invariant defined in [11].",
"In [12], Mond presented formulas to calculate the invariants $C(f)$ , $T(f)$ and $\\mu (D(f))$ of a quasi-homogeneous finitely determined map germ of any corank in terms of the weights and degrees of $f$ , where $\\mu (D(f))$ denotes the Milnor number of the double point curve $D(f)$ of $f$ (see Section REF for the definition of $D(f)$ and Th.",
"REF ).",
"So, a natural question is: Question 1: Let $f:(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^3,0)$ be a quasi-homogeneous, corank 1, finitely determined map germ.",
"Can the invariant $J$ be calculated in terms of the weights and the degrees of $f$ ?",
"It follows by Propositions 3.5 and 3.10 and Corollary 4.4 of [7] that $J=\\dfrac{1}{2}\\bigg (\\mu (D(f))-C(f)-1\\bigg )-3T(f)+m(f(D(f))$ , where $m(f(D(f))$ denotes the multiplicity of the image of the double point curve $D(f)$ .",
"So, using Mond's formulas for $C(f)$ , $T(f)$ and $\\mu (f(D(f))$ (Th.",
"REF ) we conclude that Question 1 has a positive answer if and only if there is a formula to calculate the invariant $m(f(D(f))$ in terms of the weights and the degrees of $f$ .",
"So, another natural question is: Question 2: Let $f$ be as in Question 1.",
"Can the invariant $m(f(D(f))$ be calculated in terms of the weights and degrees of $f$ ?",
"In [13], Ruas and the author provided answers to both questions above in the case where $f$ is homogeneous.",
"In this work, using a normal form for $f$ (Lemma REF ), we present a positive answer for both questions without any restriction on the weights and degrees of $f$ .",
"More precisely, we present in Theorem REF formulas to calculate both invariants, $m(f(D(f))$ and $J$ , in terms of the weights and the degrees of $f$ .",
"We finish this work by calculating the invariants $m(f(D(f))$ and $J$ in some examples to illustrate our formulas."
],
[
"Preliminaries",
"$ \\ \\ \\ \\ $ Throughout this paper, given a finite map $f:\\mathbb {C}^2\\rightarrow \\mathbb {C}^3$ , $(x,y)$ and $(X,Y,Z)$ are used to denote systems of coordinates in $\\mathbb {C}^2$ (source) and $\\mathbb {C}^3$ (target), respectively.",
"Also, $\\mathbb {C} \\lbrace x_1,\\cdots ,x_n \\rbrace \\simeq \\mathcal {O}_n$ denotes the local ring of convergent power series in $n$ variables.",
"The letters $U,V$ and $W$ are used to denote open neighborhoods of 0 in $\\mathbb {C}^2$ , $\\mathbb {C}^3$ and $\\mathbb {C}$ , respectively.",
"We also use the standard notation of singularity theory as the reader can find in Wall's survey paper [18]."
],
[
"Double point curves for corank 1 map germs",
"$ \\ \\ \\ \\ $ In this section, we deal only with of corank 1 maps from $\\mathbb {C}^2$ to $\\mathbb {C}^3$ .",
"For the general definition of double point spaces, see for instance [5], [6] and [11].",
"Consider a finite and holomorphic map $f: U\\rightarrow \\mathbb {C}^3 $ , where $U$ is an open neighbourhood of 0 in $\\mathbb {C}^2$ .",
"The double point space of $f$ , denoted by $D(f)$ , is defined (as a set) by $D(f):=\\lbrace (x,y) \\in U \\ : \\ f^{-1}(f(x,y))\\ne \\lbrace (x,y)\\rbrace \\rbrace \\cup \\Sigma (f)$ , where $\\Sigma (f)$ is the ramification set of $f$ .",
"We also consider the lifting of the $D(f)$ in $U \\times U$ , denoted by $D^2(f)$ , given by the pairs $((x,y),(x^{^{\\prime }},y^{^{\\prime }}))$ such that either $f(x,y)=f(x^{^{\\prime }},y^{^{\\prime }})$ with $(x,y) \\ne (x^{^{\\prime }},y^{^{\\prime }})$ or $(x,y)=(x^{^{\\prime }},y^{^{\\prime }})$ with $(x,y) \\in \\Sigma (f)$ .",
"We need to choose convenient analytic structures for the double point space $D(f)$ and the lifting of the double point space $D^2(f)$ .",
"As we said in Introduction, when $f$ has corank 1, local coordinates can be chosen so that these map germs can be written in the form $f(x,y)=(x, \\tilde{p}(x,y),\\tilde{q}(x,y))$ , for some function germs $\\tilde{p}, \\tilde{q} \\in m_2^2$ , where $m_2$ is the maximal ideal of $\\mathcal {O}_2$ .",
"In this case, we define the lifting of the double point space $D^2(f)$ , (as a complex space) by $D^2(f)=V \\displaystyle \\left( x-x^{^{\\prime }},\\dfrac{\\tilde{p}(x,y)-\\tilde{p}(x,y^{^{\\prime }})}{y-y^{^{\\prime }}}, \\dfrac{\\tilde{q}(x,y)-\\tilde{q}(x,y^{^{\\prime }})}{y-y^{^{\\prime }}} \\right)$ where $(x,y,x^{^{\\prime }},y^{^{\\prime }})$ are coordinates of $\\mathbb {C}^2 \\times \\mathbb {C}^2$ .",
"Once the lifting $D^2(f) \\subset U \\times U$ is defined as a complex analytic space, we now consider its image $D(f)$ (also as a complex analytic space) on $U$ by the projection $\\pi : U \\times U \\rightarrow U$ onto the first factor, which will be considered with the structure given by Fitting ideals.",
"We also consider the double point space in the target, that is, the image of $D(f)$ by $f$ , denoted by $f(D(f))$ , which will also be consider with the structure given by Fitting ideals.",
"We remark that given a finite morphism of complex spaces $h:X\\rightarrow Y$ the push-forward $h_{\\ast }\\mathcal {O}_{X}$ is a coherent sheaf of $\\mathcal {O}_{Y}-$ modules (see [3]) and to it we can (as in [10]) associate the Fitting ideal sheaves $\\mathcal {F}_{k}(h_{\\ast }\\mathcal {O}_{X})$ .",
"Notice that the support of $\\mathcal {F}_{0}(h_{\\ast }\\mathcal {O}_{X})$ is just the image $h(X)$ .",
"Analogously, if $h:(X,x)\\rightarrow (Y,y)$ is a finite map germ then we denote by $ F_{k}(h_{\\ast }\\mathcal {O}_{X})$ the kth Fitting ideal of $\\mathcal {O}_{X,x}$ as $\\mathcal {O}_{Y,y}-$ module.",
"In this way, we have the following definition.",
"Definition 2.1 Let $f:U \\rightarrow V$ be a finite mapping, where $U$ and $V$ are open neighbourhoods of 0 in $\\mathbb {C}^2$ and $\\mathbb {C}^3$ , respectively.",
"(a) Let ${\\pi }|_{D^2(f)}:D^2(f) \\subset U \\times U \\rightarrow U$ be the restriction to $D^2(f)$ of the projection $\\pi $ .",
"The double point space of $f$ is the complex space $D(f)=V(\\mathcal {F}_{0}({\\pi }_{\\ast }\\mathcal {O}_{D^2(f)}))$ .",
"Set theoretically we have the equality $D(f)=\\pi (D^{2}(f))$ .",
"(b) The double point space of $f$ in the target is the complex space $f(D(f))=V(\\mathcal {F}_{1}(f_{\\ast }\\mathcal {O}_2))$ .",
"Notice that the underlying set of $f(D(f))$ is the image of $D(f)$ by $f$ .",
"(c) Given a finite map germ $f:(\\mathbb {C}^{2},0)\\rightarrow (\\mathbb {C}^3,0)$ , the germ of the double point space of $f$ is the germ of complex space $D(f)=V(F_{0}(\\pi _{\\ast }\\mathcal {O}_{D^2(f)}))$ .",
"The germ of the double point space of $f$ in the target is the germ of the complex space $f(D(f))=V(F_{1}(f_{\\ast }\\mathcal {O}_2))$ .",
"Remark 2.2 If $f:U \\subset \\mathbb {C}^2 \\rightarrow V \\subset \\mathbb {C}^3 $ is finite and generically 1-to-1, then $D^2(f)$ is Cohen-Macaulay and has dimension 1 (see [6]).",
"Hence, $D^2(f)$ , $D(f)$ and $f(D(f))$ are complex analytic curves.",
"In this case, without any confusion, we also call these complex spaces by the “lifting of the double point curve”, the “double point curve” and the “image of the double point curve”, respectively."
],
[
"Finite determinacy and the invariants $C(f)$ and {{formula:418139df-04c6-40db-8c8e-76b7c04ab246}}",
"Definition 2.3 (a) Two map germs $f,g:(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^3,0)$ are $\\mathcal {A}$ -equivalent, denoted by $g\\sim _{\\mathcal {A}}f$ , if there exist map germs of diffeomorphisms $\\eta :(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^2,0)$ and $\\xi :(\\mathbb {C}^3,0)\\rightarrow (\\mathbb {C}^3,0)$ , such that $g=\\xi \\circ f \\circ \\eta $ .",
"(b) A map germ $f:(\\mathbb {C}^2,0) \\rightarrow (\\mathbb {C}^3,0)$ is finitely determined (or $\\mathcal {A}$ -finitely determined) if there exists a positive integer $k$ such that for any $g$ with $k$ -jets satisfying $j^kg(0)=j^kf(0)$ we have $g \\sim _{\\mathcal {A}}f$ .",
"Consider a finite map germ $f:(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^3,0)$ .",
"By Mather-Gaffney criterion ([18]), $f$ is finitely determined if and only if there is a finite representative $f:U \\rightarrow V$ , where $U\\subset \\mathbb {C}^2$ , $V \\subset \\mathbb {C}^3$ are open neighbourhoods of the origin, such that $f^{-1}(0)=\\lbrace 0 \\rbrace $ and the restriction $f:U \\setminus \\lbrace 0 \\rbrace \\rightarrow V \\setminus \\lbrace 0 \\rbrace $ is stable.",
"This means that the only singularities of $f$ on $U \\setminus \\lbrace 0 \\rbrace $ are cross-caps (or Whitney umbrellas), transverse double and triple points.",
"By shrinking $U$ if necessary, we can assume that there are no cross-caps nor triple points in $U$ .",
"Then, since we are in the nice dimensions of Mather ([8]), we can take a stabilization of $f$ , $F:U \\times D \\rightarrow \\mathbb {C}^4$ , $F(z,s)=(f_{s}(z),s)$ , where $D$ is a neighbourhood of 0 in $\\mathbb {C}$ .",
"Definition 2.4 We define $T(f):= \\sharp $ of triple points of $f_s$ $ \\ \\ \\ $ and $ \\ \\ \\ $ $C(f):= \\sharp $ of cross-caps of $f_s$ , where $s\\ne 0$ .",
"It is well known that the numbers $T(f)$ and $C(f)$ are independent of the particular choice of the stabilization and they are also analytic invariants of $f$ (see for instance [9]).",
"We remark that the space $D(f)$ plays a fundamental role in the study of the finite determinacy.",
"In [5], Marar and Mond presented necessary and sufficient conditions for a map germ $f:(\\mathbb {C}^n,0)\\rightarrow (\\mathbb {C}^p,0)$ with corank 1 to be finitely determined in terms of the dimensions of $D^2(f)$ and other multiple points spaces.",
"When $(n,p)=(2,3)$ , in [6], Marar, Nuño-Ballesteros and Peñafort-Sanchis extended this criterion of finite determinacy to the corank 2 case.",
"More precisely, they proved the following result.",
"Theorem 2.5 ([6]) Let $f:(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^{3},0)$ be a finite and generically 1-to-1 map germ.",
"Then $f$ is finitely determined if and only if $\\mu (D(f))$ is finite (equivalently, $D(f)$ is a reduced curve)."
],
[
"Identification and Fold components of $D(f)$",
"$ \\ \\ \\ \\ $ When $f:(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^3,0)$ is finitely determined, the restriction of a representative of $f$ to $D(f)$ is finite.",
"In this case, $f_{|D(f)}$ is generically 2-to-1 (i.e; 2-to-1 except at 0).",
"On the other hand, the restriction of $f$ to an irreducible component $D(f)^i$ of $D(f)$ can be generically 1-to-1 or 2-to-1.",
"This motivates us to give the following definition which is from [15] (see also [13] and [14]).",
"Definition 2.6 Let $f:(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^3,0)$ be a finitely determined map germ.",
"Let $f:U\\rightarrow V$ be a representative, where $U$ and $V$ are neighbourhoods of 0 in $\\mathbb {C}^2$ and $\\mathbb {C}^3$ , respectively.",
"Consider an irreducible component $D(f)^j$ of $D(f)$ .",
"(a) If the restriction ${f_|}_{D(f)^j}:D(f)^j\\rightarrow V$ is generically 1-to-1, we say that $D(f)^j$ is an identification component of $D(f)$ .",
"In this case, there exists an irreducible component $D(f)^i$ of $D(f)$ , with $i \\ne j$ , such that $f(D(f)^j)=f(D(f)^i)$ .",
"We say that $D(f)^i$ is the associated identification component to $D(f)^j$ or that the pair $(D(f)^j, D(f)^i)$ is a pair of identification components of $D(f)$ .",
"(b) If the restriction ${f_|}_{D(f)^j}:D(f)^j\\rightarrow V$ is generically 2-to-1, we say that $D(f)^j$ is a fold component of $D(f)$ .",
"(c) We define the sets $IC(D(f))=\\lbrace $ identification components of $D(f) \\rbrace $ and $FC(D(f))= \\lbrace $ fold components of $D(f) \\rbrace $ .",
"And we define the numbers $r_i(D(f)):=\\sharp IC(D(f))$ and $r_f(D(f)):= \\sharp FC(D(f))$ .",
"Remark 2.7 Let $f$ and $g$ be finitely determined map germs from $(\\mathbb {C}^2,0)$ to $(\\mathbb {C}^3,0)$ .",
"Suppose that $g\\sim _{\\mathcal {A}}f$ and write $g=\\xi \\circ f \\circ \\eta $ as in Definiton REF .",
"Consider representatives $f,g:U\\rightarrow V$ of $f$ and $g$ .",
"Let $D(g)^i$ be an irreducible component of $D(g)$ and consider its corresponding image by $\\eta $ , $D(f)^i:=\\eta (D(g)^i)$ , which is an irreducible component of $D(f)$ .",
"Note that , ${f_|}_{D(f)^j}:D(f)^i\\rightarrow V$ is generically $k$ -to-1 if and only if ${g_|}_{D(g)^i}:D(g)^i\\rightarrow V$ is generically $k$ -to-1, where $k=1,2$ .",
"Hence $r_i(D(f))=r_i(D(g))$ $ \\ \\ \\ $ and $ \\ \\ \\ $ $r_f(D(f))=r_f(D(g))$ .",
"The following example illustrates the two types of irreducible components of $D(f)$ presented in Definition REF .",
"Example 2.8 Let $f(x,y)=(x,y^2,xy^3-x^7y)$ be the singularity $C_7$ of Mond's list ([11]).",
"In this case, $D(f)=V(xy^2-x^7)$ .",
"Then $D(f)$ has three irreducible components given by $D(f)^1=V(y-x^3), \\ \\ \\ $ $D(f)^2=V(y+x^3) \\ \\ $ and $ \\ \\ D(f)^3=V(x)$ .",
"Notice that $(D(f)^1$ , $D(f)^2)$ is a pair of identification components and $D(f)^3$ is a fold component.",
"Hence, we have that $r_i(D(f))=2$ and $r_f(D(f))=1$ .",
"We have also that $f(D(f)^3)=V(X,Z)$ and $f(D(f)^1)=f(D(f)^2)=V(Y-X^6,Z)$ (see Figure REF ).",
"Figure: Identification and fold components of D(f)D(f) (real points)Remark 2.9 In the Example REF, we have made use of the software Surfer [17]."
],
[
"Quasi-homogeneous map germs and Mond's formulas",
"Definition 2.10 A polynomial $p(x_1,\\cdots ,x_n)$ is quasi-homogeneous if there are positive integers $w_1,\\cdots ,w_n$ , with no common factor and an integer $d$ such that $p(k^{w_1}x_1,\\cdots ,k^{w_n}x_x)=k^dp(x_1,\\cdots ,x_n)$ .",
"The number $w_i$ is called the weight of the variable $x_i$ and $d$ is called the weighted degree of $p$ .",
"We also write $w(p)$ to denote the weighted degree of $p$ .",
"In this case, we say $p$ is of type $(d; w_1,\\cdots ,w_n)$ .",
"This definition extends to polynomial map germs $f:(\\mathbb {C}^n,0)\\rightarrow (\\mathbb {C}^p,0)$ by just requiring each coordinate function $f_i$ to be quasi-homogeneous of type $(d_i; w_1,\\cdots ,w_n)$ .",
"In particular, when $f:(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^3,0)$ is quasi-homogeneous, we say that $f$ is quasi-homogeneous of type $(d_1,d_2,d_3; a,b)$ , where here we change the classical notation $w_1,w_2$ of the weights of $x$ and $y$ by $a,b$ , for simplicity.",
"The following lemma describes a normal form for a class of quasi-homogeneous map germs from $(\\mathbb {C}^2,0)$ to $(\\mathbb {C}^3,0)$ which will be considered in this work.",
"Lemma 2.11 (Normal form lemma) Let $g(x,y)=(g_1(x,y),g_2(x,y),g_3(x,y))$ be a quasi-homogeneous, corank 1, finitely determined map germ of type $(d_1,d_2,d_3; a,b)$ .",
"Then $g$ is $\\mathcal {A}$ -equivalent to a quasi-homogeneous map germ $f$ with type $(d_{i_1}=a,d_{i_2},d_{i_3};a,b)$ , which is written as $f(x,y)=(x, y^n+xp(x,y), \\alpha y^m+ xq(x,y)),$ for some integers $n,m\\ge 2$ , $\\alpha \\in \\mathbb {C}$ , $p,q \\in \\mathcal {O}_2$ , $p(x,0)=q(x,0)=0$ , where $(d_{i_1},d_{i_2},d_{i_3})$ is a permutation of $(d_1,d_2,d_3)$ such that $d_{i_2}\\le d_{i_3}$ .",
"Proof.",
"Since $g$ has corank 1, $g_i$ is a regular in $x$ or $y$ for some $i$ .",
"Without lose of generality, suppose that $g_1$ is regular in $x$ .",
"Thus $g_1(x,y)=\\gamma x + g^{^{\\prime }}_1(x,y)$ , where $\\gamma \\in \\mathbb {C}$ , $\\gamma \\ne 0$ .",
"We have that $g$ is quasi-homogeneous, this implies that $g^{^{\\prime }}_1(x,y)=\\theta y^a$ , where $\\theta \\in \\mathbb {C}$ .",
"Also, if $\\theta \\ne 0$ , then $b=1$ .",
"Consider the analytic isomorphisms $\\eta : (\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^2,0)$ and $\\xi :(\\mathbb {C}^3,0)\\rightarrow (\\mathbb {C}^3,0)$ defined by $\\eta (x,y)=(x-(\\gamma ^{-1}\\theta ) y^a,y)$ $ \\ \\ \\ $ and $ \\ \\ \\ $ $\\xi (X,Y,Z)=(\\gamma ^{-1}X,Y,Z)$ .",
"Note that the map $\\tilde{g}:=\\xi \\circ g \\circ \\eta $ is a quasi-homogeneous map germ of type $(d_{1},d_{2},d_{3};a,b)$ .",
"There exist integers $v_1,v_2$ , complex numbers $\\alpha _1,\\alpha _2$ and polynomials $p_1,p_2 \\in \\mathcal {O}_2$ such that $\\tilde{g}$ is written as $\\tilde{g}(x,y)=(x,\\alpha _1 y^{v_1} + xp_1(x,y), \\alpha _2 y^{v_2}+xp_2(x,y))$ , with $2\\le v_1,v_2$ .",
"After a change of coordinates (which does not change the quasi-homogeneous type of $\\tilde{g}$ ), we can assume that $p_1(x,0)=p_2(x,0)=0$ .",
"So, we need to show that $\\alpha _i \\ne 0$ for some $i$ with $w(\\alpha _iy^{v_i}+xp_i(x,y))\\le w(\\alpha _jy^{v_j}+xp_j(x,y))$ , where $i\\ne j$ and $i,j \\in \\lbrace 1,2\\rbrace $ .",
"Since $\\tilde{g}$ is also finitely determined, in particular, it is finite, so either $\\alpha _1 \\ne 0$ or $\\alpha _2 \\ne 0$ .",
"Thus we have three cases.",
"(1) $\\alpha _1, \\alpha _2 \\ne 0$ .",
"(2) $\\alpha _1 =0$ and $\\alpha _2 \\ne 0$ .",
"(3) $\\alpha _1 \\ne 0$ and $\\alpha _2 =0$ .",
"(1) Suppose that $\\alpha _1,\\alpha _2 \\ne 0$ .",
"In this case, we can suppose that $\\alpha _1=\\alpha _2=1$ (applying the change of coordinates $(X,Y,Z)\\mapsto (X,\\alpha _1^{-1}Y,\\alpha _2^{-1}Z)$ ).",
"Define $n:=min \\lbrace v_1,v_2 \\rbrace $ and $p:=p_i$ if $n=v_i$ .",
"Define also $m:=v_1$ and $q:=p_1$ if $n=v_2$ or $m:=v_2$ and $q:=p_2$ if $n=v_1$ .",
"So, $\\tilde{g}$ is $\\mathcal {A}$ -equivalent to $f(x,y)=(x,y^n+xp(x,y),y^m+xq(x,y))$ .",
"Note that $w(y^n+xp(x,y)\\le w(y^m+xq(x,y))$ .",
"(2) If $\\alpha _1 = 0$ and $\\alpha _2 \\ne 0$ , then the restriction of $\\tilde{g}$ to $V(x)$ is $v_2$ -to-1.",
"Since $\\tilde{g}$ if finitely determined and singular, we have that $v_2=2$ .",
"In this case, $\\tilde{g}(x,y)=(x,xp_1(x,y),\\alpha _2 y^2+xp_2(x,y))$ .",
"Again, after a change of coordinates, we can assume that $\\alpha _2=1$ .",
"Write $x p_1(x,y)= \\lambda _1 x^{k_1}y^{s_1}+\\cdots +\\lambda _l x^{k_l}y^{s_l}$ .",
"Statement: We have that $w(y^2+xp_2(x,y)) \\le w(xp_1(x,y))$ .",
"Proof of Statement: Note that $p_1(x,y)\\lnot \\equiv 0$ , otherwise the set of cross-caps $C(\\tilde{g})$ of $\\tilde{g}$ is not finite and hence $\\tilde{g}$ is not finitely determined.",
"Since $p_1(x,0)=0$ , we have that $s_i\\ge 1$ for all $i$ .",
"If $s_i\\ge 2$ for some $i$ , then the statement is clear and after a change of coordinates, we see that $\\tilde{g}$ is $\\mathcal {A}$ -equivalent to $f(x,y)=(x,y^2+xp(x,y),xq(x,y))$ , where $p=p_2$ , $q=p_1$ and $w(y^2+xp(x,y))\\le w(xq(x,y))$ , as desired.",
"Now, suppose that $s_i=1$ for all $i$ .",
"In this case, after a change of coordinates, we can write $\\tilde{g}$ as $\\tilde{g}(x,y)=(x,x^{k}y,y^2+xp_2(x,y))$ , for some $k\\ge 1$ .",
"We have that $D^2(\\tilde{g})=V(x-x^{^{\\prime }}, \\ x^{k} \\ , y+y^{^{\\prime }}+x(p_2(x,y)-p_2(x,y^{^{\\prime }}))/(y-y^{^{\\prime }}))$ which is not reduced if $k\\ge 2$ .",
"Since $\\tilde{g}$ is finitely determined, by Theorem REF and [6] we have that $k=1$ .",
"In this case, $D^2(\\tilde{g})=V(x,x^{^{\\prime }},y+y^{^{\\prime }})\\subset \\mathbb {C}^2 \\times \\mathbb {C}^2$ is a smooth curve and $\\tilde{g}$ does not have any triple points.",
"It follows by [5] that $\\tilde{g}$ is stable.",
"Hence, $\\tilde{g}$ is $\\mathcal {A}$ -equivalent to $f(x,y)=(x,y^2,xy)$ , which is considered with quasi-homogeneous type $(1,2,2;1,1)$ and has the desired properties, that is, $w(y^2)\\le w(xy)$ .",
"Now, the analysis of case (3) is analogous.",
"Remark 2.12 Some versions of Lemma REF are well know by specialists (see for instance [9]).",
"We include its proof for completeness.",
"Given a quasi-homogeneous, corank 1, finitely determined map germ, we will assume in the proofs throughout this paper that $f$ is written in the normal form in (REF ), presented in Normal form lemma.",
"In [12], Mond showed that if $f$ is quasi-homogeneous then the invariants $C(f)$ , $T(f)$ and also $\\mu (D(f))$ are determined by the weights and the degrees of $f$ .",
"More precisely, he showed the following result.",
"Theorem 2.13 ([12]) Let $f:(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^3,0)$ be a quasi-homogeneous finitely determined map germ of type $(d_1,d_2,d_3;a,b)$ .",
"Then $C(f)=\\dfrac{1}{ab}\\bigg ((d_2-a)(d_3-b)+(d_1-b)(d_3-b)+(d_1-a)(d_2-a)\\bigg )$ , $T(f)=\\dfrac{1}{6ab}(\\delta -\\epsilon )(\\delta -2\\epsilon )+\\dfrac{C(f)}{3} \\ \\ \\ \\ \\ $ and $ \\ \\ \\ \\ \\ \\mu (D(f))=\\dfrac{1}{ab}(\\delta -\\epsilon -a)(\\delta -\\epsilon -b)$ .",
"where $\\epsilon = d_{1}+d_{2}+d_{3}-a-b$ and $\\delta =d_{1}d_{2}d_{3}/(ab)$ ."
],
[
"Formulas for the invariants $J$ and {{formula:1b99346f-fdda-4718-afd1-df62c640a4a6}}",
"$ \\ \\ \\ \\ $ We note that by a parametrization of an irreducible complex germ of curve $(X,0) \\subset (\\mathbb {C}^n,0)$ we mean a primitive parametrization, that is, a holomorphic and generically 1-to-1 map germ $n$ from $(\\mathbb {C},0)$ to $(\\mathbb {C}^n,0)$ , such that $n(W,0)\\subset (X,0)$ (see for instance [4]).",
"Before we present our main result, we will need the following lemma.",
"Lemma 3.1 Let $f$ be a finitely determined, corank 1, quasi-homogeneous map germ of type $(d_1,d_2,d_3;a,b)$ .",
"Write $f$ as in Lemma REF , that is, $f(x,y)=(x, y^n+xp(x,y), \\alpha y^m+ xq(x,y))$ , with $d_2\\le d_3$ .",
"Then (a) If $V(y)$ is an irreducible component of $D(f)$ , then $V(y)\\in IC(D(f))$ and $a=1$ .",
"(b) $D(f)=V(\\lambda (x,y))$ , where $\\lambda (x,y)$ is a quasi-homogeneous polynomial of type $\\left(\\dfrac{d_2d_3}{b}-d_2-d_3+b;a,b\\right)$ and $\\lambda (x,y)=\\displaystyle { x^{s}\\prod _{i=1}^{r}}(y^a-\\alpha _i x^b),$ where $\\alpha _i \\in \\mathbb {C}$ are all distinct, $r=\\dfrac{(d_2-b)(d_3-b)-sab}{ab^2}\\ge 0$ and either $s =0$ or $s=1$ .",
"(c) If $a>d_2$ , then $p(x,y)=0$ .",
"That is, $f(x,y)=(x,y^n,\\alpha y^m+xq(x,y))$ .",
"(d) If $s=1$ in (REF ), that is, if $V(x)$ is an irreducible component of $D(f)$ , then it is a fold component of $D(f)$ .",
"(e) If $\\alpha =0$ , then $n=2$ .",
"(f) $s=0$ if and only if $\\alpha \\ne 0 $ and $gcd(n,m)=1$ .",
"In other words, $s=1$ if and only if either $\\alpha =0$ , or $\\alpha \\ne 0$ and $gcd(n,m)=2$ .",
"Proof.",
"Consider a representative $f:U\\rightarrow V$ of $f$ .",
"((a) and (b)) Suppose that $V(y)$ is an irreducible component of $D(f)$ .",
"Consider the parametrization of $V(y)$ given by the map $\\varphi _0:W\\rightarrow U$ , defined by $\\varphi _0(u)=(u,0)$ .",
"So, $f\\circ \\varphi _0:W\\rightarrow V$ , defined as $(f\\circ \\varphi _0)(u)=(u,0,0),$ is a parametrization of $f(V(y))$ .",
"Since $f\\circ \\varphi _0$ is 1-to-1, $V(y)$ is an identification component of $D(f)$ .",
"Since $f$ is quasi-homogeneous and finitely determined, we have that $\\lambda (x,y)$ is a quasi-homogeneous polynomial of type $\\left(\\dfrac{d_2d_3}{b}-d_2-d_3+b;a,b\\right)$ , by [12].",
"The only irreducible quasi-homogeneous polynomials with $w(x)=a$ and $w(y)=b$ in the ring of polynomials $\\mathbb {C}[x,y]$ are $x,y$ and $y^a-\\alpha _i x^b$ , with $\\alpha _i \\in \\mathbb {C}$ and $\\alpha _i \\ne 0$ .",
"Since the ring of polynomials $\\mathbb {C}[x,y]$ is an unique factorization domain, each irreducible factor of $\\lambda $ is on the form of $x,y$ or $y^a-\\alpha _i x^b$ .",
"By Theorem REF , $\\lambda $ is reduced, hence the irreducible factors of $\\lambda $ are all distinct.",
"So, $\\lambda $ can take the following form: $\\lambda (x,y)=\\displaystyle { x^{s}y^{l}\\prod _{i=1}^{r^{^{\\prime }}}}(y^a-\\alpha _i x^b),$ where $s,l \\in \\lbrace 0,1 \\rbrace $ , $r^{^{\\prime }}\\ge 0$ , $\\alpha _i$ are all distinct and $\\alpha _i \\ne 0$ for all $i$ .",
"We note that if $r^{^{\\prime }}=0$ , then $\\displaystyle { \\prod _{i=1}^{0}}(y^a-\\alpha _i x^b)=1$ (the empty product).",
"Consider the parametrization of $V(y^a-\\alpha _i x^b)$ given by the map $\\varphi _{\\alpha _i}:W\\rightarrow U$ , defined by $\\varphi _{\\alpha _i}(u)=(u^a,\\gamma _i u^b)$ , where $\\gamma _i=\\alpha _i^{1/a}$ .",
"So, $f\\circ \\varphi _{\\alpha _i}:W\\rightarrow V$ , defined as $(f\\circ \\varphi _{\\alpha _i})(u)=(u^a,\\gamma _{1,i}u^{d_2},\\gamma _{2,i}u^{d_3}),$ is a parametrization of $f(V(y^a-\\alpha _i x^b))$ , for some $\\gamma _{1,i},\\gamma _{2,i} \\in \\mathbb {C}$ .",
"Since $f(V(x))\\cap f(V(y))=\\lbrace (0,0,0) \\rbrace $ , the associated identification component of $V(y)$ is a curve $V(y^a-\\alpha _j x^b)$ for some $\\alpha _j \\ne 0$ .",
"Since, $(V(y),V(y^a-\\alpha _j x^b))$ is a pair of identification components of $D(f)$ , comparing (REF ) and (REF ) we see that $\\gamma _{1,j}=\\gamma _{2,j}=0$ and $a=1$ .",
"Consequently, the expression (REF ) can be rewritten as follows: $\\lambda (x,y)=\\displaystyle { x^{s}\\prod _{i=1}^{r}}(y^a-\\alpha _i x^b)$ , where $s \\in \\lbrace 0,1 \\rbrace $ , $r=\\dfrac{(d_2-b)(d_3-b)-sab}{ab^2}\\ge 0$ , $\\alpha _i$ are all distinct and we allow one of the $\\alpha _i^{\\prime }s$ to be zero.",
"(c) Suppose that $p(x,y)\\ne 0$ .",
"By assumption of the normal form of $f$ , we have that $p(x,0)=0$ , hence $p(x,y)$ is not a constant.",
"If $a>d_2=bn$ , then $w(xp(x,y))>a>d_2$ , a contradiction.",
"(d) Note that for all $i$ , $V(x)$ and $V(y^a-\\alpha _i x^b)$ have distinct images.",
"Hence, $V(x)\\notin IC(D(f))$ .",
"So if $V(x) \\subset D(f)$ , then we conclude that $V(x) \\in FC(D(f))$ .",
"(e) Suppose that $\\alpha =0$ , then restriction of $f$ to $V(x)$ is $n$ -to-1.",
"Since $f$ is finitely determined, either $n=1$ or $n=2$ .",
"Since $f$ has corank 1, we conclude that $n=2$ .",
"(f) By (d), we have that $V(x)$ is an irreducible component of $D(f)$ if and only if $f\\circ \\varphi $ is generically 2-to-1, where $\\varphi : W\\rightarrow U$ is the parametrization of $V(x)$ , defined by $\\varphi (u)=(0,u)$ .",
"Suppose that $V(x) \\subset D(f)$ .",
"This implies that either $\\alpha =0$ or $\\alpha \\ne 0$ and $gcd(n,m)=2$ .",
"On the other hand, if $\\alpha =0$ then by (e) we have that $n=2$ and $f\\circ \\varphi $ is generically 2-to-1.",
"Hence, in this case $V(x) \\subset D(f)$ .",
"If $\\alpha \\ne 0$ and $gcd(n,m)=2$ , then again $f\\circ \\varphi $ is generically 2-to-1 and hence $V(x) \\in FC(D(f))$ .",
"Now, suppose that $V(x)$ is not an irreducible component of $D(f)$ .",
"Since $f$ is generically 1-to-1, by (d) we have that $V(x)\\lnot \\subset D(f)$ if and only if $f\\circ \\varphi $ is generically 1-to-1 if and only if $\\alpha \\ne 0$ and $gcd(n,m)=1$ .",
"In the following result, $m(f(D(f)))$ denotes the Hilbert-Samuel multiplicity of the maximal ideal of the local ring $\\mathcal {O}_{f(D(f))}$ of $(f(D(f)),0)$ (or equivalently, the multiplicity of $f(D(f))$ at 0).",
"Also, $J$ denotes the number of tacnodes that appears in a stabilization of the transversal slice curve of $f(\\mathbb {C}^2)$ (see [7]).",
"Note that if $\\varphi :W\\subset \\mathbb {C}\\rightarrow V \\subset \\mathbb {C}^3$ , $\\varphi (u)=(u^m,\\varphi _2(u),\\varphi _3(u))$ is a Puiseux parametrization of a reduced curve in $\\mathbb {C}^3$ , then its multiplicity is $m$ (see for instance [1]).",
"We remark that given a germ of reduced curve $(C,0)\\subset (\\mathbb {C}^n,0)$ , it is not true that its irreducible components are also reduced, see for instance [2] where $(X_0,0)=(X_0^1 \\cup X_0^2,0)$ is a germ of reduced curve in $(\\mathbb {C}^3,0)$ , but $(X_0^1,0)$ is not reduced at 0.",
"Suppose that $f:(\\mathbb {C}^2,0)\\rightarrow (\\mathbb {C}^3,0)$ is finitely determined.",
"So, $D(f)$ is a reduced curve, by Theorem REF .",
"It follows by [6] that $f(D(f))$ is also a reduced curve.",
"However, given an irreducible component $f(D(f)^i)$ of $f(D(f))$ , it may contain a (embedded) zero dimensional component, and therefore may not be reduced.",
"If this is the case, we say that $f(D(f)^i)$ is a generically reduced curve.",
"Recently, the author and Snoussi showed in [16] that if $(C,0)$ is a germ of generically reduced curve and $(|C|,0)$ is its associated reduced curve, then the multiplicities of $(C,0)$ and $(|C|,0)$ at 0 are equal.",
"Hence, we also can calculate the multiplicity of $f(D(f)^i)$ considering its reduced structure and using a corresponding Puiseux parametrization for it.",
"We are now able to present our main result.",
"Theorem 3.2 Let $f$ be a finitely determined, corank 1, quasi-homogeneous map germ.",
"Write $f$ as in Lemma REF , that is, $f(x,y)=(x, y^n+xp(x,y), \\alpha y^m+ xq(x,y))$ and it is of type $(d_1=a,d_2,d_3;a,b)$ such that $d_2\\le d_3$ .",
"Then $m(f(D(f))=\\dfrac{1}{2ab^2}\\bigg ((d_2-b)(d_3-b)c+s a b (d_2-c) \\bigg )$ $ \\ \\ \\ \\ $ and $J=\\dfrac{1}{2ab^2}\\bigg ((d_2-b)(d_3-b)(c-3b)+b(\\delta -\\epsilon -a)(\\delta -\\epsilon -b)+b(\\epsilon -\\delta )(\\delta -2\\epsilon )+sab(d_2-c)-ab^2\\bigg )$ where $\\epsilon = d_{2}+d_{3}-b$ , $\\delta =d_{2}d_{3}/b$ , $c=min\\lbrace a,d_2 \\rbrace $ and $ s = \\left\\lbrace \\begin{array}{ll}0 & if \\ \\alpha \\ne 0 \\ and \\ gcd(n,m)=1, \\\\1 & otherwise.\\end{array}\\right.",
"$ Proof.",
"Take a representative $f:U\\rightarrow V$ of $f$ .",
"By Lemma REF (b), we have that $D(f)=V(\\lambda (x,y))$ , where $\\lambda (x,y)=\\displaystyle { x^{s}\\prod _{i=1}^{r}}(y^a-\\alpha _i x^b)$ , $s=0$ or 1, $\\alpha _i \\in \\mathbb {C}$ are all distinct and $r=\\dfrac{(d_2-b)(d_3-b)-sab}{ab^2}$ .",
"Set ${C}_{\\alpha _i}:=V(y^a-\\alpha _i x^b)$ .",
"As in the proof of Lemma REF , consider a parametrization $\\varphi _{\\alpha _i}: W \\rightarrow U $ of ${C}_{\\alpha _i}$ defined by $\\varphi _{\\alpha _i}(u)=(u^a,\\gamma _i u^b)$ , where $W$ is an open neighbourhood of 0 in $\\mathbb {C}$ and $\\gamma _i:=\\alpha _i^{1/a}$ .",
"So, if ${C}_{\\alpha _i}$ is an identification component of $D(f)$ , then the mapping $\\tilde{\\varphi }_{\\alpha _i}:= f\\circ \\varphi _{\\alpha _i}: W \\rightarrow V$ , defined by $\\tilde{\\varphi }_{\\alpha _i}:=(u^a,\\gamma _{1,i} u^{d_2}, \\gamma _{2,i} u^{d_3}),$ is a parametrization of $f({C}_{\\alpha _i})$ , for some $\\gamma _{1,i}, \\gamma _{2,i} \\in \\mathbb {C}$ .",
"On the other hand, if ${C}_{\\alpha _i}$ is a fold component of $D(f)$ , then the mapping $\\varphi ^{^{\\prime }}_{\\alpha _i}: W \\rightarrow V$ , defined by $\\varphi ^{^{\\prime }}_{\\alpha _i}(u):=(u^{a/2},\\gamma _{1,i}^{^{\\prime }} u^{{d_2}/2}, \\gamma _{2,i}^{^{\\prime }} u^{{d_3}/2}),$ is a parametrization of $f({C}_{\\alpha _i})$ , for some $\\gamma _{1,i}^{^{\\prime }}, \\gamma _{2,i}^{^{\\prime }} \\in \\mathbb {C}$ .",
"Set $c:=min\\lbrace a, d_2 \\rbrace $ .",
"Note that if $a>d_2$ , then $\\gamma _{1,i},\\gamma _{1,i}^{^{\\prime }}\\ne 0$ , by Lemma REF (c).",
"It follows by (REF ) and (REF ) that $ m(f({C}_{\\alpha _i})) = \\left\\lbrace \\begin{array}{ll}c & if \\ {C}_{\\alpha _i} \\in IC(D(f)), \\\\c/2 & if \\ {C}_{\\alpha _i} \\in FC(D(f)).",
"\\\\\\end{array}\\right.",
"$ Set ${C}:=V(x)$ .",
"If ${C} \\subset D(f)$ , then by Lemma REF (d) we have that it is a fold component of $D(f)$ .",
"In this case, the map $\\varphi :W\\rightarrow V$ defined by $\\varphi (u)=(0,u^{n/2},\\alpha u^{m/2})$ is a parametrization of ${C}$ .",
"It follows by (REF ) that $m({C})=n/2$ .",
"Hence, we have that $m(f(D(f)))=\\bigg (\\dfrac{r_i(D(f))}{2}\\bigg )c+(r_f(D(f))-s)\\bigg (\\dfrac{c}{2}\\bigg )+s \\bigg (\\dfrac{n}{2}\\bigg )$ .",
"By Lemma REF (b), we have that $r_i(D(f))+r_f(D(f))-s=r=\\dfrac{(d_2-b)(d_3-b)-sab}{ab^2}$ .",
"Also, note that $n=d_2/b$ .",
"Hence, $m(f(D(f)))=\\bigg (\\dfrac{(d_2-b)(d_3-b)-sab}{ab^2}\\bigg )\\bigg (\\dfrac{c}{2}\\bigg )+s \\bigg (\\dfrac{d_2}{2b}\\bigg )=\\dfrac{1}{2ab^2}\\bigg ((d_2-b)(d_3-b)c+s a b (d_2-c) \\bigg ).$ It follows by Propositions 3.5 and 3.10 and Corollary 4.4 of [7] that $J=\\dfrac{1}{2}\\bigg (\\mu (D(f))-C(f)-1\\bigg )-3T(f)+m(f(D(f)).$ Now, by Theorem REF and the expressions (REF ) and (REF ), we have that $J=\\dfrac{1}{2ab}\\bigg ((\\delta -\\epsilon -a)(\\delta -\\epsilon -b)+(\\epsilon -\\delta )(\\delta -2\\epsilon )-3(d_2-b)(d_3-b)-ab\\bigg )+\\dfrac{1}{2ab^2}\\bigg ((d_2-b)(d_3-b)c+s a b (d_2-c) \\bigg )$ $=\\dfrac{1}{2ab^2}\\bigg ((d_2-b)(d_3-b)(c-3b)+b(\\delta -\\epsilon -a)(\\delta -\\epsilon -b)+b(\\epsilon -\\delta )(\\delta -2\\epsilon )+sab(d_2-c)-ab^2\\bigg )$ , where $\\epsilon = d_{2}+d_{3}-b$ and $\\delta =d_{2}d_{3}/b$ ."
],
[
"Examples",
"$ \\ \\ \\ \\ $ When we look to the formulas in Theorem REF , we identify four cases.",
"More precisely, we identified four situations depending on the values that $c$ and $s$ assume.",
"In this section, we present examples illustrating these situations.",
"Example 4.1 (a) ($c=a$ and $s=0$ ) Consider the $F_4$ -singularity of Mond's list [11], given by $f(x,y)=(x,y^2,y^5+x^3y)$ .",
"We have that $f$ is quasi-homogeneous of type $(4,6,15; 4,3)$ .",
"In this case $c=4$ and $s=0$ .",
"By Theorem REF we have that $m(f(D(f)))=2$ and $J=3$ .",
"(b) ($c=a$ and $s=1$ ) Consider the map germ $f(x,y)=(x,y^4,y^6+x^5y-5x^3y^3+4xy^5)$ , which is quasi-homogeneous of type $(1,4,6; 1,1)$ .",
"In this case $c=1$ and $s=1$ .",
"Again by Theorem REF we have that $m(f(D(f)))=9$ and $J=39$ .",
"We remark that $f$ is from [13], where finite determinacy is proved.",
"(c) ($c=d_2$ and $s=0$ ) Consider the $H_2$ -singularity of of Mond's list, given by $f(x,y)=(x,y^3,y^5+xy)$ , which is quasi-homogeneous of type $(4,3,5; 4,1)$ .",
"Using Theorem REF we have that $m(f(D(f)))=3$ and $J=2$ .",
"(d) ($c=d_2$ and $s=1$ ) Consider the map germ $f(x,y)=(x,y^2,x^2y-xy^5)$ , which is quasi-homogeneous of type $(4,2,9; 4,1)$ .",
"We have that $D(f)=V(x(x-y^4))$ which is reduced.",
"So, by Theorem REF we have that $f$ is finitely determined.",
"Using Theorem REF we have that $m(f(D(f)))=2$ and $J=4$ .",
"Example 4.2 Inspired in Example REF(a) and (c) we finish this work presenting in Table REF values for $m(f(D(f))$ and $J$ for every map germ in Mond's list using the formulas of Theorem REF .",
"We also include in Table REF the values of $r_i(D(f))$ and $r_f(D(f))$ .",
"Table: Quasi-homogeneous map germs in Mond's list .Acknowlegments: We would like to thank Jawad Snoussi and Guillermo Peñafort-Sanchis for many helpful conversations, suggestions and comments on this work.",
"The author would like to thank CONACyT for the financial support by Fordecyt 265667 and UNAM/DGAPA for support by PAPIIT IN 113817."
]
] | 2001.03590 |
[
[
"Time-local optimal control for parameter estimation in the Gaussian\n regime"
],
[
"Abstract Information about a classical parameter encoded in a quantum state can only decrease if the state undergoes a non-unitary evolution, arising from the interaction with an environment.",
"However, instantaneous control unitaries may be used to mitigate the decrease of information caused by an open dynamics.",
"A possible, locally optimal (in time) choice for such controls is the one that maximises the time-derivative of the quantum Fisher information (QFI) associated with a parameter encoded in an initial state.",
"In this study, we focus on a single bosonic mode subject to a Markovian, thermal master equation, and determine analytically the optimal time-local control of the QFI for its initial squeezing angle (optical phase) and strength.",
"We show that a single initial control operation is already optimal for such cases and quantitatively investigate situations where the optimal control is applied after the open dynamical evolution has begun."
],
[
"Introduction",
"In analysing the accuracy of high precision metrological setups, one must take into account quantum mechanics, since it enforces an intrinsically statistical description of experiments.",
"The field of quantum parameter estimation was born to address the challenges that arise when combining concepts from classical statistics with the formalism of quantum mechanics , .",
"This line of research has lead to the realisation that peculiar properties of quantum systems can be employed to build high precision sensors, with performances not available to purely classical systems , , , , .",
"However, when the quantum system is subjected to a noisy (non-unitary) dynamics the promised advantage can easily be lost , , .",
"Several approaches have been proposed to suppress or at least mitigate the effect of noise in quantum metrology.",
"Some of those assume (partial) access to the environment causing the non-unitary dynamics , , , , , , possibly also applying measurement-based feedback.",
"Another popular way of tackling this issue is the use of error correcting codes, which have to be specifically tailored for metrological applications , , , , , , , , , .",
"The form of quantum control that is closest to experimental implementations is focused on devising optimal time-dependent pulses given an available set of control Hamiltonians .",
"In this context, optimal pulses for noisy metrology have been investigated by numerical methods , , .",
"Dynamical decoupling , is a particularly useful approach in choosing pulses to protect a quantum system from an environment, and it has also been studied in the context of quantum metrology , .",
"We mention that the interplay between quantum metrology and quantum control is a vast subject that goes well beyond the aim of counteracting and mitigating the effect of noise.",
"Optimal control pulses have been employed to achieve a nonclassical time-scaling for the precision in the estimation of parameters characterizing time-dependent fields , , , and also to enhance the performance of remote parameter estimation .",
"In some physical systems, it is reasonable to assume that the control operations can act essentially instantaneously (relative to the timescales of the free dynamics).",
"Under this assumption, we can think of the controlled dynamics as the free nonunitary evolution interspersed by unitary operations; this is the framework we will be working in.",
"The use of such control unitaries was shown to be useful to restore a nonclassical time scaling when sensing nontrivial Hamiltonian parameters , , , and the same idea has been applied to Gaussian multimode interferometry .",
"In this paper, we focus on a particular flavour of optimal control that is “time-local', meaning that the time-derivative of some figure of merit is optimised over instantaneous unitary operations.",
"This approach has been applied to thermodynamical quantities, both in finite-dimensional and Gaussian quantum systems , and also to counteract the decay of entanglement in two-mode Gaussian systems .",
"Here we follow the same approach but with the aim of counteracting the decay of the metrological usefulness of a quantum state, quantified by the quantum Fisher information (QFI).",
"In particular, we focus on a simple but paradigmatic model: a single bosonic mode evolving in a Markovian thermal environment.",
"Furthermore, we concentrate on Gaussian states and Gaussian control operations, since they are readily implementable in many physical platforms , , , .",
"We consider a situation in which the encoding of the unknown parameter happens before the free open dynamics of the system.",
"This is a reasonable model when also the parameter encoding happens very quickly compared to the time scale of the free evolution.",
"In other words, the noisy dynamics can be considered as an unavoidable part of the detection stage, which does not affect the parameter encoding stage.",
"This approach is different from considering the encoding on the parameter as part of the dynamical evolution, as done in many of the studies we have mentioned.",
"One of our main results and our starting point for this analysis is a compact analytical formula for the time derivative of the QFI.",
"We find that the information about a parameter encoded in the first moments is unaffected by the control strategies we consider, and thus we restrict to parameters encoded in the covariance matrix (CM), i.e., angle and strength of squeezing.",
"Not surprisingly, the estimation of squeezing with Gaussian states has been considered by various authors , , , , , , .",
"Here, we find the optimal unitary controls to preserve the QFI associated with both the squeezing angle (i.e., an optical phase parameter) and the squeezing strength.",
"Interestingly, the optimal strategy we find is analogous to the one found in , for noiseless Hamiltonian estimation, i.e., applying the inverse of the parameter encoding unitary.",
"This inverse transformation also appears when studying optimal measurements for noiseless multiparameter estimation , .",
"Since these optimal operations depend on the unknown true value of the parameter, their effectiveness only make sense in the context of an adaptive estimation scheme .",
"Remarkably, we also find that repeated control operations are unnecessary and the optimal control operation is equivalent to an optimal encoding of the state before it undergoes the open dynamics.",
"We also study the effect of applying the optimal control operations with some delay after the start of the open evolution of the state.",
"Applying an initial control operation is equivalent to optimally encoding information before the action of a noisy channel, an idea that has been proposed to preserve the multipartite entanglement and the QFI of multiqubit systems , and recently tested experimentally .",
"In a continuous variables setting, one may, along similar lines, use squeezing to minimise the decoherence of non-Gaussian states evolving in a Gaussian environment, which is formally equivalent to squeezing the state of the environment , , , .",
"This paper is structured as follows.",
"In Section , we introduce Gaussian systems and the dynamical model we will consider in the following.",
"In Section , we revise the basics of quantum parameter estimation, in particular applied to Gaussian systems.",
"Section contains our results about time-local optimal control applied to the preservation of the QFI of Gaussian quantum states.",
"Section ends the paper with some remarks and possible directions for future studies."
],
[
"System and dynamics",
"We shall consider a system of one bosonic, mode associated to a vector of canonical operators $\\hat{r} = (\\hat{x},\\hat{p})^{\\sf T}$ obeying the canonical commutation relations (CCR) $[\\hat{x}, \\hat{p}] = I\\hat{\\mathbb {1}} $ , where we have set $\\hbar =1$ .",
"One may also express the CCR in terms of the symmetrised commutator $[\\hat{r}, \\hat{r}^{\\sf T}] = \\hat{r} \\hat{r}^{\\sf T} - (\\hat{r}\\hat{r}^{\\sf T})^{\\sf T} =i\\Omega $ , where $\\Omega $ is a $2 \\times 2 $ matrix known as the symplectic form: $\\Omega = \\begin{pmatrix} 0 & 1 \\\\-1 & 0\\end{pmatrix}\\; .$ For a quantum state $\\hat{\\varrho }$ , the expectation value of the observable $\\hat{x}$ is given by $\\langle \\hat{x} \\rangle = \\text{Tr}[\\hat{\\varrho } \\hat{x}]$ .",
"Using vector notation, this can be generalised to give the first and second statistical moments of a state: $r= \\text{Tr}[\\hat{\\varrho } \\hat{r}] \\quad ,\\quad \\sigma = \\text{Tr}[\\lbrace (\\hat{r} - r), (\\hat{r} - r)^{\\sf T}\\rbrace \\hat{\\varrho }] \\, .$ The above definition leads to a real, symmetric CM ${\\sigma }$ , satisfying ${\\sigma }\\ge I\\Omega $ .",
"We will consider the encoding of classical information (a real parameter's value) in Gaussian states, which may in general be defined as the ground and thermal states of quadratic Hamiltonians.",
"Such states are fully characterised by first and second statistical moments, as defined above.",
"Unitary operations which map Gaussian states into Gaussian states are those generated by a quadratic Hamiltonian.",
"The effect of such operations on the vector of operators is a symplectic transformation $\\hat{r} \\rightarrow S \\hat{r}$ where $S$ is a $2 \\times 2$ real matrix belonging to the real symplectic group $\\mathrm {Sp}_{2,{\\mathbb {R}}}$ , i.e.",
"$S\\Omega S^{\\sf T} = \\Omega $ .",
"The corresponding effect on the CM of the system is the transformation $ \\sigma \\xrightarrow{} S \\sigma S^{\\sf T}$ , while first moments are transformed according to $r\\mapsto S r$ .",
"It will be convenient to parametrise single-mode symplectic transformations though their singular value decomposition : $S = \\left(\\begin{array}{cc}\\cos \\varphi &\\sin \\varphi \\\\-\\sin \\varphi &\\cos \\varphi \\end{array}\\right)\\left(\\begin{array}{cc}z & 0 \\\\0& \\frac{1}{z} \\end{array}\\right)\\left(\\begin{array}{cc}\\cos \\chi &\\sin \\chi \\\\-\\sin \\chi &\\cos \\chi \\end{array}\\right)\\, , $ for $\\chi ,\\varphi \\in [0,2\\pi [$ and $z\\ge 1$ .",
"From the normal mode decomposition of ${\\sigma }$ and Eq.",
"(REF ), it follows that the most general CM of a single-mode system may be written as a rotated and squeezed thermal state of the free Hamiltonian $(\\hat{x}^2+\\hat{p}^2)$ : ${\\sigma }= \\nu \\left(\\begin{array}{cc}\\cos \\theta &\\sin \\theta \\\\-\\sin \\theta &\\cos \\theta \\end{array}\\right)\\left(\\begin{array}{cc}y^2 &0\\\\0&\\frac{1}{y^2} \\end{array}\\right)\\left(\\begin{array}{cc}\\cos \\theta &-\\sin \\theta \\\\\\sin \\theta &\\cos \\theta \\end{array}\\right) \\, ,$ with squeezing parameter $y\\ge 1$ , optical phase $\\theta \\in [0,2\\pi [$ , we will also use the terms squeezing strength for the parameter $r$ , defined as $y=\\mathrm {e}^{r}$ , and squeezing angle $\\theta $ .",
"These parameters appear in the unitary transformation $\\hat{S}_{\\xi } = \\mathrm {e}^{ \\frac{1}{4} \\xi ^*\\hat{a}^2 - \\frac{1}{4} \\xi \\hat{a}^{\\dag 2}}$ , with the complex parameter $\\xi = r \\mathrm {e}^{I\\theta }$ , where we have introduced the bosonic annihilation operator $\\hat{a} = (\\hat{x} + I\\, \\hat{p})/\\sqrt{2}$ .",
"The symplectic eigenvalue $\\nu \\ge 1$ captures the temperature and purity of the state; states with $\\nu =1$ are pure, while the ground state of the free Hamiltonian (the “vacuum” state) is obtained by setting $y=\\nu =1$ , so that ${\\sigma }=\\mathbb {1}$ .",
"The free, uncontrolled dynamics of our system will be the diffusion induced by contact with a white noise (Markovian) environment at finite temperature, described by a Lindblad master equation $\\dot{\\hat{\\varrho }} = \\mathcal {L}_{\\bar{N}} \\hat{\\varrho } = \\left( \\bar{N} + 1\\right) \\mathcal {D} [ \\hat{a} ] \\hat{\\varrho } + \\bar{N} \\mathcal {D} [ \\hat{a}_i^\\dag ] \\hat{\\varrho }\\, , $ where we have introduced the superoperator $\\mathcal {D} \\left[ \\hat{o} \\right] \\!",
"\\hat{\\varrho } = \\hat{o} \\hat{\\rho } \\hat{o}^\\dag - \\left\\lbrace \\hat{o}^\\dag \\hat{o} , \\hat{\\rho } \\right\\rbrace $ and the Lindbladian $\\mathcal {L}_{\\bar{N}}$ , i.e.",
"the generator of the dynamical semigroup , .",
"We work with dimension-less time, expressed in units of the inverse loss rate, that thus does not appear in (REF ); $\\bar{N}$ is the mean number of excitations in the environment, related to its inverse temperature $\\beta $ by the Bose law.",
"This dynamics describes loss in a thermal environment and is ubiquitous in quantum optics.",
"Since the generator is time-independent, the solution from a time $t_0$ to a time $t$ is formally given by the map $\\mathrm {e}^{(t-t_0) \\mathcal {L}_{\\bar{N}}}$ .",
"This evolution is also known as the quantum attenuator channel.",
"At the level of Gaussian states this dynamics is described by the following equations of motion for first and second moments: $\\dot{{\\sigma }} &= -{\\sigma }+ N\\mathbb {1}\\; , \\\\\\dot{r} & = -\\frac{r}{2} \\; ,$ where $N=(2\\bar{N}+1)$ .",
"For a more general and detailed description of Gaussian quantum systems and dynamics similar to the one presented here see , .",
"We shall assume the ability to intersperse such an open dynamics with instantaneous Gaussian (IG) unitary operations $\\hat{U}_\\mathsf {IG}$ that will enact our control, which correspond to symplectic transformations in the phase-space description suited to Gaussian states.",
"In practice, this is a reasonable assumption if the inverse loss rate is much larger than the time it takes to perform a control operation.",
"Albeit such times may vary widely in specific cases, there exist optical set-ups with typical decoherence times around 10 $\\mu {\\rm s}$ and operating times around 1 ${\\rm ns}$ , where the hypothesis of instantaneous controls is very reasonable."
],
[
"Quantum parameter estimation",
"Let $\\hat{\\varrho }_\\theta $ be a set of quantum states whose exact form depends on an unknown parameter $\\theta $ .",
"The problem of the optimal estimation of $\\theta $ (i.e., of obtaining an estimator with minimal variance) through a fixed POVM $\\Pi = \\lbrace \\hat{\\Pi }_x \\ge 0 \\, , \\, \\sum _x \\hat{\\Pi }_x = \\mathbb {1}\\rbrace $ , is a classical problem associated with the probability distribution $p(x|\\theta )=\\mathrm {Tr}[\\hat{\\Pi }_x \\hat{\\varrho }_\\theta ]$ .",
"The optimal solution is given by the classical Fisher information $I_{\\Pi ,\\theta }$ (where we emphasise the dependence on the chosen POVM in the context pictured above), given by $I_{\\Pi ,\\theta } = \\sum _{\\mu } p(\\mu |\\theta ) \\left[\\partial _{\\theta }\\ln \\left(p(\\mu |\\theta )\\right)\\right]^2 =\\sum _{\\mu } \\frac{\\left(p^{\\prime }(\\mu |\\theta )\\right)^2}{p(\\mu |\\theta )} \\;$ (where the prime denotes the partial derivative with respect to the parameter $\\theta $ ), and by the associated Cramér–Rao bound, which may be saturated by unbiased estimators: $\\Delta \\theta \\ge \\frac{1}{\\sqrt{n I_{\\Pi ,\\theta }}} \\; , $ where $\\Delta \\theta $ is the standard deviation on the estimate of $\\theta $ and $n$ is the number of measurements carried out (in the language of statistics this corresponds to the sample size).",
"We stress that, while an unbiased estimator might not exist for finite $n$ , it is usually possible to find an estimator that saturates (REF ) in the limit of large $n$ , .",
"The optimisation of the classical Fisher information over all possible POVMs gives rise to the QFI $I_{\\theta }$ , , , : $I_{\\theta } = \\max _{\\Pi } I_{\\Pi ,\\theta } = \\lim _{\\epsilon \\rightarrow 0} \\frac{ 8 \\left(1 - F \\left[ \\hat{\\varrho }_\\theta , \\hat{\\varrho }_{\\theta + \\epsilon } \\right] \\right) }{ \\epsilon ^2 } \\; , $ where, as above, the symbol $\\Pi $ stands for the whole set of POVMs and the QFI is expressed in terms of the fidelity $F \\left[ \\varrho , \\sigma \\right] = \\left\\Vert \\sqrt{\\varrho } \\sqrt{\\sigma } \\right\\Vert _1$ , where $\\Vert A \\Vert _1 = \\mathrm {Tr} \\left[ \\sqrt{A A^\\dag } \\right]$ is the trace norm.",
"One can prove that the QFI is monotonically decreasing when parameter independent channels are applied to the state, while it is invariant if the transformation is a unitary , .",
"As one would expect, the QFI enters the quantum Cramér–Rao bound $\\Delta \\theta \\ge \\frac{1}{\\sqrt{ n I_{\\theta }}} \\; .",
"$ Since this inequality may be shown to be achievable, it represents the ultimate bound to quantum parameter estimation.",
"The QFI is therefore the most fundamental quantity in assessing the sensitivity of a system to a certain parameter, which might reflect environmental or technical factors.",
"Just as in the classical case, the inequality (REF ) is usually saturated only in the asymptotic limit.",
"However the optimal POVM $\\mathrm {arg \\,max}_{\\Pi } I_{\\Pi ,\\theta }$ might depend on the true unknown value $\\theta $ , thus some kind of adaptive strategy is needed in general .",
"In other words, the information quantified by the QFI makes sense in a local estimation scenario, where we assume to have prior knowledge about the parameter value and to be in a neighbourhood of the true value of the parameter.",
"If the quantum states $\\hat{\\varrho }_{\\theta }$ are all Gaussian states, and therefore the dependence on $\\theta $ is entirely contained in the CM ${\\sigma }$ and in the vector of first moments $r$ , one may obtain an analytical formula for the QFI.",
"Such a formula is particularly wieldy for single-mode Gaussian states , , , , , on which we shall focus in this paper: $I_{\\theta } = \\frac{1}{2}\\frac{{\\rm Tr}[({\\sigma }^{-1}{\\sigma }^{\\prime })^2]}{1+\\mu ^2} +\\frac{2\\mu ^{\\prime 2}}{1-\\mu ^4}+ 2{r}^{\\prime \\sf T}{\\sigma }^{-1}r^{\\prime } \\; ,$ where $\\mu =\\mathrm {Tr}[\\hat{\\varrho }_{\\theta }^2]=1/\\sqrt{\\mathrm {Det}{\\sigma }}=1/\\nu $ is the purity of the quantum states and the prime $^{\\prime }$ denotes differentiation with respect to the parameter $\\theta $ .",
"Notice that, in all of the formulae above, the derivatives are taken at the `true' value $\\bar{\\theta }$ of the parameter $\\theta $ , so that both the QFI and the optimal POVM will in general depend on $\\bar{\\theta }$ (more on this issue later).",
"In the following, it will be convenient to make the distinction between $\\theta $ and $\\bar{\\theta }$ explicit and clear."
],
[
"Locally optimal control to protect the quantum Fisher information",
"Later on, we shall assume Gaussian states depending on an unknown optical phase or squeezing parameter and subject to the diffusive dynamics (REF ,), and assess the performance of instantaneous control symplectics towards the task of parameter estimation.",
"We would like therefore to determine controls that maximise the evolving QFI associated with phase estimation.",
"Notice that, although the QFI is obviously invariant under unitary, and hence symplectic, operations, its time-derivative under the free open dynamics we are considering need not be, and in fact is not.",
"Therefore, symplectic controls may enhance the QFI during the time evolution.",
"Our first step is then to obtain a general expression for the time-derivative of the QFI (REF ) under Eqs.",
"(REF ) and (), which is derived in and turns out to be rather compact: $& \\dot{I_{\\theta }} = \\frac{\\mu ^2{\\rm Tr}[({\\sigma }^{-1}{\\sigma }^{\\prime })^2](N{\\rm Tr}[{\\sigma }^{-1}]-2)}{2(1+\\mu ^2)^2}-\\frac{N{\\rm Tr}[({\\sigma }^{-1}{\\sigma }^{\\prime })^2{\\sigma }^{-1}]}{1+\\mu ^2} \\nonumber \\\\&- \\frac{\\mu ^2 {\\rm Tr}[{\\sigma }^{-1}{\\sigma }^{\\prime }]\\left((N{\\rm Tr}[{\\sigma }^{-1}]-2){\\rm Tr}[{\\sigma }^{-1}{\\sigma }^{\\prime }] +2N {\\rm Tr}[{\\sigma }^{-2}{\\sigma }^{\\prime }] \\right)}{2(1-\\mu ^4)} \\nonumber \\\\& +\\frac{\\mu ^6}{(1-\\mu ^4)^2}{\\rm Tr}[{\\sigma }^{-1}{\\sigma }^{\\prime }]^2(2-N{\\rm Tr}[{\\sigma }^{-1}]) - 2N{\\bf r}^{\\prime \\sf T} {\\sigma }^{-2}{\\bf r}^{\\prime } \\; .",
"$ The “locally” (in time) optimal control will be determined by letting ${\\sigma }\\mapsto S {\\sigma }S^{\\sf T}$ (and likewise for ${\\sigma }^{\\prime }$ ) and ${\\bf r}\\mapsto S\\mathbf {r}$ in the expression above, and by maximising it with respect to the parameters $\\chi $ , $\\varphi $ and $z$ that parametrise the control $S$ as per Eq.",
"(REF ).",
"Inspection reveals that the controlled $\\dot{I_{\\theta }}$ does not depend on the first rotation in the singular value decomposition of $S$ , so that we can set $\\chi =0$ in what follows without loss of generality.",
"Also note that the control can never help preserving the QFI associated with first moments, since the only term which depends on them is clearly invariant under any $S$ .",
"We will therefore neglect first moments hereafter."
],
[
"Estimation of the squeezing angle",
"As a first case study, let us consider the estimation of the optical phase $\\theta $ of a squeezed state with first moments independent from $\\theta $ and CM of the form of Eq.",
"(REF ).",
"Notice that this also encompasses the case of a state undergoing the dynamics described by Eqs.",
"(REF ,) for any initial transient time, since the dynamics is phase-covariant, i.e.",
"it commutes with the action of the phase shifter imprinting the dependence on the parameter $\\theta $ .",
"In other words, the only difference between considering the instantaneous control acting at the beginning or at some intermediate time is reflected in different values for the parameters $\\nu $ and $y$ , but the optimization problem to be solved remains identical.",
"It is now convenient to define $R_{\\theta } = \\left(\\begin{array}{cc}\\cos \\theta &\\sin \\theta \\\\-\\sin \\theta &\\cos \\theta \\end{array}\\right)$ and observe that $R_{\\theta }^{\\prime } &= R_{\\bar{\\theta }} \\Omega = \\Omega R_{\\bar{\\theta }} \\; , \\\\R_{\\theta }^{\\sf T \\prime } &= R_{-\\theta }^{\\prime } = - R_{-\\bar{\\theta }} \\Omega = -\\Omega R_{-\\bar{\\theta }} \\;$ (recall that all the derivatives with respect to the parameter $\\theta $ must be taken at the true value $\\bar{\\theta }$ ), where $\\Omega $ is the $2\\times 2$ symplectic form.",
"Let us also define the diagonal squeezing matrices $Y^{2}={\\rm diag}(y^2,1/y^2)$ , which characterises the initial state, and $Z={\\rm diag}(z,1/z)$ , characterising the control operation.",
"Including the transformation (REF ) for $\\chi =0$ , one has the following expressions for the controlled ${\\sigma }^{-1}$ and ${\\sigma }^{\\prime }$ associated to the estimation of the phase of squeezing: ${\\sigma }^{-1} &= \\nu ^{-1} Z^{-1} R_{(\\bar{\\theta }+\\varphi )} Y^{-2} R_{-(\\bar{\\theta }+\\varphi )} Z^{-1} \\; , \\\\{\\sigma }^{\\prime } &= \\nu Z R_{(\\bar{\\theta }-\\varphi )} [\\Omega ,Y^2] R_{(\\varphi -\\bar{\\theta })} Z \\nonumber \\\\ &=\\nu \\left(\\frac{1}{y^2}-y^2\\right)Z R_{(\\bar{\\theta }-\\varphi )} \\sigma _x R_{(\\varphi -\\bar{\\theta })} Z \\, ,$ where $\\sigma _x$ is the Pauli $x$ matrix.",
"Whence ${\\sigma }^{-1}{\\sigma }^{\\prime } = \\left(\\frac{1}{y^2}-y^2\\right) Z^{-1} R_{(\\bar{\\theta }+\\varphi )} Y^{-2} \\sigma _x R_{-(\\bar{\\theta }+\\varphi )} Z$ and therefore, in this case, ${\\rm Tr}[{\\sigma }^{-1}{\\sigma }^{\\prime }] = \\nu ^{-1} \\left(\\frac{1}{y^2}-y^2\\right) {\\rm Tr}[Y^{-2}\\sigma _x] = 0 \\,,$ which simplifies our task greatly, since it sets to zero the third and fourth term in Eq.",
"(REF ) which, noticing that ${\\rm Tr}[({\\sigma }^{-1}{\\sigma }^{\\prime })^2]= 2\\left(\\frac{1}{y^2}-y^2\\right)$ and ${\\rm Tr}[({\\sigma }^{-1}{\\sigma }^{\\prime })^2{\\sigma }^{-1}]=\\left(\\frac{1}{y^2}-y^2\\right) {\\rm Tr}[{\\sigma }^{-1}]$ , reduces to: $\\dot{I_{\\theta }} = -\\left(\\frac{1}{y^2}-y^2\\right) \\frac{2\\nu ^2 + N\\nu ^2(2\\nu ^2 + 1){\\rm Tr}[{\\sigma }^{-1}]}{2(\\nu ^2+1)^2} \\; .$ The optimal control operation is therefore the one that minimises ${\\rm Tr}[{\\sigma }^{-1}]$ , whose coefficient above is negative.",
"It is apparent from the general expression of a single-mode CM (REF ) that, up to the symplectic eigenvalue that is not affected by a symplectic transformation, ${\\rm Tr}{\\sigma }^{-1}$ is minimised, obtaining the value $2/\\nu $ , by making the CM proportional to the identity, which can always be done through a symplectic control that undoes the initial rotation $\\bar{\\theta }$ and squeezing $z$ , i.e.",
"by choosing $\\varphi = - \\bar{\\theta }$ and $z=1/y$ .",
"Notice that such an optimal control depends on the unknown value $\\bar{\\theta }$ .",
"This is however not too worrisome in practice, as such a value may be estimated with a first run of measurements without controls, which would provide the hypothetical experimentalist with an approximation of the optimal transformation.",
"This situation is common for local quantum estimation theory and it is analogous to the well known fact that the optimal POVM, attaining the quantum Cramér–Rao bound, also depends on $\\bar{\\theta }$ .",
"Moreover, very remarkably, once the initial squeezing is undone and the state is brought in the optimal form, no further control is required to maintain optimality, since the free diffusive dynamics does not change the form of the (unsqueezed) CM: a single manipulation is therefore optimal among all possible control strategies.",
"Figure: Schematic representation of the control protocol we consider.A squeezed vacuum state ϱ ^ ξ =S ^ ξ |0〉〈0|S ^ ξ † \\hat{\\varrho }_\\xi = \\hat{S}_\\xi |0 \\rangle \\langle 0 |\\hat{S}_\\xi ^\\dag is subjected to the open dynamics describing loss in a thermal environment and obeys the master equation (), or equivalently equation () for the CM, since the state has zero first moments.At time t c t_c (in units of the inverse loss rate) a control operation is applied to the system, in the form of a instantaneous Gaussian unitary 𝒰 𝖨𝖦 ϱ ^=U ^ 𝖨𝖦 ϱ ^U ^ 𝖨𝖦 † \\mathcal {U}_{\\mathsf {IG}} \\hat{\\varrho } = \\hat{U}_\\mathsf {IG} \\hat{\\varrho } \\hat{U}_\\mathsf {IG}^\\dag , with the aim of slowing down the decrease in metrological usefulness of the state, quantified by the QFI.The evolution then proceeds to the final time tt and the evolved state ϱ ^ ξ,t \\hat{\\varrho }_{\\xi ,t} is measured.The metrological task is to estimate the value of either φ\\phi (squeezing angle) or rr (squeezing strength), where ξ=re iφ \\xi = r \\mathrm {e}^{i \\phi }.Summing up, we have shown that the sensitivity of squeezed states to their optical phase is enhanced if the squeezing is undone through a single control operation.",
"In order to quantify the advantage granted by optimal instantaneous symplectic controls, it is expedient to assume that, at the time $t=0$ , an initial pure squeezed vacuum state $\\hat{\\rho }_\\xi = \\hat{S}_{\\xi } |0\\rangle \\langle 0 | \\hat{S}_{\\xi }^\\dag $ , with any given $\\bar{\\theta }$ and $y$ and $\\nu =1$ , starts being subject to the diffusive dynamics (REF ) and then undergoes optimal control at a time $t_c\\ge 0$ .",
"This setup is schematically represented in Fig.",
"REF .",
"Before the control is applied (i.e., for $t<t_c$ ), one has ${\\sigma }&= {\\rm e}^{-t} R_{\\bar{\\theta }} Y^2 R_{-\\bar{\\theta }} + (1-{\\rm e}^{-t})N\\mathbb {1}\\; , \\\\{\\sigma }^{\\prime }&= {\\rm e}^{-t} \\left(\\frac{1}{y^2}-y^2\\right) R_{\\bar{\\theta }} \\sigma _x R_{-\\bar{\\theta }} \\; ,$ which may be inserted into Eq.",
"(REF ) to obtain the QFI of the evolved state (as shown in , the second term on the RHS of Eq.",
"(REF ) is proportional to ${\\rm Tr}[{\\sigma }^{-1}{\\sigma }^{\\prime }]$ and hence vanishes in our case, and so does the third as the first moments do not contribute to the estimation).",
"After the control is applied, for $t\\ge t_c$ , one has instead ${\\sigma }&= {\\rm e}^{-t} Y_{c}^{-1} R_{(\\bar{\\theta }-\\varphi _c)} Y^2 R_{-(\\bar{\\theta }-\\varphi _c)} Y_{c}^{-1} +(1-{\\rm e}^{-t})N Y_{c}^2\\nonumber \\\\&= \\left({\\rm e}^{-(t-t_c)}\\nu _c +(1-{\\rm e}^{-(t-t_c)})N \\right) \\mathbb {1}\\; , \\\\{\\sigma }^{\\prime }& = {\\rm e}^{-t}{\\rm e}^{-t} \\left(\\frac{1}{y^2}-y^2\\right) Y_{c}^{-1} \\sigma _x Y_{c}^{-1} ={\\rm e}^{-t} \\left(\\frac{1}{y^2}-y^2\\right) \\sigma _x \\; ,$ where $\\varphi _c=\\bar{\\theta }$ (the distinction between the two has been maintained to make the derivation of ${\\sigma }^{\\prime }$ clearer), the diagonal squeezing transformation $Y_{c}$ is chosen ad hoc to make the CM proportional to the identity, and $\\nu _c = \\sqrt{\\Big [{\\rm e}^{-t_c}y^2+(1-{\\rm e}^{-t_c})N\\Big ]\\Big [\\frac{{\\rm e}^{-t_c}}{y^2}+(1-{\\rm e}^{-t_c})N\\Big ]}$ is the symplectic eigenvalue of the evolving state at the moment the control is enacted.",
"Figure: The QFI for the estimation of the squeezing angle θ\\theta as a function of time (both dimensionless, the latter in units of inverse loss rate), shown for different times at which the locally optimal control is applied:the top (blue) curve depicts the case where the control is applied at t=0t=0; at each subsequent curve, from top to bottom, the control is applied 0.050.05 inverse loss rates later; the bottom curve (green) depicts the uncontrolled QFI.In the upper panel we consider a system with initial squeezing y=3y=3 and an open dynamics with N=1N=1 (“pure loss”), while in the lower panel we consider y=10y=10 and N=2N=2.The calculations detailed in the two previous paragraphs yield the following expression for the “controlled” QFI as a function of time: $I_{\\theta } = \\left\\lbrace \\begin{array}{ll} \\frac{{\\rm e}^{-2t} \\left(\\frac{1}{y^2}-y^2\\right)^2}{\\Big [{\\rm e}^{-t}y^2+(1-{\\rm e}^{-t})N\\Big ] \\Big [\\frac{{\\rm e}^{-t}}{y^2}+(1-{\\rm e}^{-t})N\\Big ]+1} \\; , &\\; t<t_c\\, , \\\\\\\\\\frac{{\\rm e}^{-2t} \\left(\\frac{1}{y^2}-y^2\\right)^2}{ \\left({\\rm e}^{-(t-t_c)} \\nu _c + (1-{\\rm e}^{-(t-t_c)})N\\right)^2 +1} \\; , &\\; t\\ge t_c \\, .\\end{array} \\right.$ The effect of such a control scheme on the QFI is illustrated in Fig.",
"REF , where one may appreciate the advantage gained by activating the control protocol after different intervals from the initial time.",
"By comparing the two panels we can also appreciate on a qualitative level that an initial control operation is more useful when the initial squeezing is high, since in this case the uncontrolled QFI drops more steeply."
],
[
"Estimation of the squeezing strength",
"The same analysis carried out above for the squeezing phase may be repeated for the squeezing strength of the initial state.",
"To this aim, it is convenient to re-parametrise the state, setting $y^2 = \\mathrm {e}^{r}$ , so that $Y^{2}={\\rm e}^{r \\sigma _z}$ ($\\sigma _z$ being the Pauli $z$ matrix), and consider the estimation of the parameter $r$ .",
"The problem becomes then completely phase-invariant, so we can omit the rotation in the evolving state without loss of generality, and just assume the CM $\\nu {\\rm e}^{r \\sigma _z}$ , for $r>0$ and $\\nu \\ge 1$ .",
"The controlled ${\\sigma }^{-1}$ and ${\\sigma }^{-1}$ associated to the estimation of the squeezing parameter $r$ take the form ${\\sigma }^{-1} &= \\nu ^{-1} Z R_{-\\varphi } {\\rm e}^{-r\\sigma _z} R_{\\varphi } Z \\; , \\\\{\\sigma }^{\\prime } &= \\nu Z^{-1} R_{-\\varphi } {\\rm e}^{r\\sigma _z} \\sigma _z R_{\\varphi } Z^{-1} \\; .$ Whence ${\\sigma }^{-1}{\\sigma }^{\\prime } = Z R_{-\\varphi } \\sigma _z R_{\\varphi } Z^{-1} \\; ,$ such that, once again, ${\\rm Tr}[{\\sigma }^{-1}{\\sigma }^{\\prime }] = 0$ , and ${\\rm Tr}[({\\sigma }^{-1}{\\sigma }^{\\prime })^2]= 2$ , ${\\rm Tr}[({\\sigma }^{-1}{\\sigma }^{\\prime })^2{\\sigma }^{-1}]={\\rm Tr}[{\\sigma }^{-1}]$ , leading to $\\dot{I_r} = - \\frac{2\\nu ^2 + N\\nu ^2(2\\nu ^2 + 1){\\rm Tr}[{\\sigma }^{-1}]}{2(\\nu ^2+1)^2} \\; .$ In this case too, the optimal control operation is the one that minimises ${\\rm Tr}[{\\sigma }^{-1}]$ that, as above, is obtained by setting $Z = {\\rm e}^{r\\sigma _z}$ and $\\varphi =0$ .",
"The optimal strategy requires a single control, since the free dynamics preserve the optimal form of the CM (of minimal ${\\rm Tr}[{\\sigma }^{-1}]$ upon symplectic action).",
"Figure: The QFI for the estimation of the squeezing strength as a function of time (both dimensionless, the latter in units of inverse loss rate), shown for different times at which the locally optimal control is applied:the top (blue) curve depicts the case where the control is applied at t=0t=0; at each subsequent curve, from top to bottom, the control is applied 0.050.05 inverse loss rates later; the bottom curve (green) depicts the uncontrolled QFI.In the upper panel we consider a system with initial squeezing y=3y=3 and an open dynamics with N=1N=1 (“pure loss”), while in the lower panel we consider y=10y=10 and N=2N=2.Notice that the formulae for the estimation of the squeezing parameter are identical to what found above, except for the absence of the factor $(y^{-2}-y^2)$ in ${\\sigma }^{\\prime }$ (clearly, for the optical phase, the QFI is always equal to 0 for $y=1$ , because the initial state is then rotationally invariant; this is not an issue for squeezing, since such an operation admits no invariant states).",
"Assuming, like above, to start with a pure squeezed vacuum state $\\hat{\\rho }_\\xi = \\hat{S}_{\\xi } |0\\rangle \\langle 0 | \\hat{S}_{\\xi }^\\dag $ , and that the control is applied at some time $t_c\\ge 0$ , one has the rescaled formula for the QFI as a function of time: $I_r = \\left\\lbrace \\begin{array}{ll} \\frac{{\\rm e}^{-2t}}{\\big [{\\rm e}^{-t}y^2+(1-{\\rm e}^{-t})N\\big ]\\big [\\frac{{\\rm e}^{-t}}{y^2}+(1-{\\rm e}^{-t})N\\big ]+1} \\; , &\\; t<t_c\\, , \\\\\\\\\\frac{{\\rm e}^{-2t}}{ \\big [{\\rm e}^{-(t-t_c)} \\nu _c + (1-{\\rm e}^{-(t-t_c)})N\\big ]^2+1} \\; , &\\; t\\ge t_c \\, .\\end{array} \\right.$ with $\\nu _c^2 = \\big [{\\rm e}^{-t_c}y^2+(1-{\\rm e}^{-t_c})N\\big ]\\big [\\frac{{\\rm e}^{-t_c}}{y^2}+(1-{\\rm e}^{-t_c})N\\big ]$ , exactly as in the previous section.",
"We plot this quantity in Fig.",
"REF : the behaviour in time is qualitatively the same as the previous case of phase estimation, albeit the scale is different because of the absence of the factor $(y^{-2}-y^2)^2$ in the expression for $I_r$ .",
"In this case the initial QFI at $t=0$ is independent from the parameter and equal to $1/2$ .",
"While the phase estimation considered in the previous section is not possible with an unsqueezed state, the strength can instead be estimated at the true value $r=0$ .",
"However, as one would expect, in this particular case there is no need for control operations since the state is already thermal; this can also be checked explicitly from (REF ) by setting $y=1$ ."
],
[
"Conclusions and remarks",
"We have only started to uncover the usefulness of time-local quantum control for the task of parameter estimation.",
"In particular, Gaussian systems proved to be a useful test bed for these strategies, since their simplicity allowed us to derive closed form expressions for the relevant quantities at play.",
"Essentially, we have shown that, for the considered open dynamics, the locally-optimal (in time) way to delay the decay of QFI about a parameter unitarily encoded in the CM of a single-mode Gaussian state is to unsqueeze it, thereby transforming it into a thermal state.",
"This means that such a thermal state is best suited to withstand a lossy evolution in a thermal environment, which is mathematically described by a phase-covariant channel.",
"Intriguingly, our results might be related to the fact that the minimum output entropy of a phase-covariant Gaussian channel is achieved by a thermal input state , , although we lack a deeper understanding of this connection.",
"Several extensions of the ideas we have proposed here can be envisioned, most notably dropping the assumption of fast parameter encoding.",
"This means that, instead of only delaying the demise of the QFI of an initial state, we should consider the situation where the parameter is encoded simultaneously to the open dynamics, e.g., by adding an Hamiltonian term in the Lindblad master equation or estimating parameters of the non-unitary part of the Gaussian dynamics, see, e.g., , , , , , , .",
"In such scenarios time-local control would be used to increase the rate at which information about the parameter is acquired during the dynamics.",
"Finally, let us briefly mention that there exist dynamical decoupling schemes tailored for continuous variable systems , that could in principle be used to completely remove the effect of the environment.",
"However, in dynamical decoupling the control operations must be not only instantaneous, but they have to be applied with a rate greater than the environment cut-off frequency and are thus very hard to implement in practice.",
"On the other hand our proposed strategy, while only capable of mitigating the effect of noise, can be readily implemented on experimental platforms through a single control operation."
],
[
"Acknowledgements",
"We are grateful to D. Branford, A. Datta and M. G. Genoni for useful discussions.",
"FA acknowledges financial support from the UK National Quantum Technologies Programme (EP/M013243/1) and from the National Science Center (Poland) grant No.",
"2016/22/E/ST2/00559."
],
[
"Derivation of the time derivative of the QFI",
"The starting point of this derivation is Eq.",
"(REF ) for the QFI of a single mode state together with the equations of motion of the first an second statistical moments of the Gaussian state (REF ) and ().",
"The time derivative of the QFI (REF ) is $\\begin{split}\\dot{I}_{\\theta } = & \\frac{1}{2} \\frac{(1+\\mu ^2) \\mathrm {Tr}[ \\partial _t \\left( {\\sigma }^{-1} {\\sigma }\\right)^2 ] - \\mathrm {Tr}[ \\left( {\\sigma }^{-1} {\\sigma }\\right) ] \\partial _t \\mu ^2 }{(1+\\mu ^2)^2} \\\\& + \\frac{2 (1 - \\mu ^4) \\partial _t ({\\mu ^{\\prime }}^2) + 2 \\mu ^{\\prime 2} \\partial _t \\mu ^4 }{(1-\\mu ^4)^2} \\\\&+ 2 (\\partial _t r^{\\prime })^^{-1} r^{\\prime } + 2 r^{\\prime {\\sigma }^{-1} (\\partial _t r)+ 2 r^{\\prime (\\partial _t {\\sigma }^{-1}) r^{\\prime }}}\\end{split}Finally, we need the equation of motions for the derivatives of the first moment vector and of the CM, obtained by differentiating~(\\ref {difsig}) and~(\\ref {difr}) with respect to the parameter:{\\begin{@align}{1}{-1}\\dot{{\\sigma }^{\\prime }}= - {\\sigma }^{\\prime } \\\\\\dot{r^{\\prime }}= - \\frac{r^{\\prime }}{2} .\\end{@align}}$ In this calculation we will use the the following formula for the derivative of the determinant of an invertible matrix $A(t)$ : $\\frac{\\mathrm {d}}{\\mathrm {d} t} \\mathrm {Det} A(t) = \\mathrm {Det} A(t) \\mathrm {Tr}\\left[ A(t)^{-1}\\frac{\\mathrm {d}}{\\mathrm {d}t} A(t) \\right] $ , as well as the formula for the derivative of the inverse matrix $\\frac{\\mathrm {d}}{\\mathrm {d} t} A(t)^{-1} =- A(t)^{-1} \\left( \\frac{\\mathrm {d}}{\\mathrm {d}t} A(t) \\right) A(t)^{-1}$ ."
],
[
"First term",
"We first start by noticing that $\\mathrm {Tr}\\left[ \\partial _t \\left( {\\sigma }^{-1} {\\sigma }^{\\prime } \\right)^2 \\right] = 2 \\mathrm {Tr}\\left[ ( {\\sigma }^{-1} {\\sigma }^{\\prime }) \\partial _t ( {\\sigma }^{-1} {\\sigma }^{\\prime } ) \\right] ,$ where we have used the cyclicity of the trace.",
"This term can be simplified as follows $\\begin{split}& ( {\\sigma }^{-1} {\\sigma }^{\\prime }) \\partial _t ( {\\sigma }^{-1} {\\sigma }^{\\prime } ) = ( {\\sigma }^{-1} {\\sigma }^{\\prime }) (\\partial _t {\\sigma }^{-1} {\\sigma }^{\\prime } + {\\sigma }^{-1} \\partial _t {\\sigma }^{\\prime } ) \\\\&= ( {\\sigma }^{-1} {\\sigma }^{\\prime }) (-{\\sigma }^{-1} (\\partial _t {\\sigma }) {\\sigma }^{-1} {\\sigma }^{\\prime } + {\\sigma }^{-1} \\partial _t {\\sigma }^{\\prime } ) \\\\&= ( {\\sigma }^{-1} {\\sigma }^{\\prime }) (-{\\sigma }^{-1} (-{\\sigma }+ N \\mathbb {1}) {\\sigma }^{-1} {\\sigma }^{\\prime } + {\\sigma }^{-1} (-{\\sigma }^{\\prime }) ) \\\\& = - N {\\sigma }^{-1} {\\sigma }^{\\prime } {\\sigma }^{-2} {\\sigma }^{\\prime } \\, ,\\end{split}$ so that we obtain $\\mathrm {Tr}\\left[ \\partial _t \\left( {\\sigma }^{-1} {\\sigma }^{\\prime } \\right)^2 \\right] = -2 N \\mathrm {Tr}\\left[ ({\\sigma }^{-1}{\\sigma }^{\\prime })^2 {\\sigma }^{-1}\\right] \\,.$ The second part of the first term's numerator can be expanded using the following identity $\\begin{split}\\partial _t \\mu ^2 &= \\partial _t ( \\mathrm {Det} {\\sigma })^{-1} = - \\frac{1}{(\\mathrm {Det} {\\sigma })^2} ( \\partial _t \\mathrm {Det}{\\sigma }) \\\\& = - \\frac{1}{\\mathrm {Det} {\\sigma }} \\mathrm {Tr}\\left[ {\\sigma }^{-1} \\partial _t {\\sigma }\\right] \\\\& = - \\frac{1}{\\mathrm {Det} {\\sigma }} \\mathrm {Tr}\\left[ {\\sigma }^{-1} (-{\\sigma }+ N \\mathbb {1}) \\right] \\\\& = - \\mu ^2 \\left( N \\mathrm {Tr}[ {\\sigma }^{-1} ] -2 \\right) \\,.\\end{split}$ Using Eqs.",
"(REF ) and (REF ), from the first term in (REF ) we obtain the first line of Eq.",
"(REF ).",
"First of all, using the definition of the purity $\\mu =(\\mathrm {Det}{\\sigma })^{-1/2}$ we find that $\\mu ^{\\prime } = - \\frac{1}{2} (\\mathrm {Det}{\\sigma })^{-1/2} \\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime } \\right] = - \\frac{\\mu }{2} \\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime } \\right]$ and analogously $\\partial _t \\mu = - \\frac{\\mu }{2} \\mathrm {Tr}\\left[ {\\sigma }^{-1} \\partial _t {\\sigma }\\right] \\,.$ The first term we need to evaluate from the second line of Eq.",
"(REF ) is the following $\\partial _t ({\\mu ^{\\prime }}^2) = 2 \\mu ^{\\prime } (\\partial _t \\mu ^{\\prime }) = -\\mu \\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime } \\right] (\\partial _t \\mu ^{\\prime }),$ where now we need to evaluate this last term $\\begin{split}\\partial _t \\mu ^{\\prime } &= -\\frac{1}{2} (\\partial _t \\mu ) \\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime }\\right] -\\frac{1}{2} \\mu \\left( \\partial _t \\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime }\\right] \\right) \\\\& = \\frac{\\mu }{4} \\mathrm {Tr}\\left[ {\\sigma }^{-1} \\partial _t {\\sigma }\\right] \\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime }\\right] \\\\& - \\frac{\\mu }{2} \\left( -\\mathrm {Tr}\\left[ {\\sigma }^{-1} (\\partial _t {\\sigma }) {\\sigma }^{-1} {\\sigma }^{\\prime }\\right] + \\mathrm {Tr}\\left[ {\\sigma }^{-1} (\\partial _t {\\sigma }^{\\prime }) \\right] \\right) \\\\&= \\frac{\\mu }{4} \\mathrm {Tr}\\left[ {\\sigma }^{-1} (-{\\sigma }+ N \\mathbb {1}) \\right] \\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime }\\right] \\\\& - \\frac{\\mu }{2} \\left( -\\mathrm {Tr}\\left[ {\\sigma }^{-1} (-{\\sigma }+ N \\mathbb {1}) {\\sigma }^{-1} {\\sigma }^{\\prime }\\right] + \\mathrm {Tr}\\left[ {\\sigma }^{-1} (-{\\sigma }^{\\prime }) \\right] \\right) \\\\&= \\frac{\\mu }{4} \\left\\lbrace \\left( N \\mathrm {Tr}\\left[ {\\sigma }^{-1} \\right] -2 \\right)\\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime }\\right] + 2 N \\mathrm {Tr}\\left[ {\\sigma }^{-2} {\\sigma }^{\\prime }\\right] \\right\\rbrace \\end{split}$ We thus get to $\\begin{split}\\partial _t ({\\mu ^{\\prime }}^2) = -\\frac{\\mu ^2}{4} \\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime } \\right] &\\biggl \\lbrace \\left( N \\mathrm {Tr}\\left[ {\\sigma }^{-1} \\right] -2 \\right)\\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime }\\right] \\\\& + 2 N \\mathrm {Tr}\\left[ {\\sigma }^{-2} {\\sigma }^{\\prime }\\right] \\biggr \\rbrace \\end{split}$ For the next part of the second line of Eq.",
"(REF ) we need to evaluate the following term $\\begin{split}\\mu ^{\\prime 2} \\partial _t \\mu ^4 &= \\left(- \\frac{\\mu }{2} \\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime } \\right] \\right)^2 4 \\mu ^3 \\left( -\\mu \\mathrm {Tr}\\left[ {\\sigma }^{-1} \\partial _t {\\sigma }\\right] \\right) \\\\& = - \\mu ^6 \\mathrm {Tr}\\left[ {\\sigma }^{-1} {\\sigma }^{\\prime } \\right] \\left( N \\mathrm {Tr}\\left[ {\\sigma }^{-1} \\right] - 2 \\right) .\\end{split}$ From Eqs.",
"(REF ), (REF ) and (REF ) we obtain the first term on the third line of Eq.",
"(REF ).",
"We consider the three terms on the last line of (REF ) together and we find $\\begin{split}&2 (\\partial _t r^{\\prime })^^{-1} r^{\\prime } + 2 r^{\\prime {\\sigma }^{-1} (\\partial _t r)+ 2 r^{\\prime } (\\partial _t {\\sigma }^{-1}) r^{\\prime } \\\\&= - 2 {r^{\\prime }}^^{-1} r^{\\prime } - 2 r^{\\prime {\\sigma }^{-1} (\\partial _t {\\sigma }) {\\sigma }^{-1} r^{\\prime } \\\\& = - 2 {r^{\\prime }}^^{-1} r^{\\prime } - 2 r^{\\prime {\\sigma }^{-1} (-{\\sigma }+ N \\mathbb {1}) {\\sigma }^{-1} r^{\\prime } \\\\& = - 2 N r^{\\prime {\\sigma }^{-2} r^{\\prime } \\; ,}}which is the last term in Eq.~(\\ref {idiot}).",
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[
[
"A Differentiable Recurrent Surface for Asynchronous Event-Based Data"
],
[
"Abstract Dynamic Vision Sensors (DVSs) asynchronously stream events in correspondence of pixels subject to brightness changes.",
"Differently from classic vision devices, they produce a sparse representation of the scene.",
"Therefore, to apply standard computer vision algorithms, events need to be integrated into a frame or event-surface.",
"This is usually attained through hand-crafted grids that reconstruct the frame using ad-hoc heuristics.",
"In this paper, we propose Matrix-LSTM, a grid of Long Short-Term Memory (LSTM) cells that efficiently process events and learn end-to-end task-dependent event-surfaces.",
"Compared to existing reconstruction approaches, our learned event-surface shows good flexibility and expressiveness on optical flow estimation on the MVSEC benchmark and it improves the state-of-the-art of event-based object classification on the N-Cars dataset."
],
[
"Implementation",
"The Matrix-LSTM feature extraction process can be implemented efficiently by means of two order-aware reshape operations.",
"These two operations, namely groupByPixel and groupByTime, allow event streams to be split based on the pixel location and temporal bin.",
"After being reshaped, the input is ready to be processed by a single LSTM network, implementing parameter sharing across all pixel locations and temporal windows.",
"In the following we give a detailed overview of the two reshape operators.",
"This operation translates from event-sequences to pixel-sequences.",
"Let $X$ be a tensor of shape $N \\times T_{max} \\times F$ , representing the features $f^{(x,y)}_{n,i}$ of a batch of $N$ samples, where $T_{max}$ is the length of the longest sequence in the batch.",
"We define the groupByPixel mapping on $X$ as an order-aware reshape operation that rearranges the events into a tensor of pixel-sequences of shape $P \\times T^{(x,y)}_{max} \\times F$ where $T^{(x,y)}_{max}$ is the length of the longest pixel sequence $\\mathcal {E}^{(x,y)}_n$ and $P$ is the number of active pixels (i.e., having at least one event) in the batch, which equals $N \\cdot H \\cdot W$ only in the worst case.",
"Pixel-sequences shorter than $T^{(x,y)}_{max}$ are padded with zero events to be processed in parallel.",
"The tensor thus obtained is then processed by the LSTM cell that treats samples in the first dimension independently, effectively implementing parameter sharing and applying the transformation in parallel over all the pixels.",
"The LSTM output tensor, which has the same shape of the input one, is then sampled by taking the output corresponding to the last event in each pixel-sequence $\\mathcal {E}^{(x,y)}_n$ , ignoring values computed on padded values, and the obtained values are then used to populate the dense representation.",
"To improve efficiency, for each pixel-sequence $\\mathcal {E}^{(x,y)}_n$ , groupByPixel keeps also track of the original spatial position $(x,y)$ , the index of the sample inside the batch and the length of the pixel-sequence $T^{(x,y)}_n$ , namely the index of the last event before padding.",
"Given this set of indexes, the densification step can be performed as a simple slicing operation.",
"See Figure REF for visual clues.",
"groupByPixel is implemented as a custom CUDA kernel that processes each sample in parallel and places each event feature in the output tensor maintaining the original temporal order."
],
[
"GroupByTime",
"The Matrix-LSTM variant that operates on temporal bins performs a similar pre-processing step.",
"Each sample in the batch is divided into a fixed set of intervals.",
"The groupByTime cuda kernel pre-processes the input events generating a $N * B \\times T_{max}^b \\times F$ tensor where the $B$ bins are grouped in the first dimension and taking care of properly padding intervals ($T_{max}^b$ is the length of the longest bin in the batch).",
"The Matrix-LSTM mechanism is then applied as usual and the resulting ${N*B \\times H \\times W \\times C}$ tensor is finally reshaped into a ${N \\times H \\times W \\times B*C}$ event-surface.",
"In Figure 2a of the paper we report a comparison between ConvLSTM and Matrix-LSTM using the Ev2Vid–ResNet18 configuration, i.e., the configuration of choice for all comparisons in the paper.",
"For completeness, in Table REF we extend the evaluation also to EST, using the 16 channels and 1 bin setting, which is one of our best performing EST configuration on N-Cars.",
"Matrix-LSTM performs better on all experiments, highlighting its capabilities to better handle asynchronous event-based data if compared to ConvLSTM.",
"We also highlight that the performance improvement is greater on delay relative temporal features than on ts absolute ones.",
"ConvLSTM, indeed, processes temporal slices containing potentially uncorrelated events that happened at different time instants.",
"While delays are always consistent within each pixel sequence, they are not within the ConvLSTM kernel receptive field.",
"Using an absolute temporal encoding alleviates this issue on both Ev2Vid and EST architectures, while still performing worst than Matrix-LSTM.",
"Our extraction layer, indeed, preserves the original events arrival order within each receptive field during features extraction, which allows to achieve better performance both on ts absolute and delay relative input features.",
"Moreover, structured delay relative features perform better on Matrix-LSTM than simple absolute features."
],
[
"Time Performance",
"While the performance reported in Figure 3b of the paper are computed on each sample independently to enable a fair comparison with the other methods, in Figure REF *fig:chansvsbatch and Figure REF *fig:binvsbatch we study instead how the mean time required to process a sample over all the N-Cars training dataset and the corresponding events throughput change as a function of the batch size.",
"Both performance dramatically increase when multiple samples are processed simultaneously in batch.",
"This is crucial at training time, when optimization techniques greatly benefit from batch computation.",
"Furthermore, while increasing the number of output channels, for the same choice of batch size, increases the time required to process each sample (since the resulting Matrix-LSTM operates on a larger hidden state), increasing the number of bins has an opposite behaviour.",
"Multiple intervals are indeed processed independently and in parallel by the Matrix-LSTM that has to process a smaller number of events in each spatial location, sequentially.",
"In both configurations, finally, increasing the batch size reduces the mean processing time."
],
[
"Classification",
"We perform additional experiments on the N-MNIST dataset and on the newly introduced ASL-DVS dataset.",
"On N-MNIST we directly compare with the Ev2Vid reconstruction procedure, where the custom convolutional network proposed in is used as backbone, while we compare with the EST surface on ASL-DVS, making use of ResNet50 as backbone.",
"On both cases, Matrix-LSTM performs better than other event-surface mechanisms and also outperforms alternative classification architectures."
],
[
"Ev-FlowNet Baseline Results",
"We performed optical flow experiments starting from the publicly available Ev-FlowNet codebase and replacing the original hand-crafted features with the proposed Matrix-LSTM layer.",
"We first made sure to revert the baseline architecture to the original configuration, checking that we were able to replicate the paper results.",
"Indeed, the public code contains minor upgrades over the paper version.",
"We contacted the authors that provided us with the needed modifications.",
"These consist of removing the batch normalization layers, setting to 2 the number of output channels of the layer preceding the optical flow prediction layer, and disabling random rotations during training.",
"For completeness, we report the results we obtained by training the baseline from scratch with these fixes in Table REF .",
"To test how the network adapts to different flow magnitudes, the Ev-FlowNet was tested on two evaluation settings for each test sequence: with input frames and corresponding events that are one frame apart (denoted as dt=1), and with frames and events four frames apart (denoted as dt=4).",
"While we were able to closely replicate the results of the first configuration (dt=1), with a minor improvement in the indoor_flying2 sequence, the performance we obtain on the dt=4 setup is instead worse on all sequences, as reported on the first two rows of Table REF .",
"Despite this discrepancy, which prevents the Matrix-LSTM performance on dt=4 settings to be directly compared with the results reported on the Ev-FlowNet paper, we can still evaluate the benefits of our surface on larger flow magnitudes.",
"Indeed, this work evaluates the Matrix-LSTM layer based on the relative performance improvement obtained by substituting the original features with our layer.",
"Using our Ev-FlowNet results as baseline, we show that Matrix-LSTM is able to improve the optical flow quality even on the dt=4 setting, highlighting the capability of the layer to adapt to different sequence lengths and movement conditions.",
"We report an improvement of up to $30.426\\%$ on dt=1 settings and up to $39.546\\%$ on dt=4 settings using our results as baseline.",
"Table: Effect of adding a Squeeze-and-Excitation layer on the optical flow prediction task"
],
[
"Squeeze-and-Excitation Layer",
"Optical flow prediction is a complex task that requires neural networks to extract accurate features precisely describing motion inside the scene.",
"An event aggregation mechanism is therefore required to extract rich temporal features from the events.",
"In Section $4.2$ of the paper we show that time resolution is a key factor for extracting effective feature with Matrix-LSTM.",
"In particular, increasing the number of bins has great impact on the predicted flow and allows the network to retain temporal information over long sequences.",
"Here we focus, instead, on the effect of correlating temporal features by adding a SELayer to the Matrix-LSTM output.",
"Table REF reports the performance obtained using this additional layer on the MVSEC task.",
"The results we obtained show that adding an SELayer only improves performance on the 4 bins configuration for the dt=4 benchmark, while it consistently helps reducing the AEE metric on the dt=1 setting.",
"By comparing features obtained from subsequent intervals, the SELayer adaptively recalibrates features and helps modelling interdependencies between time instants, which is crucial for predicting optical flow.",
"We believe that a similar approach can also be applied to other event aggregation mechanisms based on voxel-grids of temporal bins to improve their performance, especially those employing data driven optimization mechanisms ."
],
[
"Qualitative Results",
"The event aggregation process performed by the Matrix-LSTM layer is incremental.",
"Events in each pixel location are processed sequentially; state and output of the LSTM are updated each time.",
"We propose to visualize the Matrix-LSTM surface as an RGB image by using the ResNet18-Ev2Vid configuration and interpreting the 3 output channels as RGB color.",
"A video of such visualization showing the incremental frame reconstruction on N-Caltech101 samples is provided at this url: https://marcocannici.github.io/matrixlstm.",
"We use a similar visualization technique to show optical flow predictions for indoor_flying sequences.",
"Since we use our best performing model that uses 2 temporal bins, we decide to only show the first 3 channels of each temporal interval.",
"Moreover, instead of visualizing how the event representation builds as new events arrive, we only show the frame obtained after having processed each window of events."
]
] | 2001.03455 |
[
[
"Biophysical models of cis-regulation as interpretable neural networks"
],
[
"Abstract The adoption of deep learning techniques in genomics has been hindered by the difficulty of mechanistically interpreting the models that these techniques produce.",
"In recent years, a variety of post-hoc attribution methods have been proposed for addressing this neural network interpretability problem in the context of gene regulation.",
"Here we describe a complementary way of approaching this problem.",
"Our strategy is based on the observation that two large classes of biophysical models of cis-regulatory mechanisms can be expressed as deep neural networks in which nodes and weights have explicit physiochemical interpretations.",
"We also demonstrate how such biophysical networks can be rapidly inferred, using modern deep learning frameworks, from the data produced by certain types of massively parallel reporter assays (MPRAs).",
"These results suggest a scalable strategy for using MPRAs to systematically characterize the biophysical basis of gene regulation in a wide range of biological contexts.",
"They also highlight gene regulation as a promising venue for the development of scientifically interpretable approaches to deep learning."
],
[
"The adoption of deep learning techniques in genomics has been hindered by the difficulty of mechanistically interpreting the models that these techniques produce.",
"In recent years, a variety of post-hoc attribution methods have been proposed for addressing this neural network interpretability problem in the context of gene regulation.",
"Here we describe a complementary way of approaching this problem.",
"Our strategy is based on the observation that two large classes of biophysical models of cis-regulatory mechanisms can be expressed as deep neural networks in which nodes and weights have explicit physiochemical interpretations.",
"We also demonstrate how such biophysical networks can be rapidly inferred, using modern deep learning frameworks, from the data produced by certain types of massively parallel reporter assays (MPRAs).",
"These results suggest a scalable strategy for using MPRAs to systematically characterize the biophysical basis of gene regulation in a wide range of biological contexts.",
"They also highlight gene regulation as a promising venue for the development of scientifically interpretable approaches to deep learning.",
"Deep learning – the use of large multi-layer neural networks in machine learning applications – is revolutionizing information technology [1].",
"There is currently a great deal of interest in applying deep learning techniques to problems in genomics, especially for understanding gene regulation [2], [3], [4], [5].",
"These applications remain somewhat controversial, however, due to the difficulty of mechanistically interpreting neural network models trained on functional genomics data.",
"Multiple attribution strategies, which seek to extract meaning post-hoc from neural networks that have rather generic architectures, have been proposed for addressing this interpretability problem [6], [7], [8].",
"However, there remains a substantial gap between the outputs of such attribution methods and fully mechanistic models of gene regulation.",
"Here we advocate for a complementary approach: the inference of neural network models whose architecture reflects explicit biophysical hypotheses for how cis-regulatory sequences function.",
"This strategy is based on two key observations.",
"First, two large classes of biophysical models can be formulated as three-layer neural networks that have a stereotyped form and in which nodes and weights have explicit physiochemical interpretations.",
"This is true of thermodynamic models, which rely on a quasi-equilibrium assumption [9], [10], [11], [12], [13], [14], as well as kinetic models, which are more complex but do not make such assumptions [15], [16], [17].",
"Second, existing deep learning frameworks such as TensorFlow [18] are able to rapidly infer such models from the data produced by certain classes of MPRAs.",
"The idea of using neural networks to model the biophysics of gene regulation goes back to at least [19].",
"To our knowledge, however, the fact that all thermodynamic models of gene regulation can be represented by a simple three-layer architecture has not been previously reported.",
"We are not aware of any prior work that uses neural networks to represent kinetic models or models involving King-Altman diagrams.",
"There is a growing literature on modeling MPRA data using neural networks [20], [21], [22], [23], [24].",
"However, such modeling has yet to be advocated for MPRAs that dissect the mechanisms of single cis-regulatory sequences, such as in [25], [26], [27], [28], [29], which is our focus here.",
"Finally, some recent work has used deep learning frameworks to infer parametric models of cis-regulation [30], [31].",
"These studies did not, however, infer the type of explicit biophysical quantities, such as $\\Delta G$ values, that our approach recovers.",
"Figure: A thermodynamic model of transcriptional regulation.",
"(A) Transcriptional activation at the E. coli lac promoter is regulated by two proteins, CRP and 70 ^{70} RNA polymerase (RNAP).",
"CRP is a transcriptional activator that up-regulates transcription by stabilizing RNAP-DNA binding.",
"ΔG C \\Delta G_C and ΔG R \\Delta G_R respectively denote the Gibbs free energies of the CRP-DNA and RNAP-DNA interactions, while ΔG I \\Delta G_I denotes the Gibbs free energy of interaction between CRP and RNAP.",
"(B) Like all thermodynamic models of gene regulation, this model consists of a set of states, each state having an associated Gibbs free energy and activity.",
"The probability of each state is assumed to follow the Boltzmann distribution.",
"(C) The corresponding activity predicted by such thermodynamic models is the state-specific activity averaged together using these Boltzmann probabilities.",
"(D) This model formulated as a three-layer neural network.",
"First layer nodes represent interaction energies, second layer nodes represent state probabilities, and third layer nodes represent transcriptional activity.",
"The values of weights are indicated; gray lines correspond to zero weights.",
"The second layer has a softmin activation, while the third has a linear activation.",
"All thermodynamic models of cis-regulation can be represented using this general three-layer form.All thermodynamic models of cis-regulation can be represented as three-layer neural networks as follows.",
"First one defines a set of molecular complexes, or “states”, which we index using $s$ .",
"Each state has both a Gibbs free energy $\\Delta G_s$ and an associated activity $\\alpha _s$ .",
"These energies determine the probability $P_s$ of each state occurring in thermodynamic equilibrium via the Boltzmann distribution,To reduce notational burden, all $\\Delta G$ values are assumed to be in thermal units.",
"At $37^\\circ $ C, one thermal unit is $1\\, k_B T = 0.62~\\mathrm {kcal/mol}$ , where $k_B$ is Boltzmann's constant and $T$ is temperature.",
"$P_s = \\frac{e^{-\\Delta G_s}}{\\sum _{s^{\\prime }} e^{-\\Delta G_{s^{\\prime }}}}.$ The energy of each state is, in turn, computed using integral combinations of the individual interaction energies $\\Delta G_j$ that occur in that state.",
"We can therefore write $\\Delta G_s = \\sum _j \\omega _{sj} \\Delta G_j$ , where $\\omega _{sj}$ is the number of times that interaction $j$ occurs in state $s$ .",
"The resulting activity predicted by the model is given by the activities $\\alpha _s$ of the individual states averaged over this distribution, i.e., $t = \\sum _s \\alpha _s P_s$ .",
"Fig.",
"REF illustrates a thermodynamic model for transcriptional activation at the E. coli lac promoter.",
"This model involves two proteins, CRP and RNAP, as well as three interaction energies: $\\Delta G_C$ , $\\Delta G_R$ , and $\\Delta G_I$ .",
"The rate of transcription $t$ is further assumed to be proportional to the fraction of time that RNAP is bound to DNA (Fig.",
"REF A).",
"This model is summarized by four different states, two of which lead to transcription and two of which do not (Fig.",
"REF B).",
"Fig.",
"REF C shows the resulting formula for $t$ in terms of model parameters.",
"This model is readily formulated as a feed-forward neural network (Fig.",
"REF D).",
"Indeed, all thermodynamic models of cis-regulation can be formulated as three-layer neural networks: layer 1 represents molecular interaction energies, layer 2 (which uses a softmin activation) represents state probabilities, and layer 3 (using linear activation) represents the biological activity of interest, which in this case is transcription rate.",
"Figure: Inference of a thermodynamic model from MPRA data.",
"(A) Schematic of the sort-seq MPRA of .",
"A 75 bp region of the E. coli lac promoter was mutagenized at 12% per nucleotide.",
"Variant promoters were then used to drive the expression of GFP.",
"Cells carrying these expression constructs were then sorted using FACS, and the variant sequences in each bin were sequenced.",
"This yielded data on about 5×10 4 5 \\times 10^4 variant promoters across 10 bins.",
"(B) The neural network from Fig.",
"D, but with ΔG C \\Delta G_C and ΔG R \\Delta G_R expressed as linear functions of the DNA sequence x →\\vec{x}, as well as a dense feed-forward network mapping activity tt to bins via a probability distribution p( bin |t)p(\\mathrm {bin}|t).",
"Gray lines indicate weights fixed at 0.",
"The weights linking nodes P 3 P_3 and P 4 P_4 to node tt were constrained to have the same value t sat t_{sat}.",
"(C) The parameter values inferred from the MPRA data of .",
"Shown are the CRP energy matrix θ → C \\vec{\\theta }_C, the RNAP energy matrix θ → R \\vec{\\theta }_R, and the CRP-RNAP interaction energy ΔG I \\Delta G_I.",
"Since increasingly negative energy corresponds to stronger binding, they yy-axis in the logo plots is inverted.",
"Logos were generated using Logomaker .We can infer thermodynamic models like these for a cis-regulatory sequence of interest (the wild-type sequence) from the data produced by an MPRA performed on an appropriate sequence library [26].",
"Indeed, a number of MPRAs have been performed with this explicit purpose in mind [26], [27], [33], [34], [35].",
"To this end, such MPRAs are generally performed using libraries that consist of sequence variants that differ from the wild-type sequence by a small number of single nucleotide polymorphisms (SNPs).",
"The key modeling assumption that motivates using libraries of this form is that the assayed sequence variants will form the same molecular complexes as the wild-type sequence, but with Gibbs free energies and state activities whose values vary from sequence to sequence.",
"By contrast, variant libraries that contain insertions, deletions, or large regions of random DNA (e.g.",
"[20], [21], [23], [24], [31]) are unlikely to satisfy this modeling assumption.",
"Fig.",
"REF A summarizes the sort-seq MPRA described in [26].",
"Lac promoter variants were used to drive GFP expression in E. coli, cells were sorted into 10 bins using fluorescence-activated cell sorting, and the variant promoters within each bin were sequenced.",
"This yielded data comprising about $5 \\times 10^4$ variant lac promoter sequences, each associated with one of 10 bins.",
"The authors then fit the biophysical model shown in Fig.",
"REF C, but under the assumption that $\\Delta G_C = \\vec{\\theta }_C \\cdot \\vec{x} + \\mu _C$ and $\\Delta G_R = \\vec{\\theta }_R \\cdot \\vec{x} + \\mu _R$ , where $\\vec{x}$ is a one-hot encoding of promoter DNA sequence.",
"Here we used TensorFlow to infer the same model formulated as a deep neural network.",
"Specifically, we augmented the network in Fig.",
"REF D by making $\\Delta G_C$ and $\\Delta G_R$ sequence-dependent as in [26].",
"To link $t$ to the MPRA measurements, we introduced a feed-forward network with one hidden layer and a softmax output layer corresponding to the 10 bins into which cells were sorted.",
"Model parameters were then fit to the MPRA dataset using stochastic gradient descent and early stopping.",
"The results agreed well with those reported in [26].",
"In particular, the parameters in the energy matrices for CRP ($\\vec{\\theta }_C$ ) and RNAP ($\\vec{\\theta }_R$ ) exhibited respective Pearson correlation coefficients of 0.986 and 0.994 with those reported in [26].",
"The protein-protein interaction energy that we found, $ \\Delta G_I = -2.9$ kcal/mol, was also compatible with the previously reported value $\\Delta G_I =-3.3 \\pm 0.4 $ kcal/mol.",
"A major difference between our results and those of [26] is the ease with which they were obtained.",
"Training of the network in Fig.",
"REF B consistently took about 15 minutes on a standard laptop computer.",
"The model fitting procedure in [26], by contrast, relied on a custom Parallel Tempering Monte Carlo algorithm that took about a week to run on a multi-node computer cluster (personal communication), and more recent efforts to train biophysical models on MPRA data have encountered similar computational bottlenecks [34], [35].",
"Also of note is the fact that in [26] the authors inferred models using information maximization.",
"Specifically, the authors fit the parameters of $t(\\vec{x})$ by maximizing the mutual information $I[t; \\mathrm {bin}]$ between model predictions and observed bins.",
"One difficulty with this strategy is the need to estimate mutual information.",
"Instead, we used maximum likelihood to infer the parameters of $t(\\vec{x})$ as well as the experimental transfer function (i.e., noise model) $p(\\mathrm {bin}|t)$ , which was modeled by a dense feed-forward network with one hidden layer.",
"These two inference methods, however, are essentially equivalent: in the large data regime, the parameters of $t$ that maximize $I[t; \\mathrm {bin}]$ are the same as the parameters one obtains when maximizing likelihood over the parameters of both $t$ and $p(\\mathrm {bin} | t)$ ; see [36], [37], [38].",
"Figure: A kinetic model for transcriptional initiation by E. coli RNAP.",
"(A) In this model, promoter DNA can participate in three complexes: unbound, closed, and open , .",
"Transitions between these complexes are governed by four rate constants: k 1 k_1, k -1 k_{-1}, k 2 k_2, and k 3 k_3.",
"(B) A formula for the steady-state rate of mRNA production can be obtained using King-Altman diagrams , .",
"(C) This formula can be represented using the three-layer neural network shown, where layer 1 represents log transition rates, layer 2 (softmax activation) represents normalized King-Altman diagrams, and layer 3 (linear activation) represents promoter activity.",
"Black lines indicate weight 1; gray lines indicate weight 0.",
"Note that the single nonzero weight connecting layer 2 to layer 3 (orange) is actually the transition rate k 3 k_3 from layer 1.",
"All kinetic models of cis-regulation share this general three-layer form.A shortcoming of thermodynamic models is that they ignore non-equilibrium processes.",
"Kinetic models address this problem by providing a fully non-equilibrium characterization of steady-state activity.",
"Kinetic models are specified by listing explicit state-to-state transition rates rather than Gibbs free energies.",
"Fig.",
"REF A shows a three-state kinetic model of transcriptional initiation consisting of unbound promoter DNA, an RNAP-DNA complex in the closed conformation, and the RNAP-DNA complex in the open conformation [39], [40].",
"The rate $k_3$ going from the open state to the unbound state represents transcript initiation.",
"The overall transcription rate in steady state is therefore $k_3$ times the occupancy of the open complex.",
"King-Altman diagrams [41], [42], a technique from mathematical enzymology, provide a straight-forward way to compute steady-state occupancy in kinetic models.",
"Specifically, each state's occupancy is proportional to the sum of directed spanning trees (a.k.a.",
"King-Altman diagrams) that flow to that state, where each spanning tree's value is given by the product of rates comprising that tree.",
"Fig.",
"REF B illustrates this procedure for the kinetic model in Fig.",
"REF A.",
"Every such kinetic model can be represented by a three-layer neural network (e.g., Fig.",
"REF C) in which first layer nodes represent log transition rates, second layer nodes (after a softmax activation) represent normalized King-Altman diagrams, and third layer nodes represent the activities of interest.",
"Here we have shown how both thermodynamic and kinetic models of gene regulation can be formulated as three-layer deep neural networks in which nodes and weights have explicit biophysical meaning.",
"This represents a new strategy for interpretable deep learning in the study of gene regulation, one complementary to existing post-hoc attribution methods.",
"We have further demonstrated that a neural-network-based thermodynamic model can be rapidly inferred from MPRA data using TensorFlow.",
"This was done in the context of a well-characterized bacterial promoter because previous studies of this system have established concrete results against which we could compare our inferred model.",
"The same modeling approach, however, should be readily applicable to a wide variety of biological systems amenable to MPRAs, including transcriptional regulation and alternative mRNA splicing in higher eukaryotes.",
"Code Availability and Acknowledgements: The neural network model shown in Fig.",
"2C, as well as the scripts used to infer it from the data of [26], are available at https://github.com/jbkinney/19_mlcb.",
"We thank Anand Murugan, Yifei Huang, Peter Koo, Alan Moses, and Mahdi Kooshkbaghi for helpful discussions, as well as and three anonymous referees for providing constructive criticism.",
"This project was supported by NIH grant 1R35GM133777 and a grant from the CSHL/Northwell Health partnership."
]
] | 2001.03560 |
[
[
"Centrifugal acceleration of protons by a supermassive black hole"
],
[
"Abstract The centrifugal acceleration is due to the rotating poloidal magnetic field in the magnetosphere creates the electric field which is orthogonal to the magnetic field.",
"Charged particles with finite cyclotron radii can move along the electric field and receive energy.",
"Centrifugal acceleration pushes particles to the periphery, where their azimuthal velocity reaches the light speed.",
"We have calculated particle trajectories by numerical and analytical methods.",
"The maximum obtained energies depend on the parameter of the particle magnetization $ \\kappa $, which is the ratio of rotation frequency of magnetic field lines in the magnetosphere $ \\Omega_F $ to non-relativistic cyclotron frequency of particles $ \\omega_c $, $ \\kappa = \\Omega_F /\\omega_c << 1 $, and from the parameter $ \\alpha $ which is the ratio of toroidal magnetic field $ B_T $ to the poloidal one $ B_P $, $ \\alpha = B_T / B_P $.",
"It is shown that for small toroidal fields, $ \\alpha <\\kappa^{1/4} $, the maximum Lorentz factor $ \\gamma_m $ is only the square root of magnetization, $ \\gamma_m = \\kappa^{-1/2} $, while for large toroidal fields, $ \\alpha >\\kappa^{1/4} $, the energy increases significantly, $ \\gamma_m = \\kappa^{-2/3} $.",
"However, the maximum possible acceleration, $ \\gamma_m = \\kappa^{-1} $, is not achieved in the magnetosphere.",
"For a number of active galactic nuclei, such as M87, maximum values of Lorentz factor for accelerated protons are found.",
"Also for special case of Sgr.",
"A* estimations of the maximum proton energy and its energy flux are obtained.",
"They are in agreement with experimental data obtained by HESS Cherenkov telescope."
],
[
"Introduction",
"At present, it is believed that the most powerful sources of gamma radiation, such as relativistic jets of blazars, Lac, quasars and radio galaxies, are the most efficient particle accelerators (Blandford, Meier and Readhead, (2019)).",
"Following the classic paradigm of active galactic nuclei (AGN) (Robson, (1996)), the base of these objects contains a central object in the form of a supermassive black hole surrounded by an accretion disk; its matter accrets onto the surface of the event horizon.",
"For radio galaxies, blazars and Lac, a part of the accreted matter does not fall onto the centre, but leaves along the axis of rotation outwards, perpendicular to the accretion disk plane, forming a collimated plasma flow (jet).",
"The radiation of jets is non-thermal, variable and lies in a wide range of the spectrum of electromagnetic waves, from radio to X-ray.",
"A number of Cherenkov astronomy observatories of (such as VERITAS, HESS, MAGIC) over the past few years have registered sources of gamma rays of extremely high energies up beyond the TeV range.",
"The most striking sources of such energy are M87, NGC 5128, 1ES 2344+514.",
"In addition to extragalactic sources, a significant value of radiation energy (up to $ 10^{15} eV $ ) was discovered in Sgr.",
"A*, the centre of our Galaxy (Abramowski et al., 2016).",
"The main carriers of such high energies are both electrons with energies up to TeV units (Ghisellini et al., 1998) and protons with energies up to $ \\simeq 10^{21} $ eV.",
"To explain the origin of particles with energies in this range, the centrifugal acceleration mechanism was proposed as the main acceleration mechanism (Rieger and Mannheim, 2000); (Rieger and Aharonian, 2008).",
"Many works considered the acceleration of electrons.",
"However, much of electrons' energy is lost are due to synchrotron radiation, so their acceleration is not as efficient as the acceleration of protons.",
"We are interested in the possibility of reaching the limiting energy achieved during the acceleration of protons, the main component of the cosmic ray spectrum (Ginzburg, 1957).",
"The acceleration of particles in the AGN can occur both in the accretion disk and in the black hole magnetosphere (Istomin and Sol, 2009).",
"The acceleration in the disk is due to the presence of a turbulent electromagnetic field generated by the turbulent motion of the matter of the accretion disk, which leads to accretion.",
"In the magnetosphere, acceleration of charged particles is associated with the presence of an electric field proportional to the angular velocity of rotation of the magnetospheric plasma that is brought into the rotation by the rotating black hole (Blandford and Znajek, 1977), and its accretion disk (Blandford and Payne, 1982).",
"Also, acceleration can be induced by shock waves and in regions of reconnection of magnetic field lines.",
"However, the first two mechanisms are always present and are stationary or quasi-stationary, while, in general, the processes leading to the appearance of shock waves and regions with a reverse magnetic field may not appear.",
"Here we consider only the acceleration of charged particles in the magnetosphere of a black hole.",
"The acceleration of particles in the plasma disk by a stochastic electromagnetic field was studied in detail by Istomin and Sol, (2009).",
"Due to interactions of energetic particles with the photon field of the disk (the disk temperature is about 10 eV), the maximum Lorentz factor of protons cannot reach large values.",
"In this case, the acceleration in the disk can be considered as the initial process of pre-acceleration of particles injected from disk to the magnetosphere.",
"This work is this continuation of part of the work started by Istomin and Sol, (2009), where the acceleration of particles in a black hole magnetosphere was considered in the vicinity the accretion disk, neglecting the toroidal magnetic field.",
"The magnetic sield structure is described in section .",
"The equations for charged particles motion are given in section , along with the approximate analytical expression for maximum energies.",
"Section presense calculations for a range of real AGNs in terms of proton acceleration.",
"The results are discussed in section ."
],
[
"Magnetic field structure",
"Due to rather high temperature ($T\\simeq 10 \\, eV$ ) and the relatively low plasma density of the disk ($n\\le 10^7 \\, cm^{-3}$ ) the plasma fluid can be described by equations of ideal magnetic hydrodynamics (MHD).",
"This is due to the fact that particles (protons and electrons) are magnetized, i.e.",
"the cyclotron frequency of protons in a magnetic field $B$ , $\\omega _{ci}\\simeq 10^4(B/1\\,G) \\, s^{-1}$ much larger than frequency of proton-proton collisions $\\nu \\simeq 10^{-6}(n/1 \\, cm^{-3})(T/1 \\, eV)^{-3/2} s^{-1}$ .",
"Therefore, the electric field in the disk $ {\\bf E} $ is connected with the magnetic field $ {\\bf B} $ by the relation ${\\bf E}=-\\frac{1}{c}{\\bf u}\\times {\\bf B},$ where ${\\bf u}$ is the velocity of the disk plasma.",
"For the thin disk it means that $E_\\phi = u_\\rho B_z$ .",
"Here $E_\\phi $ is the toroidal electric field, $u_\\rho $ is the radial velocity of the disk matter, which does not equal to zero during accretion, $B_z$ is the component of the magnetic field orthogonal to the disk plane.",
"The toroidal electric field $ E_\\phi $ for stationary or quasistationary accretion is zero as the electric field is potential.",
"Therefore, there should be no vertical magnetic field in the disk.",
"The disk plasma should be polarized so as to push out the $ B_z $ component.",
"Therefore at the boundary of the accretion disk the magnetic field in the magnetosphere can only have a radial, along with a spherical radius $ r $ , component $ B_r $ and the toroidal magnetic field $ B_\\phi $ .",
"By virtue of axial symmetry, i.e.",
"non-dependence of all quantities on the azimuth angle $ \\phi $ , the radial field $ B_r \\propto r^{-2} $ .",
"It is the monopoly field.",
"But due to divergence-free, the radial magnetic field must have different signs in different half spaces, $ z> 0 $ and $ z <0 $ .",
"Thus, $ B_r \\propto s / r^2 $ , where the value of $ s $ is the sign of $ z $ , $ s = \\mathrm {sign}(z) $ .",
"This configuration of the magnetic field is called the split monopoly.",
"A toroidal field $ B_\\phi $ , superimposed onto the radial magnetic field, is created by radial electric currents $ j_r (j_\\phi = j_\\theta = 0) $ , flowing along the radial magnetic field in the disk and in the magnetosphere in the immediate vicinity of the black hole.",
"The forward and reverse currents are closed in the jet along the axis of rotation of the black hole, perpendicular to the accretion disk.",
"A toroidal magnetic field in the general case has the form $ B_\\phi \\propto r^{-1} F (\\theta ) $ , where $ F (\\theta ) $ is some function of the polar angle $ \\theta $ .",
"It is determined by the dependence of the radial current $ j_r (\\theta ) $ on the polar angle, i.e.",
"how the jets electric current is closed in the black hole vicinity, $ j_r \\propto \\partial (\\sin \\theta B_\\phi ) /\\partial \\theta / r \\sin \\theta $ .",
"In the simplest case, $ F (\\theta ) = const (\\theta ) $ .",
"Since the electric currents of the jet above the disk and under the disk run in different directions, the toroidal field, as well as the poloidal field, is proportional to the value of $ s $ , $ B_\\phi \\propto s / r $ .",
"The configuration of magnetic field lines, defined by the ratio $ dr / B_r = rd \\phi / B_\\phi $ , is expanding spirals $ r = r_1 \\phi + r_0 $ , where the values of $ r_0 $ and $ r_1 $ are constants.",
"Figure REF shows the magnetic field lines of this configuration in the magnetosphere.",
"It should be noted that the absence of the vertical magnetic field $ B_z $ in the disk refers to a stationary, or quasi-stationary, magnetic field.",
"It means that due to the magnetization of the plasma, charged particles (protons and electrons) cannot pass through vertical magnetic field during accretion.",
"However, if electric currents in the disk do not have time to change so that $ B_z \\simeq 0 $ , then a non-zero vertical magnetic field will be captured by the accreating plasma flux going to the centre and be amplified due to the magnetic flux conservation.",
"This can lead to a decrease of the accretion rate, up to its cessation.",
"Also can appear of non-stationary electromagnetic fields in the disk and particle acceleration.",
"However, numerical calculations of the structure of disks and their surroundings near the black hole using general relativity magneto hydro dynamics (GRMHD) show that arising inside the disk vortices strongly elongated along the disk with mainly horizontal magnetic field (Mishra et al., 2016).",
"This means that indeed, although the field is not stationary, the electric currents in the plasma of the disk are designed so that the magnetic field created by them does not prevent accretion.",
"For this, it is necessary that the plasma conductivity $ \\sigma _c $ be small enough to suppress short vortices of the order of the disk thickness $ H $ , in which the vertical component of the magnetic field is large, $ \\sigma _c <v / H $ .",
"Here the velocity $ v $ is the velocity of matter in a vortex.",
"The vortex revolution time is $ \\simeq H / v $ .",
"Apparently, the numerical magnetic viscosity (inversely to the conductivity) in the calculations satisfies this condition.",
"Also, the real conductivity of the turbulent matter is anomalous and significantly less than the classical one.",
"Because of this, the magnetic lines above the disk are directed tangentially to the disk (Morales Teixeira, Avara, & McKinney, 2018) and the poloidal magnetic field in magnetosphere is close to radial one due to its strong compression by the plasma pressure of the disk near the central object (Kathirgamaraju et al., 2019; Liska et al., 2019).",
"Figure: Scheme of picture of magnetic field lines in the magnetosphere in the vicinity of a rotating black holeFor split monopole magnetic field lines directly exit from the vicinity of the central black hole.",
"They are involved into the rotation by the black hole and rotate with an angular velocity of $ \\Omega _F $ , which for optimal matching is half the angular velocity of rotation of a black hole $ \\Omega _H $ , $ \\Omega _F = \\Omega _H / 2 $ (Blandford and Znajek, 1977).",
"The rotating radial magnetic field excites in the magnetosphere a $\\theta $ component of the electric field, $ E_\\theta $ , which, by virtue of Faraday's law, is $E_\\theta dl=-\\frac{1}{c}\\frac{d\\Phi }{dt}=-\\frac{r\\sin \\theta B_r}{c}\\frac{d\\phi }{dt}dl.$ Here the element of length $ dl $ is taken along the $ \\theta $ direction.",
"We get $E_\\theta =-\\frac{r\\Omega _F\\sin \\theta B_r}{c}.$ The relation (REF ) looks like a condition (REF ) in ideal MHD, where instead of the fluid velocity $ {\\bf u} $ the azimuthal rotation velocity appears, $ {\\bf u} = {\\bf e_\\phi } r \\sin \\theta \\Omega _F $ .",
"The electric field (REF ) in the magnetosphere arises near the black hole horizon due to its rotation and is transmitted along the radial magnetic field to the magnetosphere, which is located from the centre right up to the light cylinder surface $ r = c/ \\Omega _F \\sin \\theta $ .",
"The polar field $ E_\\theta $ in the magnetosphere is the agent that transfers the black hole rotation to the rotation of plasma of the magnetosphere and, when charges are displaced in the polar direction, it transfers energy to them.",
"We emphasize here that the polar electric field $ E_\\theta $ (REF ) is the result the central object rotation, and is not a consequence of MHD, which may not be satisfied in the magnetosphere.",
"This topic is discussed in the section 'Discussion'.",
"Thus, in the magnetosphere there is an electromagnetic field with components $B_r=s B_0\\left(\\frac{r}{r_L}\\right)^{-2},$ $B_\\phi =\\alpha s B_0\\sin \\theta \\left(\\frac{r}{r_L}\\right)^{-1}, \\,$ $E_\\theta =-s B_0\\sin \\theta \\left(\\frac{r}{r_L}\\right)^{-1}.$ Here, $ r_L = c / \\Omega _F $ is the natural size of the magnetosphere in the transverse direction.",
"This is the radius of the light cylinder, at which the electric field is compared with the magnetic one.",
"For a spherical distance $ r $ , the light surface is at the distance $ r = r_L / \\sin \\theta $ .",
"The value of $ B_0 $ is the value of the radial magnetic field at the distance $ r = r_L $ .",
"At the light cylinder the value of the radial field falls from the value of $ B_0 $ near the accretion disk $ (\\theta \\simeq \\pi / 2) $ to $ B_0 \\sin ^2 \\theta $ .",
"The coefficient $ \\alpha $ is the ratio of the toroidal magnetic field to the poloidal one at the distance $ r = r_L $ from the centre.",
"Since the split monopole field and the toroidal field differently depend on the radial distance $ r $ , then at 'small' distances $ r <\\alpha ^{-1} r_L $ the poloidal magnetic field prevails, while at 'large' distances $ r> \\alpha ^{-1} r_L $ the field becomes predominantly toroidal.",
"So for $ \\alpha > 1 $ the magnetic field near the light surface, $ r = r_L / \\sin \\theta $ , where, as we will see, the main acceleration of charged particles occurs, the field is mainly toroidal.",
"For small values of $ \\alpha $ , $ \\alpha <1 $ , the field is only at large distances from the centre, $ r> (\\alpha \\sin \\theta )^{-1} r_L $ , is close to the toroidal one, while near the field centre it is close to the poloidal one."
],
[
"Particle acceleration",
"Motion of particles of mass $ m $ and charge $ q $ in the electromagnetic field of the black hole magnetosphere is described by equations $\\frac{d{\\bf p}}{dt}=q\\left({\\bf E}+\\frac{1}{c}\\left[{\\bf v,B}\\right]\\right), \\,$ $\\frac{d{\\bf r}}{dt}={\\bf v}=\\frac{{\\bf p}}{m\\gamma }, \\,$ $\\gamma ^2=1+\\frac{p^2}{m^2 c^2}.$ Here $ {\\bf r} $ and $ {\\bf p} $ are the coordinate and the momentum of a particle, $ \\gamma $ is its Lorentz factor.",
"It is convenient for us to introduce dimensionless time, coordinates, velocity and momentum, $t^{\\prime }=\\frac{\\omega _c t}{\\gamma _i}, \\,{\\bf r^{\\prime }}=\\frac{{\\bf r}}{r_L}, \\, {\\bf v^{\\prime }}=\\frac{{\\bf v}}{c}, \\,{\\bf p^{\\prime }}=\\frac{{\\bf p}}{mc\\gamma _i}.$ The initial value of the Lorentz factor is $ \\gamma _i $ , the nonrelativistic cyclotron frequency of a particle rotation in the $ B_0 $ field is $ \\omega _c = qB_0 / mc $ .",
"Let us also introduce the value of the Lorentz factor relative to the initial energy, $ \\gamma ^{\\prime }= \\gamma / \\gamma _i $ .",
"In these variables, the equations of motion (REF ) in spherical coordinates $ r, \\theta , \\phi $ (primes are omitted) have the form $&&\\frac{dp_r}{dt}=\\frac{\\kappa }{r\\gamma }\\left(p_\\theta ^2+p_\\phi ^2\\right)+\\frac{s\\alpha }{r\\gamma }p_\\theta , \\nonumber \\\\&&\\frac{dp_\\theta }{dt}=-\\frac{\\kappa }{r\\gamma }\\left(p_r p_\\theta -p_\\phi ^2 \\cot \\theta \\right)-\\frac{s}{r}\\sin \\theta +\\frac{s}{r^2\\gamma }p_\\phi -\\frac{s\\alpha }{r\\gamma }p_r, \\nonumber \\\\&&\\frac{dp_\\phi }{dt}=-\\frac{\\kappa }{r\\gamma }\\left(p_r+p_\\theta \\cot \\theta \\right)p_\\phi - \\frac{s}{r^2\\gamma }p_\\theta , \\\\&&\\frac{dr}{dt}=\\frac{\\kappa }{\\gamma } p_r, \\nonumber \\\\&&\\frac{d\\theta }{dt}=\\frac{\\kappa }{r\\gamma }p_\\theta .",
"\\nonumber $ The equations (REF ) contain two dimensionless constants, $ \\alpha $ and $ \\kappa $ .",
"The value of $ \\alpha $ , introduced in the previous section, is equal to the ratio of the toroidal magnetic field to the poloidal one at the distance $ r = r_L $ , $ \\alpha = B_ \\phi / B_r |_{r = r_L} $ .",
"The value of $ \\kappa $ is equal to $\\kappa =\\frac{c\\gamma _i}{r_L\\omega _c}=\\frac{r_c}{r_L}=\\frac{\\Omega _F}{\\omega _c/\\gamma _i}$ and is actually the parameter of a charged particle magnetization, that is the ratio the cyclotron radius of a particle, $ r_c = c \\gamma _i / \\omega _c $ , to the radius of the light cylinder, or the ratio of the rotation frequency of magnetic field lines to the cyclotron frequency.",
"It is clear that in the magnetosphere only magnetized particles can be accelerated, $ \\kappa <1 $ , $ \\gamma _i <\\gamma _0 = r_L \\omega _c / c $ , which conforms the Hillas criterion (Hillas, 1984), which is geometric in nature: it is impossible to accelerate a particle in a magnetic field whose scale is smaller than the cyclotron radius.",
"The acceleration of 'cold' particles, $ \\gamma _i \\simeq 1 $ is considered.",
"In this case, $ \\kappa << 1 $ .",
"The question is to find the energy maximum $ \\gamma _m $ when proton reaches the light surface $ r = r_L / \\sin \\theta $ .",
"First of all need to solve a system of equations (REF ).",
"The numerical calculation of the proton trajectory starting from the magnetosphere internal region, $ r < r_L $ , is show in Figures (REF ) and (REF ).",
"Figure: The azimuthal momentum of the particle p φ p_\\phi versus the radial distance rsinθr \\sin \\theta for κ=10 −2 \\kappa = 10^{−2} and α=10 −2 \\alpha = 10^{−2}.",
"The light cylinder surface is located at 10 on the abscissa axis.Figure: The polar momentum of the particle p θ p_\\theta versus the radial distance rsinθr \\sin \\theta for the same parameters as in Fig.",
"2.Figure: The Lorentz factor of the particle γ \\gamma versus the radial distancersinθr \\sin \\theta for same parameters as in Fig.",
"2First, it can be seen that the behavior of the toroidal momentum $ p_ \\phi $ near the light surface, $ r \\simeq 1 / \\sin \\theta $ , practically does not differ from the behavior of the Lorentz factor $\\gamma $ (energy).",
"This is due to the fact that the system (REF ) has two integrals of motion: energy $ {\\cal E} = const $ and angular momentum $ {\\cal L} = const $ , ${\\cal E}=\\gamma - \\frac{s}{\\kappa }\\cos \\theta , \\, {\\cal L}=rp_\\phi \\sin \\theta -\\gamma .$ As well as energy consisting of particle energy and the work of an electric field, the angular momentum is the sum of the mechanical momentum and the momentum of the electromagnetic field.",
"Since the initial radius $ r_0 $ , from which the particle starts, is less than unity, $ r_0 << 1 $ , then, as it follows from the conservation of the angular momentum (REF ), $ p_{\\phi _{max}} = \\gamma _m-1 $ .",
"The value of $ \\gamma _m $ is equal to the maximum Lorentz factor of a particle achieved at the light surface.",
"In addition, $ \\gamma ^2 = p_r^2 + p_ \\theta ^2 + p_\\phi ^2 + 1 / \\gamma _i^2 $ , and $ (p_r^2 + p_\\theta ^2) |_{max} = 2 \\gamma _m \\, (\\gamma _m >> 1) $ .",
"Thus, particle acceleration occurs in the azimuthal direction, $ p_\\phi \\simeq \\gamma $ , while the poloidal components of a particle are much smaller, $ p_r \\simeq p_\\theta \\simeq \\gamma ^{1/2} $ .",
"This is the centrifugal acceleration.",
"Secondly, the angle $ \\theta $ varies strongly only in the vicinity of the light surface, practically without changing on the main part of the trajectory (Fig.",
"(REF )).",
"As we will see, always $ \\gamma << \\kappa ^{-1}, \\, \\kappa << 1 $ , and it follows from the energy conservation law (REF ) that $ \\cos \\theta - \\cos \\theta _0 << 1 $ .",
"Since the main acceleration occurs near the light surface (Fig.",
"(REF )), then $ \\Delta \\theta = \\theta - \\theta _0 = - \\sigma \\kappa \\gamma _m / \\sin \\theta $ .",
"On the other hand, $ d \\theta / dr = p_\\theta \\sin \\theta / p_r = \\Delta \\theta / \\Delta r $ , where $ \\Delta r $ is the size of the region near the light surface $ r = 1 / \\sin \\theta $ , where the main acceleration of particles occurs $\\Delta r=-\\frac{s\\kappa \\gamma _m}{\\sin ^2\\theta }\\frac{p_r}{p_\\theta }\\mid _{r=1/\\sin \\theta }.$ Small angles $ \\theta \\simeq 0 $ are not considered because the possible region of acceleration in this case is far from the centre, where the fields are small and the acceleration is not efficient.",
"Figure: The polar angle versus the radial distance rsin(θ)r \\sin (\\theta ) for κ=10 -4 \\kappa = 10^{-4} Passing in the system (REF ) from time derivatives to derivatives over coordinate $r, \\, d/dt=d/dr(p_r/\\gamma )$ , and replacing derivatives by fractions $d{\\bf p}/dr\\simeq {\\bf p}|_{r=1/\\sin \\theta }/\\Delta r$ , we obtain an algebraic system of equations that determines the values of ($\\gamma _m, {\\bf p})|_{r=1/\\sin \\theta }$ at the light surface $r=1/\\sin \\theta $ $&&p_r^2+\\frac{\\alpha \\gamma _m}{\\sin \\theta }p_r-\\frac{\\gamma _m(\\sin ^2\\theta +\\kappa \\gamma _m^2|\\cos \\theta |)}{\\sin ^2\\theta }=0; \\nonumber \\\\&&p_\\theta =-p_r\\frac{s\\kappa \\gamma _m^2\\sin \\theta }{\\sin ^2\\theta +\\kappa \\gamma _m^2|\\cos \\theta |}; \\\\&&p_\\theta ^2+p_r^2=2\\gamma _m.",
"\\nonumber $ First consider the case of small values of $ \\alpha $ .",
"Put (REF ) $ \\alpha = 0 $ in the system of equations we find $\\gamma _m=\\kappa ^{-1/2}\\sin \\theta .$ This result coincides with the expression obtained in (Istomin and Sol, 2009) for the case of $ \\theta = \\pi / 2 $ , where the acceleration of particles in the magnetosphere was considered only near the accretion disk.",
"The expression $ \\gamma _m = \\kappa ^{-1/2} (\\sin \\theta = 1) $ was obtained from the analysis of numerical calculations of particle trajectories, while here we use the analytical approximation.",
"We show in Figure (REF ) the comparison of numerical calculations (points) with the analytical expression (REF ).",
"There are in a good agreement.",
"Figure: The Lorentz factor of a particle at the light surface γ m \\gamma _m versus the polar angle θ \\theta for κ=10 -4 \\kappa = 10^{-4} .",
"Points are the result of numerical calculations, the curve is the dependance ().To obtain a general expression for $ \\gamma _m $ for the arbitrary value of $ \\alpha $ , we need to solve the quadratic equation, the first equation of the system (REF ), with respect to the value of $ p_r $ .",
"However, the resulting expression is inconvenient for analysis, so we will find an approximate expression for the value of $ p_r $ in the case of large values of $ \\alpha , \\, \\alpha ^2 \\gamma _m >> 1, $ $ p_r\\simeq \\frac{\\sin ^2\\theta +\\kappa \\gamma _m^2|\\cos \\theta |}{\\alpha \\sin \\theta }.$ The result is $\\gamma _m\\simeq 2^{1/3}\\left(\\alpha \\kappa ^{-1}\\right)^{2/3}\\sin ^{2/3}\\theta .$ By comparing expressions (REF ) and (REF ) obtain the value of $ \\alpha = \\alpha _1 $ , at which the dependence of the maximum Lorentz factor $ \\gamma _m $ on the magnetization parameter $ \\kappa $ at small toroidal fields, $ \\alpha <\\alpha _1 $ , transits to the dependence at large toroidal fields, $ \\alpha > \\alpha _1 $ , $\\alpha _1 =2^{-1/2}\\kappa ^{1/4}\\sin ^{1/2}\\theta \\simeq \\kappa ^{1/4}.$ Figure (REF ) depicts the dependence of $ \\gamma _m $ on the toroidal magnetic field magnitude.",
"The dots correspond the values calculated numerically using the equations of motion of particles in the magnetosphere (REF ).",
"Clearly visible is the transition from independent $ \\alpha $ at small values of $ \\alpha $ relation to the relation of $ \\gamma _m \\propto \\alpha ^ {2/3} $ in the region $ \\alpha > \\alpha _1 $ .",
"Figure: The maximum Lorentz factor of particle γ m \\gamma _m versus the value of α \\alpha for κ=10 -4 \\kappa = 10^{-4} .",
"The slope of the curve for α>10 -1 \\alpha > 10^{-1} just corresponds to the dependence γ m ∝α 2/3 \\gamma _m\\propto \\alpha ^{2/3} ().The result of our studies of particle acceleration shows that the acceleration depends not only on the magnitude of the magnetic field - the magnetization parameter $ \\kappa $ (REF ), but also on the topology of the magnetic field, on the ratio between the poloidal and toroidal magnetic fields.",
"The acceleration of particles in the magnetosphere of the rotating black hole is carried out, by the electric field $ E_ \\theta $ (REF ).",
"This field arises when magnetic field lines rotate with the angular velocity of $ \\Omega _F $ , which is proportional to the angular velocity of a black hole rotation.",
"The particle, making cyclotron rotation, and drifting in crossed fields in the direction of rotation due to polarization drift (increasing its toroidal momentum) shifts from the original magnetic surface and acquires additional energy in the electric field.",
"The strongest deviation occurs near the light surface at the distance $ \\Delta r $ (REF ) from it.",
"Here, accelerating, the particle increases its cyclotron radius and accelerates even more until it reaches the light surface.",
"In the toroidal magnetic field near the light surface, the particle, in contrast to movement in the poloidal field, also experiences centrifugal and gradient drifts directed along the electric field, which deviate it from the given magnetic surface and produces more efficient acceleration.",
"Here the values of $ \\gamma $ and $ \\gamma _m $ are values of the Lorentz factor and the maximum Lorentz factor relative to the value of $ \\gamma _i $ , which is the initial Lorentz factor.",
"Actually expressions $\\gamma _m\\simeq \\kappa ^{-1/2}=\\gamma _0^{1/2}, \\,\\alpha <\\gamma _0^{-1/4}; \\,$ $ \\gamma _m\\simeq \\kappa ^{-2/3}=\\gamma _0^{2/3}, \\,\\alpha >\\gamma _0^{-1/4},$ are valid only for the acceleration of 'cold' particles, $ \\gamma _i \\simeq 1 $ .",
"The value of $ \\gamma _0 = \\omega _c / \\Omega _F $ is the maximum possible value of the energy of accelerated particles (the Hillas criterion).",
"When $ \\gamma = \\gamma _0 $ , the cyclotron radius of the particle becomes equal to the radius of the light cylinder.",
"In fact, the acceleration is not so strong, the powers are $ 1/2 $ and $ 2/3 $ , not 1, in the real field of the magnetosphere of the rotating black hole with an accretion disk.",
"But the presence of the jet, i.e.",
"the toroidal magnetic field generated by an electric current flowing in a jet, makes acceleration more efficient (the power of $ 2/3 $ ).",
"If particles are accelerated not from 'cold' state, but from the Lorentz factor $ \\gamma _i> 1 $ , then the expressions (REF ) for them take the form $&&\\gamma _m\\simeq (\\gamma _i\\gamma _0)^{1/2}, \\, \\alpha <(\\gamma _0/\\gamma _i)^{-1/4}; \\nonumber \\\\&&\\gamma _m\\simeq \\gamma _i^{1/3}\\gamma _0^{2/3}, \\, \\alpha >(\\gamma _0/\\gamma _i)^{-1/4}.$ If particles are pre-accelerated in the black hole magnetosphere or in a turbulent accretion disk to energies $ \\gamma _i >> 1 $ , as it was shown by (Istomin and Sol, 2009), then the maximum energy $ \\gamma _m $ , obtained in the process of centrifugal acceleration, can significantly approach the value of $\\gamma _0$ .",
"The dependence of $ \\gamma _m $ on the angle $ \\theta $ , see formulas (REF , REF ), means that at different heights $ z $ above the equatorial plane the particle energy is also different.",
"Accelerated particles, leaving the vicinity of the black hole, will form a single energy spectrum.",
"Summing them, crossing the light surface $ r = r_L / \\sin \\theta $ and acquiring energy $ \\gamma _m $ , that depends on the angle $ \\theta $ , we can obtain the distribution function of fast particles $ f (\\gamma _m) $ .",
"As $ z = r_L \\cot \\theta , \\, dz = -d \\theta / \\sin ^2 \\theta $ , and $ f (\\gamma _m) d \\gamma _m\\propto \\pi r_L^2 ndz $ , where $ n $ is the density of the 'primary' particles in the magnetosphere inside the light surface ($ n \\simeq const $ ), we get $&&f(\\gamma _m)\\propto \\gamma _m^{-2}, \\, \\gamma _m<\\gamma _0^{1/2}, \\, \\alpha <\\gamma _0^{-1/4}; \\nonumber \\\\&&f(\\gamma _m)\\propto \\gamma _m^{-2.5}, \\, \\gamma _m<\\gamma _0^{2/3}, \\, \\alpha >\\gamma _0^{-1/4}.$ Here we consider that angles $ \\theta $ differ from some small value of $\\theta _1 $ , determined by the height of the magnetosphere, to $\\theta _2<\\pi /2$ .",
"The angle $ \\theta _2 $ separates region of the magnetosphere adjacent to the accretion disk, where the matter rotates with the speed close to the speed of rotation of the disk, and the region where the matter rotates with the speed close to the speed of rotation of the black hole.",
"Therefore, when getting the expression (REF ), one can put $ \\cos \\theta \\simeq 1 $ .",
"The value of $ \\sin \\theta $ basically takes small values, which corresponds to the energies $ \\gamma _m <\\gamma _0 ^ {1/2} $ for small toroidal fields, and $ \\gamma _m <\\gamma _0^{2/3} $ for large ones.",
"It should be noted that, as we have seen, for large toroidal fields, large values of $ \\alpha $ , particle acceleration is more efficient than in the absence of a toroidal field.",
"Generally speaking, there may exist such magnetic field topologies for which it is still possible more efficient acceleration, $ \\gamma _m = \\gamma _0 $ .",
"However, in case a split monopole magnetiс fields to combined with a toroidal field, the acceleration at large toroidal fields does not depend on the strength of the poloidal field, $ \\gamma _m = (\\alpha \\kappa ^{-1})^{2/3} $ .",
"It's evident the geometry of the poloidal field that such topologies, if they exist, should rarely be realized."
],
[
"AGN",
"Centrifugal acceleration occurs in the magnetosphere of a black hole directly in its vicinity of the size of the radius of the light cylinder, $ r \\simeq r_L = c / \\Omega _F $ .",
"For optimal matching of the rotation of a magnetosphere with the rotation of a black hole the value of $ \\Omega _F$ is $ \\Omega _F = \\Omega _H / 2 $ , where $ \\Omega _H $ is the angular velocity of rotation of the black hole, $&&\\Omega _H=\\frac{2c}{r_g}\\frac{a}{1+(1-a^2)^{1/2}}, \\nonumber \\\\&&r_g=\\frac{2GM}{c^2}.$ The value of $ r_g $ is the gravitational radius of a black hole of the $ M $ mass, $ r_g = 3 \\cdot 10^5 (M / M_\\odot ) \\, = 3 \\cdot 10^{14} M_9 \\, cm $ .",
"Here the value of $ M_9 $ is the mass of a black hole measured in units of $ 10^9M_\\odot $ .",
"We used masses of black hole$ M_9 $ given in referred papers: in the center of our Galaxy Sgr.",
"A* (Ghez, et al., 2008) and six AGN; NGC 1365 (Fazeli, et al., 2019), NGC 4051 (Seifina, Chekhtman & Titarchuk, 2018), NGC 4151 (Bentz, et al., 2006), NGC 4486 (M 87) (Broderick, et al., 2015), NGC 6166 (Maggorian, et al., 1998) and NGC 7469 (Seifina, Titarchuk & Ugolkova, 2018).",
"See the table.",
"Parameter $ a $ characterizes the rotation of a black hole, and is equal to its specific angular momentum, $ a <1 $ .",
"So, $\\Omega _F=10^{-4}aM_9^{-1}/[1+(1-a^2)^{1/2}] \\, s^{-1}$ .",
"The magnetic field strength in the magnetosphere $ B $ is one of characteristics of the central black hole, it plays a key role in the process of particle acceleration and underlies the calculation of $ \\gamma _{m} $ .",
"The data presented in the work of Daly (2019) contain magnetic field values for some of the most famous AGN.",
"The magnetic field strength near the horizon for six active galaxies is given by the Table REF .",
"The value of $ B_4 $ is the magnitude of the magnetic field in units of $ 10^4 \\, G $ .",
"We see that given values of the magnetic field near the horizon $ B_4 $ are less than $M_9^{-1/2}$ .",
"This is due to the fact that the maximum magnetic field can be estimated as follows.",
"Suppose that for a central object with the Eddington luminosity, $ L \\simeq L_ {Edd} $ , the magnetic field energy density $ B^2/8 \\pi $ near the horizon is comparable with the energy density of the accreting matter (equipartition).",
"Then $ L_{Edd} \\simeq 4 \\pi cr_g^2 (B^2/8 \\pi ) $ , and one can introduce the value of $ B_{Edd} = (2 L_{Edd} / c r_g^2)^{1 / 2} \\simeq 1.7\\cdot 10^4 M_9^{-1/2} \\, $ G. This value, on the one hand, is the characteristic value of the magnetic field in AGN and, on the other hand, is its upper limit, $ B = bB_{Edd}, \\, b <1 $ .",
"The coefficient $ b $ depends on the mass of the black hole $ M $ , on the specific angular momentum of the black hole $ a $ , on the kinetic luminosity of the jet $ L_j $ , on the bolometric luminosity of the disk $ L_{bol}$ and on the value of the accretion rate $ {\\dot{M}} $ .",
"Therefore, to determine $ b $ , it is necessary to know the model of the accretion disk, the jet model, etc.",
"Using physical considerations and empirical dependences, Daly (2016, 2019) developed a procedure of determining the value of $ B_4 $ , as well as $ a $ .",
"The Daly's values of the specific angular momentum $ a $ for many AGNs turned out to be close to those estimated by other methods.",
"Thus, the magnetic fields $ B_4 $ estimated by him can be considered quite reliable.",
"Since the cyclotron rotation frequency for protons is equal to $ \\omega _c = 10^4(B/1 \\, G) \\, s^{-1}$ , then the value of the maximum possible Lorentz factor $ \\gamma _0 =\\omega _c / \\Omega _F $ is $&&\\gamma _0=4\\cdot 10^{12}M_9B_4\\frac{1+(1-a^2)^{1/2}}{a}\\simeq 4\\cdot 10^{12}M_9B_4, \\nonumber \\\\&&a\\simeq 1,$ which corresponds the proton energy $ E_ {max}^{(1)} = 3.8 \\cdot 10^{21} M_9 B_4 \\, eV $ .",
"However, as it follows from our calculations, the split monopole configuration allows to accelerating particles to values smaller than $\\gamma _0$ , $\\gamma _m=\\gamma _0^{2/3}=2.5\\cdot 10^{8}(M_9B_4)^{2/3}$ .",
"For protons this is $ E_{max}^{(2/3)} = 2.3 \\cdot 10^{17} (M_9B_4)^{2/3} \\, eV $ .",
"In the case of a weak toroidal field, acceleration is even less efficient, $\\gamma _m=\\gamma _0^{1/2}=2\\cdot 10^{6}(M_9B_4)^{1/2}, \\, E_{max}^{(1/2)}=1.9\\cdot 10^{15}(M_9B_4)^{1/2} \\, eV$ .",
"Here we consider the example of the galaxy M87 (NGC4486).",
"It has a powerful jet through which electric current flows.",
"Let us estimate the magnitudes of the poloidal and the toroidal magnetic fields in its magnetosphere.",
"Jet luminosity is $ L = 10^{44} \\, $ erg/s (Broderick et al., 2015).",
"Equate it to the electric power released in the external part of the electric current loop.",
"With the optimal matching, the resistance of the external electrical circuit is equal to the horizon resistance $ R_H = 4 \\pi / c = 377 \\, $ Ohm Thorne et al., (1986), $ L = R_H I ^ 2 $ .",
"Then we estimate the value of the electric current $ I $ in the jet, $ I \\simeq 1.6 \\cdot 10 ^ {17} \\, A $ .",
"This current, flowing in the jet in the forward direction in the centre and in the opposite direction at the periphery, closes in the accretion disk, flowing from the outer regions of the disc to the inner region near the black hole horizon.",
"It creates a toroidal magnetic field $ B_\\phi $ in the magnetosphere, $ B_\\phi \\simeq 2I / ch $ , where the value of $ h $ is half the height of the disk on its inner edge.",
"So, we get $ B_ \\phi \\simeq 10^2 (M_9 h / r_g)^{-1} \\, G $ .",
"Given $ h / r_g \\simeq 3\\cdot 10^{-3} $ , we find that the estimated toroidal magnetic field near the horizon is of the order of magnitude given in Table 1 for М87.",
"Let us now estimate the value of the polodal field $ B_P $ in the vicinity of the horizon.",
"The electric current is generated by the dynamo machine created by a rotating poloidal magnetic field.",
"The generated voltage is $ U = B_Pr_g^2 \\Omega _H / 2c $ (Landau and Lifshitz, 1984; Thorne et al., 1986).",
"On the other hand, $ U = 2R_HI $ .",
"From here we get $ B_P = 16 \\pi I / cr_g = 8 \\pi B_\\phi (h / r_g) $ .",
"Thus, the ratio of the toroidal magnetic field to the poloidal one, i.e.",
"$ \\alpha $ , equals $ \\alpha = r_g / 8 \\pi h \\simeq 10> 1 $ .",
"Therefore, for M87 the energy of accelerated protons corresponds to the estimated value in Table REF .",
"The nucleus of our galaxy Sgr.",
"A* contains the black hole with the mass of $ 4.3 \\cdot 10^6 M_\\odot $ .",
"It is not active, its luminosity is only $ \\simeq 10^{-9} $ of the Eddington luminosity.",
"There is no jet, and one can think that there is no toroidal magnetic field, $ B_\\phi = 0 $ .",
"Therefore, the maximum Lorentz factor of particles is given by the expression $ \\gamma _m = \\gamma _0^{1/2} \\simeq 8 \\cdot 10^4 $ , which corresponds to the proton energy of $ E_{max}^{(1/2)} \\simeq 7.4 \\cdot 10^{13} \\, eV $ .",
"However, the observed spectrum of gamma photons from Sgr.",
"A* is the power law with the exponent $ \\simeq -2.3 $ (Abramowski et al., 2016).",
"It is an intermediate value between the values of -2 and -2.5, given by the expression (REF ) for weak and strong toroidal magnetic field respectively.",
"It should be assumed that a weak toroidal magnetic field is still present in the magnetosphere, $ \\alpha \\simeq \\gamma _0^{-1/4} \\simeq 6.3 \\cdot 10^{-3} $ .",
"Plasma accretion must be accompanied by the appearance of an electric current, and hence generation of a toroidal magnetic field, the way it occurs in case of accretion onto a neutron star (Istomin and Haensel, 2013).",
"Therefore, the maximum possible energies of accelerated protons lie within interval $ E_{max}^{(1/2)}<E_{max} <E_{max}^{(2/3)} $ , i.e.",
"$ 7.4 \\cdot 10^{13} \\, eV <E_{max} <3.2 \\cdot 10^{15} \\, eV $ , which is consistent with the observations of the Galactic centre by the HESS Cherenkov telescope (Abramowski et al., 2016).",
"As for the power $ W $ radiated into accelerated particles, it is only part of the total power lost by the black hole, $S = R_H I^2 = U^2/4 R_H = (B_P^2/16 \\pi )cr_L^2$ .",
"This part is the ratio of the maximum energy of the accelerated protons $ \\gamma _m $ to the maximum possible particle energy obtained under complete transformation of the energy of rotation of the black hole into the energy of particles, $ \\kappa ^{-1} $ , that is $ W = S \\gamma _m \\kappa = S \\gamma _m / \\gamma _0 $ .",
"For Sgr.",
"A* the value of $ W $ lies within $ 1.5 \\cdot 10^{35} $ erg/s <$ W $ <$ 7 \\cdot 10^{36} $ erg/s, since $ \\kappa ^{-1/2 } <\\gamma _m <\\kappa ^{-2/3} $ .",
"Table: Upper limits for proton energy.",
"Here, M 9 ≡M BH /10 9 M ⊙ M_9 \\equiv M_{BH} / 10^9 M_\\odot , B 4 ≡B φ /10 4 G B_4 \\equiv B_\\phi / 10 ^ 4 \\, G , proton energy E max (1) =m p c 2 γ 0 ,E max (1/2) =m p c 2 γ 0 1/2 ,E max (2/3) =m p c 2 γ 0 2/3 E_{max}^{(1)}=m_pc^2\\gamma _0,\\, E_{max}^{(1/2)}= m_pc^2\\gamma _0^{1/2}, \\, E_{max}^{(2/3)}=m_pc^2\\gamma _0^{2/3} given in units of eV eV Black hole masses have been updated for Sgr A∗ (Ghez et al.",
"2008), NGC 1365 (Fazeli et al.",
"2019), NGC 4051 (Seifina et al.",
"2018), NGC 4151 (Bentz et al.",
"2006), NGC 4258 González-Lópezlira R. A., et al., 2017), NGC 4486 (Broderick et al.",
"2015), NGC 6166 (Maggorian.",
"1998), NGC 7469 (Seifina et al.",
"2018).",
"It should be noted that we used to estimate the value of the black hole rotation parameter $a \\simeq 1 $ , which is true for Sgr.",
"A* and M87.",
"For other galaxies, the rotation may not be so fast, $ a << 1 $ .",
"In this case, the size of the magnetosphere increases, $ r_L \\propto a^{-1} $ , the magnetization parameter $ \\kappa $ decreases by the same value, which leads to acceleration to higher energies, $ \\gamma _0 \\propto a^{-1}$ .",
"We would like to emphasize that obtained estimates of $ \\gamma _{m} $ are the Lorentz factor achieved by a 'cold' particle, $ \\gamma _i \\simeq 1 $ , as a result of acceleration in the magnetosphere.",
"If the particle has undergone preliminary acceleration, $ \\gamma _i> 1 $ , then the maximum possible energies increase substantially (REF ).",
"For example, some particles can be accelerated in a turbulent disk (Istomin and Sol, 2009)).",
"In addition to the above, we would like to add that the results obtained here can also be used for microquasars such as SS 433 (Abeysekara, 2018)."
],
[
"Discussion",
"We have shown that in the magnetosphere of the rotating black hole up to the light surface of $r <c/\\Omega _F\\sin \\theta $ , occurs an acceleration of charged particles.",
"It is bound to the fact that the pole electric field $E_\\theta $ makes possible the particle deviation for cyclotron radius of $ r_c = c \\gamma _i/\\omega _c$ from an initial magnetic surface $\\theta =const$ .",
"The cyclotron radius grows in proportion to energies of a particle, $\\gamma $ .",
"Therefore, acceleration is the most effective near the light surface, where $E_\\theta \\simeq B_r$ .",
"As a result, the work of the electric field $e c\\gamma B_0/\\omega _c$ is just equal to the energy derived by $mc^2\\gamma $ particle.",
"Thus, accounting of final cyclotron radius is very important for acceleration of a charged particle in a magnetosphere especially as it grows with the increase in the particle energy.",
"The solution of the particle acceleration problem presented here is based on the integration of the equations of particles motion in an electromagnetic field, which is equivalent to the solution of kinetic equations for individual components - protons and electrons (we consider the electron-proton plasma in the magnetosphere rather than the electron-positron plasma, which is also possible under efficient production of pairs (Beskin, Istomin & Pariev, 1992)).",
"The problem of plasma acceleration by the rotating neutron star or black hole was also considered in the MHD approximation in many works, starting from the works of Michel (1969), (1974).",
"Here we explain in details what the MHD approximation is.",
"In general, a neutral fluid, consisting of positive and negative charges, is described by unified equations of hydrodynamics type, in which the interaction of a fluid with the magnetic field is taken into account.",
"This suggests (I) that protons and electrons move together as a single liquid, i.e.",
"$ u_i - u_e << u_i = u + m_e (u-u_e) / m_i \\simeq u $ .",
"Here, $ u $ is the average mass fluid velocity.",
"In addition (II), the individual characteristics of the particles disappear, in particular the cyclotron radii of electrons and protons, which, generally speaking, are very different from each other, $r_{ci}>>r_{ce}$ .",
"This means that MHD equations work in regions whose characteristic size $ L $ significantly exceeds the cyclotron radius of protons, $ L >> r_{ci} $ .",
"In MHD equations, which are hydrodynamics type equations, the ion cyclotron radius plays the role of the mean free path $ l $ in ordinary hydrodynamics.",
"Therefore, even in weakly collisional systems, $ l \\simeq L $ , MHD is true, if $ L >> r_{ci} $ .",
"Under ideal conditions, i.e.",
"neglecting dissipation, the MHD approximation uses the condition of freezing of magnetic field into a liquid, $ {\\bf E} = - ({\\bf u}\\times {\\bf B}) / c $ (REF ).",
"It follows from Ohm's law $ {\\bf j} = \\sigma _c [{\\bf E} + ({\\bf u}\\times {\\bf B}) / c] $ under the condition of high conductivity, $ \\sigma _c \\rightarrow \\infty $ .",
"However, the very Ohm's law, called the generalized Ohm's law (Gurnett & Bhattacharjee, 2004), obtained after protons and electrons having been considered, is as follows: ${\\bf j}=\\sigma _c\\left\\lbrace \\left[{\\bf E}+\\frac{1}{c}[{\\bf uB}]\\right]+\\frac{1}{enc}[{\\bf jB}] +\\frac{1}{enc}\\nabla P_e\\right\\rbrace .$ Here the quantity $ P_e $ is the electron pressure.",
"Since $ {\\bf j} = en ({\\bf u}_i - {\\bf u}_e) $ , the second term on the right-hand side of the generalized Ohm's law is small compared to the first one if the condition $ {\\bf u}_i - {\\bf u}_e << {\\bf u} \\simeq {\\bf u}_i $ .",
"In reality, in our problem of centrifugal particle acceleration in the black hole magnetosphere ($ r <r_L / \\sin \\theta $ ), the polar electric field $ E_\\theta $ (REF ) does work on both electrons and protons, $ u_{i \\theta } <0$ (Fig.",
"REF ), $ u_{e \\theta }> 0 $ .",
"Therefore, $ |(u_i-u_e)_\\theta | = |(u_{i \\theta } - |u_{e \\theta }|)|> |u_{i \\theta }|\\simeq |u_\\theta | $ .",
"Thus, we see that the frozen-in condition (REF ), which is one of the basic postulates of ideal MHD, is not fulfilled in our problem.",
"It is obvious that if $ {\\bf E} = - {\\bf u}\\times {\\bf B} / c $ , then $ {\\bf E} = - {\\bf u}_i\\times {\\bf B} / c $ with the accuracy of the small ratio $ m_e / m_i << 1 $ , and the electric field does not work on protons, $ {\\bf u_iE} = 0 $ .",
"In terms of physical interpreting the ideal MHD does not work here.",
"The ideality implies not only the absence of Joule heating, but also absence of direct transfer of energy from the electric field to charged particles (the liquid is neutral).",
"The acceleration of the flow in MHD can be only accelerated by two means: the transfer of thermal energy of the matter into kinetic energy, which is carried out in hydrodynamics, as well as the transformation of the energy of the magnetic field into the energy of the flow.",
"This can be seen from the Euler equation in the MHD approximation: $n\\frac{d{\\bf p}}{dt}=-\\nabla P-\\frac{1}{8\\pi }\\nabla B^2\\frac{1}{4\\pi }({\\bf B\\nabla }){\\bf B}.$ Here, the acceleration occurs due to the pumping of the toroidal magnetic field by the central object rotation.",
"Indeed, the radial field rapidly decreases with distance, $ \\propto r^{-2} $ , but the toroidal field generation, $ B_\\phi \\propto r^{-1} $ , takes place due to the freezing-in of the magnetic field under differential rotation of the fluid.",
"With the relation $ E_\\theta = -r \\Omega _F \\sin \\theta B_r / c = (u_r B_\\phi -u_\\phi B_r) / c $ , which is the $ \\theta $ component of the equation (REF ), arising toroidal field is $B_\\phi = B_r\\frac{u_\\phi -\\Omega _Fr\\sin \\theta }{u_r}.$ Calculations in the MHD approximation show that the flow acceleration occurs at large distances from the centre, $ r >> r_L $ , and up to the Lorentz factor $ \\gamma _m = \\sigma ^{1/3} $ (Michel (1969), (1974); (Beskin, Kuznetsova & Rafikov, (1998)).",
"Here the value of $ \\sigma $ is the so-called Michel's parameter, $ \\sigma = e\\Phi \\Omega _F / mc^3 $ .",
"The flux of the radial magnetic field $\\Phi $ is, $ \\Phi = \\pi B_0r_L^2 $ .",
"It is easy to see that $ \\sigma = \\pi \\omega _c / \\Omega _F >> 1 $ .",
"In fact, this is reciprocal to the magnetization parameter, $ \\sigma \\simeq \\kappa ^{-1}$ (REF ).",
"It should be noted that MHD acceleration is almost isotropic, $ u_\\rho \\simeq u_\\phi $ , while the centrifugal acceleration inside the magnetosphere ($ r <r_L $ ) discussed in this article, is strongly anisotropic, $ p_\\phi >> p_\\rho , \\, p_\\theta $ .",
"Thus, we see that centrifugal acceleration, which is not described in the MHD approximation, is more effective for the case of a radial magnetic field (split monopole), $ \\gamma _m \\simeq \\sigma ^{1/2}, \\, \\sigma ^{2 / 3} >> \\sigma ^{1/3} $ .",
"However this is valid for the radial configuration of the magnetic field.",
"In case are magnetic surfaces of parabolic configuration, MHD acceleration there can reach to maximum values, $ \\gamma _m \\simeq \\sigma $ (Beskin & Nokhrina, (2006); Tchekovskoy, McKinney & Narayan, (2009)).",
"Here the acceleration is affected by the flow shape, the way it occurs in hydrodynamics (nozzle shape).",
"However, in this case MHD calculations should be performed starting from the light surface with the boundary conditions specified by the internal flow, $ r<r_L $ ."
],
[
"Acknowledgments",
"This work was supported by Russian Foundation for Fundamental Research, grant number 17-02-00788."
]
] | 2001.03555 |
[
[
"On Optimal Load-Memory Tradeoff of Cache-Aided Scalar Linear Function\n Retrieval"
],
[
"Abstract Coded caching has the potential to greatly reduce network traffic by leveraging the cheap and abundant storage available in end-user devices so as to create multicast opportunities in the delivery phase.",
"In the seminal work by Maddah-Ali and Niesen (MAN), the shared-link coded caching problem was formulated, where each user demands one file (i.e., single file retrieval).",
"This paper generalizes the MAN problem so as to allow users to request scalar linear functions of the files.",
"This paper proposes a novel coded delivery scheme that, based on MAN uncoded cache placement, is shown to allow for the decoding of arbitrary scalar linear functions of the files (on arbitrary finite fields).",
"Interestingly, and quite surprisingly, it is shown that the load for cache-aided scalar linear function retrieval depends on the number of linearly independent functions that are demanded, akin to the cache-aided single-file retrieval problem where the load depends on the number of distinct file requests.",
"The proposed scheme is optimal under the constraint of uncoded cache placement, in terms of worst-case load, and within a factor 2 otherwise.",
"The key idea of this paper can be extended to all scenarios which the original MAN scheme has been extended to, including demand-private and/or device-to-device settings."
],
[
"Introduction",
"Information theoretic coded caching was originally proposed by Maddah-Ali and Niesen (MAN) in [1] for the shared-link caching systems containing a server with a library of ${\\mathsf {N}}$ equal-length files, which is connected to ${\\mathsf {K}}$ users through a noiseless shared-link.",
"Each user can store ${\\mathsf {M}}$ files in their local cache.",
"Two phases are included in the MAN caching system: i) cache placement phase: content is pushed into each cache without knowledge of future demands; ii) delivery phase: each user demands one file, and according to the cache contents, the server broadcasts coded packets to all the users.",
"The objective is to minimize the transmitted load (i.e., number of transmitted bits normalized by the length of a single file) to satisfy the all the user' demands.",
"The MAN coded caching scheme proposed in [1], uses a combinatorial design in the placement phase (referred to as MAN cache placement), such that in the delivery phase binary multicast messages (referred to as MAN multicast messages) can simultaneously satisfy the demands of users.",
"Under the constraint of uncoded cache placement (i.e., each user directly caches a subset of the library bits), the MAN scheme can achieve the minimum worst-case load among all possible demands when ${\\mathsf {N}}\\ge {\\mathsf {K}}$ [2].",
"On the observation that if if there are files demanded multiple times, some MAN multicast messages can be obtained as a binary linear combination of other MAN multicast messages, Yu, Maddah-Ali, and Avestimehr (YMA) proposed an improved delivery scheme in [3].",
"The YMA delivery, with MAN placement, achieves the minimum worst-case load under the constraint of uncoded cache placement.",
"The cost of uncoded cache placement compared to coded cache placement was proved in [4] to be at most 2.",
"MAN coded caching [1] has been extended to numerous models, such as Device-to-Device (D2D) caching systems [5], private coded caching [6], [7], coded distributed computing [8], and coded data shuffling [9], [10], [11] – just to name a few.",
"A common point of these models is that each user requests one file – some allow for users to request (the equivalent of) multiple files [12], [13], [8], [9], [10], [11] which however does not change much the nature of the problem.",
"In general, linear and multivariate polynomial operations are widely used fundamental primitives for building the complex queries that support on-line analytics and data mining procedures.",
"For example, linear operations are critical in modern deep learning and artificial intelligence algorithms, where matrix-matrix or matrix-vector multiplications are at the core of iterative optimization algorithms; while algebraic polynomial queries naturally arise in engineering problems such as those described by differential equations and distributed machine learning algorithms [14], [15].",
"In those scenarios, it may be too resource-consuming (in terms of bandwidth, or execution time, or storage space) to download locally all the input variables in order to compute the desired output value.",
"Instead, it is desirable to directly download the result of the desired output function.",
"This paper studies the fundamental tradeoff between local storage and network load when users are interested in retrieving a function of the dataset available at the server.",
"The question we ask in this paper is, compared to the original MAN caching problem, whether the optimal worst-case load is increased when the users are allowed to request scalar linear functions of the files – the first non-trivial extension of the MAN single-file-retrieval problem, on the way to understand the problem of retrieving general functions.",
"The original MAN shared-link caching problem in [1] where each user request one file is a special case of the formulated shared-link cache-aided scalar linear function retrieval problem.",
"In addition to the novel problem formulation, our main results are as follows: Achievable scheme for demanded functions on the binary field.",
"We start by considering the case of scalar linear functions on $\\mathbb {F}_2$ .",
"Based on the YMA delivery, which uses an “interference cancellation” idea on $\\mathbb {F}_2$ , we propose a novel delivery scheme whose key idea is to deliver only the largest set of linearly independent functions, while the remaining ones can be reconstructed by proper linear combinations of those already retrieved.",
"This can be thought of as the generalization of the idea to only deliver the files requested by the “leader users” in the YMA delivery.",
"Generalization to demanded functions on arbitrary finite field.",
"We then generalize the proposed scheme to the case where the demands are scalar linear functions on an arbitrary finite field $\\mathbb {F}_{{\\mathsf {q}}}$ .",
"To the best of our knowledge, even for the originally MAN coded caching problem, no caching scheme is known in the literature for arbitrary finite fields.",
"Compared to the YMA delivery scheme, we use different encoding (based on a finite field interference alignment idea) and decoding procedures that work on an arbitrary finite field.",
"Interestingly, the achieved load by the proposed scheme only depends on the number of linearly independent functions that are demanded, akin to the YMA's cache-aided single-file retrieval scheme where the load depends on the number of distinct file requests.",
"Optimality.",
"On observation that the converse bound for the original MAN caching problem in [2], [3] is also a converse in the considered cache-aided function retrieval problem, we prove that the proposed scheme achieves the optimal worst-cast load under the constraint of uncoded cache placement.",
"Moreover, the achieved worst-case load of the proposed scheme is also proved to be order optimal in general within a factor of 2.",
"From the results in this paper, we can answer the question we asked at the beginning of this paper: the optimal worst-case load under the constraint of uncoded cache placement is not increased when the users are allowed to request scalar linear functions of the files.",
"The rest of this paper is organized as follows.",
"Section formulates the cache-aided function retrieval problem and introduces some related results in the literature.",
"Section provides and discusses the main results in this paper.",
"Section and Section describe the proposed achievable caching schemes on the binary field and on arbitrary finite field, respectively.",
"Section concludes the paper.",
"Some of the proofs are given in the Appendices."
],
[
"Notation Convention",
"Calligraphic symbols denote sets, bold symbols denote vectors, and sans-serif symbols denote system parameters.",
"We use $|\\cdot |$ to represent the cardinality of a set or the length of a vector; $[a:b]:=\\left\\lbrace a,a+1,\\ldots ,b\\right\\rbrace $ and $[n] := [1,2,\\ldots ,n]$ ; $\\oplus $ represents bit-wise XOR; $\\mathbb {E}[\\cdot ]$ represents the expectation value of a random variable; $[a]^+:=\\max \\lbrace a,0\\rbrace $ ; $a!=a\\times (a-1) \\times \\ldots \\times 1$ represents the factorial of $a$ ; $\\mathbb {F}_q$ represents a finite field with order $q$ ; $\\text{rank}_q(\\mathbb {A})$ represents the rank of matrix $\\mathbb {A}$ on field $\\mathbb {F}_q$ ; $\\text{det}(\\mathbb {A})$ represents the determinant matrix $\\mathbb {A}$ ; $\\mathbb {A}_{{\\mathcal {S}},{\\mathcal {V}}}$ represents the sub-matrix of $\\mathbb {A}$ by selecting from $\\mathbb {A}$ , the rows with indices in ${\\mathcal {S}}$ and the columns with indices in ${\\mathcal {V}}$ .",
"we let $\\binom{x}{y}=0$ if $x<0$ or $y<0$ or $x<y$ .",
"In this paper, for each set of integers ${\\mathcal {S}}$ , we sort the elements in ${\\mathcal {S}}$ in an increasing order and denote the $i^{\\text{th}}$ smallest element by ${\\mathcal {S}}(i)$ , i.e., ${\\mathcal {S}}(1)<\\ldots <{\\mathcal {S}}(|{\\mathcal {S}}|)$ ."
],
[
"System Model",
"A $({\\mathsf {K}},{\\mathsf {N}},{\\mathsf {M}},{\\mathsf {q}})$ shared-link cache-aided scalar linear function retrieval problem is defined as follows.",
"A central server has access to a library of ${\\mathsf {N}}$ files.",
"The files are denoted as $F_1,\\ldots , F_{{\\mathsf {N}}}$ .",
"Each file has ${\\mathsf {B}}$ independent and uniformly distributed symbols over a finite field $\\mathbb {F}_{{\\mathsf {q}}}$ , for some prime-power ${\\mathsf {q}}$ .",
"The central server is connected to ${\\mathsf {K}}$ users through an error-free shared-link.",
"Each user is equipped with a cache that can store up to ${\\mathsf {M}}{\\mathsf {B}}$ symbols, where ${\\mathsf {M}}\\in [0,{\\mathsf {N}}]$ .",
"The system operates in two phases.",
"Cache Placement Phase.",
"During the cache placement phase, each user stores information about the ${\\mathsf {N}}$ files in its local cache without knowledge of future users' demands, that is, there exist placement functions $\\phi _k, \\ k\\in [{\\mathsf {K}}]$ , such that $\\phi _k &: [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {B}}{\\mathsf {N}}} \\rightarrow [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {B}}{\\mathsf {M}}},$ We denote the content in the cache of user $k\\in [{\\mathsf {K}}]$ by $Z_{k}=\\phi _k(F_1, \\ldots , F_{\\mathsf {N}})$ .",
"Delivery Phase.",
"During the delivery phase, each user requests one scalar linear function of the files.",
"The demand of user $k \\in [{\\mathsf {K}}]$ is represented by the row vector ${\\mathbf {y}}_k=(y_{k,1}, \\ldots , y_{k,{\\mathsf {N}}}) \\in [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {N}}}$ , which means that user $k$ wants to retrieve $y_{k,1} F_1 +\\ldots + y_{k,{\\mathsf {N}}} F_{{\\mathsf {N}}}$ .",
"We denote the demand matrix of all users by $\\mathbb {D}=[{\\mathbf {y}}_1;\\ldots ;{\\mathbf {y}}_{{\\mathsf {K}}}] \\in [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {K}}{\\mathsf {N}}}.$ Given the demand matrix $\\mathbb {D}$ , the server broadcasts the message $X = \\psi (\\mathbb {D}, F_1, \\ldots , F_{\\mathsf {N}})$ to each user $k\\in [{\\mathsf {K}}]$ , where the encoding function $\\psi $ is such that $\\psi &: [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {K}}{\\mathsf {N}}} \\times [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {B}}{\\mathsf {N}}} \\rightarrow [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {B}}{\\mathsf {R}}},$ for some non-negative ${\\mathsf {R}}$ .",
"Decoding.",
"Each user $k\\in [{\\mathsf {K}}]$ decode its desired function from $(\\mathbb {D},Z_k,X)$ .",
"In other words, there exist decoding functions $\\xi _k, \\ k\\in [{\\mathsf {K}}]$ , such that $\\xi _k &: [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {K}}{\\mathsf {N}}} \\times [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {B}}{\\mathsf {M}}} \\times [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {B}}{\\mathsf {R}}} \\rightarrow [\\mathbb {F}_{{\\mathsf {q}}}]^{{\\mathsf {B}}},\\\\\\xi _k(\\mathbb {D},Z_k,X) &= y_{k,1} F_1 +\\ldots + y_{k,{\\mathsf {N}}} F_{{\\mathsf {N}}}.",
"$ Objective.",
"For a given memory size ${\\mathsf {M}}\\in [0,{\\mathsf {N}}]$ , our objective is to determine the minimum worst-case load among all possible demands, defined as the smallest ${\\mathsf {R}}$ such that there exist placement functions $\\phi _k, k\\in [{\\mathsf {K}}],$ encoding function $\\psi $ , and decoding functions $\\xi _k, k\\in [{\\mathsf {K}}],$ satisfying all the above constraints.",
"The optimal load is denoted as $R^{\\star }$ .",
"If each user directly copies some symbols of the ${\\mathsf {N}}$ files into its cache, the cache placement is said to be uncoded.",
"The minimum worst-case load under the constraint of uncoded cache placement is denoted by ${\\mathsf {R}}^{\\star }_{\\text{u}}$ ."
],
[
"Review of the MAN {{cite:8c8176e3a76c45fd420b3c6b4fb564978f9ba995}} and YMA {{cite:45cee1e6a96db2a029a5d9578c7582f779e4088d}} Coded Caching Schemes",
"In the following, we review the MAN and YMA coded caching schemes, which are on the binary field $\\mathbb {F}_2$ , for the shared-link caching problem, where each user requests one file."
],
[
"MAN Scheme",
"File Split.",
"Let $t\\in [0:{\\mathsf {K}}]$ .",
"Partition each file $F_i, i\\in [{\\mathsf {N}}]$ , into $\\binom{{\\mathsf {K}}}{t}$ equal-length subfiles denoted as $F_i=\\lbrace F_{i,{\\mathcal {W}}}:{\\mathcal {W}}\\subseteq [{\\mathsf {K}}], |{\\mathcal {W}}|=t\\rbrace .$ Placement Phase.",
"User $k\\in [{\\mathsf {K}}]$ caches $F_{i,{\\mathcal {W}}}, i\\in [{\\mathsf {N}}]$ , if $k\\in {\\mathcal {W}}$ .",
"Hence, each user caches ${\\mathsf {N}}\\binom{ {\\mathsf {K}}-1}{t-1}$ subfiles, each of which contains $\\frac{{\\mathsf {B}}}{ \\binom{{\\mathsf {K}}}{t}}$ symbols, which requires ${\\mathsf {M}}=\\frac{{\\mathsf {N}}t}{{\\mathsf {K}}}.$ Delivery Phase.",
"User $k\\in [{\\mathsf {K}}]$ requests the file with index $d_k \\in [{\\mathsf {N}}]$ .",
"The server then broadcasts the following MAN multicast messages: for each ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {S}}|=t+1$ , the server sends $W_{{\\mathcal {S}}}= \\underset{k\\in {\\mathcal {S}}}{\\oplus } F_{d_k,{\\mathcal {S}}\\setminus \\lbrace k\\rbrace }.$ Decoding.",
"The multicast message $W_{{\\mathcal {S}}}$ in (REF ) is useful to each user $k \\in {\\mathcal {S}}$ , since this user caches all subfiles contained by $W_{{\\mathcal {S}}}$ except for the desired subfile $F_{d_k, {\\mathcal {S}}\\setminus \\lbrace k\\rbrace }$ .",
"Considering all multicast messages, each user can recover all uncached subfiles and thus recover its demanded file.",
"Load.",
"The achieved memory-load tradeoff of the MAN scheme is the lower convex envelop of the following points $({\\mathsf {M}}, {\\mathsf {R}})= \\left(\\frac{{\\mathsf {N}}t}{{\\mathsf {K}}}, \\frac{\\binom{{\\mathsf {K}}}{t+1}}{\\binom{{\\mathsf {K}}}{t}} \\right), \\forall t\\in [0:{\\mathsf {K}}].",
"$ File splitting and cache placement are as for the MAN scheme.",
"Delivery Phase.",
"The main idea of the YMA delivery is that, when a file is demanded by multiple users, some MAN multicast messages in (REF ) can be obtained as a linear combinations of others.",
"Thus the load of the MAN scheme in (REF ) can be further reduced by removing the redundant MAN multicast messages.",
"More precisely, for each demanded file, randomly choose one user among all users demanding this file and designate it as the “leader user” for this file.",
"Let ${\\mathcal {D}}:= \\cup _{k\\in [{\\mathsf {K}}]} \\lbrace d_k\\rbrace $ be the set of all distinct files that are demanded, and ${\\mathcal {L}}$ be the set of $|{\\mathcal {D}}|$ leader users.",
"The server only sends those multicast message $W_{{\\mathcal {S}}}$ in (REF ) that are useful for the leader users, that is, if ${\\mathcal {S}}\\cap {\\mathcal {L}}\\ne \\emptyset $ , thus saving $\\binom{{\\mathsf {K}}- |{\\mathcal {D}}|}{t+1}$ transmissions.",
"Decoding.",
"Clearly, all leaders users can decode their demanded files as per the MAN scheme.",
"The non-leader users appear to miss the multicast message $W_{{\\mathcal {A}}}$ for each ${\\mathcal {A}}\\subseteq [{\\mathsf {K}}]$ where ${\\mathcal {A}}\\cap {\\mathcal {L}}=\\emptyset $ and $|{\\mathcal {A}}|=t+1$ .",
"It was proved in [3] that $\\underset{{\\mathcal {F}}\\in {F}_{{\\mathcal {B}}}}{\\oplus } W_{{\\mathcal {B}}\\setminus {\\mathcal {F}}}=0,$ where ${\\mathcal {B}}={\\mathcal {A}}\\cup {\\mathcal {L}}$ , and ${F}_{{\\mathcal {B}}}$ is the family of subsets ${\\mathcal {F}}\\subseteq {\\mathcal {B}}$ , where each file in ${\\mathcal {D}}$ is requested by exactly one user in ${\\mathcal {F}}$ .",
"The key observation is that in $\\underset{{\\mathcal {F}}\\in {F}_{{\\mathcal {B}}} }{\\oplus } W_{{\\mathcal {B}}\\setminus {\\mathcal {F}}}$ each involved subfile appears exactly twice (i.e., contained into two MAN multicast messages)In this paper, $A$ `appears' in a linear combination means that in the linear combination, there exists some term in the linear combination including $A$ .",
"A linear combination `contains' $B$ means that in the linear combination, the total coefficient of $B$ is not 0.",
"For example, we say $A$ appears in the linear combination $(A\\oplus B)\\oplus (A \\oplus C)$ , but the linear combination does not contain $A$ ., whose contribution on $\\mathbb {F}_2$ is thus zero.",
"From (REF ), we have $W_{{\\mathcal {A}}} =\\underset{{\\mathcal {F}}\\in {F}_{{\\mathcal {B}}}: {\\mathcal {F}}\\ne {\\mathcal {L}}}{\\oplus } W_{{\\mathcal {B}}\\setminus {\\mathcal {F}}}.$ In other words, the multicast message $W_{{\\mathcal {A}}}$ can be reconstructed by all users from the delivery phase.",
"Load.",
"The YMA scheme requires the load of $\\frac{\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}- |{\\mathcal {D}}|}{t+1}}{\\binom{{\\mathsf {K}}}{t}},$ if the set of the demanded files is ${\\mathcal {D}}$ .",
"The worst-case load is attained for $|{\\mathcal {D}}|=\\min ({\\mathsf {N}},{\\mathsf {K}})$ , thus the achieved memory-load tradeoff of the YMA scheme is the lower convex envelop of the following points $({\\mathsf {M}}, {\\mathsf {R}})= \\left(\\frac{{\\mathsf {N}}t}{{\\mathsf {K}}}, \\frac{\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}- \\min ({\\mathsf {N}},{\\mathsf {K}}) }{t+1}}{\\binom{{\\mathsf {K}}}{t}} \\right), \\forall t\\in [0:{\\mathsf {K}}].",
"$"
],
[
"Main Results and Discussion",
"In this section, we summarize the main results in this paper.",
"The proposed caching scheme in Section (for ${\\mathsf {q}}=2$ ) and Section (for general prime-power ${\\mathsf {q}}$ ), achieves the following load.",
"Theorem 1 (Achievability) For the $({\\mathsf {K}},{\\mathsf {N}},{\\mathsf {M}},{\\mathsf {q}})$ shared-link cache-aided scalar linear function retrieval problem, the YMA load in (REF ) is an achievable worst-case load.",
"More precisely, for cache size ${\\mathsf {M}}= \\frac{{\\mathsf {N}}t}{{\\mathsf {K}}}$ , with $t\\in [0:{\\mathsf {K}}]$ , and for demand matrix $\\mathbb {D}$ , the load ${\\mathsf {R}}(\\mathbb {D}) := \\frac{\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}-\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D}) }{t+1}}{ \\binom{{\\mathsf {K}}}{t}}$ is achievable.",
"The worst-case load is attained by $\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D}) =\\min ({\\mathsf {N}},{\\mathsf {K}})$ .",
"$\\square $ Remark 1 (Dependance on the rank of the demand matrix) The load in (REF ) is a generalization of the load in (REF ) achieved by the YMA scheme.",
"More precisely, if each user $k\\in [{\\mathsf {K}}]$ requests one file (i.e., ${\\mathbf {y}}_k\\in [0:1]^{{\\mathsf {N}}}$ with a unit norm), $\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D})$ is exactly the number of demanded files, and thus the proposed scheme achieves the load in (REF ) as the YMA scheme.",
"Interestingly, the load of the proposed scheme only depends on the rank of the demand matrix of all users, instead of on the specifically demanded functions.",
"$\\square $ Remark 2 (High-level ideas to derive the load in Theorem REF ) We partition the “symbol positions\" set $[{\\mathsf {B}}]$ as follows $[{\\mathsf {B}}] = \\lbrace {\\mathcal {I}}_{\\mathcal {W}}: {\\mathcal {W}}\\subseteq [{\\mathsf {K}}], |{\\mathcal {W}}|=t \\rbrace \\ \\text{such that} \\ |{\\mathcal {I}}_{\\mathcal {W}}| = {\\mathsf {B}}/\\binom{{\\mathsf {K}}}{t}.$ Then, with a Matlab-inspired notation, we let $F_{i,{\\mathcal {W}}} = F_i({\\mathcal {I}}_{\\mathcal {W}}), \\ \\forall {\\mathcal {W}}\\subseteq [{\\mathsf {K}}] : |{\\mathcal {W}}|=t, \\ \\forall i\\in [{\\mathsf {N}}],$ representing the set of symbols of $F_i$ whose position is in ${\\mathcal {I}}_{\\mathcal {W}}$ .",
"As in the MAN placement, user $k\\in [{\\mathsf {K}}]$ caches $F_{i,{\\mathcal {W}}}$ if $k\\in {\\mathcal {W}}$ .",
"By doing so, any scalar linear function is naturally partitioned into “blocks” as follows $y_{k,1} F_1 +\\ldots + y_{k,{\\mathsf {N}}} F_{{\\mathsf {N}}}&=\\lbrace \\underbrace{y_{k,1} F_1({\\mathcal {I}}_{\\mathcal {W}}) +\\ldots + y_{k,{\\mathsf {N}}} F_{{\\mathsf {N}}}({\\mathcal {I}}_{\\mathcal {W}})}_{\\text{$:=B_{k,{\\mathcal {W}}}$ is the ${\\mathcal {W}}$-th block of the $k$-th demanded function}}: {\\mathcal {W}}\\subseteq [{\\mathsf {K}}], |{\\mathcal {W}}|=t\\rbrace .$ Some blocks of the demanded functions can thus be computed based on the cache content available at each user while the remaining ones need to be delivered by the server.",
"With this specific file split (and corresponding MAN cache placement), we operate the MAN delivery scheme over the blocks instead of over the subfiles; more precisely, instead of (REF ) we transmit $W_{{\\mathcal {S}}}= \\sum _{k\\in {\\mathcal {S}}} \\alpha _{{\\mathcal {S}},k} B_{k,{\\mathcal {S}}\\setminus \\lbrace k\\rbrace }, \\forall {\\mathcal {S}}\\subseteq [{\\mathsf {K}}] : |{\\mathcal {S}}|=t+1,$ for some $\\alpha _{{\\mathcal {S}},k} \\in \\mathbb {F}_{{\\mathsf {q}}} \\setminus \\lbrace 0\\rbrace $ and where $B_{k,{\\mathcal {W}}}$ was defined in (REF ).",
"Clearly, this scheme achieves the same load as in (REF ) (and works on any finite field and any $\\alpha _{{\\mathcal {S}},k} \\in \\mathbb {F}_{{\\mathsf {q}}} \\setminus \\lbrace 0\\rbrace $ ).",
"The questions is, whether with (REF ) we can do something similar to the YMA delivery scheme.",
"More specifically, what is a suitable definition of the leader user set ${\\mathcal {L}}$ ; what is a suitable choice of $\\alpha _{{\\mathcal {S}},k}$ 's in (REF ); and assuming we only send the multicast messages in (REF ) that are useful for the leader users (i.e., $ W_{{\\mathcal {S}}}$ where ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ , $|{\\mathcal {S}}|=t+1$ , and ${\\mathcal {S}}\\cap {\\mathcal {L}}\\ne \\emptyset $ ), what is the counterpart of (REF ); here for each ${\\mathcal {A}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {A}}|=t+1$ and ${\\mathcal {A}}\\cap {\\mathcal {L}}=\\emptyset $ , we seek $W_{{\\mathcal {A}}}= \\sum _{{\\mathcal {S}}\\subseteq [{\\mathsf {K}}]:|{\\mathcal {S}}|=t+1, {\\mathcal {S}}\\cap {\\mathcal {L}}\\ne \\emptyset } \\beta _{{\\mathcal {A}},{\\mathcal {S}}} W_{{\\mathcal {S}}}.$ The novelty of our scheme lays in the answers to these questions as follows: we first choose $\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D})$ leaders (the set of leader users is denoted by ${\\mathcal {L}}$ ), where the demand matrix of the leaders is full-rank.",
"When ${\\mathsf {q}}=2$ (i.e., on the binary field), lets $\\alpha _{{\\mathcal {S}},k} =1$ .",
"When ${\\mathsf {q}}$ is a prime-power, the proposed scheme in Section separates the demanded blocks by the leaders and non-leaders in $W_{{\\mathcal {S}}}$ in (REF ) as $\\sum _{k\\in {\\mathcal {S}}} \\alpha _{{\\mathcal {S}},k} B_{k,{\\mathcal {S}}\\setminus \\lbrace k\\rbrace } = \\sum _{k_1\\in {\\mathcal {S}}\\cap {\\mathcal {L}}} \\alpha _{{\\mathcal {S}},k_1} B_{k_1,{\\mathcal {S}}\\setminus \\lbrace k_1\\rbrace } +\\sum _{k_2\\in {\\mathcal {S}}\\setminus {\\mathcal {L}}} \\alpha _{{\\mathcal {S}},k_2} B_{k_2,{\\mathcal {S}}\\setminus \\lbrace k_2\\rbrace };$ we then alternate the coefficients of the desired blocks by the leaders (i.e., users in ${{\\mathcal {S}}\\cap {\\mathcal {L}}}$ ) between $+1$ and $-1$ , i.e., the coefficient of the desired block of the first leader is $+1$ , the coefficient of the desired block of the second leader is $-1$ , the coefficient of the desired block of the third leader is $+1$ , etc; similarly, we alternate the coefficients of the desired blocks by the non-leaders (i.e., users in ${{\\mathcal {S}}\\setminus {\\mathcal {L}}}$ ) between $+1$ and $-1$ .This type of code was originally proposed in [16] for the private function retrieval problem, where there is a memory-less user aiming to retrieval a scalar linear function of the files in the library from multiple servers (each server can access to the whole library), while preserving the demand of this user from each server.",
"With the above encoding scheme, we can compute the decoding coefficients (as $ \\beta _{{\\mathcal {A}},{\\mathcal {S}}}$ in (REF )) such that (REF ) holds for each ${\\mathcal {A}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {A}}|=t+1$ and ${\\mathcal {A}}\\cap {\\mathcal {L}}=\\emptyset $ .",
"In other words, each user can recover all multicast messages $W_{{\\mathcal {S}}}$ where ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ and $|{\\mathcal {S}}|=t+1$ , and thus it can recover its desired function.",
"$\\square $ Since the setting where each user demands one file is a special case of the considered cache-aided scalar linear function retrieval problem, the converse bounds in [2], [3], [4] for the original shared-link coded caching problem is also a converse in our considered problem, thus we have: Theorem 2 (Optimality) For the $({\\mathsf {K}},{\\mathsf {N}},{\\mathsf {M}},{\\mathsf {q}})$ shared-link cache-aided scalar linear function retrieval problem, under the constraint of uncoded cache placement, the optimal worst-case load-memory tradeoff is the lower convex envelop of $({\\mathsf {M}}, {\\mathsf {R}}^{\\star }_{\\text{u}}) = \\left( \\frac{{\\mathsf {N}}t}{{\\mathsf {K}}}, \\frac{\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}-\\min \\lbrace {\\mathsf {K}},{\\mathsf {N}}\\rbrace }{t+1}}{ \\binom{{\\mathsf {K}}}{t}} \\right), \\ \\forall t\\in [0:{\\mathsf {K}}].",
"$ Moreover, the achieved worst-case load in (REF ) is optimal within a factor of 2 in general.",
"$\\square $ Remark 3 (Extensions) We discuss three extensions of the proposed caching scheme in Theorem REF in the following.",
"Optimal average load under uncoded and symmetric cache placement.",
"We define uncoded and symmetric cache placement as follows, which is a generalization of file split in (REF )-(REF ).",
"We partition the “symbol positions\" set $[{\\mathsf {B}}]$ as $[{\\mathsf {B}}] = \\lbrace {\\mathcal {I}}_{\\mathcal {W}}: {\\mathcal {W}}\\subseteq [{\\mathsf {K}}] \\rbrace ,$ and let $F_{i,{\\mathcal {W}}} = F_i({\\mathcal {I}}_{\\mathcal {W}})$ as in (REF ).",
"Each user $k\\in [{\\mathsf {K}}]$ caches $F_{i,{\\mathcal {W}}}$ if $k\\in {\\mathcal {W}}$ .",
"Hence, in the delivery phase, user $k$ needs to recover $B_{k,{\\mathcal {W}}}$ (defined in (REF )) where ${\\mathcal {W}}\\subseteq [{\\mathsf {K}}]\\setminus \\lbrace k\\rbrace $ .",
"By directly using [4] in the caching converse bound under uncoded cache placement in [2], [3], we can prove that the proposed caching scheme in Theorem REF achieves the minimum average load over uniform demand distribution under the constraint of uncoded and symmetric cache placement cross files.",
"Corollary 1 [Optimal average load] For the $({\\mathsf {K}},{\\mathsf {N}},{\\mathsf {M}},{\\mathsf {q}})$ shared-link cache-aided scalar linear function retrieval problem, under the constraint of uncoded and symmetric cache placement, the minimum average load over uniform demand distribution is the lower convex envelop of $({\\mathsf {M}}, {\\mathsf {R}}) = \\left( \\frac{{\\mathsf {N}}t}{{\\mathsf {K}}}, \\mathbb {E}_{\\mathbb {D}}\\left[\\frac{\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}-\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D}) }{t+1}}{ \\binom{{\\mathsf {K}}}{t}}\\right] \\right), \\ \\forall t\\in [0:{\\mathsf {K}}].$ Notice that an uncoded and asymmetric cache placement can be treated as a special case of the inter-file coded cache placement in the originally MAN caching problem.",
"It is one of the on-going works to derive the converse bound under the constraints of uncoded cache placement for the considered cache-aided function retrieval problem.",
"Device-to-Device (D2D) cache-aided scalar linear function retrieval.",
"Coded caching was originally used in Device-to-Device networks in [5], where in the delivery phase each user broadcasts packets as functions of its cached content and the users' demands, to all other users.",
"The authors in [17] extended the YMA scheme to D2D networks by dividing the D2D networks into ${\\mathsf {K}}$ shared-link networks, and used the YMA scheme in each shared-link network.",
"Hence, when users request scalar linear functions, we can use the same method as in [17] to divide the D2D networks into ${\\mathsf {K}}$ shared-link networks, and then use the proposed caching scheme in Theorem REF in each shared-link network.",
"Corollary 2 [D2D cache-aided scalar linear function retrieval] For the $({\\mathsf {K}},{\\mathsf {N}},{\\mathsf {M}},{\\mathsf {q}})$ D2D cache-aided scalar linear function retrieval problem, the minimum worse-case load is upper bounded by the lower convex envelop of $({\\mathsf {M}}, {\\mathsf {R}}) = \\left( \\frac{{\\mathsf {N}}t}{{\\mathsf {K}}}, \\max _{\\mathbb {D}} \\frac{\\binom{{\\mathsf {K}}-1}{t}- \\frac{1}{{\\mathsf {K}}} \\sum _{k\\in [{\\mathsf {K}}]} \\binom{{\\mathsf {K}}-1-\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D}_{[{\\mathsf {K}}]\\setminus \\lbrace k\\rbrace }) }{t}}{ \\binom{{\\mathsf {K}}-1}{t-1}} \\right), \\ \\forall t\\in [{\\mathsf {K}}].$ Cache-aided private scalar linear function retrieval.",
"For the successful decoding of the proposed scheme in Theorem REF , users need to be aware of the demands of other users, which is not private.",
"To preserve the privacy of the demand of each user against other users, we can generate virtual users as in [6], such that each of all possible demanded functions (the total number of possible demanded functions is ${\\mathsf {N}}^{\\prime }:=\\frac{{\\mathsf {q}}^{{\\mathsf {N}}}-1}{{\\mathsf {q}}-1 }$ ) is demanded exactly ${\\mathsf {K}}$ times.",
"Thus there are totally ${\\mathsf {N}}^{\\prime } {\\mathsf {K}}$ real or virtual users in the system.",
"Then the proposed scheme in Theorem REF can be used to satisfy the demands of all real or virtual users.",
"Since each user cannot distinguish other real users from virtual users, the resulting scheme does not leak any information on the demands of real users.",
"Corollary 3 [Cache-aided private scalar linear function retrieval] For the $({\\mathsf {K}},{\\mathsf {N}},{\\mathsf {M}},{\\mathsf {q}})$ D2D cache-aided scalar linear function retrieval problem, the minimum load is upper bounded by the lower convex envelop of $({\\mathsf {M}}, {\\mathsf {R}}) = \\left(\\frac{t}{{\\mathsf {N}}^{\\prime } {\\mathsf {K}}}{\\mathsf {N}},\\frac{\\binom{{\\mathsf {N}}^{\\prime } {\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {N}}^{\\prime } {\\mathsf {K}}- {\\mathsf {N}}}{t+1} }{ \\binom{{\\mathsf {N}}^{\\prime } {\\mathsf {K}}}{t} } \\right) , \\ \\forall t\\in [{\\mathsf {N}}^{\\prime } {\\mathsf {K}}].$ $\\square $"
],
[
"Achievable Scheme in Theorem ",
"In the following, we describe the proposed scheme when the demands are scalar linear functions on $\\mathbb {F}_2$ .",
"We start with the following example."
],
[
"Example",
"Consider the $({\\mathsf {K}},{\\mathsf {N}},{\\mathsf {M}},{\\mathsf {q}})=(6,3,1,2)$ shared-link cache-aided scalar linear function retrieval problem, where $t={\\mathsf {K}}{\\mathsf {M}}/{\\mathsf {N}}=2 $ .",
"In the cache placement, each file is partitioned into $\\binom{{\\mathsf {K}}}{t}=15$ equal-length subfiles.",
"We use the file split in (REF )-(REF ), resulting in the demand split in (REF ).",
"In the delivery phase, we assume that $&\\text{user $1$ demands $F_1$;}\\\\&\\text{user $2$ demands $F_2$;}\\\\&\\text{user $3$ demands $F_3$;}\\\\&\\text{user $4$ demands $F_1 \\oplus F_2$;}\\\\&\\text{user $5$ demands $F_1 \\oplus F_3$;}\\\\&\\text{user $6$ demands $F_1 \\oplus F_2 \\oplus F_3$;}$ i.e., the demand matrix is $\\mathbb {D}=\\left[\\begin{array}{ccc}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\\\1 & 1 & 0\\\\1 & 0 & 1\\\\1 & 1 & 1\\end{array}\\right].$ On the observation that $\\text{rank}_2(\\mathbb {D})=3$ , we choose 3 users as leaders, where the demand matrix of these 3 leaders is also full-rank.",
"Here, we choose ${\\mathcal {L}}=[3]$ .",
"Encoding.",
"For each set ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {S}}|=t+1=3$ , we generate a multicast message with $\\alpha _{{\\mathcal {S}},k} =1$ in (REF ).",
"Hence, we have $& W_{\\lbrace 1,2,3\\rbrace }= F_{1,\\lbrace 2,3\\rbrace } \\oplus F_{2,\\lbrace 1,3\\rbrace } \\oplus F_{3,\\lbrace 1,2\\rbrace }; \\\\&W_{\\lbrace 1,2,4\\rbrace }= F_{1,\\lbrace 2,4\\rbrace } \\oplus F_{2,\\lbrace 1,4\\rbrace } \\oplus (F_{1,\\lbrace 1,2\\rbrace }\\oplus F_{2,\\lbrace 1,2\\rbrace }); \\\\& W_{\\lbrace 1,2,5\\rbrace }= F_{1,\\lbrace 2,5\\rbrace } \\oplus F_{2,\\lbrace 1,5\\rbrace } \\oplus (F_{1,\\lbrace 1,2\\rbrace }\\oplus F_{3,\\lbrace 1,2\\rbrace }); \\\\& W_{\\lbrace 1,2,6\\rbrace }= F_{1,\\lbrace 2,6\\rbrace } \\oplus F_{2,\\lbrace 1,6\\rbrace } \\oplus (F_{1,\\lbrace 1,2\\rbrace }\\oplus F_{2,\\lbrace 1,2\\rbrace }\\oplus F_{3,\\lbrace 1,2\\rbrace }); \\\\& W_{\\lbrace 1,3,4\\rbrace }= F_{1,\\lbrace 3,4\\rbrace } \\oplus F_{3,\\lbrace 1,4\\rbrace } \\oplus (F_{1,\\lbrace 1,3\\rbrace }\\oplus F_{2,\\lbrace 1,3\\rbrace }); \\\\& W_{\\lbrace 1,3,5\\rbrace }= F_{1,\\lbrace 3,5\\rbrace } \\oplus F_{3,\\lbrace 1,5\\rbrace } \\oplus (F_{1,\\lbrace 1,3\\rbrace }\\oplus F_{3,\\lbrace 1,3\\rbrace }); \\\\& W_{\\lbrace 1,3,6\\rbrace }= F_{1,\\lbrace 3,6\\rbrace } \\oplus F_{3,\\lbrace 1,6\\rbrace } \\oplus (F_{1,\\lbrace 1,3\\rbrace }\\oplus F_{2,\\lbrace 1,3\\rbrace }\\oplus F_{3,\\lbrace 1,3\\rbrace }); \\\\& W_{\\lbrace 1,4,5\\rbrace }= F_{1,\\lbrace 4,5\\rbrace } \\oplus (F_{1,\\lbrace 1,5\\rbrace }\\oplus F_{2,\\lbrace 1,5\\rbrace }) \\oplus (F_{1,\\lbrace 1,4\\rbrace }\\oplus F_{3,\\lbrace 1,4\\rbrace }); \\\\&W_{\\lbrace 1,4,6\\rbrace }= F_{1,\\lbrace 4,6\\rbrace } \\oplus (F_{1,\\lbrace 1,6\\rbrace }\\oplus F_{2,\\lbrace 1,6\\rbrace }) \\oplus (F_{1,\\lbrace 1,4\\rbrace }\\oplus F_{2,\\lbrace 1,4\\rbrace } \\oplus F_{3,\\lbrace 1,4\\rbrace }); \\\\& W_{\\lbrace 1,5,6\\rbrace }=F_{1,\\lbrace 5,6\\rbrace } \\oplus (F_{1,\\lbrace 1,6\\rbrace } \\oplus F_{3,\\lbrace 1,6\\rbrace }) \\oplus (F_{1,\\lbrace 1,5\\rbrace } \\oplus F_{2,\\lbrace 1,5\\rbrace } \\oplus F_{3,\\lbrace 1,5\\rbrace }); \\\\&W_{\\lbrace 2,3,4\\rbrace }= F_{2,\\lbrace 3,4\\rbrace } \\oplus F_{3,\\lbrace 2,4\\rbrace } \\oplus (F_{1,\\lbrace 2,3\\rbrace } \\oplus F_{2,\\lbrace 2,3\\rbrace }); \\\\& W_{\\lbrace 2,3,5\\rbrace }= F_{2,\\lbrace 3,5\\rbrace } \\oplus F_{3,\\lbrace 2,5\\rbrace } \\oplus (F_{1,\\lbrace 2,3\\rbrace } \\oplus F_{3,\\lbrace 2,3\\rbrace }); \\\\& W_{\\lbrace 2,3,6\\rbrace }= F_{2,\\lbrace 3,6\\rbrace } \\oplus F_{3,\\lbrace 2,6\\rbrace } \\oplus (F_{1,\\lbrace 2,3\\rbrace } \\oplus F_{2,\\lbrace 2,3\\rbrace } \\oplus F_{3,\\lbrace 2,3\\rbrace }); \\\\&W_{\\lbrace 2,4,5\\rbrace }= F_{2,\\lbrace 4,5\\rbrace } \\oplus (F_{1,\\lbrace 2,5\\rbrace }\\oplus F_{2,\\lbrace 2,5\\rbrace }) \\oplus (F_{1,\\lbrace 2,4\\rbrace } \\oplus F_{3,\\lbrace 2,4\\rbrace }); \\\\& W_{\\lbrace 2,4,6\\rbrace }= F_{2,\\lbrace 4,6\\rbrace } \\oplus (F_{1,\\lbrace 2,6\\rbrace }\\oplus F_{2,\\lbrace 2,6\\rbrace }) \\oplus (F_{1,\\lbrace 2,4\\rbrace } \\oplus F_{2,\\lbrace 2,4\\rbrace } \\oplus F_{3,\\lbrace 2,4\\rbrace }); \\\\& W_{\\lbrace 2,5,6\\rbrace }= F_{2,\\lbrace 5,6\\rbrace } \\oplus (F_{1,\\lbrace 2,6\\rbrace }\\oplus F_{3,\\lbrace 2,6\\rbrace }) \\oplus (F_{1,\\lbrace 2,5\\rbrace } \\oplus F_{2,\\lbrace 2,5\\rbrace } \\oplus F_{3,\\lbrace 2,5\\rbrace }); \\\\& W_{\\lbrace 3,4,5\\rbrace }= F_{3,\\lbrace 4,5\\rbrace } \\oplus (F_{1,\\lbrace 3,5\\rbrace }\\oplus F_{2,\\lbrace 3,5\\rbrace }) \\oplus (F_{1,\\lbrace 3,4\\rbrace } \\oplus F_{3,\\lbrace 3,4\\rbrace }); \\\\&W_{\\lbrace 3,4,6\\rbrace }= F_{3,\\lbrace 4,6\\rbrace } \\oplus (F_{1,\\lbrace 3,6\\rbrace }\\oplus F_{2,\\lbrace 3,6\\rbrace }) \\oplus (F_{1,\\lbrace 3,4\\rbrace } \\oplus F_{2,\\lbrace 3,4\\rbrace } \\oplus F_{3,\\lbrace 3,4\\rbrace }); \\\\& W_{\\lbrace 3,5,6\\rbrace }= F_{3,\\lbrace 5,6\\rbrace } \\oplus (F_{1,\\lbrace 3,6\\rbrace }\\oplus F_{3,\\lbrace 3,6\\rbrace }) \\oplus (F_{1,\\lbrace 3,5\\rbrace } \\oplus F_{2,\\lbrace 3,5\\rbrace } \\oplus F_{3,\\lbrace 3,5\\rbrace }); \\\\&W_{\\lbrace 4,5,6\\rbrace }= (F_{1,\\lbrace 5,6\\rbrace } \\oplus F_{2,\\lbrace 5,6\\rbrace }) \\oplus (F_{1,\\lbrace 4,6\\rbrace }\\oplus F_{3,\\lbrace 4,6\\rbrace }) \\oplus (F_{1,\\lbrace 4,5\\rbrace } \\oplus F_{2,\\lbrace 4,5\\rbrace } \\oplus F_{3,\\lbrace 4,5\\rbrace }).",
"$ Delivery.",
"The server broadcasts $W_{{\\mathcal {S}}}$ for each ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {S}}|=t+1=3$ and ${\\mathcal {S}}\\cap {\\mathcal {L}}\\ne \\emptyset $ .",
"In other words, the server broadcasts all the multicast messages in (REF ) except for $W_{\\lbrace 4,5,6\\rbrace }$ .",
"Decoding.",
"We show that the untransmitted multicast message $W_{\\lbrace 4,5,6\\rbrace }$ can be reconstructed by the transmitted multicast messages.",
"For each set of users ${\\mathcal {B}}\\subseteq [{\\mathsf {K}}]$ , we define ${V}_{{\\mathcal {B}}}$ as the family of subsets ${\\mathcal {V}}\\subseteq {\\mathcal {B}}$ , where $|{\\mathcal {V}}|=|{\\mathcal {L}}| $ and $\\text{rank}_2(\\mathbb {D}_{{\\mathcal {V}}})=|{\\mathcal {L}}|.$ It can be seen that ${V}_{{\\mathcal {B}}}$ is the generalization of ${F}_{{\\mathcal {B}}}$ defined in the YMA scheme described in Section REF .",
"When ${\\mathcal {B}}={\\mathcal {L}}\\cup \\lbrace 4,5,6\\rbrace =[6]$ , we have ${V}_{[6]} = \\Big \\lbrace &\\lbrace 1,2,3\\rbrace ,\\lbrace 1,2,5\\rbrace ,\\lbrace 1,2,6\\rbrace ,\\lbrace 1,3,4\\rbrace ,\\lbrace 1,3,6\\rbrace ,\\lbrace 1,4,5\\rbrace ,\\lbrace 1,4,6\\rbrace ,\\lbrace 1,5,6\\rbrace ,\\nonumber \\\\&\\lbrace 2,3,4\\rbrace ,\\lbrace 2,3,5\\rbrace ,\\lbrace 2,3,6\\rbrace ,\\lbrace 2,4,5\\rbrace ,\\lbrace 2,4,6\\rbrace ,\\lbrace 3,4,5\\rbrace ,\\lbrace 3,5,6\\rbrace ,\\lbrace 4,5,6\\rbrace \\Big \\rbrace .$ From the above definition, we focus on the following sum of multicast messages $&\\underset{{\\mathcal {V}}\\in {V}_{[6]} }{\\oplus } W_{[6] \\setminus {\\mathcal {V}}}=0,$ where (REF ) is because on the LHS of (REF ), among all subfiles $F_{i,{\\mathcal {W}}}$ where $i\\in [3]$ , ${\\mathcal {W}}\\subseteq [6]$ , and $|{\\mathcal {W}}|=2$ , the coefficient of each of $F_{2,\\lbrace 2,4\\rbrace }$ , $F_{2,\\lbrace 2,6\\rbrace }$ , $F_{2,\\lbrace 4,6\\rbrace }$ , $F_{3,\\lbrace 3,5\\rbrace }$ , $F_{3,\\lbrace 3,6\\rbrace }$ , $F_{3,\\lbrace 5,6\\rbrace }$ is 0, $F_{1,\\lbrace 2,3\\rbrace }$ appears 4 times and other subfiles appear 2 times.",
"Hence, the sum is equivalent to 0 on $\\mathbb {F}_2$ .",
"Notice that in the YMA delivery scheme, the coefficient of each subfile appearing in the sum $\\underset{{\\mathcal {F}}\\in {F}_{{\\mathcal {B}}} }{\\oplus } W_{{\\mathcal {B}}\\setminus {\\mathcal {F}}}$ is 2.",
"We can write (REF ) as $W_{\\lbrace 4,5,6\\rbrace }=\\underset{{\\mathcal {V}}\\in {V}_{[6]} : {\\mathcal {V}}\\ne {\\mathcal {L}}}{\\oplus } W_{[6] \\setminus {\\mathcal {V}}}.$ In other words, the untransmitted multicast message $W_{\\lbrace 4,5,6\\rbrace }$ can be reconstructed by the transmitted multicast messages.",
"Thus each user can recover all the multicast messages in (REF ), and then recover its desired function.",
"Performance.",
"In total we transmit $\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}-\\text{rank}_2 (\\mathbb {D}) }{t+1}=\\binom{6}{3}-\\binom{3}{3}=19$ multicast messages, each of which contains $\\frac{{\\mathsf {B}}}{20}$ bits.",
"Hence, the transmitted load is $\\frac{19}{20}$ , which coincides with the optimal worst-case load in Theorem REF ."
],
[
"General Description",
"We use the file split in (REF )-(REF ), resulting in the demand split in (REF ).",
"In the delivery phase, the demand matrix $\\mathbb {D}$ is revealed where each element in $\\mathbb {D}$ is either 0 or 1.",
"Among the ${\\mathsf {K}}$ users we first choose $\\text{rank}_2(\\mathbb {D})$ leaders (assume the set of leaders is ${\\mathcal {L}}=\\lbrace {\\mathcal {L}}(1),\\ldots ,{\\mathcal {L}}(|{\\mathcal {L}}|)\\rbrace $ ), where $|{\\mathcal {L}}|= \\text{rank}_2(\\mathbb {D}_{{\\mathcal {L}}})= \\text{rank}_2(\\mathbb {D} ).$ Encoding.",
"We focus on each set ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {S}}|=t+1$ , and generate the multicast message in (REF ) with $\\alpha _{{\\mathcal {S}},k} =1$ .",
"Delivery.",
"The server broadcasts $W_{{\\mathcal {S}}}$ for each ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {S}}|=t+1$ and ${\\mathcal {S}}\\cap {\\mathcal {L}}\\ne \\emptyset $ .",
"Decoding.",
"For each set of users ${\\mathcal {B}}\\subseteq [{\\mathsf {K}}]$ , recall that ${V}_{{\\mathcal {B}}}$ is the family of subsets ${\\mathcal {V}}\\subseteq {\\mathcal {B}}$ , where $|{\\mathcal {V}}|=|{\\mathcal {L}}|$ and $ \\text{rank}_2(\\mathbb {D}_{{\\mathcal {V}}})=|{\\mathcal {L}}|.$ We now consider each set ${\\mathcal {A}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {A}}|=t+1$ and ${\\mathcal {A}}\\cap {\\mathcal {L}}=\\emptyset $ , and focus on the binary sum $\\underset{{\\mathcal {V}}\\in {V}_{{\\mathcal {B}}} }{\\oplus } W_{{\\mathcal {B}}\\setminus {\\mathcal {V}}} ,$ where ${\\mathcal {B}}= {\\mathcal {L}}\\cup {\\mathcal {A}}$ .",
"A subfile $F_{i,{\\mathcal {W}}}$ appears in the sum (REF ) if and only if ${\\mathcal {W}}\\subseteq {\\mathcal {B}}$ and there exists some user $k\\in {\\mathcal {B}}\\setminus {\\mathcal {W}}$ such that $\\text{rank}_2(\\mathbb {D}_{ {\\mathcal {B}}\\setminus ({\\mathcal {W}}\\cup \\lbrace k\\rbrace ) })= |{\\mathcal {L}}|$ (i.e., $\\mathbb {D}_{ {\\mathcal {B}}\\setminus ({\\mathcal {W}}\\cup \\lbrace k\\rbrace ) }$ is full-rank) and $y_{k,i} \\ne 0$ .",
"We then provide the following Lemma, proved in Appendix .",
"Lemma 1 If $F_{i,{\\mathcal {W}}}$ appears in the sum (REF ), the number of multicast messages in the sum which contains $F_{i,{\\mathcal {W}}}$ is even.Notice that in the YMA scheme for the original MAN caching problem, each subfile in (REF ) is contained by two multicast messages in (REF ).",
"Hence, Lemma REF is also a generalization of [3] for the YMA scheme.",
"$\\square $ From Lemma REF , it can be seen that each subfile in the sum (REF ) appears an even number of times, and thus the coefficient of this subfile in the sum is 0, which leads to $W_{{\\mathcal {A}}}= \\underset{{\\mathcal {V}}\\in {V}_{{\\mathcal {B}}} :{\\mathcal {V}}\\ne {\\mathcal {L}}}{\\oplus } W_{{\\mathcal {B}}\\setminus {\\mathcal {V}}}.$ In other words, $W_{{\\mathcal {A}}}$ can be reconstructed by the transmitted multicast messages.",
"As a result, each user $k$ can recover each multicast message $W_{{\\mathcal {S}}}$ where ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ and $|{\\mathcal {S}}|=t+1$ , and thus it can decode its desired function.",
"Performance.",
"In total, we transmit $\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}-\\text{rank}_2 (\\mathbb {D}) }{t+1}$ multicast messages, each of which contains $\\frac{{\\mathsf {B}}}{ \\binom{{\\mathsf {K}}}{t}}$ bits.",
"Hence, the transmitted load is $\\frac{\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}-\\text{rank}_2 (\\mathbb {D}) }{t+1}}{ \\binom{{\\mathsf {K}}}{t}}.$ For the worst-case demands where $\\text{rank}_2 (\\mathbb {D})$ is full-rank, we have $\\text{rank}_2 (\\mathbb {D}) =\\min \\lbrace {\\mathsf {K}},{\\mathsf {N}}\\rbrace $ , and we achieve the worst-case load in (REF )."
],
[
"Achievable Scheme in Theorem ",
"In the following, we generalize the proposed caching scheme in Section to the case where the demands are scalar linear functions on arbitrary finite field $\\mathbb {F}_{\\mathsf {q}}$ .",
"All the operations in the proposed scheme are on $\\mathbb {F}_{\\mathsf {q}}$ .",
"We again start with an example."
],
[
"Example",
"Consider the $({\\mathsf {K}},{\\mathsf {N}},{\\mathsf {M}},{\\mathsf {q}})=(5,3,3/5,{\\mathsf {q}})$ shared-link cache-aided scalar linear function retrieval problem, where ${\\mathsf {q}}$ is a prime-power.",
"In this case, we have $t={\\mathsf {K}}{\\mathsf {M}}/{\\mathsf {N}}=1$ .",
"Hence, in the cache placement, each file is partitioned into $\\binom{{\\mathsf {K}}}{t}= 5$ equal-length subfiles.",
"We use the file split in (REF )-(REF ), resulting in the demand split in (REF ).",
"In the delivery phase, we assume that $&\\text{user $1$ demands $F_1$;}\\\\&\\text{user $2$ demands $F_2$;}\\\\&\\text{user $3$ demands $F_3$;}\\\\&\\text{user $4$ demands $y_{4,1}F_1 + y_{4,2} F_2+y_{4,3} F_3$;}\\\\&\\text{user $5$ demands $y_{5,1}F_1 + y_{5,2} F_2+y_{5,3} F_3$;}$ i.e., the demand matrix is $\\mathbb {D}=\\left[\\begin{array}{ccc}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\\\y_{4,1} & y_{4,2} & y_{4,3}\\\\y_{5,1} & y_{5,2} & y_{5,3}\\end{array}\\right]\\in [\\mathbb {F}_{{\\mathsf {q}}}]^{5 \\times 3}.$ We choose the set of leaders ${\\mathcal {L}}=[3]$ , since $\\text{rank}_{{\\mathsf {q}}}(\\mathbb {D}_{[3]})=3$ .",
"Each user $k\\in [{\\mathsf {K}}]$ should recover each block $B_{k,{\\mathcal {W}}}= y_{k,1} F_{1,{\\mathcal {W}}} +y_{k,2} F_{2,{\\mathcal {W}}}+ y_{k,3} F_{3,{\\mathcal {W}}}$ in the delivery phase, where ${\\mathcal {W}}\\in [5]\\setminus \\lbrace k\\rbrace $ and $|{\\mathcal {W}}|=1$ .",
"Encoding.",
"For each set ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {S}}|=t+1=2$ , recall that the multicast messages are given in (REF ) and we separate it as $W_{{\\mathcal {S}}}&= \\sum _{k_1\\in {\\mathcal {S}}\\cap {\\mathcal {L}}} \\alpha _{{\\mathcal {S}},k_1} B_{k_1,{\\mathcal {S}}\\setminus \\lbrace k_1\\rbrace } +\\sum _{k_2\\in {\\mathcal {S}}\\setminus {\\mathcal {L}}} \\alpha _{{\\mathcal {S}},k_2} B_{k_2,{\\mathcal {S}}\\setminus \\lbrace k_2\\rbrace } .",
"$ We first alternate the coefficients (either 1 or $-1$ ) of the desired blocks of the leaders in ${\\mathcal {S}}$ , and then alternate the coefficients (either 1 or $-1$ ) of the desired blocks of the non-leaders in ${\\mathcal {S}}$ .",
"For example, if ${\\mathcal {S}}=\\lbrace 1,2\\rbrace $ , we have $W_{\\lbrace 1,2\\rbrace }= F_{1,\\lbrace 2\\rbrace }- F_{2,\\lbrace 1\\rbrace }$ ; if ${\\mathcal {S}}=\\lbrace 1,4\\rbrace $ , we have $W_{\\lbrace 1,4\\rbrace }= F_{1,\\lbrace 4\\rbrace }+(y_{4,1} F_{1,\\lbrace 1\\rbrace } +y_{4,2} F_{2,\\lbrace 1\\rbrace }+ y_{4,3} F_{3,\\lbrace 1\\rbrace })$ ; if ${\\mathcal {S}}=\\lbrace 4,5\\rbrace $ , we have $W_{\\lbrace 4,5\\rbrace }=(y_{4,1} F_{1,\\lbrace 5\\rbrace } +y_{4,2} F_{2,\\lbrace 5\\rbrace }+ y_{4,3} F_{3,\\lbrace 5\\rbrace })-(y_{5,1} F_{1,\\lbrace 4\\rbrace } +y_{5,2} F_{2,\\lbrace 4\\rbrace }+ y_{5,3} F_{3,\\lbrace 4\\rbrace })$ .",
"With this, we can list all the multicast messages as $&W_{\\lbrace 1,2\\rbrace }= F_{1,\\lbrace 2\\rbrace }- F_{2,\\lbrace 1\\rbrace }; \\\\&W_{\\lbrace 1,3\\rbrace }= F_{1,\\lbrace 3\\rbrace }- F_{3,\\lbrace 1\\rbrace }; \\\\&W_{\\lbrace 1,4\\rbrace }= F_{1,\\lbrace 4\\rbrace }+(y_{4,1} F_{1,\\lbrace 1\\rbrace } +y_{4,2} F_{2,\\lbrace 1\\rbrace }+ y_{4,3} F_{3,\\lbrace 1\\rbrace }); \\\\&W_{\\lbrace 1,5\\rbrace }= F_{1,\\lbrace 5\\rbrace }+(y_{5,1} F_{1,\\lbrace 1\\rbrace } +y_{5,2} F_{2,\\lbrace 1\\rbrace }+ y_{5,3} F_{3,\\lbrace 1\\rbrace }); \\\\&W_{\\lbrace 2,3\\rbrace }= F_{2,\\lbrace 3\\rbrace }- F_{3,\\lbrace 2\\rbrace }; \\\\&W_{\\lbrace 2,4\\rbrace }= F_{2,\\lbrace 4\\rbrace }+(y_{4,1} F_{1,\\lbrace 2\\rbrace } +y_{4,2} F_{2,\\lbrace 2\\rbrace }+ y_{4,3} F_{3,\\lbrace 2\\rbrace }); \\\\&W_{\\lbrace 2,5\\rbrace }= F_{2,\\lbrace 5\\rbrace }+(y_{5,1} F_{1,\\lbrace 2\\rbrace } +y_{5,2} F_{2,\\lbrace 2\\rbrace }+ y_{5,3} F_{3,\\lbrace 2\\rbrace }); \\\\&W_{\\lbrace 3,4\\rbrace }= F_{3,\\lbrace 4\\rbrace }+(y_{4,1} F_{1,\\lbrace 3\\rbrace } +y_{4,2} F_{2,\\lbrace 3\\rbrace }+ y_{4,3} F_{3,\\lbrace 3\\rbrace }); \\\\&W_{\\lbrace 3,5\\rbrace }= F_{3,\\lbrace 5\\rbrace }+(y_{5,1} F_{1,\\lbrace 3\\rbrace } +y_{5,2} F_{2,\\lbrace 3\\rbrace }+ y_{5,3} F_{3,\\lbrace 3\\rbrace }); \\\\&W_{\\lbrace 4,5\\rbrace }=(y_{4,1} F_{1,\\lbrace 5\\rbrace } +y_{4,2} F_{2,\\lbrace 5\\rbrace }+ y_{4,3} F_{3,\\lbrace 5\\rbrace })-(y_{5,1} F_{1,\\lbrace 4\\rbrace } +y_{5,2} F_{2,\\lbrace 4\\rbrace }+ y_{5,3} F_{3,\\lbrace 4\\rbrace }).$ Delivery.",
"The server broadcasts $W_{{\\mathcal {S}}}$ for each ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {S}}|=t+1=2$ and ${\\mathcal {S}}\\cap {\\mathcal {L}}\\ne \\emptyset $ .",
"In other words, the server broadcasts all the multicast messages in (REF ) except for $W_{\\lbrace 4,5\\rbrace }$ .",
"Decoding.",
"We first show the untransmitted multicast message $W_{\\lbrace 4,5\\rbrace }$ can be reconstructed by the transmitted multicast messages.",
"More precisely, we aim to choose the decoding coefficients $\\beta _{\\lbrace 4,5\\rbrace ,{\\mathcal {S}}} \\in \\mathbb {F}_{{\\mathsf {q}}}$ for each ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {S}}|=t+1$ and ${\\mathcal {S}}\\cap {\\mathcal {L}}\\ne \\emptyset $ , such that $W_{\\lbrace 4,5\\rbrace }= \\sum _{{\\mathcal {S}}\\subseteq [{\\mathsf {K}}]:|{\\mathcal {S}}|=t+1, {\\mathcal {S}}\\cap {\\mathcal {L}}\\ne \\emptyset } \\beta _{\\lbrace 4,5\\rbrace ,{\\mathcal {S}}} W_{{\\mathcal {S}}}.$ Since on the RHS of (REF ) $F_{1,\\lbrace 4\\rbrace }$ only appears in $W_{\\lbrace 1,4\\rbrace }$ and on the LHS of (REF ) the coefficient of $F_{1,\\lbrace 4\\rbrace }$ is $-y_{5,1}$ , in order to have the same coefficient of $F_{1,\\lbrace 4\\rbrace }$ on both sides of (REF ), we let $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,4\\rbrace }= -y_{5,1}= -\\text{det}([y_{5,1}]).$ Similarly, we let $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 2,4\\rbrace }= -y_{5,2}=-\\text{det}([y_{5,2}]), $ such that the coefficients of $F_{2,\\lbrace 4\\rbrace }$ on both sides of (REF ) are the same; let $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 3,4\\rbrace }= -y_{5,3}= -\\text{det}([y_{5,3}]), $ such that the coefficients of $F_{3,\\lbrace 4\\rbrace }$ on both sides of (REF ) are the same; let $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,5\\rbrace }= y_{4,1}= \\text{det}([y_{4,1}]), $ such that the coefficients of $F_{1,\\lbrace 5\\rbrace }$ on both sides of (REF ) are the same; let $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 2,5\\rbrace }= y_{4,2}= \\text{det}([y_{4,2}]), $ such that the coefficients of $F_{2,\\lbrace 5\\rbrace }$ on both sides of (REF ) are the same; let $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 3,5\\rbrace }= y_{4,3}= \\text{det}([y_{4,3}]), $ such that the coefficients of $F_{3,\\lbrace 5\\rbrace }$ on both sides of (REF ) are the same.",
"Next we focus $ F_{1,\\lbrace 1\\rbrace }$ , which appears in $W_{\\lbrace 1,4\\rbrace }$ and $W_{\\lbrace 1,5\\rbrace }$ .",
"Since $ \\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,4\\rbrace }=-y_{5,1}$ and $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,5\\rbrace }= y_{4,1}$ , the coefficient of $ F_{1,\\lbrace 1\\rbrace }$ on the RHS of (REF ) is $y_{4,1} \\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,4\\rbrace } +y_{5,1}\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,5\\rbrace }=0.$ Similarly, the coefficient of $ F_{2,\\lbrace 2\\rbrace }$ on the RHS of (REF ), which appears in $ W_{\\lbrace 2,4\\rbrace } $ and $ W_{\\lbrace 2,5\\rbrace } $ , is 0.",
"The coefficient of $ F_{3,\\lbrace 3\\rbrace }$ on the RHS of (REF ), which appears in $ W_{\\lbrace 3,4\\rbrace } $ and $ W_{\\lbrace 3,5\\rbrace } $ , is 0, Now we focus on $ F_{1,\\lbrace 2\\rbrace }$ , which appears in $W_{\\lbrace 1,2\\rbrace }$ , $W_{\\lbrace 2,4\\rbrace }$ , and $W_{\\lbrace 2,5\\rbrace }$ .",
"Since $ \\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 2,4\\rbrace }=-y_{5,2}$ and $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 2,5\\rbrace }= y_{4,2}$ , in order to let the coefficient of $ F_{1,\\lbrace 2\\rbrace }$ on the RHS of (REF ) be 0, we let $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,2\\rbrace }= y_{4,1} y_{5,2}-y_{5,1} y_{4,2}=\\text{det}([y_{4,1},y_{4,2};y_{5,1},y_{5,2}]).$ In addition, $F_{2,\\lbrace 1\\rbrace }$ appears in $W_{\\lbrace 1,2\\rbrace }$ , $W_{\\lbrace 1,4\\rbrace }$ , and $W_{\\lbrace 1,5\\rbrace }$ .",
"The coefficient of $F_{2,\\lbrace 1\\rbrace }$ on the RHS of (REF ) is $-\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,2\\rbrace }+ y_{4,2} \\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,4\\rbrace }+ y_{5,2} \\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,5\\rbrace }=0.$ Similarly, we let $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,3\\rbrace }=y_{4,1} y_{5,3} -y_{5,1}y_{4,3}= \\text{det}([y_{4,1},y_{4,3};y_{5,1},y_{5,3}]),$ such that the coefficients of $F_{1,\\lbrace 3\\rbrace }$ and $F_{3,\\lbrace 1\\rbrace }$ on the RHS of (REF ) are 0.",
"We let $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 2,3\\rbrace }= y_{4,2} y_{5,3} - y_{5,2}y_{4,3}= \\text{det}([y_{4,2},y_{4,3};y_{5,2},y_{5,3}]),$ such that the coefficients of $F_{2,\\lbrace 3\\rbrace }$ and $F_{3,\\lbrace 2\\rbrace }$ on the RHS of (REF ) are 0.",
"With the above choice of decoding coefficients, on the RHS of (REF ), the coefficients of all the subfiles which is not contained by $W_{\\lbrace 4,5\\rbrace }$ are 0.",
"In addition, the coefficients of each subfile contained by $W_{\\lbrace 4,5\\rbrace }$ are the same on both sides of (REF ).",
"Thus we prove (REF ).",
"In conclusion, each user can recover all multicast messages in (REF ), and then recover its demanded function.",
"Performance.",
"In total we transmit $\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}-\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D}) }{t+1}=\\binom{5}{2}-\\binom{2}{2}=9$ multicast messages, each of which contains $\\frac{{\\mathsf {B}}}{5}$ symbols.",
"Hence, the transmitted load is $\\frac{9}{5}$ , which coincides with the optimal worst-case load in Theorem REF ."
],
[
"General Description",
"We use the file split in (REF )-(REF ), resulting in the demand split in (REF ).",
"In the delivery phase, after the demand matrix $\\mathbb {D}$ is revealed, among the ${\\mathsf {K}}$ users we first choose $\\text{rank}_{{\\mathsf {q}}}(\\mathbb {D})$ leaders (assume the set of leaders is ${\\mathcal {L}}=\\lbrace {\\mathcal {L}}(1),\\ldots ,{\\mathcal {L}}(|{\\mathcal {L}}|)\\rbrace $ ), where $|{\\mathcal {L}}|= \\text{rank}_{{\\mathsf {q}}}(\\mathbb {D}_{{\\mathcal {L}}})= \\text{rank}_{{\\mathsf {q}}}(\\mathbb {D} ).$ For each $i\\in [|{\\mathcal {L}}|]$ , we also define that the leader index of leader ${\\mathcal {L}}(i)$ is $i$ .",
"From (REF ), we can represent the demands of non-leaders by the linear combinations of the demands of leaders.",
"More precisely, we define $F^{\\prime }_i:= y_{{\\mathcal {L}}(i),1} F_1 +\\ldots + y_{{\\mathcal {L}}(i),{\\mathsf {N}}} F_{{\\mathsf {N}}}, \\ \\forall i\\in [|{\\mathcal {L}}|], $ and represent the demand of each user $k\\in [{\\mathsf {K}}]$ by $y_{k,1} F_1 +\\ldots + y_{k,{\\mathsf {N}}} F_{{\\mathsf {N}}} = x_{k,1} F^{\\prime }_1 +\\ldots + x_{k,|{\\mathcal {L}}|} F^{\\prime }_{|{\\mathcal {L}}|}.$ Clearly, for each leader ${\\mathcal {L}}(i)$ where $i\\in [|{\\mathcal {L}}|]$ , $ {\\mathbf {x}}_{{\\mathcal {L}}(i)}$ is an $|{\\mathcal {L}}|$ -dimension unit vector where the $i^{\\text{th}}$ element is 1.",
"The transformed demand matrix $\\mathbb {D}^{\\prime }$ is defined as follows, $\\mathbb {D}^{\\prime }=[x_{1,1},\\ldots ,x_{1,|{\\mathcal {L}}|}; \\ldots ; x_{{\\mathsf {K}},1},\\ldots ,x_{{\\mathsf {K}},|{\\mathcal {L}}|}].$ In addition, for each $i\\in [|{\\mathcal {L}}|]$ and each ${\\mathcal {W}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {W}}|=t$ , we define $F^{\\prime }_{i,{\\mathcal {W}}}:=y_{{\\mathcal {L}}(i),1} F_{1,{\\mathcal {W}}} +\\ldots + y_{{\\mathcal {L}}(i),{\\mathsf {N}}} F_{{\\mathsf {N}},{\\mathcal {W}}},$ refer $ F^{\\prime }_{i,{\\mathcal {W}}}$ to as a transformed subfile, and refer $B^{\\prime }_{k,{\\mathcal {W}}}=x_{k,1} F^{\\prime }_{1,{\\mathcal {W}}} +\\ldots + x_{k,|{\\mathcal {L}}|} F^{\\prime }_{|{\\mathcal {L}}|,{\\mathcal {W}}}$ to as a transformed block.",
"Encoding.",
"For each ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ , we denote the set of leaders in ${\\mathcal {S}}$ by ${\\mathcal {L}}_{{\\mathcal {S}}}:={\\mathcal {S}}\\cap {\\mathcal {L}},$ and the set of non-leaders in ${\\mathcal {S}}$ by ${\\mathcal {N}}_{{\\mathcal {S}}}:= {\\mathcal {S}}\\setminus {\\mathcal {L}}.$ We also denote the leader indices of leaders in ${\\mathcal {S}}$ by $\\text{Ind}_{{\\mathcal {S}}}:= \\lbrace i\\in [|{\\mathcal {L}}|]: {\\mathcal {L}}(i)\\in {\\mathcal {S}}\\rbrace ,$ For example, if ${\\mathcal {L}}=\\lbrace 2,4,5\\rbrace $ and ${\\mathcal {S}}=\\lbrace 1,2,5\\rbrace $ , we have ${\\mathcal {L}}_{{\\mathcal {S}}}=\\lbrace 2,5\\rbrace $ , ${\\mathcal {N}}_{{\\mathcal {S}}}=\\lbrace 1\\rbrace $ , and $\\text{Ind}_{{\\mathcal {S}}}=\\lbrace 1,3\\rbrace $ .",
"Now we focus on each set ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {S}}|=t+1$ , and generate the multicast message $W_{{\\mathcal {S}}}=\\sum _{i\\in [|{\\mathcal {L}}_{{\\mathcal {S}}}|]} (-1)^{i-1} B^{\\prime }_{{\\mathcal {L}}_{{\\mathcal {S}}}(i), {\\mathcal {S}}\\setminus \\left\\lbrace {\\mathcal {L}}_{{\\mathcal {S}}}(i)\\right\\rbrace } +\\sum _{j\\in [|{\\mathcal {N}}_{{\\mathcal {S}}}|]}(-1)^{j-1} B^{\\prime }_{{\\mathcal {N}}_{{\\mathcal {S}}}(j),{\\mathcal {S}}\\setminus \\lbrace {\\mathcal {N}}_{{\\mathcal {S}}}(j)\\rbrace }.",
"$ The construction of $W_{{\\mathcal {S}}}$ can be explained as follows.",
"The coefficient of each transformed block in $W_{{\\mathcal {S}}}$ is either 1 or $-1$ .",
"We divide the transformed blocks in $W_{{\\mathcal {S}}}$ into two groups, demanded by leaders and non-leaders, respectively.",
"We alternate the sign (i.e., the coefficient 1 or $-1$ ) of each transformed block demanded by leaders, and alternate the sign of each transformed block demanded by non-leaders, respectively.",
"We then sum the resulting summations of these two groups.",
"For each $i\\in [|{\\mathcal {L}}_{{\\mathcal {S}}}|$ , by the construction in (REF ), we have $B^{\\prime }_{{\\mathcal {L}}_{{\\mathcal {S}}}(i), {\\mathcal {S}}\\setminus \\left\\lbrace {\\mathcal {L}}_{{\\mathcal {S}}}(i)\\right\\rbrace } = F^{\\prime }_{\\text{Ind}_{{\\mathcal {S}}}(i), {\\mathcal {S}}\\setminus \\left\\lbrace {\\mathcal {L}}_{{\\mathcal {S}}}(i)\\right\\rbrace }.$ Delivery.",
"The server broadcasts $W_{{\\mathcal {S}}}$ for each ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {S}}|=t+1$ and ${\\mathcal {S}}\\cap {\\mathcal {L}}\\ne \\emptyset $ .",
"Decoding.",
"We consider each set ${\\mathcal {A}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {A}}|=t+1$ and ${\\mathcal {A}}\\cap {\\mathcal {L}}= \\emptyset $ .",
"We define that the non-leader index of non-leader ${\\mathcal {A}}(i)$ is $i$ , where $i\\in [t+1]$ .",
"For each ${\\mathcal {S}}\\subseteq {\\mathcal {A}}\\cup {\\mathcal {L}}$ , recall that $\\text{Ind}_{{\\mathcal {S}}}$ defined in (REF ) represents the leader indices of leaders in ${\\mathcal {S}}$ and that ${\\mathcal {N}}_{{\\mathcal {S}}}$ defined in (REF ) represents the set of non-leaders in ${\\mathcal {S}}$ .",
"By definition, we have ${\\mathcal {N}}_{{\\mathcal {S}}} \\subseteq {\\mathcal {A}}$ .",
"In addition, with a slight abuse of notation we denote the non-leader indices of non-leaders in ${\\mathcal {A}}\\setminus {\\mathcal {S}}$ by $\\overline{\\text{Ind}}_{{\\mathcal {S}}}=\\lbrace i\\in [t+1]: {\\mathcal {A}}(i) \\notin {\\mathcal {S}}\\rbrace .$ For example, if ${\\mathcal {A}}=\\lbrace 4,5,6\\rbrace $ and ${\\mathcal {S}}=\\lbrace 1,2,5\\rbrace $ , we have $\\overline{\\text{Ind}}_{{\\mathcal {S}}}=\\lbrace 1,3\\rbrace $ .",
"For any set ${\\mathcal {X}}$ and any number $y$ , we define $\\text{Tot}({\\mathcal {X}})$ as the sum of the elements in ${\\mathcal {X}}$ , i.e., $&\\text{Tot}({\\mathcal {X}}):=\\sum _{i\\in |{\\mathcal {X}}|}{\\mathcal {X}}(i);.$ For example, if ${\\mathcal {X}}=\\lbrace 1,3,4,5\\rbrace $ , we have $\\text{Tot}({\\mathcal {X}})=1+3+4+5=13$ .",
"Recall that $\\mathbb {A}_{{\\mathcal {S}},{\\mathcal {V}}}$ represents the sub-matrix of $\\mathbb {A}$ by selecting from $\\mathbb {A}$ , the rows with indices in ${\\mathcal {S}}$ and the columns with indices in ${\\mathcal {V}}$ .",
"It will be proved in Appendix that $&W_{{\\mathcal {A}}}= \\sum _{{\\mathcal {S}}\\subseteq {\\mathcal {A}}\\cup {\\mathcal {L}}:|{\\mathcal {S}}|=t+1, {\\mathcal {S}}\\ne {\\mathcal {A}}} \\beta _{{\\mathcal {A}},{\\mathcal {S}}} W_{{\\mathcal {S}}}, \\\\& \\beta _{{\\mathcal {A}},{\\mathcal {S}}}= (-1)^{1 +\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {S}}})} \\text{det}(\\mathbb {D}^{\\prime }_{{\\mathcal {A}}\\setminus {\\mathcal {S}},\\text{Ind}_{{\\mathcal {S}}} }).$ In other words, each user $k\\in [{\\mathsf {K}}]$ can recover all messages $W_{{\\mathcal {S}}}$ where ${\\mathcal {S}}\\subseteq [{\\mathsf {K}}]$ and $|{\\mathcal {S}}|=t+1$ .",
"For each desired transformed block $B^{\\prime }_{k,{\\mathcal {W}}}$ , where ${\\mathcal {W}}\\subseteq ([{\\mathsf {K}}] \\setminus \\lbrace k\\rbrace )$ and $|{\\mathcal {W}}|=t$ , user $k$ can recover it in $W_{{\\mathcal {W}}\\cup \\lbrace k\\rbrace }$ , because it knows all the other transformed blocks in $W_{{\\mathcal {W}}\\cup \\lbrace k\\rbrace }$ .",
"Hence, user $k$ can recover $ x_{k,1} F^{\\prime }_1 +\\ldots + x_{k,|{\\mathcal {L}}|} F^{\\prime }_{|{\\mathcal {L}}|}$ , which is identical to its demand.",
"Performance.",
"In total, we transmit $\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}-\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D}) }{t+1}$ multicast messages, each of which contains $\\frac{{\\mathsf {B}}}{ \\binom{{\\mathsf {K}}}{t}}$ symbols.",
"Hence, the transmitted load is $\\frac{\\binom{{\\mathsf {K}}}{t+1}-\\binom{{\\mathsf {K}}-\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D}) }{t+1}}{ \\binom{{\\mathsf {K}}}{t}}.$ For the worst-case demands where $\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D})$ is full-rank, we have $\\text{rank}_{{\\mathsf {q}}} (\\mathbb {D}) =\\min \\lbrace {\\mathsf {K}},{\\mathsf {N}}\\rbrace $ , and we achieve the worst-case load in (REF )."
],
[
"Conclusions",
"In this paper, we introduced a novel problem, cache-aided function retrieval, which is a generalization of the classic coded caching problem and allows users to request scalar linear functions of files.",
"We proposed a novel scheme for the demands functions on arbitrary finite field.",
"The proposed scheme was proved to be optimal under the constraint of uncoded cache placement.",
"In addition, for any demand, the achieved load only depends on the rank of the demand matrix.",
"From the results in this paper, we showed that compared to the original MAN caching problem, the optimal worst-case load of coded caching under the constraint of uncoded cache placement, is not increased when users request scalar linear functions.",
"Further works include the extension of the proposed caching scheme to the case where the demanded functions are non-linear or vectorial, and finding novel caching schemes for the cache-aided function retrieval problem with coded cache placement."
],
[
"Proof of Lemma ",
"To prove Lemma REF , it is equivalent to prove that the number of users $k\\in {\\mathcal {B}}\\setminus {\\mathcal {W}}$ satisfying the following constraints is even, Constraint 1: $\\text{rank}_2(\\mathbb {D}_{ {\\mathcal {B}}\\setminus ({\\mathcal {W}}\\cup \\lbrace k\\rbrace ) })= |{\\mathcal {L}}|$ ; Constraint 2: $y_{k,i} \\ne 0$ .",
"We assume that user $k_1$ satisfies the above constraints.",
"Hence, $\\mathbb {D}_{ {\\mathcal {B}}\\setminus ({\\mathcal {W}}\\cup \\lbrace k_1\\rbrace ) }$ is full-rank, and $y_{k_1,i} \\ne 0$ .",
"We let ${\\mathcal {Y}}=\\lbrace {\\mathcal {Y}}(1),\\ldots , {\\mathcal {Y}}(|{\\mathcal {L}}|)\\rbrace ={\\mathcal {B}}\\setminus ({\\mathcal {W}}\\cup \\lbrace k_1\\rbrace ) $ .",
"In the following, we operate a linear space transformation.",
"More precisely, we let $G_j= {\\mathbf {y}}_{{\\mathcal {Y}}(j)}[F_1;\\ldots ;F_{{\\mathsf {N}}}], \\ \\forall j\\in [|{\\mathcal {L}}|].$ From (REF ), we can re-write the demand of each user ${\\mathcal {Y}}(j)$ as $G_j={\\mathbf {y}}^{\\prime }_{j} [G_1;\\ldots ;G_{|{\\mathcal {L}}|}],$ where ${\\mathbf {y}}^{\\prime }_{j}$ is the $|{\\mathcal {L}}|$ -dimension unit vector whose $j^{\\text{th}}$ element is 1.",
"The transformed demand matrix of the users in ${\\mathcal {Y}}$ is $\\mathbb {D}^{\\prime }_{{\\mathcal {Y}}}=[{\\mathbf {y}}^{\\prime }_{1};\\ldots ;{\\mathbf {y}}^{\\prime }_{|{\\mathcal {L}}|}],$ which is an identity matrix.",
"In addition, we can also re-write the demand of user $k_1$ as ${\\mathbf {y}}^{\\prime } [G_1;\\ldots ;G_{|{\\mathcal {L}}|}],$ where ${\\mathbf {y}}^{\\prime }$ is an $|{\\mathcal {L}}|$ -dimension vector on $\\mathbb {F}_2$ .",
"Notice that if the $p^{\\text{th}}$ element in ${\\mathbf {y}}^{\\prime }$ is 1 and $G_p$ contains $F_{i}$ , $F_{i}$ appears one time in $ {\\mathbf {y}}^{\\prime } [G_1;\\ldots ;G_{|{\\mathcal {L}}|}]$ .",
"Since $y_{k_1,i} \\ne 0$ , it can be seen that $ {\\mathbf {y}}^{\\prime } [G_1;\\ldots ;G_{|{\\mathcal {L}}|}]$ contains $F_{i}$ .",
"Hence, the number of $p \\in [|{\\mathcal {L}}|]$ where the $p^{\\text{th}}$ element in ${\\mathbf {y}}^{\\prime }$ is 1 and $G_p$ contains $F_{i}$ , is odd.",
"For each of such $p$ , if we replace the $p^{\\text{th}}$ row of $\\mathbb {D}^{\\prime }_{{\\mathcal {Y}}}$ by ${\\mathbf {y}}^{\\prime }$ , the resulting matrix is still full-rank, because the $p^{\\text{th}}$ element in ${\\mathbf {y}}^{\\prime }$ is 1.",
"Since the resulting matrix is full-rank, it can be seen that $ \\mathbb {D}_{ {\\mathcal {B}}\\setminus ({\\mathcal {W}}\\cup \\lbrace {\\mathcal {Y}}(p)\\rbrace ) } $ is also full-rank.",
"In addition, since $G_p$ contains $F_{i}$ , we can see that $y_{{\\mathcal {Y}}(p),i} \\ne 0$ .",
"Hence, user ${\\mathcal {Y}}(p)$ also satisfies the two constraints.",
"Moreover, for any $s \\in [|{\\mathcal {L}}|]$ , if the $s^{\\text{th}}$ element in ${\\mathbf {y}}^{\\prime }$ is not 1, user ${\\mathcal {Y}}(s)$ does not satisfy Constraint 1; if $G_{s}$ does not contain $F_{s}$ , user ${\\mathcal {Y}}(s)$ does not satisfy Constraint 2.",
"As a result, besides user $k_1$ , the number of users in ${\\mathcal {B}}\\setminus {\\mathcal {W}}$ satisfying the two constraints is odd.",
"In conclusion, by taking user $k_1$ into consideration, the number of users in ${\\mathcal {B}}\\setminus {\\mathcal {W}}$ satisfying the two constraints is even.",
"Thus Lemma REF is proved."
],
[
"Proof of (",
"We focus on one set of non-leaders ${\\mathcal {A}}\\subseteq [{\\mathsf {K}}]$ where $|{\\mathcal {A}}|=t+1$ and ${\\mathcal {A}}\\cap {\\mathcal {L}}= \\emptyset $ .",
"For any positive integer $n$ , $\\text{Perm}(n)$ represents the set of all permutations of $[n]$ .",
"For any set ${\\mathcal {X}}$ and any number $y$ , we define $\\text{Card}({\\mathcal {X}},y)$ as the number of elements in ${\\mathcal {X}}$ which is smaller than $y$ , i.e., $\\text{Card}({\\mathcal {X}},y):= |\\lbrace i\\in {\\mathcal {X}}:i<y\\rbrace |.",
"$ For example, if ${\\mathcal {X}}=\\lbrace 1,3,4,5\\rbrace $ and $y=4$ , we have $\\text{Card}({\\mathcal {X}},y)=|\\lbrace 1,3\\rbrace |=2$ .",
"Our objective is to prove $&W_{{\\mathcal {A}}}= \\sum _{{\\mathcal {S}}\\subseteq {\\mathcal {A}}\\cup {\\mathcal {L}}:|{\\mathcal {S}}|=t+1, {\\mathcal {S}}\\ne {\\mathcal {A}}} \\beta _{{\\mathcal {A}},{\\mathcal {S}}} W_{{\\mathcal {S}}} , \\\\& \\beta _{{\\mathcal {A}},{\\mathcal {S}}}= \\sum _{\\begin{array}{c}{\\mathbf {u}}=(u_1,\\ldots ,u_{|\\text{Ind}_{{\\mathcal {S}}}|})\\\\ \\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {S}}}|)\\end{array}} (-1)^{1 +\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {S}}})+\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {S}}}|]}{\\sum } \\text{Card}([|\\text{Ind}_{{\\mathcal {S}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) } \\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {S}}}|]} x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {S}}}(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {S}}}(i_2)},$ where () is obtained from expand the determinant in ().",
"Let us go back to the illustrated example in Section REF , where we choose ${\\mathcal {L}}= [3]$ .",
"When ${\\mathcal {A}}=\\lbrace 4,5\\rbrace $ and ${\\mathcal {S}}=\\lbrace 1,2\\rbrace $ , from the definition in (REF ) we have $\\text{Ind}_{{\\mathcal {S}}}=[2]$ and from the definition in (REF ) we have $\\overline{\\text{Ind}}_{{\\mathcal {S}}}=[2]$ .",
"In addition, $\\text{Perm}(|\\text{Ind}_{{\\mathcal {S}}}|)=\\text{Perm}(2)\\lbrace (1,2),(2,1)\\rbrace $ .",
"Hence, when ${\\mathbf {u}}=(u_1,u_2)=(1,2)$ , in () we have the term $&(-1)^{1+\\text{Tot}([2])+\\text{Card}([2]\\setminus \\lbrace 1\\rbrace , 1) + \\text{Card}([2]\\setminus \\lbrace 1,2\\rbrace , 2)} x_{4,1} x_{5,2} \\nonumber \\\\&=(-1)^{1+\\text{Tot}([2])+\\text{Card}([2]\\setminus \\lbrace 1\\rbrace , 1) + \\text{Card}([2]\\setminus \\lbrace 1,2\\rbrace , 2)} y_{4,1} y_{5,2} \\nonumber \\\\&=y_{4,1} y_{5,2},$ where (REF ) is because in the example we have $F_{i}=F^{\\prime }_i$ for each $i\\in [{\\mathsf {N}}]$ , and thus ${\\mathbf {x}}_k={\\mathbf {y}}_k$ for each $k \\in [{\\mathsf {K}}]$ .",
"Similarly, when ${\\mathbf {u}}=(2,1)$ , in () we have the term $&(-1)^{1+\\text{Tot}([2])+\\text{Card}([2]\\setminus \\lbrace 2\\rbrace , 2) + \\text{Card}([2]\\setminus \\lbrace 1,2\\rbrace , 1)} x_{5,1} x_{4,2} \\nonumber \\\\&=(-1)^{1+\\text{Tot}([2])+\\text{Card}([2]\\setminus \\lbrace 2\\rbrace , 2) + \\text{Card}([2]\\setminus \\lbrace 1,2\\rbrace , 1)} y_{5,1} y_{4,2} \\nonumber \\\\&=-y_{5,1} y_{4,2}.$ Hence, in () we have $\\beta _{\\lbrace 4,5\\rbrace ,\\lbrace 1,2\\rbrace }= y_{4,1} y_{5,2}-y_{5,1} y_{4,2}$ , which coincides (REF ) in the illustrated example.",
"By the definition of $W_{{\\mathcal {S}}}$ in (REF ), it is obvious to check that in (REF ), there only exist the transformed subfiles $F_{i,{\\mathcal {W}}}$ where $i\\in [{\\mathsf {N}}]$ , $|{\\mathcal {W}}|\\subseteq ({\\mathcal {A}}\\cup {\\mathcal {L}})$ , and $|{\\mathcal {W}}|=t$ .",
"Now we divide such transformed subiles into hierarchies, where we say a transformed subfile $F^{\\prime }_{i,{\\mathcal {W}}}$ appearing in (REF ) is in Hierarchy $h\\in [0:t]$ , if $|{\\mathcal {W}}\\cap {\\mathcal {L}}|=h$ .",
"In addition, on the LHS of (REF ), only transformed subfiles in Hierarchy 0 exist.",
"We consider the following three cases, Case 1: $F^{\\prime }_{i,{\\mathcal {W}}}$ is in Hierarchy 0.",
"In Appendix REF , we will prove that the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ on the RHS of (REF ) is equal to the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ on the LHS of (REF ).",
"Case 2: $F^{\\prime }_{i,{\\mathcal {W}}}$ is in Hierarchy $h>0$ and ${\\mathcal {L}}(i)\\in {\\mathcal {W}}$ .",
"In Appendix REF , we will prove that the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ on the RHS of (REF ) is 0.",
"Case 3: $F^{\\prime }_{i,{\\mathcal {W}}}$ is in Hierarchy $h>0$ and ${\\mathcal {L}}(i)\\notin {\\mathcal {W}}$ .",
"In Appendix REF , we will prove that the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ on the RHS of (REF ) is 0.",
"Hence, after proving the above three cases, (REF ) can be directly derived.",
"In the illustrated example in Section REF , since $F_{i}=F^{\\prime }_i$ for each $i\\in [{\\mathsf {N}}]$ , it can be seen that $F^{\\prime }_{i,{\\mathcal {W}}}=F_{i,{\\mathcal {W}}}$ for each $i\\in [{\\mathsf {N}}]$ , $|{\\mathcal {W}}|\\subseteq [{\\mathsf {K}}]$ , and $|{\\mathcal {W}}|=t$ .",
"For each subfile $F_{i,{\\mathcal {W}}}$ , it is in one of the following three cases, Case 1: $F_{i,{\\mathcal {W}}}$ is in Hierarchy 0.",
"In this case, we have the subfiles $F_{i,\\lbrace 4\\rbrace }$ , $F_{i,\\lbrace 5\\rbrace }$ for $i\\in [3]$ .",
"Case 2: $F_{i,{\\mathcal {W}}}$ is in Hierarchy 1 and ${\\mathcal {L}}(i)\\in {\\mathcal {W}}$ .",
"In this case, we have the subfiles $F_{i,\\lbrace i\\rbrace }$ for $i\\in [3]$ .",
"Case 3: $F_{i,{\\mathcal {W}}}$ is in Hierarchy 1 and ${\\mathcal {L}}(i)\\notin {\\mathcal {W}}$ .",
"In this case, we have the subfiles $F_{i,\\lbrace j\\rbrace }$ for $i\\in [3]$ and $j \\in [3]\\setminus \\lbrace i\\rbrace $ ."
],
[
"Case 1",
"If $F^{\\prime }_{i,{\\mathcal {W}}}$ is in Hierarchy 0, we have ${\\mathcal {W}}\\subseteq {\\mathcal {A}}$ .",
"Since $|{\\mathcal {A}}|-|{\\mathcal {W}}|=1$ , we assume that $\\lbrace {\\mathcal {A}}(k)\\rbrace = {\\mathcal {A}}\\setminus {\\mathcal {W}}$ .",
"On the LHS of (REF ), $F^{\\prime }_{i,{\\mathcal {W}}}$ appears in $W_{{\\mathcal {A}}}$ , where from (REF ) we have $W_{{\\mathcal {A}}}=\\sum _{j\\in [t+1]}(-1)^{j-1} (x_{{\\mathcal {A}}(j),1} F^{\\prime }_{1,{\\mathcal {S}}\\setminus \\lbrace {\\mathcal {A}}(j)\\rbrace } +\\ldots + x_{{\\mathcal {A}}(j),|{\\mathcal {L}}|} F^{\\prime }_{|{\\mathcal {L}}|,{\\mathcal {S}}\\setminus \\lbrace {\\mathcal {A}}(j)\\rbrace } ).$ Hence, the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ in $W_{{\\mathcal {A}}}$ is $(-1)^{k-1}x_{{\\mathcal {A}}(k), i}$ .",
"Let us then focus on the RHS of (REF ).",
"$F^{\\prime }_{i,{\\mathcal {W}}}$ appears in $W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ .",
"Since ${\\mathcal {L}}(i)$ is the only leader in ${\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace $ (i.e., $\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }=\\lbrace i\\rbrace $ ), the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ in $W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ is $(-1)^{1-1}=1$ .",
"In addition, by computing $\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }=\\lbrace k\\rbrace $ , we have $\\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }=(-1)^{1+k+0}x_{{\\mathcal {A}}(k),i}=(-1)^{k+1} x_{{\\mathcal {A}}(k),i}=(-1)^{k-1} x_{{\\mathcal {A}}(k),i}.$ Hence, the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ on the RHS of (REF ) (i.e., in $ \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace } W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ ) is $(-1)^{k-1} x_{{\\mathcal {A}}(k),i} \\times 1=(-1)^{k-1} x_{{\\mathcal {A}}(k),i},$ which is the same as the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ on the LHS of (REF )."
],
[
"Case 2",
"Now we focus on one transformed subfile $F^{\\prime }_{i,{\\mathcal {W}}}$ in Hierarchy $h>0$ where ${\\mathcal {L}}(i) \\in {\\mathcal {W}}$ .",
"By definition, we have $|{\\mathcal {W}}\\cap {\\mathcal {L}}|=h$ .",
"On the RHS of (REF ), since ${\\mathcal {L}}(i) \\in {\\mathcal {W}}$ , $F^{\\prime }_{i,{\\mathcal {W}}}$ only appears in $W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }$ , where $k\\in \\overline{\\text{Ind}}_{{\\mathcal {W}}}$ .",
"We define that $\\text{the $\\left( \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k)\\right)^{\\text{th}}$ smallest element in $\\overline{\\text{Ind}}_{{\\mathcal {W}}}$ is $k$.}",
"$ We focus on one $k\\in \\overline{\\text{Ind}}_{{\\mathcal {W}}}$ .",
"${\\mathcal {A}}(k)$ is the $k^{\\text{th}}$ element in ${\\mathcal {A}}$ , and in ${\\mathcal {A}}\\setminus {\\mathcal {W}}$ there are $ \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k)-1$ elements smaller than ${\\mathcal {A}}(k)$ .",
"Hence, in ${\\mathcal {N}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }$ there are $k-1- \\left( \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k)-1 \\right)=k- \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k)$ elements smaller than ${\\mathcal {A}}(k)$ .",
"So from (REF ), it can be seen that the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ in $W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }$ is $(-1)^{k- \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k)} x_{{\\mathcal {A}}(k),i}.$ In addition, we have $& \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }= \\sum _{\\begin{array}{c}{\\mathbf {u}}=(u_1,\\ldots ,u_{|\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }|})\\\\ \\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }|)\\end{array}} (-1)^{1 +\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace })+ \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) } \\nonumber \\\\& \\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }|]} x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(i_2)}\\\\& =\\sum _{\\begin{array}{c}{\\mathbf {u}}=(u_1,\\ldots ,u_{|\\text{Ind}_{{\\mathcal {W}}}|})\\\\ \\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {W}}}|)\\end{array}} (-1)^{1 +\\left(\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}})-k\\right)+\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) } \\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|]} x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}}(i_2)}.$ From (REF ) and (REF ), the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ in $\\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace } W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }$ is $(-1)^{k- \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k)} x_{{\\mathcal {A}}(k),i} \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }$ .",
"In the following, we will prove $\\sum _{k\\in \\overline{\\text{Ind}}_{{\\mathcal {W}}}} (-1)^{k- \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k)} x_{{\\mathcal {A}}(k),i} \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }=0,$ such that the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ on the RHS of (REF ) is 0.",
"Let us focus on one $k \\in \\overline{\\text{Ind}}_{{\\mathcal {W}}}$ and one permutation ${\\mathbf {u}}=(u_1,\\ldots ,u_{|\\text{Ind}_{{\\mathcal {W}}}|}) \\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {W}}}|) $ .",
"The term in (REF ) caused by $k $ and ${\\mathbf {u}}$ is $&(-1)^{k - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k )} x_{{\\mathcal {A}}(k ),i} \\hspace{0.0pt}\\left\\lbrace \\hspace{0.0pt}(-1)^{1 +\\left(\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}})-k\\right)+\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|]} \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}}(i_2)} \\hspace{0.0pt}\\right\\rbrace \\\\&=(-1)^{ - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k ) +1+\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}}) +\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) }\\hspace{0.0pt}\\left\\lbrace \\hspace{0.0pt}x_{{\\mathcal {A}}(k ),i} \\hspace{0.0pt}\\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}\\hspace{0.0pt}x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}}(i_2)}\\right\\rbrace .",
"$ Notice that in the product $x_{{\\mathcal {A}}(k ),i} \\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|]} x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}}(i_2)}, $ there is one term whose second subscript is $i^{\\prime }$ for each $i^{\\prime } \\in \\text{Ind}_{{\\mathcal {W}}}\\setminus \\lbrace i\\rbrace $ , and there are two terms whose second subscript is $i$ .",
"We define that $\\text{the $\\left( \\text{Ind}_{{\\mathcal {W}}}^{-1}(i)\\right)^{\\text{th}}$ smallest element in $\\text{Ind}_{{\\mathcal {W}}}$ is $i$.",
"}$ Hence, the two terms in (REF ) whose second subscript is $i$ are $x_{{\\mathcal {A}}(k ),i}$ and $x_{ {\\mathcal {A}}(k^{\\prime }) ,i}$ , where $k^{\\prime }:= \\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(u_{ \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) }) $ .",
"In addition, the combination $k^{\\prime }$ and ${\\mathbf {u}}^{\\prime }=(u^{\\prime }_1,\\ldots ,u^{\\prime }_{|\\text{Ind}_{{\\mathcal {W}}}|})$ also causes a term in (REF ) which has the product $x_{{\\mathcal {A}}(k^{\\prime }),i} \\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|]} x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k^{\\prime })\\rbrace }(u^{\\prime }_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}}(i_2)}.",
"$ The products in (REF ) and (REF ) are identical if ${\\mathbf {u}}^{\\prime }$ is as follows, for $j \\in [|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) \\rbrace $ , we have ${\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k^{\\prime })\\rbrace }(u^{\\prime }_j)\\big )= {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(u_{j})\\big ) ;$ such that $x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k^{\\prime })\\rbrace }(u^{\\prime }_{j})\\big ) ,\\text{Ind}_{{\\mathcal {W}}}(j)}= x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(u_{j})\\big ) ,\\text{Ind}_{{\\mathcal {W}}}(j)};$ for $j=\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)$ , we have ${\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k^{\\prime })\\rbrace }(u^{\\prime }_j)\\big )={\\mathcal {A}}(k);$ such that $x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k^{\\prime })\\rbrace }(u^{\\prime }_{j})\\big ) ,\\text{Ind}_{{\\mathcal {W}}}(j)}=x_{{\\mathcal {A}}(k),i}.$ It is obvious to check that there does not exist any other combination of $k^{\\prime \\prime } \\in \\overline{\\text{Ind}}_{{\\mathcal {W}}}$ and ${\\mathbf {u}}^{\\prime \\prime } \\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {W}}}|)$ , causing a term on the LHS of (REF ) which has the product in (REF ), except the two above combinations.",
"In Appendix , we will prove that $& (-1)^{ - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k ) +1+\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}}) +\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) } + \\nonumber \\\\& (-1)^{ - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) +1+\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}}) +\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i^{\\prime }_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u^{\\prime }_1,\\ldots ,u^{\\prime }_{i^{\\prime }_1} \\rbrace ,u^{\\prime }_{i^{\\prime }_1}) } =0,$ such that the coefficient of the product in (REF ) on the LHS of (REF ) is 0.",
"In other words, for each combination of $k$ and ${\\mathbf {u}}$ on the LHS of (REF ), there is exactly one term caused by the combination of $k^{\\prime }$ and ${\\mathbf {u}}^{\\prime }$ , such that the sum of these two caused terms is 0.",
"Thus (REF ) is proved."
],
[
"Case 3",
"Lastly we focus on one transformed subfile $F^{\\prime }_{i,{\\mathcal {W}}}$ in Hierarchy $h>0$ where ${\\mathcal {L}}(i) \\notin {\\mathcal {W}}$ .",
"By definition, we have $|{\\mathcal {W}}\\cap {\\mathcal {L}}|=h$ .",
"On the RHS of (REF ), since ${\\mathcal {L}}(i) \\notin {\\mathcal {W}}$ , $F^{\\prime }_{i,{\\mathcal {W}}}$ appears in $W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ .",
"In addition, $F^{\\prime }_{i,{\\mathcal {W}}}$ also appears in $W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }$ , where $k\\in \\overline{\\text{Ind}}_{{\\mathcal {W}}}$ .",
"Let us first focus on $W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ .",
"Recall that the $\\left( \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)\\right)^{\\text{th}}$ element in $\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ is $i$ .",
"From (REF ), it can be seen that the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ in $W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ is $(-1)^{ \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1 } .$ In addition, we have $& \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace } =\\sum _{\\begin{array}{c}{\\mathbf {u}}=(u_1,\\ldots ,u_{|\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }|})\\\\ \\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }|)\\end{array}} (-1)^{1+\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace })+ \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) } \\nonumber \\\\& \\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }|]} x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }(i_2)}\\\\&= \\sum _{\\begin{array}{c}{\\mathbf {u}}=(u_1,\\ldots ,u_{|\\text{Ind}_{{\\mathcal {W}}}|+1})\\\\ \\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {W}}}|+1)\\end{array}} \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}(-1)^{ 1 +\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}})+ \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]} \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}}(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }(i_2)}$ Let us then focus on $W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }$ , where $k\\in \\overline{\\text{Ind}}_{{\\mathcal {W}}}$ .",
"It was proved in (REF ) that the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ in $W_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }$ is $(-1)^{k- \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k)} x_{{\\mathcal {A}}(k),i}.$ In addition, it was proved in (REF ) that $& \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace } = \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\sum _{\\begin{array}{c}{\\mathbf {u}}=(u_1,\\ldots ,u_{|\\text{Ind}_{{\\mathcal {W}}}|})\\\\ \\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {W}}}|)\\end{array}} \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}(-1)^{1+\\left(\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}})-k\\right)+\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|]} \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}}(i_2)}.$ In the following, we will prove $(-1)^{ \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1 } \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace } + \\sum _{k\\in \\overline{\\text{Ind}}_{{\\mathcal {W}}}} (-1)^{k- \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k)} x_{{\\mathcal {A}}(k),i} \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }=0,$ such that the coefficient of $F^{\\prime }_{i,{\\mathcal {W}}}$ on the RHS of (REF ) is 0.",
"Notice that there are $t-|\\text{Ind}_{{\\mathcal {W}}}|$ non-leaders in ${\\mathcal {W}}$ .",
"Since there are totally $t+1$ non-leaders in ${\\mathcal {A}}$ , we have $|\\overline{\\text{Ind}}_{{\\mathcal {W}}}|=t+1-(t-|\\text{Ind}_{{\\mathcal {W}}}|)=|\\text{Ind}_{{\\mathcal {W}}}|+1.$ Let us focus on one permutation ${\\mathbf {u}}=(u_1,\\ldots ,u_{|\\text{Ind}_{{\\mathcal {W}}}|+1}) \\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {W}}}|+1)$ in $\\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ .",
"The term in (REF ) caused by ${\\mathbf {u}}$ is $&(-1)^{ \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1 } \\hspace{0.0pt}\\left\\lbrace (-1)^{ 1 +\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}})+ \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]} \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}}(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }(i_2)} \\right\\rbrace \\\\&=(-1)^{ \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) +\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}})+ \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) } \\left\\lbrace \\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]} \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}}(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }(i_2)} \\right\\rbrace .",
"$ We can rewrite the product term in (REF ) as follows (recall again that the $\\left( \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)\\right)^{\\text{th}}$ element in $\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ is $i$ ), $&\\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]} \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}}(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }(i_2)} \\nonumber \\\\&=x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}}(u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) })\\big ) , i }\\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1] \\setminus \\lbrace \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)\\rbrace } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}}(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }(i_2)}\\\\&=x_{ {\\mathcal {A}}(\\widetilde{k} ) , i }\\prod _{i_2\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1] \\setminus \\lbrace \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)\\rbrace } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}x_{ {\\mathcal {A}}\\big (\\overline{\\text{Ind}}_{{\\mathcal {W}}}(u_{i_2})\\big ) ,\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }(i_2)},$ where we define $\\widetilde{k} :=\\overline{\\text{Ind}}_{{\\mathcal {W}}}(u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) })$ .",
"Hence, on the LHS of (REF ), besides $(-1)^{ \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1 } \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ , only the caused term by the combination of $\\widetilde{k}$ and $\\widetilde{{\\mathbf {u}}}=(\\widetilde{u}_1,\\ldots ,\\widetilde{u}_{|\\text{Ind}_{{\\mathcal {W}}}|})$ has the product in (REF ), where $&\\widetilde{{\\mathbf {u}}}=(g(u_1),\\ldots ,g(u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1}),g(u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)+1}),\\ldots , g(u_{|\\text{Ind}_{{\\mathcal {W}}}|+1}) ),\\\\& g(u_j):={\\left\\lbrace \\begin{array}{ll} u_j, & \\text{ if } u_j< u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) } \\\\u_j-1 & \\text{ if } u_j> u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) }\\end{array}\\right.",
"}.$ In Appendix , we will prove that $&(-1)^{ \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) +\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}})+ \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) } + \\nonumber \\\\& (-1)^{ - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(\\widetilde{k} ) +1+\\text{Tot}(\\overline{\\text{Ind}}_{{\\mathcal {W}}}) +\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{\\widetilde{i}_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace \\widetilde{u}_1,\\ldots ,\\widetilde{u}_{\\widetilde{i}_1} \\rbrace ,\\widetilde{u}_{\\widetilde{i}_1}) } =0,$ such that the coefficient of the product in (REF ) on the LHS of (REF ) is 0.",
"Hence, for each permutation ${\\mathbf {u}}\\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {W}}}|+1)$ , there is exactly one term caused by the combination of $\\widetilde{k} \\in \\overline{\\text{Ind}}_{{\\mathcal {W}}} $ and $\\widetilde{{\\mathbf {u}}} \\in \\text{Perm}(|\\text{Ind}_{{\\mathcal {W}}}|) $ , such that the sum of these two caused terms are 0.",
"In addition, on the LHS of (REF ), there are $(|\\text{Ind}_{{\\mathcal {W}}}|+1)!$ terms in $(-1)^{ \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1 } \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }$ .",
"Recall that in (REF ), we proved $|\\overline{\\text{Ind}}_{{\\mathcal {W}}}|=|\\text{Ind}_{{\\mathcal {W}}}|+1$ .",
"Hence, on the LHS of (REF ), there are $|\\text{Ind}_{{\\mathcal {W}}}|!",
"\\times (|\\text{Ind}_{{\\mathcal {W}}}|+1)= (|\\text{Ind}_{{\\mathcal {W}}}|+1)!$ terms in $ \\sum _{k\\in \\overline{\\text{Ind}}_{{\\mathcal {W}}}} (-1)^{k- \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k)} x_{{\\mathcal {A}}(k),i} \\beta _{{\\mathcal {A}},{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }$ .",
"In conclusion, we prove (REF )."
],
[
"Proof of (",
"To prove (REF ), it is equivalent to prove $(-1)^{ -\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k ) - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) +\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1})+\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i^{\\prime }_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u^{\\prime }_1,\\ldots ,u^{\\prime }_{i^{\\prime }_1} \\rbrace ,u^{\\prime }_{i^{\\prime }_1}) }=-1.$ Let us focus on $\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1})$ .",
"By the definition of the function $\\text{Card}(\\cdot )$ in (REF ), we have $& \\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1})\\nonumber \\\\&=\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]: i_1 \\ne \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) }{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}\\big (([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}\\rbrace ) \\setminus \\lbrace u_1,\\ldots ,u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)-1},u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)+1},\\ldots , u_{i_1} \\rbrace ,u_{i_1} \\big )\\nonumber \\\\&+|\\lbrace i_2 \\in [ \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) -1] : u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}< u_{i_2} \\rbrace |+|\\lbrace i_3\\in [\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)+1:|\\text{Ind}_{{\\mathcal {W}}}| ]: u_{i_3}<u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)} \\rbrace |\\\\&=\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]: i_1 \\ne \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) }{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}\\big (([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}\\rbrace ) \\setminus \\lbrace u_1,\\ldots ,u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)-1},u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)+1},\\ldots , u_{i_1} \\rbrace ,u_{i_1} \\big )\\nonumber \\\\&+ (\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)-1- |\\lbrace i_2 \\in [ \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) -1] : u_{i_2}<u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)} \\rbrace |) +|\\lbrace i_3\\in [\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)+1:|\\text{Ind}_{{\\mathcal {W}}}| ]: u_{i_3}<u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)} \\rbrace |\\\\&=\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]: i_1 \\ne \\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}\\big (([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}\\rbrace ) \\setminus \\lbrace u_1,\\ldots ,u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)-1},u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)+1},\\ldots , u_{i_1} \\rbrace ,u_{i_1} \\big )\\nonumber \\\\&+ (\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)-1- |\\lbrace i_2 \\in [ \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) -1] : u_{i_2}<u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)} \\rbrace |) \\nonumber \\\\&+\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|] , u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)})-|\\lbrace i_2 \\in [ \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) -1] : u_{i_2}<u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)} \\rbrace |.$ Similarly, for $\\underset{i^{\\prime }_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u^{\\prime }_1,\\ldots ,u^{\\prime }_{i^{\\prime }_1} \\rbrace ,u^{\\prime }_{i^{\\prime }_1})$ , from the same derivation as (REF ), we have $& \\underset{i^{\\prime }_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u^{\\prime }_1,\\ldots ,u^{\\prime }_{i^{\\prime }_1} \\rbrace ,u^{\\prime }_{i^{\\prime }_1}) \\nonumber \\\\&=\\underset{i^{\\prime }_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]: i^{\\prime }_1 \\ne \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) }{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}\\big (([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}\\rbrace ) \\setminus \\lbrace u^{\\prime }_1,\\ldots ,u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)-1},u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)+1},\\ldots , u^{\\prime }_{i_1} \\rbrace ,u^{\\prime }_{i_1} \\big )\\nonumber \\\\&+ (\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)-1- |\\lbrace i^{\\prime }_2 \\in [ \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) -1] : u^{\\prime }_{i^{\\prime }_2}<u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)} \\rbrace |) \\nonumber \\\\&+\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|], u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)})-|\\lbrace i^{\\prime }_2 \\in [ \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) -1] : u^{\\prime }_{i^{\\prime }_2}<u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)} \\rbrace |.$ In addition, from (REF ), it can be seen that $&\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]: i_1 \\ne \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) }{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}\\big (([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}\\rbrace ) \\setminus \\lbrace u_1,\\ldots ,u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)-1},u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)+1},\\ldots , u_{i_1} \\rbrace ,u_{i_1} \\big )\\nonumber \\\\&=\\underset{i^{\\prime }_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]: i^{\\prime }_1 \\ne \\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}\\big (([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}\\rbrace ) \\setminus \\lbrace u^{\\prime }_1,\\ldots ,u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)-1},u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)+1},\\ldots , u^{\\prime }_{i_1} \\rbrace ,u^{\\prime }_{i_1} \\big ).",
"$ From (REF )-(REF ), and the fact that $(-1)^{2 a}=(-1)^0$ for any integer $a$ , we have $&(-1)^{\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1})+ \\underset{i^{\\prime }_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u^{\\prime }_1,\\ldots ,u^{\\prime }_{i^{\\prime }_1} \\rbrace ,u^{\\prime }_{i^{\\prime }_1})} \\nonumber \\\\&=(-1)^{\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|], u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)})+\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|], u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}) }.$ Without loss of generality, we assume $k<k^{\\prime }$ .",
"Recall that $ \\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }(u_{ \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) })=k^{\\prime } $ .",
"By the definition in (REF ), we can see that in $\\overline{\\text{Ind}}_{{\\mathcal {W}}}$ , there are $\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) -1$ elements smaller than $k^{\\prime }$ .",
"By the assumption, $k<k^{\\prime }$ .",
"Hence, in $\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k)\\rbrace }$ , there are $\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) -2$ elements smaller than $k^{\\prime }$ .",
"In other words, $u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}= \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) -1,$ which leads to $\\text{Card}(\\lbrace u_1,\\ldots ,u_{|\\text{Ind}_{{\\mathcal {W}}}|}\\rbrace , u_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)})=\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) -2.$ Similarly, recall that $ \\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k^{\\prime })\\rbrace }(u^{\\prime }_{ \\text{Ind}_{{\\mathcal {W}}}^{-1}(i) })=k $ .",
"In $\\overline{\\text{Ind}}_{{\\mathcal {W}}}$ , there are $\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k) -1$ elements smaller than $k$ .",
"By the assumption, $k<k^{\\prime }$ .",
"Hence, in $\\overline{\\text{Ind}}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {A}}(k^{\\prime })\\rbrace }$ , there are $\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k ) -1$ elements smaller than $k $ .",
"In other words, $u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)}=\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k ),$ which leads to $\\text{Card}(\\lbrace u^{\\prime }_1,\\ldots ,u^{\\prime }_{|\\text{Ind}_{{\\mathcal {W}}}|}\\rbrace , u^{\\prime }_{\\text{Ind}_{{\\mathcal {W}}}^{-1}(i)})=\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k) -1.$ We take (REF ) and (REF ) into (REF ) to obtain, $&(-1)^{\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1})+ \\underset{i^{\\prime }_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u^{\\prime }_1,\\ldots ,u^{\\prime }_{i^{\\prime }_1} \\rbrace ,u^{\\prime }_{i^{\\prime }_1})} \\nonumber \\\\&=(-1)^{\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) -2+\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k) -1}\\\\&=(-1)^{\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) + \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k) +1}.",
"$ Finally, we have $&(-1)^{ -\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k ) - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) +\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1})+\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i^{\\prime }_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace u^{\\prime }_1,\\ldots ,u^{\\prime }_{i^{\\prime }_1} \\rbrace ,u^{\\prime }_{i^{\\prime }_1}) } \\nonumber \\\\&=(-1)^{ -\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k ) - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) +\\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k^{\\prime } ) + \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(k) +1}\\\\&=-1,$ which proves (REF )."
],
[
"Proof of (",
"To prove (REF ), it is equivalent to prove $(-1)^{1+ \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(\\widetilde{k} )+\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1})+ \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{\\widetilde{i}_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace \\widetilde{u}_1,\\ldots ,\\widetilde{u}_{\\widetilde{i}_1} \\rbrace ,\\widetilde{u}_{\\widetilde{i}_1}) }=-1.$ Let us focus on $\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1})$ .",
"By the definition of the function $\\text{Card}(\\cdot )$ in (REF ), we have $& \\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1}) \\nonumber \\\\&= \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{\\begin{array}{c} i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]:\\\\ i_1 \\ne \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) \\end{array}}{\\sum } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}\\big (([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)} \\rbrace ) \\setminus \\lbrace u_1,\\ldots ,u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1}, u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)+1},\\ldots , u_{i_1} \\rbrace ,u_{i_1} \\big ) \\nonumber \\\\&+|\\lbrace i_2 \\in [\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1]:u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)}< u_{i_2} \\rbrace | \\nonumber \\\\ & + |\\lbrace i_3 \\in [\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)+1:|\\text{Ind}_{{\\mathcal {W}}}|+1 ]:u_{i_3}< u_{|\\text{Ind}_{{\\mathcal {W}}}|+1} \\rbrace |\\\\&= \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{\\begin{array}{c} i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]:\\\\ i_1 \\ne \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) \\end{array}}{\\sum } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}\\big (([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)} \\rbrace ) \\setminus \\lbrace u_1,\\ldots ,u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1}, u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)+1},\\ldots , u_{i_1} \\rbrace ,u_{i_1} \\big ) \\nonumber \\\\&+ \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1- |\\lbrace i_2 \\in [\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1]: u_{i_2}< u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)} \\rbrace | \\nonumber \\\\& + \\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|+1] ,u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)} )-|\\lbrace i_2 \\in [\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1]:u_{i_2}< u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)} \\rbrace |$ From the construction of $\\widetilde{{\\mathbf {u}}}$ in (REF ), we have $&\\underset{\\begin{array}{c} i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]:\\\\ i_1 \\ne \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) \\end{array}}{\\sum } \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}\\big (([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)} \\rbrace ) \\setminus \\lbrace u_1,\\ldots ,u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)-1}, u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)+1},\\ldots , u_{i_1} \\rbrace ,u_{i_1} \\big ) \\nonumber \\\\&= \\underset{\\widetilde{i}_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace \\widetilde{u}_1,\\ldots ,\\widetilde{u}_{\\widetilde{i}_1} \\rbrace ,\\widetilde{u}_{\\widetilde{i}_1}).$ From (REF ) and (REF ), and the fact that $(-1)^{2 a}=(-1)^0$ for any integer $a$ , we have $& (-1)^{1+ \\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(\\widetilde{k} )+\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{i_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|+1]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|+1]\\setminus \\lbrace u_1,\\ldots ,u_{i_1} \\rbrace ,u_{i_1})+ \\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\underset{\\widetilde{i}_1\\in [|\\text{Ind}_{{\\mathcal {W}}}|]}{\\sum }\\hspace{0.0pt}\\hspace{0.0pt}\\hspace{0.0pt}\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|]\\setminus \\lbrace \\widetilde{u}_1,\\ldots ,\\widetilde{u}_{\\widetilde{i}_1} \\rbrace ,\\widetilde{u}_{\\widetilde{i}_1}) }\\nonumber \\\\&=(-1)^{ - \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(\\widetilde{k} )+\\text{Card}([|\\text{Ind}_{{\\mathcal {W}}}|+1],u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)} )}\\\\&=-1 ,$ where (REF ) comes from that $\\widetilde{k} :=\\overline{\\text{Ind}}_{{\\mathcal {W}}}(u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i) })$ , and thus $u_{\\text{Ind}_{{\\mathcal {W}}\\cup \\lbrace {\\mathcal {L}}(i)\\rbrace }^{-1}(i)} = \\overline{\\text{Ind}}^{-1}_{{\\mathcal {W}}}(\\widetilde{k})$ ."
]
] | 2001.03577 |
[
[
"Exploratory Study on User's Dynamic Visual Acuity and Quality Perception\n of Impaired Images"
],
[
"Abstract In this paper we assess the impact of head movement on user's visual acuity and their quality perception of impaired images.",
"There are physical limitations on the amount of visual information a person can perceive and physical limitations regarding the speed at which our body, and as a consequence our head, can explore a scene.",
"In these limitations lie fundamental solutions for the communication of multimedia systems.",
"As such, subjects were asked to evaluate the perceptual quality of static images presented on a TV screen while their head was in a dynamic (moving) state.",
"The idea is potentially applicable to virtual reality applications and therefore, we also measured the image quality perception of each subject on a head mounted display.",
"Experiments show the significant decrease in visual acuity and quality perception when the user's head is not static, and give an indication on how much the quality can be reduced without the user noticing any impairments."
],
[
"Introduction",
"Improving the resolution of head mounted devices is currently one of the main areas of focus in the field of virtual reality (VR).",
"This was not surprising, especially since the amount of visual information displayed by VR displays is much smaller than the human viewing capacity.",
"VR displays deliver around 15 pixels per degree, with a $90^{\\circ }$ field of view and a fixed depth of focus of two meters [1], [2].",
"This is in large contrast to TV screens where the resolution starts to match human perception [3].",
"Humans are, capable of perceiving around 120 pixels per degree, a field of view between $220^{\\circ }$ and $230^{\\circ }$ , and a depth of focus which can vary.",
"Although it will take some time to fully exploit the human visual system, VR experiences will certainly head towards improving in that area.",
"Such a poor visual system performance not only yields lower quality immersive experiences, but it is also linked to greater VR sickness, which poses a comfort problem to users [4].",
"VR sickness or simulator sickness, also known as ‘Virtual Reality Induced Symptoms and Effects’ (VRISE) or cybersickness, refers to a constellation of oculomotor and nausea related symptoms that users experience during and after participating in a simulated environment [5], [6].",
"Similar to motion sickness, simulator sickness is caused by a conflict between the perceived visual information and the bodily senses.",
"Amongst the several factors which have been found to contribute to VRISE, there is also the issue of lag.",
"Lag occurs when there is a delay or latency between the actions of a user, for instance head motion, and the system’s response [4].",
"Considering a bad visual performance induces simulator sickness, pursuing a better view in VR will become crucial in the future [4].",
"Such a view expansion with a higher resolution and a wider field of view would, consequently, demand a substantial amount of bandwidth and storage.",
"For this reason, blurring the points outside of the human vision range as a way to minimize streaming data will become a more evident solution.",
"This blurring in VR could occur through foveated rendering if the VR installation is equipped with eye-tracking technology or through system-feedback on the user's position and movement [7].",
"In the current study, we have explored the possibilities of the latter in order to minimize streaming data.",
"As users' bodies are frequently moving around in VR, we want to learn how substantially their head movement affects their visual acuity in such a way that they see little or no detail.",
"The goal of the current study was to learn users' changes in (a) visual acuity performance and (b) perception of impaired image quality when their heads are stationary (static) versus when their heads are in motion (dynamic).",
"According to our knowledge, there is no other study on dynamic visual acuity targeting the improvement of multimedia applications.",
"From the literature we learn that visual acuity declines when horizontal movement of the head increases due to an imperfect pursuit of eye movements, resulting in a continued image motion on the retina [8], [9], [10].",
"For television (TV) and VR, we expected to find a similar pattern.",
"Consequently, as users in motion will see less detail, we anticipated that they would rate the impairment of images as less perceptible than when observing statically.",
"The main contribution of this work is the realization of the perception loss of the subjects quantitatively, when their head is in a dynamic state.",
"This is already crucial information for being able to control the delivery of high quality video at lower bitrates in the case of a dynamic viewing environment (e.g., in VR).",
"Additionally, the results give us an indication for potentially studying cybersickness in VR, as well as useful insights for providing quality VR experiences at lower data rates.",
"Visual acuity is the ability of the eye to perceive details.",
"It depends on optical factors and neural factors.",
"Visual acuity can be described as static and dynamic.",
"Static visual acuity, the common measure of visual acuity, is defined as the slightest detail that the human eye can distinguish in a stationary, high contrast target (typically an eye chart with black letters on a white background).",
"The standard Sloan chart [11] for measuring visual acuity is shown in Fig.",
"REF .",
"Dynamic visual acuity is the ability of the eye to visually discern fine detail either in a moving object, or a stationary object while the actual subject is in motion, or both.",
"In other words, dynamic visual acuity is the ability to distinguish fine detail when there is relative motion between the object and the observer.",
"While static visual acuity is the common measure, for scenarios where motion is present such as driving licensing, dynamic visual acuity should also be considered.",
"Similarly, dynamic visual acuity would seem to be more relevant when subjectively evaluating VR content, or any other content present in a dynamic viewing environment.",
"So far, practical dynamic visual acuity test methods are very limited and serving only medical purposes [12], [13].",
"However, in the next sub-section we elaborate on the current status of the relevant research and literature (according to our findings).",
"First, we focus on the research on visual acuity in respect of a vision research point of view, and then, we focus more on the research as perceived from a Quality of Experience (QoE) point of view.",
"Figure: Typical Sloan visual acuity test used for clustering the subjects and setting weights on opinion scores.",
"It ranges between 0 and 2, where 2 is the best score achieved for reading the bottom set of characters."
],
[
"Related work",
"There have been attempts in the past to increase our knowledge on perception aspects under movement both from a vision research point of view and from a QoE point of view.",
"With respect to vision research, research has been expanded towards movement starting at the basis, our eyes.",
"The eye is limited in its visual capacity and these limits become more stringent with increased freedom of movement.",
"When considering the eye by itself, only the foveola (approx.",
"$1^{\\circ }$ of vision) and the fovea (approx.",
"$5^{\\circ }$ of vision) provide us with a high acuity of vision.",
"When moving further from our gaze direction, our visual acuity almost linearly decreases up to the mid-peripheral vision (approx.",
"$30^{\\circ }$ of vision), from where visual acuity drops even faster [14].",
"In addition to that, lateral masking phenomena make our peripheral visual acuity decline when more information is present [15].",
"When the eye starts moving and exploring the scene, one talks about saccades.",
"Express saccades take around 100 ms and regular saccades around 150 ms corresponding to almost 14 displayed pictures considering a screen refresh rate of 90 Hz [16].",
"Because the eye is in motion, perception loss is inevitable [17].",
"When moving our head, visual perception starts to interact with vestibular compensation.",
"The vestibular system in the inner ear senses body motion (mainly the head), and uses this to control movement of the eye.",
"The vestibulo-ocular reflex (VOR) moves the eyes contrary to the head, enabling gaze stabilization for both linear and angular head motion.",
"To measure visual acuity loss between stationary head and a rotating head, different dynamic visual acuity tests have been created [18], [19].",
"However, in this area (vision research), the state of the art has a strict focus on visual acuity rather than error visibility of visual coding related quality loss.",
"In this research domain, most tests involve unnatural structures such as letters and symbols to measure visual acuity and discard any realistically looking scenes.",
"Such assumptions limit the possibility to translate such research findings explicitly to the domain of visual representation and interaction.",
"With respect to QoE for visual representation and coding, multiple studies have tried to push the state of the art in perception in the direction of plenoptic VR content.",
"These studies can be classified in the QoE tests purely based on graphics and 3D meshes [20], and the ones based on camera captured-content [21].",
"For most works, the camera-captured VR content is transformed back to regular 2D video, which eliminates the interaction possibilities and the physical movements one could make [22].",
"First efforts to investigate VR quality of experience are made in ITU recommendations under development such as the ITU-T SG12, Q13, G.QoE-VR and P.360-VR, but these are mainly limited to $360^{\\circ }$ video.",
"Irrespective of the fact that these works are limited to only a subset of the freedom to move (only rotation of the head), they are mainly focused on overall quality of experience rather than perception capabilities on the fundamental level.",
"Additional to the perception restrictions of moving eyes there are the physical limitations involving movement of our body.",
"Kinematics of the human body has been extensively investigated and modelled, but never used for the investigation of dynamic visual acuity and the actual latency to achieve a gaze direction, for all possible directions.",
"The efforts needed for predicting that latency, which is associated with a different gaze direction, can be complemented by the research on visual saliency.",
"Saliency research tries to model the region of interest for the observer and tries to answer the question on where the observer will be looking in certain content.",
"The state of the art in this area seems promising since it is already expanding from 2D video towards $360^{\\circ }$ VR experiences[23], [24]."
],
[
"Experimental setup",
"Dynamic visual acuity is the acuity which is obtained during relative motion of either the optotypes (standardized symbols for testing vision) or the observer [9].",
"Contrary to the state of the art, the optotypes in this experiment remained stationary in both the dynamic and static tests.",
"The dynamic aspect in our experiment was introduced by the subject's head movement.",
"We chose to do this as we are interested in the visual performance of users when they move their heads, and not when the images are moving around them, because this is how viewers in VR typically interact with content.",
"To ensure reliability, one test supervisor stood behind the subjects and turned the participants’ heads at a pre-determined pace ($3.5$ Hz) and angle ($30^{\\circ }$ ), similar to a method described in [25].",
"To facilitate this, participants remained seated during the entire duration of the experiment (see Fig.",
"REF ).",
"One subject who felt nauseous due to VR sickness was unable to perform the dynamic tests and was, thus, excluded from further testing.",
"In total 22 people participated.",
"Each visual acuity test took approximately 3 to 4 minutes, while the entire test session was kept under the maximum testing limit of 30 minutes, as recommended by [26].",
"This study used the consumer version of Oculus Rift (https://www.oculus.com/rift) as head-mounted display to project the test interface using the Virtual Desktop application (https://www.vrdesktop.net).",
"The participants did not use the Touch controllers during the virtual reality projection.",
"Oculus Rift delivers a resolution of $1080\\times 1200$ pixels per eye, a frame rate of 90 fps, and a latency between $15.3$ ms and $19.7$ ms.",
"The TV monitor, which was used for comparison, was a Philips 9000 series Smart-LED TV (46PFL9705k/02) with a resolution of $1280\\times 1024$ pixels.",
"It had a 46 inch diagonal dimension, and was thus satisfying the minimum requirement of 14 inch to be used in visual quality assessment tests [27].",
"Figure: The dynamic aspect of the experiment was introduced by a supervisor standing behind the subject.",
"The participant's head was turned horizontally at a pre-determined pace of 3.53.5 Hz and a width of 30 ∘ 30^{\\circ } ."
],
[
"Subjective test",
"The recruitment and the study occurred at a knowledge and innovation center in Ghent-Belgium.",
"All participants were recruited through voluntary participation.",
"As compensation for their time and effort, participants were given one consumption of 5 on site.",
"In total 22 people participated in the experiments, of which 15 were female and 7 were male.",
"This exceeds the minimum-norm of 15 as recommended by ITU-T [27].",
"We solely recruited non-experts who were not involved in the study process, and thus, had no preconceptions about the goal of the experiment.",
"The ages of the participants varied between 20 and 54 with an average age of $28.45$ .",
"If the respondents normally wore sight corrections, these were also worn during the test.",
"Before the actual experiments took place, a pilot study was organized.",
"The first pilot consisted of a small scale trial run of the actual experiment in order to assess the efficacy of the experiment design, the clarity of instructions, the instruments, and the total testing time.",
"At the start of the experiment, the subjects were informed about the procedure and they filled in a consent form.",
"All subjects participated in a visual acuity test and an impaired image quality rating (IIQR) under four conditions: on a TV and on a head mounted display (HMD), while the user’s head was either moving or static.",
"All subjects viewed the same set of images with counterbalanced quality levels, environment (HMD or TV), and state of the user's heads (static or dynamic).",
"Before starting the VR test, each participant was given enough time to ensure the headset was tightly fitted around their head, and to adjust the lens slider on the bottom of the headset.",
"Participants were seated three meters from the TV screen.",
"To obtain a similar distance-to-chart on the HMD screen, the display settings within Virtual Desktop were changed to fish-eye 30o.",
"Subjects were given instructions to read each letter from the displayed Sloan chart (see Fig.",
"REF ).",
"When participants read a line incorrectly, their visual acuity value was the score of the previous line.",
"In case participants read the last line correctly, they received the highest score possible.The Sloan charts used for this experiment were the 2000 Series Revised ETDRS Chart 1, 2 and 3 published by Precision Vision [11].",
"Each conducted impaired image quality test consisted of two phases: the training test, and the main test.",
"The training phase introduced the subjects to the test setup and allowed them to practice the assessment tasks.",
"The content used for the training was similar to the main test content.",
"To keep the total testing time under 30 minutes, we only selected two images of the same resolution ($2200\\times 1500$ pixels).",
"The first image consisted of a woman's head and upper body sitting in front of a textured background, while the second image showed the landscape of a big city consisting of buildings and a blue sky as a background.",
"The introduced impairment was a result of 1) downscaling to different resolutions and 2) JPEG coding using different quantization parameters.",
"The evaluation technique was a Double Stimulus Impairment Scale (DSIS) method, where the reference images and the impaired images were shown side by side (see Fig.",
"REF ).",
"Subjects were aware of the position of the reference image, and they were asked to answer the following question: \"How do you rate the impairment of the image on the right in comparison to the image on the left?\"",
"on a six-point impairment scale [28]: imperceptible just perceptible definitely perceptible but not annoying somewhat objectionable definitely objectionable very objectionable This is similar to the latest ITU picture assessment standardized 5-grade impairment scale [29].",
"Each subject viewed a total of ten images per test condition: five reference images, and five impaired images in various resolutions and compression sizes of either test set 1 (woman) or test set 2 (city).",
"Every participant received 10 seconds to view each pair of images and assess the quality by voting from 1 to 6 according to the above described impairment scale.",
"The average score for each case is the mean opinion score (MOS) [30] used for visualizing the results of this work.",
"Figure: Example of the evaluation method (double stimulus impairment scale) used in the actual tests.",
"The viewer sees the original picture on the left side and the impaired picture on the right side (Top: test set woman, bottom: test set city).Table: Visual acuity results of the test subjects –Maximum possible acuity score: 2 –Format: Mean(Standard Deviation)."
],
[
"Results",
"The visual acuity test scores are presented in Table REF .",
"As expected, subjects scored significantly higher on the static acuity test than on the dynamic visual acuity test both on TV and in VR.",
"Table REF summarizes our results.",
"For each level of impairment we introduced, for every scenario (Static/Dynamic, TV/HMD), for each test set, we calculated the mean (MOS) and its standard deviation.",
"For evaluating relevant correlations, we calculated the effect sizes between static and dynamic scenarios for each test set.",
"Pearson's $r$ is often used as effect size when paired quantitative data are available and it is calculated as follows.",
"Given pairs of data $(x_1,y_1),...,(x_n,y_n)$ consisting of $n$ number of pairs, the Pearson's $r$ coefficient is defined as: $r = \\frac{{}\\sum _{i=1}^{n} (x_i - \\overline{x})(y_i - \\overline{y})}{\\sqrt{\\sum _{i=1}^{n} (x_i - \\overline{x})^2(y_i - \\overline{y})^2}},$ where $n$ is the number of the paired samples, $x_i,y_i$ are the sample points, and $\\overline{x}=\\frac{1}{n}\\sum _{i=1}^{n}x_i$ is the mean of the samples (same for $\\overline{y}$ ).",
"A larger absolute value indicates a stronger effect (larger correlation).",
"Dynamic visual acuity scores correlated well with the dynamic IIQR results.",
"More specifically, on TV: $r = 0.46, p < 0.05$ , and on HMD: $r = 0.56, p < 0.01$ ), where $r$ is the correlation coefficient and $p$ is the significance level.",
"Typically, if $p$ is lower than the conventional $5\\%$ ($p < 0.05$ ) the correlation coefficient is called statistically significant.",
"In other words, during the dynamic IIQR respondents with a better dynamic sight rated the impaired images as more annoying than other participants who had lower dynamic visual acuity test scores.",
"For the static and dynamic conditions on HMD, we see a medium effect size ($r = 0.33$ ) at the lowest impairment level (level 1) for the test set woman, whereas for the test set city there is a medium effect size ($r = 0.27$ ) at the impairment level 3.",
"When comparing to the static and dynamic conditions on TV, we validate that impairment levels 2 and 4 provide medium and large effect sizes for the woman test set.",
"Similarly, for the city test set, the range of impairment levels $2-5$ yields medium and large effect sizes.",
"In Fig.",
"REF and Fig.",
"REF , those effect sizes for different impairment levels are presented in terms of the corresponding impairment scale.",
"More explicitly, for the TV test scenario, stronger compression results in subjects perceiving more impairments.",
"However, when the subjects are in a dynamic state, their vision is less sensitive and the impairments are less perceptible.",
"In that way, we can get an indication of how much the quality could be reduced, without being noticeable by the majority of the subjects.",
"For the HMD test scenario, it can be seen that only level 4 compression and higher results in impairment that gets definitely noticeable and objectionable.",
"This might be caused by the inadequate resolution of the HMD which does not allow the user to perceive the scene in high detail.",
"On the other hand, we can draw similar conclusion for how much the quality can be dropped (for the specific setup) without noticeable compression artifacts.",
"Figure: Perceived impairment scale vs. level of impairment (compression/downscaling) for the test image woman.",
"Level 1 - 5 denotes different impairment levels scale, where level 1 is high resolution and light compression while level 5 is low resolution and heavy compression.Figure: Perceived impairment scale vs. level of impairment (compression/downscaling) for the test image city.",
"Level 1 - 5 denotes different impairment levels scale, where level 1 is high resolution and light compression while level 5 is low resolution and heavy compression.Table: Summarized results of the subjective test including means, standard deviations, and effect sizes (Pearson's correlation) of the mean opinion scores for different levels of impairment and different conditions."
],
[
"Discussion",
"From this experiment we can draw the following three conclusions which are applicable to both the TV and VR scenarios.",
"Firstly, users performed better at the static visual acuity test than at the dynamic test which is in line with past studies [8], [31].",
"Secondly, dynamic acuity correlated with perception of impairment.",
"People who scored higher in dynamic sight tests, were able to see the differences between the reference image and impaired images during the dynamic impairment rating.",
"Finally, although the results were statistically insignificant in the impairment ratings, we found promising effect sizes (Pearson's correlation) between static and dynamic conditions for the different impairment levels.",
"Unlike significance tests, effect size is independent of sample size.",
"Statistical significance, on the other hand, depends upon both sample size and effect size.",
"For this reason, $p$ values are considered to be confounded because of their dependence on sample size.",
"Sometimes a statistically significant result means only that a huge sample size was used [32], [33].",
"From all the above, we can assume that the impairment perception of the human visual system in a static and a dynamic environment is linearly correlated, something that can be exploited to create practical QoE bandwidth management solutions.",
"Future work of this research consists of extending the tests to regular video, 360o video, and light field video.",
"Additionally, a more realistic motion scheme for the user is a work in progress, where the dynamic visual acuity and content evaluation can be measured during actual movement of the subject.",
"That movement can be directly dependent on the viewed content which will lead the subject throughout the process.",
"In other words, the subject will evaluate various VR content while freely roaming in space, which is a more realistic scenario than forced horizontal head movement.",
"Finally, while a link between poor visual system performance and cybersickness was confirmed in the past [4], future extensions of this work could probe whether or not there is a connection between user's visual acuity and cybersickness."
],
[
"Conclusions",
"We investigated the correlation of (dynamic) visual acuity and impairment rating of 2D images on TV and on HMD.",
"Results showed that visual acuity and impairment ratings are indeed correlated, while users' scores on TV and on HMD for the same content are not necessarily correlated.",
"Although using distorted images and maintaining a good user experience might seem mutually exclusive, our experiment shows that, when in motion, users rate certain impairment levels of images as less perceptible than static users.",
"To the best of our knowledge, this study was the first to combine and compare static and dynamic visual acuity and impaired image quality tests both on TV and on HMD.",
"The identified range of quality tipping points can be used as a baseline for subsequent research using a dataset consisting of various types of video content used on HMDs."
]
] | 2001.03542 |
[
[
"The peculiar emission line spectra of core-Extremely Red BOSS Quasars at\n $z\\sim$2-3: orientation and/or evolution?"
],
[
"Abstract Core-extremely red quasars (core-ERQ) have been proposed to represent an intermediate evolutionary phase in which a heavily obscured quasar is blowing out the circumnuclear interstellar medium with very energetic outflows prior to becoming an optical quasar.",
"We revise the general UV and optical emission line properties of core-ERQ in the context of the AGN orientation-based unification scenario.",
"We use diagnostic diagrams based on ultraviolet (UV) emission line ratios and UV-optical line kinematic information to compare the physical and kinematic gas properties of core-ERQ with those of other luminous narrow and broad line AGN.",
"In particular, we provide a revised comparison of the [OIII] kinematics in 21 core-ERQ with other samples of quasars matched in luminosity with the aim of evaluating whether core-ERQ host the most extreme [OIII] outflows.",
"The UV line ratios suggest that the physical properties (for instance, density, metallicity) of the ionised gas in core-ERQ are similar to those observed in the BLR of blue Nitrogen-loud QSOs.",
"The [OIII] outflow velocities of core-ERQ are, on average, consistent with those of very luminous blue QSO1, although extreme outflows are much more frequent in core-ERQ.",
"These similarities can be explained in the context of the AGN unification model, assuming that core-ERQ are viewed with an intermediate orientation between type 2 (edge-on) and type 1 (face-on) QSOs.",
"We propose that core-ERQ are very luminous but otherwise normal quasars viewed at an intermediate orientation.",
"Such orientation allows a direct view of the outer part of the large BLR, from which core-ERQ UV emission originates; the extreme [OIII] outflow velocities are instead a consequence of the very high luminosity of core-ERQ."
],
[
"Introduction",
"Ross et al.",
"[81] discovered a population of extremely red quasars (ERQ) in Data Release 10 (DR10) of the Baryon Oscillation Sky Survey (BOSS; Ross et al.",
"[80], Dawson et al.",
"[25]) in the Sloan Digital Sky Survey-III (SDSS-III; Eisenstein et al.",
"[30]).",
"They show very red colours (SDSS $r$ band to WISE $W4$ band) similar to dust obscured galaxies (DOG).",
"The authors identified 65 ERQ which span a redshift range of 0.28$<z<$ 4.36 with a bimodal distribution, with peaks at $z\\sim $ 0.8 and $z\\sim $ 2.5.",
"Most objects are type 2 quasars (QSO2) or heavily reddened type 1 quasars (QSO1), but there is a subsample of 12 objects presenting very peculiar emission line to continuum properties which defy standard explanations based on extinction and/or orientation.",
"This subclass, which was later named core-ERQ by Hamann et al.",
"(2017, [40] hereafter), is the topic of this paper.",
"[40] enlarged the core-ERQ catalogue to 97 objects by defining a less stringent colour condition.",
"These have nearly uniform peculiar properties selected via $i-W3\\ge $ 4.6 (AB system) and rest frame equivalent width of CIV$\\lambda $ 1550 (CIV hereafter), REW$_{\\rm CIV}>$ 100 Å, at redshifts $z\\sim $ 2.0-3.4.",
"Core-ERQ have very high bolometric luminosities (median log$(L_{\\rm bol}(\\rm erg~s^{-1}))=$ 47.1$\\pm $ 0.3; see [40] and [81] for a detailed characterisation and investigation of core-ERQ properties).",
"They show unexpectedly flat UV spectra given their red UV-to-mid-IR colours and large line REW: 50% core-ERQ have REW$_{\\rm CIV}>$ 150 Å, vs. $\\ll $ 1% in normal blue type 1 quasars matched in $i$ or $W3$ magnitude.",
"They show signs of strong absorption in the X-rays with inferred column densities of $N_ {\\rm H}$ 10$^{23}$ cm$^{-2}$ [36].",
"Core-ERQ have peculiar, wingless rest-frame UV emission line profiles, with full width at half maximum (FWHM) values between between those found for very luminous type 1 and type 2 active galactic nuclei (AGN) at similar $z$ .",
"Their median FWHM$_ {\\rm CIV}$ =3050$\\pm $ 990 km s$^{-1}$ is narrower than FWHM$_ {\\rm CIV}$ =5836$\\pm $ 1576 km s$^{-1}$ for blue quasars matched in $W3$ ([40]), but significantly broader than FWHM$_ {\\rm CIV}<$ 2000 km s$^{-1}$ of QSO2 [1] and narrow line radio galaxies (NLRG, [26]) at similar $z$ .",
"Large blueshifts in excess of 2500 km s$^{-1}$ in CIV and other high-ionization UV lines compared to the HI Balmer and low-ionization permitted lines in the UV are also reported, which have been interpreted in terms of outflows [40].",
"CIV blueshifts are common in normal QSO1 (e.g.",
"[35], [92], [83], [103]).",
"They have been widely interpreted within the context of accretion disk-wind models (e.g.",
"[77]) Even more striking are the kinematic properties of the forbidden [OIII]$\\lambda \\lambda $ 4959,5007 lines ([OIII] hereafter) revealed by near infrared spectroscopy.",
"(Zakamska et al.",
"2016 ([111] hereafter), [70]).",
"Core-ERQ exhibit extremely broad and blueshifted [O III] emission, with widths ($W_{90}$ ) ranging between 2053 and 7227 km s$^{-1}$ , and maximum outflow speeds ($V_{\\rm 98}$ ) up to 6702 km s$^{-1}$ .",
"According to the authors, at least 3-5 per cent of their bolometric luminosity is being converted into the kinetic power of the observed winds, thus having the potential of affecting the entire host galaxy.",
"Core-ERQ also show peculiar line ratios.",
"For instance, Ly$\\alpha $ is often very strongly absorbed, sometimes almost completely, resulting in very low line intensities relative to other lines such as CIV or NV$\\lambda $ 1240 (hereafter, NV).",
"They are also characterised by a high NV/CIV ratio (often $>$ 1.5) and an intermediate [OIII]/H$\\beta \\sim $ 1–4 ratio ([70])These values are derived using the total line fluxes, without any differentiation between NLR and BLR.",
"between type 1 and type 2 luminous AGN.",
"Disentangling the various explanations for these puzzling core-ERQ spectral features in its own right is of great interest.",
"Moreover, these systems might also be very relevant to studies of galaxy formation and evolution.",
"They have been proposed to be near- or super-Eddington accreting obscured quasars, hosts of some of the most massive black holes (BH) at $z\\sim $ 3, capable of triggering strong galactic outflows that inhibit star formation in the early universe.",
"They may represent an intermediate phase in which a heavily obscured quasar is blowing out the circumnuclear interstellar medium (ISM) with very energetic outflows prior to becoming an optical quasar ([111], [40], [36], [70]).",
"In the unified model of AGN (e.g., [4], [96]), the orientation with respect to the observer of a dusty, obscuring central structure (torus or other) located within the BH gravitational radius of influence can explain certain differences found between an obscured and an unobscured quasar.",
"Such obscuring structure would block the view along some lines of sight towards the accretion disk and the clouds within the broad line region (BLR), so that these become partly or totally hidden.",
"Independently of the specific properties of the blocking structure and the role of other factors that can influence the diversity of quasar properties, it is clear that orientation is key to explain certain differences (e.g., [3], [74], [32], [58]).",
"We investigate in this paper the role played by orientation, extinction and the extreme luminosities of the core-ERQ on their observed emission lines and continuum spectral properties and the possible implications on the evolutive scenario.",
"We adopt H$_0$ = 71 km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _\\Lambda $ = 0.73, and $\\Omega _m$ = 0.27."
],
[
"Sample",
"The sample studied in this paper consists of 21 core-ERQ.",
"We focus on the 20 core-ERQ studied by [70] (which includes the four objects studied by [111]), for which the authors present near infrared spectroscopy in the H$\\beta $ -[OIII]$\\lambda \\lambda $ 4959,5007 region.",
"These lines, especially the [OIII] doublet will provide essential information in our argumentation.",
"We exclude core-ERQ-like objects (see [40]) to have a more homogeneous sample in terms of peculiar spectral energy distributions (SED) and emission line properties.",
"Detailed information on the sample can be found in [70].",
"We also include the core-ERQ SDSS J171420.38+414815.7 at $z=$ 2.34 (J1714+4148 hereafter) that we observed with the 10.4m Gran Telescopio Canarias (GTC) (see Sect.",
")."
],
[
"Measurement of the UV line ratios",
"We have measured UV line ratios for the sample of core-ERQ using the BOSS optical spectra.",
"As a guidance, we show in Table REF the line ratios for the four core-ERQ in [111] and for J1714+4148.",
"The UV ratios for 16 the remaining core-ERQ are shown in Appendix .",
"Through the text, we will refer to the emission lines as follows: NV for NV$\\lambda $ 1240, CIV for CIV$\\lambda $ 1550, HeII for HeII$\\lambda $ 1640, CIII] for CIII]$\\lambda $ 1909, SiV+OIV] for SiV+OIV]$\\lambda $ 1400.",
"Ly$\\alpha $ is often blended with NV (see Fig.",
"18 in [40]).",
"This effect is less severe in core-ERQ than in QSO1 because Ly$\\alpha $ is often heavily absorbed and the lines are narrower.",
"Isolating both lines could be done clearly in 16/21 objects.",
"A careful evaluation of the possible contamination of NV by Ly$\\alpha $ was necessary in five objects with severe blending (see, for instance, spectra of J0834+0159 and J2215-0056 in Fig.",
"1 of [111]).",
"To estimate this effect, we proceeded as follows.",
"Maximum contamination might be expected for the highest possible Ly$\\alpha $ flux.",
"This is, for no Ly$\\alpha $ absorption and maximum flux relative to NV.",
"As a reference to estimate this, we have used the measured CIV fluxes and the typical Ly$\\alpha $ /CIV ratios observed in AGN.",
"A range of values Ly$\\alpha $ /CIV$\\sim $ 1-20 is measured in luminous type 2 AGN (e.g.",
"[26], [105], [1]).",
"The lowest values are a consequence of prominent Ly$\\alpha $ absorption.",
"The ratio is typically Ly$\\alpha $ /CIV$$ 6 in QSO1 ([112], [54], [94]).",
"For each object we have created artificial Ly$\\alpha $ spectral profiles for which Ly$\\alpha $ /CIV=20.",
"The Ly$\\alpha $ FWHM and velocity offset relative to NV are set to be equal to those of the strongest emission lines (CIV, SiIV).",
"This \"unabsorbed\" Ly$\\alpha $ profile would result in a maximum possible contamination of NV.",
"The strongest emission lines in our sample have spectral profiles reasonably well fitted with a Gaussian.",
"Consistently, in our analysis we have assumed a Gaussian shape for the unabsorbed Ly$\\alpha $ .",
"Given that the UV lines of each target typically show a range in their FWHM and in their redshifts $z_{i}$ , we have assumed the worst case scenario and adopted as Ly$\\alpha $ FWHM the broadest value observed among the different lines.",
"It was found that uncertainties on the relative velocity ($z_{\\rm Ly\\alpha }$ ) with respect to NV did not have a significant impact on the results.",
"The uncertainties of the NV flux derived from the above evaluation were $$ 20% for four of the five objects under consideration and $\\sim $ 30% for a fifth object.",
"Since Ly$\\alpha $ is heavily absorbed in general and the worst case scenario has been adopted when estimating the Ly$\\alpha $ contamination, we can reasonably conclude that its impact is in general low.",
"The errors hereafter quoted about the line ratios involving NV will include all the above mentioned uncertainties.",
"These can therefore be qualified as the maximum errors expected.",
"Figure: GTC EMIR spectrum of J1714+4148 in the Hβ\\beta and [OIII] spectral region (black lines).",
"The flux is in units of ×\\times 10 -18 ^{-18} erg s -1 ^{-1} cm -2 ^{-2} Å -1 ^{-1}.",
"Different fits (red lines) were attempted by applying a variety of kinematic constraints to [OIII] and/or Hβ\\beta (fits 2 to 7) or by leaving the kinematic parameters free (fit 1).",
"For each fit, the constraints on the observed FWHM in Å are shown.",
"It is also specified whether Hβ\\beta was forced to have the same redshift as [OIII]λ\\lambda 5007 (λ\\lambda forced, fits 2 and 2b).",
"The reduced chi-squared χ 2 \\chi ^2 are also shown, and the best-fit profiles are presented underneath each panel."
],
[
"J1714+4148: GTC EMIR observations, data\nreduction and analysis",
"We obtained HK-band spectroscopy of the core-ERQ J1714+4148 at $z=$ 2.34 with the 10.4m Gran Telescopio Canarias (GTC) and the EMIR (Espectrógrafo Multiobjeto Infra-Rojo) instrument in long slit mode (program GTC14-19A).",
"EMIR is a near-infrared wide field imager and medium-resolution multi-object spectrograph installed at the Naysmith-A focal station.",
"It is equipped with a 2048$\\times $ 2048 Teledyne HAWAII-2 HgCdTe NIR-optimised chip with a pixel size of 0.2$$ .",
"The low resolution HK grism covers a spectral range of $\\sim $ 1.45-2.42 $\\mu $ m with a dispersion of 6.8 Å pixel$^{-1}$ .",
"The slit width used during the observations was 1.0$$ .",
"The instrumental profile measured from the OH sky lines is FWHM$_{\\rm IP}$ =31.3$\\pm $ 4.0 Å.",
"The total exposure time on source was 3840 seconds.",
"A typical ABBA nodding pattern was applied.",
"The seeing during the observations was FWHM$\\sim $ 0.7$$ .",
"The spectra were reduced using several python routines customised by GTC staff for EMIR spectroscopic data.",
"The sky background was first eliminated using consecutive A-B pairs.",
"They were subsequently flat-fielded, calibrated in wavelength and combined to obtain the final spectrum.",
"To correct for telluric absorption, we observed a telluric standard star with the same observing set up as the science target, right after the J1714+4148 observations and at similar airmass.",
"To apply the correction we used a version of Xtellcor ([97]) specifically modified to account for the atmospheric conditions of La Palma observatory ([73]).",
"Relative flux calibration was applied using the spectrum of the star, which was obtained with the same narrow 1.0$$ slit.",
"The accuracy is $\\sim $ 10%.",
"It is not clear how different this is from the absolute flux calibration, since near infrared magnitudes are not available for the object.",
"Based on previous experience, we estimate a maximum deviation of $$ 30% between the relative and absolute flux calibrations.",
"Since galactic extinction is very low ($A_{\\rm V}$ =0.06) correction for this effect was not applied.",
"We fitted H$\\beta $ and [OIII]$\\lambda \\lambda $ 4959,5007 with single Gaussian profiles to obtain an approximate quantitative characterisation of the gas kinematics.",
"The separation in $\\lambda $ and flux ratio (3:1) for the [OIII] doublet were fixed to the theoretical values.",
"Each line is parametrised with a central $\\lambda $ , observed FWHM and amplitude (and thus, the flux).",
"The two [OIII] lines were forced to have the same FWHM.",
"More complex profiles are likely to be more realistic ([70]) but this approach is not possible with our data given the low S/N ([OIII]$\\lambda $ 5007 is detected with S/N$\\sim $ 6).",
"All FWHM values in km s$^{-1}$ quoted below have been corrected for instrumental broadening in quadature.",
"We show in Fig.",
"REF (first panel) the H$\\beta $ and [OIII] spectrum and the best-fits for different sets of constraints.",
"We attempted different fits applying a variety of kinematic constraints to determine useful ranges of the lines FWHM and the shift in velocity $\\Delta v_{\\rm H\\beta -[OIII]}$ .",
"Some examples are shown in Fig.",
"REF .",
"Each panel shows the fit with the smallest mathematical errors (minimum reduced chi-squared, $\\chi ^2$ ) for that specific set of constraints.",
"The low S/N of the spectrum prevents accurate constraints on the [OIII] and H$\\beta $ parameters.",
"In fact, the visual comparison between the data and the fits shows that $\\chi ^2\\sim $ 1 is not always associated with satisfactory results (see, for instance, fits 4 and 7).",
"From this analysis, we propose two tentative results: the [OIII] lines are broad, with a most probable range 1700$$ FWHM$_{\\rm [OIII]}<$ 2300 km s$^{-1}$ (100$$ FWHM$_{\\rm [OIII]}^{\\rm obs}<$ 130Å.",
"The fits suggest that H$\\beta $ is broader than [OIII] and is redshifted, although we cannot discard that this is an artificial result due to the noise of the H$\\beta $ profile.",
"Tentatively, 2000$$ FWHM$_{\\rm H\\beta }$ 3000 km s$^{-1}$ and 900$\\Delta v _ {\\rm H\\beta -[OIII]}$ 1300 km s$^{-1}$ .",
"Such broad H$\\beta $ and [OIII] lines and prominent [OIII] blueshifts are common in core-ERQ ([70])."
],
[
"Comparison of the UV emission line ratios with blue QSO1, QSO2 and high $z$ NLRG",
"The line ratios of the four core-ERQ in [111] and J1714+4148 are shown in Table REF (see Appendix for the remaining 16 objects in the sample).",
"The UV line ratios of three composite spectra of other powerful AGN samples are also shown for comparison: NLRG, radio loud (RL) and radio quiet (RQ) QSO1 ([94]).",
"The NLRG composite ([101]) was created by combining high signal to noise Keck spectra of 9 powerful radio galaxies at $z\\sim $ 2.5.",
"The RQ and RL QSO1 composites ([94]) were created by combining Hubble Space Telescope (HST) spectra of QSO1 at 0.3$$ z$$ 3.6.",
"Most objects (139 out of 184) are at $z$ 1.5.",
"The RQ and RL composites include spectra of 107 and 77 objects respectively.",
"The median line ratios of the sample of nitrogen-loud (N-loud) QSO1 from [12] and the median line ratios of SDSS-III BOSS QSO2 candidates at 2$z$ 4.3 ([2], [90]) are also shown.",
"The location of the 21 core-ERQ in several diagnostic diagrams involving relevant UV lines is shown in Fig.",
"REF .",
"Figure: Comparison of the core-ERQ UV line ratios with nitrogen-loud (N-loud) quasars () and the median ratios of SDSS QSO2 at z>2z>2 (Group 1: CIV/HeII<<4 and Group 2: CIV/HeII>>4, ).",
"Since HeIIλ\\lambda 1640 is not detected for most core-ERQ, lower limits are shown with arrows for the ratios involving this line.",
"The minimum and maximum y-axis and x-axis values have been selected in all diagrams to span approximately the full range of values shown by most objects plotted.",
"The core-ERQ line ratios (measured and lower/upper limits) overlap partially with the area of the diagrams covered by N-loud quasars.The comparison with NLRG is interesting because these are obscured AGN with quasar like luminosities ([101]).",
"The BLR is hidden and the emission line spectrum originates in the NLR.",
"As shown by [42], the UV and optical emission line ratios of high $z$ NLRG are best explained by AGN photoionization of low density gas ($n < $ 1000 cm$^{-3}$ ).",
"The line ratios of the core-ERQ are inconsistent with NLRG (Table REF , see also [90]).",
"NLRG show much stronger HeII relative to other lines and fainter NV.",
"Typically, CIV/HeII$<$ 2, CIII]/HeII$<$ 1, NV/CIV$$ 1 and NV/HeII$$ 1 in NLRG (Villar Martín et al.",
"[104], [26], [42]; see also Table REF ).",
"As can be seen in Fig.",
"REF , in general, core-ERQ show CIV/HeII$$ 3, CIII]/HeII$$ 1, NV/CIV$$ 1 and NV/HeII$$ 3.",
"Star forming galaxies also show relatively weak HeII, with high CIV/HeII and CIII]/HeII similar to core-ERQ.",
"However, stellar photoionization can be discarded with confidence as a relevant excitation mechanism of the gas on the basis of the huge EW of the emission lines and the strength of NV (e.g.",
"[33], [63]).",
"[1] identified a sample of 145 QSO2 candidates at 2$z$ 4.3 selected from the quasar sample of the SDSS-III BOSS.",
"They have weak rest-frame UV continuum (typical continuum magnitude of $i\\sim $ 22) and strong CIV and Ly$\\alpha $ , with FWHM$$ 2000 km s$^{-1}$ .",
"[90] have recently studied the physical properties and abundances of the ionized gas in these systems based on their location in rest-frame UV diagnostic diagrams and on the comparison with AGN photoionization model predictions.",
"Compared to NLRG at similar redshifts, the QSO2 are offset to higher NV/HeII, CIV/HeII and CIII]/HeII.",
"[90] have classified the QSO2 in two groups: objects with `normal' CIV/HeII ratios of $<$ 4 (Group 1), and those with 'extreme' CIV/HeII ratios of $>$ 4 (Group 2).",
"Group 2 QSO2 also have systemically higher NV/HeII ratios for otherwise similar line ratios.",
"To explain such systems as well as the difference with Group 1 QSO2 and high z NLRG, [90] propose a combination of high gas density $n$ 10$^7$ cm$^{-3}$ and/or supersolar abundances $Z$ 4$\\times Z_{\\rm \\odot }$ with N/H$$ 16$\\times Z_{\\rm \\odot }$ assuming secondary production of N that results in a quadratic (rather than linear) increase of its N/H abundance.",
"The secondary production of C is also discussed by these authors as a possible explanation for the extreme CIV/HeII ratios seen in some QSO2s.",
"However, in the case of core-ERQ no models can explain the line ratios simultaneously.",
"Based on the value CIV/HeII$>$ 4, 17 of the 21 core-ERQ studied here are confirmed to belong to Group 2 (Fig.",
"REF , tope left panel).",
"The classification of the remaining 4 is not possible based on the CIV/HeII lower limits.",
"However, the high NV/HeII (lower limits in the range $\\sim $ 4-7) strongly suggest that they also belong to Group 2 (NV/HeII$<$ 3 for Group 1 QSO2, [90]).",
"The same classification appears to apply to core-ERQ in general, based on the visual inspection of their spectra ([40]).",
"The four core-ERQ studied by [111] roughly show the lower (J0834+0159 and J2215-0056) and upper values J1232+0912 and J2323-0100) of the range of line ratios spanned by core-ERQ (Fig.",
"REF ; let us not forget that only upper limits could be measured for HeII).",
"We highlight them with different symbols for guidance and as rough markers of the extreme locations in the diagrams of core-ERQ.",
"Most of the remaining core-ERQ overlap with them or lie in between.",
"We discuss next the optimum AGN photoionization models within the range explored by [90].",
"They consider $n$ in the range 10$^{2-8}$ cm$^{-3}$ and gas metallicity in the range $Z=$ (0.5-5.0)$\\times Z_{\\rm \\odot }$ .",
"$\\bullet $ J0834+0159 and J2215-0056.",
"They have among the lowest UV line ratios of core-ERQ.",
"Based on their location in the diagrams NV/CIV vs. NV/HeII, NV/CIV vs. CIV/CIII], CIV/HeII vs. CIV/CIII], the optimum models have the following parameters (see Fig.",
"1 in [90]): Secondary N and primary C models.",
"Models with $n\\sim $ 10$^{6-8}$ cm$^{-3}$ and 3-5 $Z_{\\odot }$ are consistent with the location of the two objects in all diagrams mentioned above.",
"These models are also consistent with NIV]/CIV$$ 0.3 measured in these two objects.",
"No adequate secondary N and secondary C models are found.",
"Only models with $n\\sim $ 10$^{2-4}$ cm$^{-3}$ and $Z\\sim $ 2$Z_{\\rm \\odot }$ can reach NV/CIV$\\sim $ 2.",
"However, these fail in other diagrams.",
"For instance, they produce NV/HeII$\\sim $ 1 (compared with the observed NV/HeII$>$ 3).",
"Also, they produce CIV/CIII]$$ 5, compared with the measured CIV/CIII]=1.7.",
"Therefore, within the range of models explored by [90], those favoured for J0834+0159 and J2215-0056 have primary production of C and secondary production of N, $n\\sim $ 10$^{6-8}$ cm$^{-3}$ and $Z\\sim $ 3-5$Z_{\\odot }$ .",
"$\\bullet $ J1232+0912 and J2323-0100.",
"Their line ratios are rather extreme.",
"NV is very strong compared with QSO2: NV/CIV$\\sim $ 2 and NV/HeII$$ 10, while QSO2 show in general NV/CIV$<$ 1 and NV/HeII$<$ 10.",
"CIV/CIII]$\\sim $ 5 is also at the high end of the range of values spanned by QSO2.",
"There are no optimum models within the parameter space explored by [90] that can explain all line ratios simultaneously.",
"On the other hand, the high NV ratios and large CIV/HeII strongly suggest high densities $n>$ 10$^6$ cm$^{-3}$ and/or high metallicities $Z$ 4$Z_ {\\rm \\odot }$ .",
"The discussion above suggests that densities $n$ 10$^6$ cm$^{-3}$ and well above solar metallicities are unavoidable in the gas responsible for emitting the UV lines in the four core-ERQ.",
"Since these ratios span roughly the most extreme values among core-ERQ line ratios (Fig.",
"REF ), we extend this conclusion to core-ERQ in general.",
"We will next argue that the UV emission lines in core-ERQ have a significant, even dominant contribution from the BLR (see also [2])."
],
[
"Comparison with QSO1",
"The relative weakness of HeII is characteristic of QSO1.",
"The UV emission lines in these systems are preferentially emitted in the BLR.",
"They show typical CIV/HeII$>$ 35, CIII]/HeII$>$ 10, NV/HeII$>$ 15 (Table REF ; see also [98]).",
"Moreover, most core-ERQ show enhanced NV in comparison with the QSO1 composites (NV/CIV$\\sim $ 0.45; see also [98]).",
"The 97 objects in [40] have median NV/CIV=1.44 and 90% show NV/CIV$>$ 0.5.",
"65% show NV/CIV$>$ 1.0 in the range $\\sim $ 1.0-2.9.",
"The enhanced NV emission is reminiscent of N-loud quasars.",
"The strong UV Nitrogen lines in these systems have been interpreted in terms of supersolar metallicities in the BLR (e.g.",
"[38], [29], [9], [62], [59]), sometimes as extreme as $Z>$ 10$Z_{\\rm \\odot }$ (e.g.",
"[9]).",
"[12] analysed the UV spectra of 41 N-loud quasars at $z\\sim $ 2.0-3.5.",
"They show median NV/CIV=0.95 and NV/CIV$\\sim $ 1.0-2.4 for half the objects.",
"The inferred metallicities are in the range $\\sim $ 1-18 $Z_{\\rm \\odot }$ with median 5.5 $Z_{\\rm \\odot }$ .",
"Figure: NV ratios and BLR metallicity.",
"The blue, red and green symbols correspond to the QSO1 bins in which distributed 2383 SDSS QSO1 at 2.3 <z<< z < 3.0 based on the M BH M_{\\rm BH} and the Eddington ratio, L bol L Edd \\frac{L_{\\rm bol}}{L_{\\rm Edd}}.",
"Each cross represents the location of one given L bol L_ {\\rm bol} bin.",
"The colours indicate ranges of increasing L bol L_{\\rm bol}.",
"The median values of QSO2 (Groups 1 and 2) are also shown for comparison with orange symbols (see text).The UV line ratios of [12] N-loud QSO1 are shown in the diagnostic diagrams of Fig.",
"REF .",
"The two [94] QSO1 composites do not appear because their large ratios relative to HeII place them outside three of the diagrams.",
"It can be seen that core-ERQ overlap partially in the diagrams with the area covered by N-loud QSO1.",
"This relation is likely to apply to the general population of core-ERQ, given the similar line ratios suggested by the visual inspection of their spectra ([40]).",
"This suggests that the UV lines in core-ERQ are emitted in metal rich BLR.",
"Different authors have reported a correlation between quasar $L_ {\\rm bol}$ and the BLR metallicity $Z$ , as traced by the Nitrogen (N$^{+2}$ , N$^{+3}$ , N$^{+4}$ ) emission line ratios ([39], [107], [85], [62], [46]).",
"Given the high $L_{\\rm bol}$ of core-ERQ, high BLR abundances (and the enhacement of the NV emission as a consequence) are similarly expected if they follow the same correlation.",
"According to [109], $Z/Z_{\\rm \\odot }\\sim $ 7-16$Z_{\\rm \\odot }$ are expected in the BLR of quasars with log$(L_{\\rm bol})\\sim $ 47.0-48.0 (i.e., similar to core-ERQ).",
"To investigate the origin of the $L_{\\rm bol}$ -$Z_{\\rm BLR}$ relation, [59] used the optical spectra of 2383 quasars at 2.3 $< z <$ 3.0 extracted from SDSS.",
"They divided the sample into bins based on $M_{\\rm BH}$ and the Eddington ratio $\\frac{L_{\\rm bol}}{L_{\\rm Edd}}$ .",
"They then extracted composite spectra for each bin of $M_ {\\rm BH}$ and $\\frac{L_{\\rm bol}}{L_{\\rm Edd}}$ and proceeded to measure the line ratios most sensitive to BLR metallicity, which include NV/CIV and NV/HeII.",
"They found that the most lumoninous QSO1 show the highest NV ratios (see Fig.",
"REF ).",
"We investigate next how NV/CIV and NV/HeII for core-ERQ behave in relation to this QSO1 correlation.",
"Because no reliable $M_ {\\rm BH}$ are available for core-ERQ (see Section REF ), we investigate the behaviour of the line ratios with $L_{\\rm bol}$ instead.",
"$L_{\\rm bol}$ for the different bins in [59] QSO1 sample have been obtained using $\\frac{L_{\\rm bol}}{L_{\\rm Edd}}$ and $M_{\\rm BH}$ in their Table 3.",
"They are in the range 46.34$\\le $ log($L_{\\rm bol}$ )$\\le $ 47.54.",
"For comparison, 46.98$\\le $ log($L_{\\rm bol}$ )$\\le $ 47.86 for the core-ERQ discussed in this work ([70]).",
"We plot the QSO1 bins and the core-ERQ ratios in the NV/CIV vs. NV/HeII diagram (Fig.",
"REF ).",
"QSO1 are represented with blue, red and green colours in order of increasing $L_{\\rm bol}$ .",
"The tight correlation between NV/CIV and NV/HeII for QSO1 reflects at least partially the lack of Baldwin effect of NV$\\lambda $ 1240 in type 1 AGN at $z$ 5 ([28]).",
"While the optical and UV broad and narrow lines of type 1 AGN show an anticorrelation between the REW and the continuum luminosity $\\lambda L_{1450 Å}$ (the so called Baldwin effect, [7]), the REW of NV$\\lambda $ 1240 remains nearly constant over 6 orders of magnitude in continuum luminosity ([28]).",
"The different behaviour of this line has been explained as a consequence of the increasing gas metallicity with AGN luminosity and the secondary production of N; this is, N/O $\\propto $ O/H ([50]).",
"Core-ERQ show a large scatter, in part due to the non-detection of HeII in most of them.",
"In spite of this, it appears clear that the NV ratios in these systems are among the highest compared to [59] QSO1, as we would expect based on their extreme $L_ {\\rm bol}$ .",
"This supports that the lines are emitted in metal-rich BLR.",
"The above comparison should be taken with caution.",
"NV ratios are very sensitive to the ionization level of the gas ([42], [12]).",
"To isolate ionization effects from those due to the gas abundances, it would be necessary to measure other lines such as NIV] and NIII] ([12]), which are in general undetected in the SDSS spectra of the core-ERQ.",
"In spite of this limitation, the comparison presented above with QSO1 strongly suggest that the UV lines in core-ERQ are emitted by metal rich BLR (see also [40] and [2]).",
"We argued above that the UV lines are emitted in the BLR of core-ERQ.",
"The supression of the UV/optical continuum, the wingless line profiles and the intermediate FWHM between QSO1 and QSO2 (see Sect.",
"REF ) suggest that we are observing the outskirts of this region.",
"The range in density in the BLR comes mainly from the estimated radii and photoionization theory (e.g.",
"[66], [34]).",
"Clouds with densities from $\\sim $ 10$^9$ cm$^{-3}$ (close to $n_{\\rm crit}\\sim $ 3.2$\\times $ 10$^9$ cm$^{-3}$ of C III]$\\lambda $ 1909) to $>10^{13}$ cm$^{-3}$ are expected.",
"The highest density clouds $n>$ 10$^{13}$ cm$^{-3}$ are continuum radiators [76].",
"Thus, we assume $n$ 10$^9$ cm$^{-3}$ .",
"The BLR in AGN is photoionized.",
"The main evidence is that the emission-line spectra change in response to changes in the continuum, with lag-times corresponding to characteristic radii of the BLR [71].",
"Assuming, thus, photoionization, the radiation pressure likely confines the ionized layer of the illuminated gas [91].",
"An implication is that the gas density near the ionization front varies with distance from the nucleus $r$ as: $n \\sim 7 \\times 10^4 ~L_{\\rm i,45} ~r_{\\rm 50}^{-2} ~~\\rm {cm^{-3}~~[eq.1]}$ where $L_{\\rm i,45}$ is the ionizing luminosity $L_{\\rm i}$ in units of 10$^{45}$ erg s$^{-1}$ and $r_{\\rm 50}$ is $r$ in units of 50 pc.",
"Knowing $n$ and $L_{\\rm i,45}$ we can thus infer distances.",
"We have estimated $L_{\\rm i}\\sim 0.35 \\times L_{\\rm bol}$ [91].",
"If the UV lines we see are preferentially emitted in a region of $n\\sim $ 10$^9$ cm$^{-3}$ , $r_ {\\rm UV}\\sim $ 2.1-6.7 pc or 0.7-2.1 pc for $n\\sim $ 10$^{10}$ cm$^{-3}$ , depending on the object in the sample of 20 core-ERQ in [70] (J1714+4148 is not considered here because $L_{\\rm bol}$ is not available).",
"The radial size of the BLR, $r_{\\rm BLR}$ of $z$ 0.3 type 1 AGN with $log(\\lambda L_{\\rm 5100})$ 45.0 correlates with the rest-frame 5100 Å luminosity $\\lambda L_{\\lambda \\rm 5100}$ as $ r_{\\rm BLR} = (22.3 \\pm 2.1) \\times \\bigg (\\frac{\\lambda L_{\\lambda \\rm 5100}}{10^{44}}\\bigg )^{0.69\\pm 0.05} {\\rm lt-days}$ ([48]).",
"This relation was expanded up to $log(\\lambda L_{\\rm 5100})\\sim $ 46.0 by [102].",
"We constrain next $r_{\\rm BLR}$ for the core-ERQ, assuming that they follow this correlation.",
"We have inferred $\\lambda L_{\\lambda \\rm 5100}$ following $log(L_{\\rm bol}) = (4.89\\pm 1.66) + (0.91\\pm 0.04) \\times log(\\lambda L_{\\rm 5100})$ [82].",
"If core-ERQ follow this correlation, we expect log($\\lambda L_{\\lambda \\rm 5100})\\sim $ 46.1-47.2 for the 20 core-ERQ.",
"These are $\\sim $ 7-45 times higher than the values inferred from the observations [70].",
"This shows that the optical continuum at 5100 Å is strongly suppressed ([40],[70]), as naturally expected in our proposed scenario.",
"The $\\lambda L_{\\lambda \\rm 5100}$ inferred from $L_ {\\rm bol}$ imply $r_{\\rm BLR}\\sim $ 0.6-3.1 pc depending on the object, which are in reasonable agreement with the $r_{\\rm UV}$ values estimated for $n\\sim $ 10$^{9-10}$ cm$^{-3}$ .",
"Because of their high luminosity, the expected BLR sizes are $\\sim $ 24-126 times larger than the median size inferred for the low $z$ type 1 AGN sample (29.2 lt-days, [48]).",
"The sizes could be somewhat larger if obscuration makes any of the near-and mid-infrared emission anisotropic, since the intrinsic luminosities would be even higher ([40]).",
"Core-ERQ are therefore expected to have large BLR due to their extreme luminosities.",
"This may favour the partial visibility at a wider range of inclinations than for less luminous quasars."
],
[
"Orientation and partial view of the broad line region",
"The impact of orientation on the properties of the SED and the emission line spectra of broad line active galaxies (for instance, quasars and broad line radio galaxies, BLRG) has been widely investigated in the literature (e.g.",
"[55], [53]).",
"Some authors have proposed that at least a fraction of BLRG are partially obscured, misdirected quasars, seen at somewhat larger angles than the quasar population.",
"In this scenario, they are quasars seen through the edge of the obscuring torus ([27], [61]) which, thus, does not have a distinct edge ([6]).",
"The role of orientation in the context of core-ERQ has also been discussed ([40]).",
"[2] highlight its essential role in the model they propose to explain the UV emission line and continuum spectropolarimetric properties of the core-ERQ SDSS J1652+1728 at $z$ =2.94.",
"The authors describe a polar outflow model seen at an intermediate orientation between type 1 (face on) and type 2 (edge-on) with the UV lines being produced on spatial scales similar or greater than the scales of the dusty broad emission and broad absorption emission line regions (see also [40]).",
"This scenario is in turn inspired on the model proposed by [100] for the nearest quasar MRK231.",
"Next we review and reinforce with new arguments the significant role that orientation plays in determining the observed properties of core-ERQ.",
"Based on all arguments combined, we propose that a significant fraction of core-ERQ are very luminous but otherwise normal quasars seen at an intermediate orientation between type 1 (face-on) and type 2 (edge-on).",
"This results in a partial view of the BLR, with the outskirts being visible, while the inner BLR and the AGN continuum remain hidden.",
"This scenario is consistent with: $\\bullet $ The suppression of the UV and optical continuum compared with blue QSO1 (Sect.",
"REF , [40]) $\\bullet $ The UV line ratios consistent with metal rich BLR (Sect.",
"REF ).",
"The continuum source is hidden from the view while the BLR is partially observable.",
"$\\bullet $ The core-ERQ intermediate UV line FWHM values between QSO1 and QSO2 at similar $z$ (Sect. )",
"and the lack of broad wings typical of blue QSO1 ([40]).",
"This can be explained if the inner BLR is obscured.",
"The inner BLR clouds would be responsible for the broad wings of the UV lines since they are closer to the SMBH and are expected to have higher velocities (gravitational and non-gravitational due to radiation driven outflows, for instance).",
"In such situation, the FWHM of the broad emission lines is not a reliable tracer of the BLR kinematics and not adequate to estimate black hole masses.",
"$\\bullet $ The large line-of-sight hydrogen column densities $N_{\\rm H}10^{23}$ cm$^{-2}$ implied by X-ray observations are unlikely due to galactic absorption and are naturally explained by the central AGN dusty obscuring structures (for instance, the classical equatorial optically thick dusty torus and polar dust; [75], [36]).",
"Moreover, the $N_{\\rm H}$ measured for typical type 1 AGN, for which a nearly face-on orientation is expected, are overall at least a factor of ten lower than those measured for core-ERQ (e.g.",
"[56], [57], [23]).",
"$\\bullet $ The intermediate REW$_{\\rm [OIII]}$ between QSO1 and QSO2.",
"Orientation seems to play a major role (although it is not the only driver) in the variance of the REW$_{\\rm [OIII]}$ in QSO1.",
"REW$_{\\rm [OIII]}$ is used as indicator of the accretion disk inclination (e.g.",
"[78], [14]).",
"Its value increases when we move from face-on to more edge-on systems, due to the decreasing contribution of the AGN continuum and FeII contamination from the BLR.",
"The [OIII]$\\lambda $ 5007 REW of [70] sample have REW$_{\\rm [OIII]}\\sim $ 42-646 Å with median 177 Å.",
"For comparison, the sample of WISE/SDSS selected hyper-luminous (WISSH) QSO1 with detected and measurable REW$_{\\rm [OIII]}\\sim $ 0.2-80 Å (median 2.8 Å), being the dispersion of these values due at least in part to orientation (e.g.",
"[13], [103]).",
"Another parameter that influences REW$_{\\rm [OIII]}$ is $L_{\\rm bol}$ , with very luminous systems showing lower REW$_{\\rm [OIII]}$ (the Baldwin effect mentioned above).",
"The $L_{\\rm bol}$ of the core-ERQ is within the range of the WISSH QSO1 ($>10^{47}$ erg s$^{-1}$ ).",
"If subestimated due to obscuration ([40]), the large REW for such high intrinsic $L_{\\rm bol}$ further support a more edge-on inclination than luminous blue QSO1.",
"$\\bullet $ NV/CIV vs. FWHM$_ {\\rm CIV}$ .",
"The general properties (ionization, density, kinematics, abundances) of the BLR and the NLR in AGN are extremely different, and this is clearly reflected on the BLR and NLR line FWHM and the UV emission line ratios (see comparison between the NLRG and the QSO1 line ratios in Fig.",
"REF ).",
"It is natural to expect that as we move gradually from an edge-on to a face-on orientation, the UV emission line properties should change from QSO2 type to QSO1 type, as the BLR emission becomes more prominent.",
"The UV line FWHM and the NV line ratios are two of the most clearly distinct parameters between type 1 and type 2 AGN spectra.",
"As explained in the previous section, high NV ratios are characteristic of the BLR, but not of the NLR.",
"Thus, we naturally expect that FWHM$_ {\\rm CIV}$ and NV/CIV both increase from type 2 to type 1 orientation, as the BLR becomes gradually more visible and its contribution to the UV spectrum becomes increasingly dominant relative to the NLR.",
"We have grouped the 97 core-ERQs in [40] in five groups according to the range of FWHM$_ {\\rm CIV}$ : 1000-2000, 2000-3000, 3000-4000, 4000-5000 and 5000-6000 km s$^{-1}$ .",
"We have calculated the median value of FWHM$_ {\\rm CIV}$ and NV/CIV for all groups and plotted them in Fig.",
"REF .",
"The scatter of the whole sample is huge, but when considering the median values, it is found that as the lines become broader, NV/CIV increases.",
"Objects with FWHM$<$ 2000 km s$^{-1}$ maybe similar in orientation to the QSO2 sample of [1], although $\\sim $ 10 times more luminous.",
"The large scatter of the data shows that orientation is not the only parameter influencing NV/CIV and FWHM$_ {\\rm CIV}$ .",
"Naturally, other parameters such as $n$ , metallicity, ionization parameter $U$ and the nature of the kinematics must be involved (as is the case in other AGN).",
"In spite of this complexity, that the median values behave as expected from a gradual change of inclination, adds support to the idea that orientation plays an important role.",
"According to this scenario, groups of increasingly broader CIV have an increasing relative contribution of the BLR to the emission line fluxes, that would result on higher NV/CIV, more typical of QSO1.",
"Figure: NV/CIV vs. FWHM CIV _{\\rm CIV} for the 97 core-ERQs in .",
"J1714+4148, the four core-ERQ from and the reamaining core-ERQ in are plotted with the same colour code as in Fig.",
"1.",
"All other core-ERQ from are plotted with small dark red diamonds.",
"The core-ERQ have been organised in five groups according to the FWHM CIV _ {\\rm CIV} range of values, indicated with the vertical dashed lines.",
"The median NV/CIV and FWHM CIV _ {\\rm CIV} in km s -1 ^{-1} of the five groups are shown with large orange circles.",
"The errorbars correspond to the standard error on the median values for each group.",
"A clear increase is observed of the NV/CIV ratio with line width.Figure: SDSS spectra of J1031+2903 (top), J1356+0730 (middle) and J1604+5633 (bottom) in the redshifted Lyα\\alpha +NV region.",
"Lyα\\alpha is severely absorbed in many core-ERQ.",
"It shows absorption across thousands of km s -1 ^{-1}, sometimes both redshifted and bluesfhited, as in these three objects.",
"The expected Lyα\\alpha and NVλ\\lambda 1240 central λ\\lambda are shown based on the CIV and SiIV+OIV] redshifts.",
"Fluxes in units of ×\\times 10 -17 ^{-17} erg s -1 ^{-1} cm -2 ^{-2} Å -1 ^{-1}.",
"Wavelength in Å.$\\bullet $ An intermediate orientation may also explain the heavily absorbed Ly$\\alpha $ , which suggests the presence of large amounts of absorbing gas.",
"The Ly$\\alpha $ profile in core-ERQ is often absorbed across most or even the entire spectral line profile (Fig.",
"REF ; see also [40]).",
"It is likely that in at least some objects, the emission at $\\sim $ 1216 Å flux is not dominated by Ly$\\alpha $ but by OV]$\\lambda \\lambda $ 1214,1218 instead ([43]).",
"The Ly$\\alpha $ absorbers span a broad range of velocities of thousands km s$^{-1}$ which are both blueshifted and redshifted.",
"This is reminiscent of the broad line absorbers (BLA) (also both blueshifted and/or redshifted depending on the lines) found by [113] in a small sample of QSO1.",
"They propose that the highly redshifted BAL arising from neutral hydrogen, helium and FeII atoms are due to inflowing gas directly feeding the accretion disk.",
"The authors propose high inclination angles, intermediate between face-on and edge-on and a location for the absorbers between the accretion disk and the dusty torus.",
"Such gas, possibly mixed with dust, may be the same reservoir behind the illuminated phase of a disk-like BLR inflated by an accretion disk wind proposed by [24] and [11]."
],
[
"The [OIII] outflows and the effects of orientation",
"The unified model proposes that the [OIII]$\\lambda $ 5007 emission in AGN originates from regions outside the central obscuring structure and should therefore be independent of orientation.",
"On the other hand, because the AGN continuum is strongly anisotropic, some [OIII] properties do indeed depend on orientation.",
"QSO1 studies have shown that not only REW$_{\\rm [OIII]}$ changes as a function of orientation, but also the prominence and observed kinematics of the outflowing gas in the NLR as traced by the [OIII] profile (e.g.",
"[88], [14], [55]).",
"The blue broad outflow component decreases both in intensity and in velocity shift relative to the reference wavelength with increasing REW$_{\\rm [OIII]}$ .",
"They interpret these results in terms of a gradual change of orientation from face-on (low REW$_{\\rm [OIII]}$ ) to more edge on (high REW$_{\\rm [OIII]}$ ) positions.",
"In an orientation based scenario, the most extreme kinematics associated with an ionised outflow are expected to be observed in face-on luminous quasars.",
"This means that extreme [OIII] outflows with velocities of several 1000s km s$^{-1}$ should be detected in QSO1 with $L_{\\rm bol}$ similar to those of core-ERQs.",
"On the contrary, [70] found that core-ERQ exhibit the broadest and more blueshifted [OIII] emission lines ever reported, with outflow velocities $W_{\\rm 90}$ about three times larger than those of luminosity-matched blue quasar samples (see also [111]).",
"These extreme and apparently unique [OIII] kinematics contradict the orientation scenario.",
"We revise next whether there is enough evidence to sustain this conclusion."
],
[
"Fitting [OIII] in luminous QSO1",
"[70] considered for the comparative QSO1 sample the five WISSH QSOs presented in [13], and the 74 luminous QSO1 in [89], which have 46.2$\\le log(L_{\\rm [OIII]}) \\le $ 48.2 and 1.5$< z <$ 3.5.",
"They re-built the artificial [OIII]$\\lambda $ 5007 lines for each QSO1, using the best-fit gaussian parameters quoted by [89] and [13].",
"They compared the inferred non-parametric velocities $W_{\\rm 90}$ with those of core-ERQs.",
"No errors have been estimated.",
"They derived an average $W_{\\rm 90}$ = $1550 \\pm 300$ km s$^{-1}$ for this QSO1 sample.",
"The first issue to take into account is that the low REW$_ {\\rm [OIII]}$ , the complex H$\\beta $ +[OIII] blend and the presence of strong and complex FeII emission in luminous QSO1 makes the determination of [OIII] spectral parameters very challenging.",
"Different works have shown that, albeit with a very large scatter, REW$_{\\rm [OIII]}$ decreases with increasing AGN luminosity (the [OIII] Baldwin effect; [18], [28], [89]).",
"This is, at the highest luminosities, the [OIII] lines, if detected, are more difficult (often impossible) to parametrise due to their low REW and the relatively stronger contamination by the broad H$\\beta $ and the underlying FeII multiplets.",
"We argue that the particular orientation of core-ERQ facilitates the detection and parametrisation of the outflows, thanks to the higher contrast of the [OIII] lines relative to the continuum, the FeII multiplets and to the broad H$\\beta $ , since these are totally or partially obscured.",
"Indeed, we note that i) about 70% of the 18 WISSH QSO1 analysed in [103] (see also [13]) shows weak/absent [OIII] emission, hindering the detection and accurate characterisation of the outflows in most of their targets, and ii) about 35% of the QSO1 in [89] is associated with [OIII] detections with $<3\\sigma $ significance.",
"Furthermore, the QSO1 in [89] are characterised by low REW, with an average REW$_{\\rm [OIII]}$ = $14\\pm 10Å$ (see Fig.",
"4 in [89]).",
"Therefore, the outflow velocities must be associated with very large uncertainties.",
"Recently, [21] re-analysed all rest-frame spectra of z>1.5 type 1 QSOs available from the literature, in order to infer the properties of ionised outflows traced by [OIII] gas with a uniform analysis strategy.",
"The authors provided new independent velocity measurements for the [89] targets, also reporting significant differences in the derived [OIII] fluxes and velocities.",
"They showed that such discrepancies are not due to the quality of the analysed near-infrared spectra, but to the systematic effects associated with the fit.",
"For instance, the chosen templates to model the FeII emission (see e.g.",
"[51], [103] and refs therein), the adopted function(s) to reproduce the BLR emission (for instnace, single or multiple Gaussian or Lorentzian functions, or broken power-laws; see e.g.",
"[62], [86]), and the shape of the underlying continuum (e.g.",
"[60], [99]) can significantly affect the analysis results.",
"All these systematic effects are more and more important for QSO1 with high $L_{\\rm bol}$ , for which we expect more extreme outflows (e.g.",
"[49], [45]) but also higher emission from the more nuclear regions relative to the [OIII] line flux (e.g.",
"[88], [89]).",
"To further highlight the difficulties in deriving robust [OIII] kinematics of high luminosity QSO1, we derive new outflow velocity measurements for a subsample of blue QSO1 presented in [89], using a different method, the multicomponent simultaneous fit analysis (e.g.",
"[19]).",
"We focus on the 13 out of 19 targets at z$>2$ (as the core-ERQs) associated with good quality NIR spectra.",
"We refer to Appendix for details regarding the fit analysis and the main differences with respect to the previous work by [89] and [21].",
"In Table REF we report the $W_{80}$ and $W_{90}$ measurements we derived from the [OIII] profiles, together with those from [89] and [21] analysis (see also Table REF in Appendix ).",
"The three sets of $W_{\\rm 80}$ values are also compared in Fig.",
"REF .",
"Our results are in better agreement with [21] than with [89], who measures lines 1.3 to 3 times narrower.",
"The differences in the analysis results between our work and these two studies highlight the difficulties in deriving robust [OIII] kinematic measurements for very luminous QSO1.",
"Such difficulty is reflected also by the large errors of our $W_ {\\rm 80}$ values.",
"These are larger than those in [21].",
"We consider our errors more realistic, since they take into account (whenever possible) all systematic effects related to the modelling of continuum, FeII, BLR and NLR emission (see Appendix ).",
"Non-parametric velocities errors are not available for [89].",
"The 13 QSO1 discussed above have $L_ {\\rm [OIII]}\\sim $ (0.8-15.0)$\\times $ 10$^{44}$ erg s$^{-1}$ .",
"We infer $W_ {\\rm 90}\\ge $ 2000 km s$^{-1}$ for 11 of them (see Table REF ).",
"All are above the median $W_ {\\rm 90}=$ 1568 km s$^{-1}$ quoted by [70] for the entire QSO1 sample from [89].",
"Out of the 18 QSO1 within the $L_ {\\rm [OIII]}$ range as the subsample fitted by us, they find only three with $W_ {\\rm 90}\\ge $ 2000 km s$^{-1}$ (see their Fig.",
"4).",
"Based on the [89] fits, they clearly obtain significantly narrower lines than core-ERQ for the blue QSO1 comparative sample.",
"With all the caveats mentioned above, it is clear that these kinematic parameters must be affected by very large uncertainties.",
"Unless errors are quoted, any comparison with core-ERQ can only be considered tentative.",
"It is also important to note that the $L_{\\rm [OIII]}$ core-ERQ values in [70] are lower limits.",
"They estimated $L_{\\rm [OIII]}$ for each object, multiplying the [O III] REW by the luminosity at 5007 Å.",
"To obtain the continuum fluxes at this $\\lambda $ , they linearly interpolated the WISE W1 (3.6$\\mu $ m) and SDSS $i$ or $z$ magnitudes and then converted the derived values at 5007 Å into fluxes.",
"As pointed out by the authors, the luminosities could be on average 6.3 times higher, since the median core-ERQ spectral energy distribution (SED) is suppressed at $\\sim $ 5000Å by about 2 mag relative to normal/blue quasars ([40]).",
"The correction factor could be even higher (Sect.",
"REF ).",
"Figure: Comparison of our [OIII]λ\\lambda 5007 W 80 W_{\\rm 80} with and for 13 luminous QSO1 at z>z>2.",
"The large error bars and the different results reflect the difficulty to obtain accurate [OIII] kinematic parameters in distant (z>z>2) luminous QSO1.Figure: Comparison of several type 1 AGN samples in the W80 [ OIII ] W80_{\\rm [OIII]} vs. L [ OIII ] L_{\\rm [OIII]} plane.",
"There is a clear correlation between W80 [ OIII ] W80_{\\rm [OIII]} and L [ OIII ] L_{\\rm [OIII]}.",
"Core-ERQ fall on it, with or without taking into account an extinction correction.",
"Some QSO1 from other samples show [OIII] as broad as core-ERQ.",
"Morevoer, there is not solid evidence to support that core-ERQ show more extreme kinematics than unobscured QSO1 of similar luminosity."
],
[
"[OIII] kinematics. Comparison between core-ERQ and luminous QSO1",
"In Fig.",
"REF we compare the [OIII] kinematics of core-ERQs as traced by $W_ {\\rm 80}$ with other samples of type 1 AGN.",
"In particular, we consider: $\\bullet $ [70] core-ERQ (the four objects from [111] are highlighted with different symbols as in Fig.",
"2).",
"$W_{\\rm 80}$ have been derived rescaling their $W_{90}$ , assuming $W_{80}\\sim $ 0.72$\\times W_{80}$ .",
"This is consistent with the value expected for a gaussian profile, and with the $W_{\\rm 80}/W_{\\rm 90}$ ratios in [110] for type 2 AGN, in [111] for core-ERQs and in [21] for QSO1; $\\bullet $ SDSS J1714+4148 (see Sect.",
"); $\\bullet $ 13 QSO1 from [89] at $z>$ 2 using the result from our new fits; $\\bullet $ Low $z$ X-ray detected SDSS type 1 AGN at $z<$ 0.8 from [68].",
"This sample allows us to highlight the correlation between $L_{\\rm [OIII]}$ and $W_{80}$ over a range of five dex, as well as its large scatter at fixed luminosity; $\\bullet $ Five WISSH QSO1 from [13].",
"We have measured $W_{\\rm 80}$ by reconstructing the [OIII] profiles with the Gaussian parameters provided by the authors in their tables; $\\bullet $ [21] sample of luminous QSO1 (45.5$<log(L_{\\rm bol})<$ 49.0) at 1.5$<z<$ 4.0.",
"The 13 QSO1 at $z>$ 2 re-analysed in this work have not been excluded from the figure (see Fig.REF and Table REF for a comparison between their fit results and ours); $\\bullet $ 22 heavily reddened quasar candidates from the UKIDSS-LAS, VISTA VHS and VIKING imaging surveys at 2.0$z$ 2.6 and 46.0$$ log($L_{\\rm bol}$ )$$ 48.5 ([95]).",
"These are the main conclusions: - core-ERQ fall on the $W_ {\\rm 80}$ vs. $L_ {\\rm [OIII]}$ correlation defined by different samples of type 1 AGN.",
"This is consistent with the scenario in which more kinematically extreme outflows are triggered by more luminous AGN.",
"Any reasonable correction factor to $L_ {\\rm [OIII]}$ due to dust extinction would maintain them within the correlation.",
"- there are very luminous non-obscured QSO1 with [OIII] as broad as core-ERQ ([89], [13], [21]).",
"9 out of the 22 heavily reddened quasar candidates ([95]) show [OIII] kinematics similar to some core-ERQ (see Fig.",
"REF ).",
"The extreme [OIII] kinematics is, therefore, not exclusive of this object class.",
"- taking into account the lower limits on $L_ {\\rm [OIII]}$ and the new results on the blue QSO1 sample of [89] it cannot be claimed that core-ERQ show in general more extreme kinematics compared with blue QSO1 of matched bolometric luminosities.",
"- Extreme [OIII] kinematics is indeed much more frequent in core-ERQ ($\\sim $ 100%) than in QSO1, as already pointed out by [70].",
"Considering all AGN samples, out of the 9 objects with the most extreme kinematics ($W_{80}$ 4000 km s$^{-1}$ ), 8 are core-ERQ.",
"Whether this is a real intrinsic difference due to truly more extreme outflows being triggered in core-ERQ or rather an artificial effect due to uncertainties resultant of all caveats mentioned above and the easier detection and parametrisation of outflows in core-ERQ is unknown.",
"Based on the above, we conclude that the extreme [OIII] kinematics of core-ERQ do not pose a problem for the orientation based scenario.",
"Table: QSO1 in at z≥z\\ge 2 for which [OIII] could be fitted.",
"The [OIII] W 80 W_{\\rm 80} and W 90 W_{\\rm 90} values inferred by us, and are compared.",
"z sys z_{\\rm sys} was measured relative to the peak of [OIII]λ\\lambda 5007.",
"According to our fits, most of these QSO1 show W 90 W_{90} in the range measured for core-ERQ."
],
[
"Kinetic power of the [OIII] outflows",
"High outflow kinetic powers of at least $\\sim $ 3-5% of $L_{\\rm bol}$ have been inferred for core-ERQ ([70]).",
"In principle, such powerful winds have the potential to affect the evolution of the host galaxies.",
"For these calculations, sizes $R$ =1 kpc and densities $n$ =200 cm$^{-3}$ were assumed.",
"The authors provide a range of uncertainties to account for a possible range $n\\sim $ 100-1000 cm$^{-3}$ and outflow sizes $R\\sim $ 0.5-7 kpc.",
"The most likely situation is that the ionized outflows are spatially unresolved ([47], [106], [44], [79], [93]).",
"The integrated spectra used for the outflow measuments are expected to be dominated by the compact NLR where a broad range densities of up to $n\\sim $ 10$^7$ cm$^{-3}$ are likely to exist (see [106] for a discussion).",
"For a broad distribution of densities, the emission of a certain forbidden line is expected to peak at gas with $n$ similar to its critical density $n_{\\rm crit}$ (e.g.",
"[91]).",
"For the [OIII]$\\lambda \\lambda $ 4959,5007 lines, $n\\sim n_{\\rm crit}$ =8.6$\\times $ 10$^5$ cm$^{-3}$ .",
"If as argued in Sect.",
"REF radiation pressure confinement defines the gas density spatial profile, we can infer the size of the [OIII] region for eacth object using equation [eq.",
"1], $n_ {\\rm crit}$ and $L_ {\\rm ion}$ .",
"The inferred sizes are $\\sim $ 73-227 pc, depending on the object.",
"Using these sizes and $n\\sim n_ {\\rm crit}$ , the kinetic powers of the outflows are reduced by a factor of $\\sim $ 300-977 depending on the object and become $\\ll $ 1% $L_{\\rm bol}$ .",
"The existence of lower $n$ gas in the outflows is of course possible (e.g.",
"[10], [69]) and it may carry a significant amount of mass, but with the existing data, nothing can be said about it.",
"High spatial resolution spectroscopy is essential to resolve the ionized outflows and characterise more accurately their sizes, geometry, and density distribution.",
"In the meantime, the kinetic powers are too uncertain to infer any useful conclusion regarding their potential to affect the evolution of the host galaxies."
],
[
"Summary and conclusions",
"Core-extremely red quasars (core-ERQ) have been proposed to represent an intermediate evolutionary phase in which a heavily obscured quasar is blowing out the circumnuclear ISM with very energetic outflows prior to becoming an optical quasar, as well as sites of extreme large scale ($$ 1 kpc) outflows that inhibit star formation.",
"Based on the revision of the general UV and optical emission line properties of core-ERQ at $z\\sim $ 2-3 we propose that at least a high fraction of core-ERQ are very luminous but otherwise normal quasars viewed at an intermediate orientation between type 1 (face-on) and type 2 (edge-on).",
"The general properties are not necessarily indicative of the particular evolutionary phase mentioned above, but are instead strongly determined by orientation and the effects of the high bolometric luminosities.",
"The calculations and results presented here rest on the assumption that core-ERQ follow many correlations identified in other AGN samples.",
"Although this is uncertain, the argumentations suggest that they host large broad line regions ($r_{\\rm BLR}\\sim $ several pc).",
"The highly inclined line of sight is such that we observe only part (the outskirts) of the super-solar metallicity BLR.",
"We propose that the inner BLR and the continuum source remain hidden by the torus.",
"An intermediate orientation is consistent with many properties of core-ERQ: the UV line ratios similar to N-loud QSO1, the large CIV$\\lambda $ 1550 and [OIII]$\\lambda $ 5007 rest-frame equivalent widths compared with QSO1 of similar $L_ {\\rm bol}$ , the intermediate FWHM of the UV lines between QSO1 and QSO2 at similar $z$ and their wingless profiles.",
"It is also consistent with the large $N_{\\rm H}$ 10$^{23}$ cm$^{-2}$ implied by X-ray observations and the heavily absorbed Ly$\\alpha $ profiles.",
"We present a revised comparison of the [OIII] kinematics in 21 core-ERQ (20 from [70] and SDSS J1714+4148, based on GTC EMIR near infrared spectroscopy) with other samples of type 1 AGN.",
"Core-ERQ fall on the $W_ {\\rm 80}$ vs. $L_ {\\rm [OIII]}$ correlation defined by different samples of type 1 AGN.",
"This is consistent with the scenario in which more kinematically extreme outflows are triggered by more luminous AGN.",
"The extreme [OIII] kinematics of core-ERQ are not exclusive of this object class, as demonstrated by our revised spectral fits of luminous non-obscured QSO1 at similar $z$ and other recent works ([13], [21], [95]).",
"Contrary to other works ([70]), we find that there is no evidence to support that core-ERQ show in general more extreme kinematics compared with blue QSO1 of matched bolometric luminosities.",
"The difficulty to detect and/or the large uncertainties affecting the parametrisation of the [OIII] profiles in the luminous QSO1 comparison sample, and the fact that both $L_{\\rm bol}$ and $L_{\\rm [OIII]}$ of core-ERQ are very uncertain (and underestimated), lie at the core of this discrepancy.",
"Extreme [OIII] kinematics are indeed much more frequent in core-ERQ ($\\sim $ 100%), as already pointed out by other authors ([70]).",
"Whether this is a real intrinsic difference due to truly more extreme outflows being triggered in core-ERQ or rather an artificial selection effect due to the difficulties to detect and characterise [OIII] outflows in luminous QSO1 in comparison with the much easier detection and parametrisation in core-ERQ is unknown.",
"High spatial resolution spectroscopy is essential to resolve the [OIII] outflows and characterise more accurately their sizes, geometry and density distribution.",
"In the meantime, the kinetic powers are too uncertain to infer any useful conclusion regarding their potential to affect the evolution of the host galaxies.",
"As proposed by other authors, core-ERQ are ideal systems to detect and characterise the most extreme AGN outflows.",
"Whether this is a consequence of the blow-out quasar phase or of a particular intermediate orientation remains to be disentangled.",
"We thank José Acosta Pulido for valuable scientific discussions and the anonymous referee for the thorough revision of the manuscript.",
"Thanks also to Yue Shen for making his QSO1 NIR spectra available.",
"Partly based on observations made with the GTC telescope and the EMIR instrument at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias (program GTC14-19A).",
"We thank the GTC staff for their support with the observations.",
"We thank Joel Vernet for providing the high $z$ NLRG composite spectrum.",
"MVM acknowledges support from the Spanish Ministerio de Ciencia, Innovación y Universidades through the grants AYA2015-64346-C2-2-P and PGC2018-094671-B-I00 (MCIU/AEI/FEDER,UE).",
"MP is supported by the Programa Atracción de Talento de la Comunidad de Madrid via grant 2018-T2/TIC-11715.",
"MP acknowledges support from the Spanish Ministerio de Economía y Competitividad through the grant ESP2017-83197-P. AH acknowledges FCT Fellowship SFRH/BPD/107919/2015; Support from European Community Programme (FP7/2007-2013) under grant agreement No.",
"PIRSES-GA-2013-612701 (SELGIFS); Support from FCT through national funds (PTDC/FIS-AST/3214/2012 and UID/FIS/04434/2013), and by FEDER through COMPETE (FCOMP-01-0124-FEDER-029170) and COMPETE2020 (POCI-01-0145-FEDER-007672).",
"S.M.",
"acknowledges financial support through grant AYA2016-76730-P (MINECO/FEDER).",
"This research has made use of: 1) the VizieR catalogue access tool, CDS, Strasbourg, France.",
"The original description of the VizieR service was published in Ochsenbein et al.",
"A&AS, 143, 23; 2) data from Sloan Digital Sky Survey.",
"Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England.",
"The SDSS Web Site is http://www.sdss.org/; 3) the Cosmology calculator by [108]; 4) the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration."
],
[
"UV line ratios of core-ERQ",
"We show in Tables REF and REF the line ratios for the 16 core-ERQ not included in Table REF .",
"For all objects, the line fluxes were measured by integrating the flux within the area defined by the line profiles.",
"This method is more accurate than fitting Gaussian profiles when the lines are highly asymmetric due to, for instance, absorption.",
"Gaussian fits were also applied when the line of interest was blended with a neighbouring line.",
"This was useful, for instance, to measure CIII]$\\lambda $ 1909 more precisely.",
"This line is often blended with SiIII]$\\lambda \\lambda $ 1883,1892.",
"In a minority of objects, when the lines are very broad, AlIII]$\\lambda $ 1857 can also contaminate this blend.",
"In such cases, different constraints were applied to get a range of possible CIII] fluxes.",
"As an example, when necessary, the FWHM of CIII] was fixed in km s$^{-1}$ to be the same as CIV and/or SiIV, taking into account the additional broadening because it is a doublet (1907 and 1909 Å).",
"The error bars take into account the range of fits that are acceptable for the different constraints applied.",
"Table: UV line ratios of the 16 core-ERQ not included in Table .",
"This and next table include different ratios.Table: More UV line ratios of the 16 core-ERQ not included in Table ."
],
[
"Multicomponent simultaneous spectral fit",
"Obtaining accurate [OIII] kinematic parameters is challenging in luminous QSO1 (see Sect.",
"REF ).",
"To evaluate the typical uncertainties on the derived [OIII] velocities, we have reanalysed the 14 out of 19 QSO1 in [89] sample at z$>2$ (as the core-ERQs) associated with good quality NIR spectra.",
"The remaining objects could not be fitted due to the weakness or absence of the [OIII] lines in the spectra.",
"We used the multicomponent simultaneous fit technique, generally adopted in the literature to reduce the degeneracies between FeII, BLR and NLR emission (e.g.",
"[19], [68], [69]).",
"We fitted simultaneously the H$\\beta $ +[OIII]+FeII complex together with the H$\\alpha $ +[NII]+[SII] system for the sources at z $\\sim 2$ , and MgII+FeII+FeIII and H$\\beta $ +[OIII]+FeII regions for the targets at higher $z$ .",
"The best-fit results are shown in Figs.",
"REF , REF , REF and in Tables REF , and REF .",
"These figures also show the comparison of the [OIII] best-fit obtained by [89] with ours (we could not do the same for [21] best fit, because [OIII] Gaussian parameters are not tabulated in their paper).",
"The spectrum of J2238-0821 is also shown in Fig.",
"REF .",
"Unlike in [89], our fits indicate that NLR emission is undetected in this source (consistent with [21])."
],
[
"Modelling rest-frame optical spectra",
"The H$\\alpha $ +[NII]+[SII] and H$\\beta $ +[OIII]+FeII systems of J0149+1501, J1421+2241, J1431+0535, J1436+6336 and J1220+0004 are redshifted in the K- and H-band, respectively.",
"We fitted simultaneously the continuum and all emission lines to reduce the degeneracies between BLR, NLR and FeII emission.",
"In short, we fitted a (single) power-law continuum, an optical FeII template ([51]) and a combination of Gaussian functions to model BLR and NLR emission lines.",
"In particular, depending on the complexity of FeII emission, we considered one or two FeII templates (see e.g.",
"[103]).",
"These are convolved with a Gaussian; the width of this Gaussian, the amplitude normalisations and velocity offset of the FeII templates are free variables in our fit (see [68] for further details).",
"Gaussian components were used to reproduce the emission lines.",
"Specifically, we used (1) a ‘systemic’ narrow (FWHM $<$ 700 km s$^{-1}$ ) Gaussian component for the [OIII] doublet, H$\\beta $ , H$\\alpha $ , [NII] and [SII] doublets associated with unperturbed NLR emission; (2) a broad (FWHM$>$ 2000 km s$^{-1}$ ) Gaussian component to fit the BLR H$\\beta $ and H$\\alpha $ emission; (3) an ‘outflow’ (FWHM $>$ 700 km s$^{-1}$ ) Gaussian component for all forbidden and permitted emission lines to reproduce prominent and asymmetric wings associated with outflowing gas.",
"One or more additional sets of Gaussian components were considered to reproduce more complex profiles, for instance, asymmetric BLR emission lines (see, for instance, J1421+2241, Fig.",
"REF ) and ‘extremely broad [OIII] profiles’ (for instance, J0843+0750, Fig.",
"REF ; see also Fig.",
"11 in [21]).",
"For each set of Gaussian functions, we constrained the wavelength separation between emission lines within a given set of Gaussian profiles in accordance with atomic physics.",
"This means that we constrained the velocity offset of the outflow Gaussians from the narrow (systemic) components to be the same for all the emission lines.",
"Moreover, each emission line within a given set have the same FWHM.",
"Finally, the relative flux of the two [N II]$\\lambda \\lambda $ 6548,5583 and [O III]$\\lambda \\lambda $ 4959,5007 components is fixed to 2.99 and the [S II] flux ratio is required to be within the range 0.44 $< f(\\lambda 6716)/f(\\lambda 6731) < $ 1.42; finally, the H$\\alpha $ /H$\\beta $ flux ratio has been constrained to be $> 2.85$ ([66]).",
"The number of sets used to model the spectra was derived on the basis of the Bayesian information criterion (BIC; [84]), which uses differences in $\\chi ^2$ that penalise models with more free parameters (see e.g.",
"[41], [22])."
],
[
"Modelling MgII, FeII, FeIII, H$\\beta $ and [OIII] ",
"For the remaining targets at $z>$ 2.5, the H$\\alpha $ +[NII]+[SII] complex is not covered by available spectra.",
"We therefore fitted H$\\beta $ +[OIII]+FeII systems together with the MgII+FeII+FeIII lines.",
"Under the assumption that the H$\\beta $ and MgII emission from BLR have the same kinematics (e.g.",
"[87], [15]), as well that the optical and UV iron emission have the same width and velocity (e.g.",
"[15]), the simultaneous fit technique allows us to reduce, also for these spectra, the degeneracies between NLR, BLR and iron emission.",
"We fitted the AGN continuum with two local power laws to the wavelength regions associated with the two systems (i.e., [$2600-3000$ ] Å and [$4500-5500$ ] Å).",
"In addition to the optical FeII templates, we also considered one or two FeII+FeIII templates from [72] to model the iron emission underlying the MgII lines.",
"The UV and optical iron templates are constrained to have the same widths and velocity shifts.",
"Also in this case, we considered a combination of Gaussian sets to best reproduce the BLR and NLR emission (see above).",
"We note that 1) the MgII$\\lambda \\lambda $ 2796,2803 doublet is fitted with a single Gaussian, since the lines are unresolved in all spectra: MgII emission is generally dominated by the BLR emission, which width is much higher than the separation between the two member lines.",
"Moreover, 2) the distinction between ’systemic’ and ‘outflow’ components in our best-fit results must be taken with caution, because of the degeneracies in the fit analysis, its dependency on the quality of the spectra (in terms of SNR and resolution), and the unknown intrinsic shape of ’systemic’ and ‘outflow’ emission line profiles (e.g.",
"[52], [5])."
],
[
"Main differences with respect to Shen (2016) and Coatman et al. (2019)",
"Emission from the optical continuum and iron lines in [21] and [89] are modelled with a local power-law plus a FeII template from [16], using the wavelength regions just outside the H$\\beta $ -[OIII] complex.",
"This model is then subtracted before fitting the BLR and NLR emission.",
"In our analysis, all possible emitting contributions are fitted simultaneously, considering the entire wavelength range covered by FeII, H$\\beta $ and [OIII] lines.",
"Moreover, for the sources at z $\\sim 2$ , we used a unique power-law to model the continuum in the H$\\beta $ -[OIII] and H$\\alpha $ -[NII] regions, in order to be less affected by degeneracies between strong FeII and continuum emission.",
"Finally, we preferred to use the FeII templates by [51], which can allow a more robust separation between [OIII] and FeII emission at $\\sim 5007 $ Å (see also e.g.",
"Appendix A in [68]).",
"In [89] and [21] the [OIII] lines are fitted with two Gaussian components, one for the core and one for the prominent blueshifted wing.",
"The only tied kinematics are those of the [OIII] and H$\\beta $ core components."
],
[
"Best-fit analysis and results",
"The spectral analysis was performed with a python routine using the cap-mpfit procedure (part of pPXF package by [20]) and performing Levenberg-Marquardt least-squares minimisation between the data and the model.",
"The best-fit results are shown in Figs.",
"REF , REF , REF and reported in Tables REF and REF .",
"In order to estimate errors associated with our measurements, we used Monte Carlo simulations.",
"For each modelled spectrum, we collected the fit results of 100 mock spectra obtained from the best-fit final models (red curves in Figs.",
"REF , REF and REF ) and adding Gaussian random noise (taking as a reference the noise spectra shown with cyan curves in the figures).",
"The errors, reported in Table REF , have been derived by taking the range that contains 68.3% of values evaluated from the obtained distributions for each Gaussian parameter/non-parametric measurement.",
"Finally, we note that since line profiles generally are non-Gaussian and much broader than the spectral resolution, we do not correct the observed profiles for instrumental effects and report all values as measured.",
"We also note that our estimated errors take into account (whenever possible) all systematic effects related to the (simultaneous) modelling of continuum, FeII, BLR and NLR emission.",
"In fact, our best-fit models are obtained, for each MC trial, starting from a random initialisation of the model parameters.",
"This allows us to model the spectrum taking into account the possible degeneracies between the different emitting components.",
"As a consequence, the errors reported in Table REF and REF are up to a few orders of magnitude larger than those reported in [21], more effectively reflecting the difficulties in such kinematic analysis.",
"For the sources for which we could fit both H$\\alpha $ and H$\\beta $ , together with a single power-law from $\\sim 4800$ to $\\sim 7000$ Å for the continuum, we can be quite confident about the contribution of BLR Balmer emission and the underlying continuum; all but J1220+0004 are also associated with weak FeII emission.",
"These sources, therefore, provide the most robust kinematic measurements for the [OIII] gas.",
"On the other hand, the sources at higher $z$ can be associated with higher systematic effects, because of the difficulties in modelling the local continuum underlying the H$\\beta $ and MgII linesFor a proper modelling of the MgII+FeII+FeIII complex with a ‘global’ power-law continuum, a Balmer continuum emission as well as the doublet nature of the MgII should be taken into account (e.g.",
"[60], [99])., and the presence of strong iron emission (at least in the UV part).",
"Overall, the differences in the analysis results between our work, [89] and [21] highlight the difficulties in deriving robust [OIII] kinematic measurements for very luminous type 1 QSOs.",
"Table: [OIII] best-fit parameters.",
"For each component, we report the amplitude (in erg/s/cm 2 ^2/ÅÅ), the centroid and the width of the best-fit Gaussian profile.Below we briefly comment of the best-fit results for each target.",
"$\\bullet $ J0149-1501: H$\\beta $ and H$\\alpha $ are redshifted at the edges of the H- and K-band respectively, where the filter transmission is very low.",
"As a result, the bluest part of the line profiles cannot be modelled properly.",
"The BLR emission is reproduced with one Gaussian component.",
"Narrow emission can be well constrained only for the [OIII] doublet, which also present very broad line wings.",
"A prominent ‘outflow’ component is required to reproduce the profiles of all emission lines (although the strong degeneracies do not allow us to well constrain this component for the Balmer lines).",
"Our reconstructed [OIII] profiles present more pronounced blue and red wings with respect to those in [89] and [21] , although all non-parametric velocities are consistent within 1$\\sigma $ .",
"$\\bullet $ J1421+2241: the BLR is modelled with two Gaussian profiles; the NLR is fitted with a ’systemic’ and an ‘outflow’ component.",
"Both blue and red wings of the [OIII] lines are blended with FeII and H$\\beta $ emission.",
"The [OIII] reconstructed profile is very similar to the ones in [89] and [21].",
"$\\bullet $ J1431+0535: the H$\\beta $ line is severely affected by sky-line residuals; the broad H$\\alpha $ from the BLR is strongly asymmetric.",
"We modelled the BLR emission with two Gaussian components.",
"Narrow emission can be well constrained only for the [OIII] doublet, which also present very broad line wings.",
"Our reconstructed OIII profiles present more pronounced blue and red wings with respect to those in [89] and [21], although the derived non-parametric velocities are consistent within 1$\\sigma $ with theirs.",
"$\\bullet $ J1436+6336: the BLR lines are modelled with two Gaussian profiles; the NLR is fitted with a ’systemic’ and an ‘outflow’ component.",
"Both blue and red wings of the [OIII] lines are mildly blended with FeII and H$\\beta $ .",
"Our reconstructed [OIII] profiles present more pronounced blue and red wings with respect to those in [89] and [21], although all non-parametric velocities are consistent within 2$\\sigma $ .",
"$\\bullet $ J1220+0004: the H$\\alpha $ +[NII] complex is severely affected by telluric features; a significant portion of the profile is therefore masked during the spectral fitting.",
"The BLR line profile is reconstructed on the basis of the observed H$\\beta $ profile, with two BLR Gaussian sets and one Lorentzian profile.",
"The NLR emission is fitted with a ’systemic’ component plus two very broad ‘outflow’ Gaussians.",
"Two FeII templates are also required to reproduce the H$\\beta $ +[OIII]+FeII complex.",
"Our reconstructed [OIII] profiles present more pronounced blue and red wings with respect to those in [89] and [21].",
"Under the assumption that H$\\alpha $ and H$\\beta $ have very similar profiles, the flux excess around $\\sim 5000$ Å cannot be associated with a H$\\beta $ red wing, because of the absence of a similar feature in H$\\alpha $ .",
"On the other hand, FeII templates are not capable to reproduce the broad emission close to the [OIII] doublet.",
"These arguments allow us to prefer a best-fit with very broad [OIII] lines.",
"We note however that strong degeneracies could be still present in our fit, and that systematic errors could be significant for this target.",
"$\\bullet $ J2238-0921: this target shows very complex profiles in the H$\\beta $ +[OIII]+FeII and MgII+FeII+FeIII regions.",
"We fit the MgII and H$\\beta $ BLR emission with two Gaussian components.",
"The iron emission is modelled with two templates.",
"In the optical region, to reproduce the strong peak at $\\sim 5250$ Å we include in the fit the FeIII and FeIV emission lines at 5270 and 5236Å (but a strong degeneracy between these lines and FeII features is present at these wavelengths).",
"No NLR emission associated with [OIII] is detected for this source, consistent with [21] and at odds with [89].",
"This source can be considered as a ‘weak [OIII]’ QSO1, with a CIV line blue-shift of $\\sim $ 5500 km $^{-1}$ and REW$_{\\rm CIV}\\sim $ 10 Å (see [103]).",
"$\\bullet $ J0250-0757: the BLR emission can be well reproduced with a single Gaussian profile; [OIII] is modelled with two Gaussians; kinematic results are consistent with [89] and [21].",
"$\\bullet $ J0844-0503: the [OIII] profiles have similar characteristics to the core-ERQs.",
"We modelled the H$\\beta $ and MgII BLR emission with a single Gaussian; the NRL emission is fitted with two very broad ‘outflow’ Gaussians.",
"Our reconstructed [OIII] profiles present more pronounced blue and red wings with respect to those in [89] and [21].",
"$\\bullet $ J0942-0422: the BLR emission can be well reproduced with a single Gaussian profile; the [OIII] is modelled with three Gaussians.",
"Our reconstructed [OIII] profiles present more pronounced red wings with respect to those in [89] and [21].",
"$\\bullet $ J0304-0008: the BLR emission can be well reproduced with a single Gaussian profile; the [OIII] is modelled with three Gaussians.",
"Our reconstructed [OIII] profiles present more pronounced wings with respect to those in [89], consistent with [21].",
"$\\bullet $ J0843+0750: the BLR emission can be well reproduced with a single Gaussian profile; the [OIII] is modelled with two Gaussians.",
"Our reconstructed [OIII] profiles present more pronounced red wings with respect to those in [89], consistent with [21].",
"$\\bullet $ J1019+0254: the BLR emission can be well reproduced with a single Gaussian profile; the [OIII] is modelled with two Gaussians.",
"Our reconstructed [OIII] profiles present more pronounced wings with respect to those in [89], consistent with [21] $\\bullet $ J0259+0011: the BLR emission can be well reproduced with a single Gaussian profile; the [OIII] is modelled with two Gaussians.",
"Our reconstructed [OIII] profiles present more pronounced wings with respect to those in [89], consistent with [21] $\\bullet $ J1034+0358: the BLR emission can be well reproduced with a single Gaussian profile; the [OIII] is modelled with two Gaussians.",
"Our reconstructed [OIII] profiles present more pronounced wings with respect to those in [89] and [21].",
"Figure: Parametrisation of the Hβ\\beta +[OIII]+FeII (left and middle panels) and Hα\\alpha +[NII]+[SII] (right) regions of z∼z\\sim 2 QSO1 from the sample of .",
"The red lines show the best-fit from multiple ’systemic’, ‘outflow and ‘broad’ Gaussian profiles to the line features, together with a single power-law to the continuum and one (two) template(s) to the FeII features (see text).",
"Each curve with a different colour represents a distinct Gaussian set (or a FeII template) with same kinematic properties, as labeled in the figure.",
"In the central panel we show, for each target, the spectrum around the Hβ\\beta +[OIII] lines after the subtraction of the continuum.",
"The red curve represents the best-fit retrieval model from Monte-Carlo (MC) fits, and the shaded area encompass 68% of the MC sample; the black curve shows the [OIII] best-fit obtained by .Figure: Parametrisation of the MgII+FeII+FeIII (left) and Hβ\\beta +[OIII]+FeII (middle and right panels) ) of z>2.5z>2.5 QSO1 from the sample of .",
"The red lines show the best-fit from multiple ’systemic’, ‘outflow and ‘broad’ Gaussian profiles to the line features, together with a two local power-laws to the continuum and one (two) template(s) to the FeII features (see text).",
"Each curve with a different colour represent a distinct Gaussian set (or a FeII template) with same kinematic properties, as labeled in the figure.",
"In the right panel we show, for each target, the spectrum around the Hβ\\beta +[OIII] lines after the subtraction of the continuum.",
"The red curve represents the best-fit retrieval model from MC fits, and the shaded area encompass 68% of the MC sample; the black curve shows the [OIII] best-fit obtained by .",
"For J2238-0921, we do not detect any [OIII] emission line, consistent with and at odds with : the peaks at ∼4900\\sim 4900, 5000 and 5150Å are associated with FeII emission (precisely, the 6 S^6S iron line group, according to the ordering in ).Figure: Continuation of Fig.",
"6"
]
] | 2001.03379 |
[
[
"Engineering nanoparticles with pure high-order multipole scattering"
],
[
"Abstract The ability to control scattering directionality of nanoparticles is in high demand for many nanophotonic applications.",
"One of the challenges is to design nanoparticles producing pure high-order multipole scattering (e.g., octopole, hexadecapole), whose contribution is usually negligible compared to strong low-order multipole scattering (i.e., dipole or quadrupole).",
"Here we present an intuitive way to design such nanoparticles by introducing a void inside them.",
"We show that both shell and ring nanostructures allow regimes with nearly pure high-order multipole scattering.",
"Experimentally measured scattering diagrams from properly designed silicon rings at near-infrared wavelengths (~ 800 nm) reproduce well scattering patterns of an electric octopole and magnetic hexadecapole.",
"Our findings advance significantly inverse engineering of nanoparticles from given complex scattering characteristics, with possible applications in biosensing, optical metasurfaces, and quantum communications."
],
[
"Scattering of sphere and shell nanoparticles",
"We begin by numerically analyzing the scattering produced by a Si nanoparticle of the simplest morphology – a sphere, where we introduce a void in the center and gradually transform the solid sphere into a shell (Figure REF ).",
"Throughout the rest of the work, the refractive index of Si is taken from measurements of a deposited amorphous Si (see Supporting information, Figure REF ), and the outer diameter of the sphere and shell particles is fixed to 400 nm.",
"Mie theory [39], [40] was used to calculate the total scattering cross-section (SCS) and multipole decomposition (see Supporting information, Figures REF -REF ).",
"We found that the contribution from each multipole (except for the electric dipole) to the total SCS has a resonance-shape dependence on the wavelength, i.e., there is a well-defined peak (see Supporting information, Figure REF ), whose position was determined for every void diameter and plotted on top of the total SCS map (Figure REF d).",
"These resonances are narrow and well separated, so one can get nearly pure high-order multipole scattering (electric and magnetic quadrupole, and magnetic octopole) for a solid Si sphere (Figure REF a).",
"A nearly pure electric octopole (EO) scattering can be found for the Si shell particle with the void diameter of 160 nm (Figure REF b, at $\\lambda \\approx 800$ nm).",
"Finally, by increasing the void diameter further to 280 nm, one can enable a magnetic octopole (MO) resonance well-separated from all other multipole resonances, thus having its dominant contribution in a broad wavelength range (Figure REF c, at $\\lambda \\approx 870$ nm).",
"The dominant multipole and its relative contribution to the total SCS for varied void diameter can be found in Supporting information, Figure REF .",
"In order to analyze the influence of the void size on the response of each multipole, we calculated their contribution to the total SCS at the resonance wavelength of the multipole, and normalized to its contribution at zero void size (Figure REF e).",
"By increasing the void size, the total SCS and contributions from each multipole at its resonance decrease with a blueshift of their peaks (Figure REF d,e).",
"This is expected, since the total volume of Si is reduced.",
"However, the scattering contributions from the low-order multipoles decrease faster compared to the high-order multipoles.",
"Additionally, the contributions from electric multipoles decrease faster compared to magnetic multipoles.",
"This can be explained as following: the multipole moment is proportional to the volume integral of $j_m \\left( kr \\right) \\Pi \\left( \\mathbf {r}^m,\\mathbf {j} \\right) / r^m$ , where $j_m$ is the $m$ -th order spherical Bessel function of the first kind, $k = 2\\pi n/\\lambda $ is the wavenumber in the surrounding environment, $\\mathbf {j}$ is the induced electric current density, $\\mathbf {r}$ is the radius vector of the observation point, $\\Pi \\left( \\mathbf {r}^m,\\mathbf {j} \\right)$ is the combination of vector and scalar product between $\\mathbf {j}$ and $m$ -times $\\mathbf {r}$ , and the integration is done over the volume of the particle [36], [38].",
"In the above expression $m = n - 1$ for electric and $m = n$ for magnetic multipoles of order $n$ ($n=1$ means dipole, $n=2$ - quadrupole, etc.).",
"Thus, the spherical Bessel function acts as a weight inside the integration, therefore the lower the multipole order, the smaller it is influenced by the suppression of the central part.",
"The zero-order spherical Bessel function is the only one having non-zero value at the origin (see Supporting information, Figure REF i), therefore it is only the electric dipole contribution, which is significantly influenced by the introduction of a small void (Figure REF e).",
"Figure: Multipole analysis of disk and ring scattering.",
"(a,d) Total SCS of the disk (solid black) and ring (dashed orange) with (a) 400/800 nm and (d) 800/1000 nm inner/outer diameter, respectively.",
"(b,c) Relative contributions from electric (blue) and magnetic multipoles (red) to the total SCS for the disk (b) and ring (c) with the inner/outer diameter of 400/800 nm.",
"(e,f) Relative contributions from electric (blue) and magnetic multipoles (red) to the total SCS for the disk (e) and ring (f) with the inner/outer diameter of 800/1000 nm.",
"The ring thickness is 80 nm, and the refractive index of surrounding is assumed to be n=1.48n = 1.48.",
"Magenta line indicates the wavelength, at which the scattering is dominated by electric octopole (c) or magnetic hexadecapole (f) contribution.In order to test the generality of our findings, we replaced a sphere with its flat analog – a disk – and transformed it into a ring by introducing a void in its center (Figure REF ).",
"It appeared that for the disk/ring structure multipole spectra does not have simple resonance shapes as for the sphere/shell particle, and an evolution of these spectra with a change of the void size is rather complicated (see Supporting Information, Figures REF -REF ).",
"However, the general trend remains the same: with the increase of the hole the contributions from low-order multipoles decrease faster, compared to the high-order multipoles.",
"Additionally, due to the symmetry and small thickness of the particle, the contributions from electric multipoles of the odd order and magnetic multipoles of the even order are negligible, simplifying the quest to find a regime with nearly pure high-order multipole scattering.",
"We found that at the wavelength around $\\lambda \\approx 800$ nm and ring thickness of 80 nm, the scattering is dominated by the electric octopole (EO) for the ring with the inner/outer diameter of 400/800 nm (Figure REF c), and it is dominated by the magnetic hexadecapole (MH) for the ring with the inner/outer diameter of 800/1000 nm (Figure REF f).",
"Here, the total scattering is calculated by numerical simulations (see Methods), from which multipole contributions are calculated, using the electric field inside the particle (in a same way as in our previous works [41], [42]).",
"Figure: (a,c) Experimental far-field extinction spectra (1-T1-T) and (b,d) simulated scattering spectra of Si rings with varied internal hole diameter D in D_{\\rm in} and external diameter of 800 nm (a,b) and 1000 nm (c,d), respectively.",
"In experiments the substrate is glass (n≈1.45n \\approx 1.45), and the superstrate is immersion oil (n=1.518n = 1.518).",
"The rings are illuminated from the glass side (illustrated with red arrow).",
"In simulations the ring is embedded in even surrounding with the refractive index of n=1.48n = 1.48.",
"Insets show SEM images of the structures, with labeled inner void diameter (panel size: 1500 nm).To confirm such selective scattering, we fabricated a series of isolated 80-nm-thick Si rings with varied inner (void) diameter and fixed outer diameter of 800 and 1000 nm.",
"The fabrication was done by deposition of amorphous Si on a glass substrate, followed by etching through the mask (see Methods).",
"First we measured the far-field transmission $T$ for each ring (see Methods), plot it as extinction $1-T$ , and compared with simulated total scattering (Figure REF ).",
"One can already note from Figure REF that the region with the dominant scattering by a single high-order multipole is not reflected in the total scattering spectrum.",
"However, there are still some distinct features in spectra (dips and peaks), therefore by comparing them in measurements and simulations (Figure REF ), one can indirectly verify the correspondence between simulations and experiments.",
"Figure: Scattering diagrams of Si rings.",
"(a,b) Experimentally measured and simulated scattering diagrams of Si rings with (a) 400/800 nm and (b) 800/1000 nm inner/outer diameter, respectively, compared with radiation diagrams of isolated (a) electric octopole (EO) and (b) magnetic hexadecapole (MH).",
"The incident polarization is illustrated with a red arrow in (a).",
"Magenta line in Fourier plane images shows the experimental limit for the collection angle and corresponds to the NA of the immersion-oil objective (1.25).Then we proceeded to direct measurements of the scattering diagram of Si rings (see Methods).",
"The experimentally measured scattering diagrams of rings with 400/800 nm and 800/1000 nm inner/outer diameter are shown in Figure REF , compared with simulated total scattering diagram and analytically calculated radiation pattern of an isolated multipole.",
"Regarding the latter, one can directly calculate contributions of the selected multipoles to the scattering diagram [38], when their multipole moment is known (traceless and symmetrical tensors of rank 3 for octopole and rank 4 for hexadecapole).",
"However, to simplify the analysis, we used direct expressions for the scattering diagrams of these multipoles for a perfect sphere in an even dielectric environment [43], [40]: ${\\sigma _{{\\rm {EO}}}}\\left( {\\theta ,\\varphi } \\right) \\propto {\\cos ^2}\\varphi {\\left[ {\\left( {5{{\\cos }^2}\\theta - 1} \\right)} \\right]^2} \\\\+ {\\sin ^2}\\varphi {\\left[ {\\cos \\theta \\left( {15{{\\cos }^2}\\theta - 11} \\right)} \\right]^2},$ ${\\sigma _{{\\rm {MH}}}}\\left( {\\theta ,\\varphi } \\right) \\propto {\\sin ^2}\\varphi {\\left[ {\\cos \\theta \\left( {7{{\\cos }^2}\\theta - 3} \\right)} \\right]^2} \\\\+ {\\cos ^2}\\varphi {\\left[ {\\left( {28{{\\cos }^4}\\theta - 27{{\\cos }^2}\\theta + 3} \\right)} \\right]^2},$ where $\\theta $ and $\\varphi $ are the polar and azimuthal angles, respectively, and $E$ ($M$ ) stands for electric (magnetic) multipole (see Supporting information, Supplementary Note 1).",
"Though dependence of multipole contributions on the wavelength and void size for disk/ring structure is quite different from the one of the sphere/shell particle, we expect similar scattering diagrams for isolated multipoles of these particles due to the same mirror symmetry of the sphere and the ring.",
"The above assumption and the validity of simulations in general are verified by a good agreement between experiment, simulations, and analytical calculations (Figure REF ).",
"Moreover, the scattering diagrams have a well-pronounced feature – scattering side-lobes along (Figure REF a) or across the orientation of the incident beam polarization (Figure REF b), which is a clear indication of the electric or magnetic multipole scattering.",
"In case of the ring with the inner/outer diameter of 400/800 nm (Figure REF a), scattering side-lobes are at around $\\rm NA \\approx 1.25$ in the Fourier plane, corresponding to the angle of $\\rm arcsin(1.25/1.48) \\approx \\pi /3$ , clearly confirming its electric octopole origin (see Supplementary Note 1).",
"As for the other ring with the inner/outer diameter of 800/1000 nm (Figure REF b), scattering side-lobes reach the maximum at around $\\rm NA \\approx 1$ in the Fourier plane, corresponding to the angle of $\\rm arcsin(1/1.48) \\approx \\pi /4$ , which is in accordance with the magnetic hexadecapole.",
"Interestingly, the number of principal scattering lobes (6 for octopole and 8 for hexadecapole) cannot be anticipated from the multipole name, but it rather follows $2n$ , where $n$ is the multipole order.",
"This is not a contradiction, because the multipole name reflects the minimum number of oscillating point charges required to produce the given multipole moment, while it doesn`t state the number of scattering lobes.",
"The deviation between scattering diagrams of each ring and the radiation pattern of the corresponding isolated multipole is due to the interference with other multipoles` contributions (EQ, MQ, MO, and MH for the ring with the inner/outer diameter of 400/800 nm, and ED, MQ, and EO for the ring with the inner/outer diameter of 800/1000 nm, as follows from Figure REF at $\\lambda = 800$ nm).",
"This interference can lead to a strong suppression of some scattering lobes, but their angular positions are only slightly affected, which is an indication of the dominating single high-order multipole contribution (see Supplementary Information, Figure S9, for more discussion).",
"The resulted asymmetry in forward/backward scattering is the so-called Mie effect [39], [44], which is a transition between symmetric Rayleigh scattering and diffraction.",
"In summary, we demonstrated that the relative contribution of high-order multipoles can be boosted by introducing the void inside a high-refractive index nanoparticle.",
"Using such a method allows finding a composition of the nanoparticle, whose scattering will be dominated by a single high-order multipole.",
"We proved this method on a disk/ring shape, using Si as a high-refractive index material with a thickness of 80 nm.",
"First we numerically found two structures producing nearly pure electric octopole and magnetic hexadecapole scattering.",
"Then we fabricated such structures, directly measured their scattering diagrams, and finally verified the dominant high-order multipole contribution.",
"Additionally to providing an intuitive understanding of the relation between shape and multipoles, our results can directly benefit applications in metasurfaces, sensing, quantum communications, and topological photonics.",
"Fabrication.",
"The rings were fabricated by etching amorphous silicon, deposited on a silica wafer.",
"First the fresh silica wafer was cleaned using a standard RCA clean, without the HF steps.",
"Then 80 nm of Si was deposited using LPCVD.",
"Standard reflectometry (FilmTek 4000TM) was used to measure Si thickness and refractive index (Supporting Information, Figure REF ).",
"After the Si deposition, AR-P 6200 resist from Allresist was spun at 200 nm, followed by thermal evaporation of 20 nm Al to be used as discharge path during the exposure.",
"The structures were defined by electron-beam exposure.",
"The next step involves Al layer removal in diluted phosphoric acid and development of the resist.",
"The patterned resist is then used as a mask for etching the Si layer using a Bosch process and then removed using low power oxygen plasma.",
"Numerical simulations.",
"Scattering spectra, diagrams, and the electric field inside the structures (for multipole decomposition) were calculated using a 3D simulation with the finite-element method (FEM) implemented in a commercial software (COMSOL Multiphysics).",
"A simulation sphere with a diameter of 2.4 $\\mu $ m was used with the perfectly-matched layer at the outer boundary and a tetrahedral mesh with a mesh size of $\\sim $ 30 nm inside the silicon and $\\sim $ 80 nm for the rest.",
"The silicon ring was excited by a normal-incident CW plane wave, whose wavelength was swept to calculate spectra.",
"For all calculations the permittivity value of Si was taken from measurements (Supporting Information, Figure REF ), the refractive index of surrounding environment was set to 1.48.",
"Measurements of the scattering diagram.",
"Schematic diagram of the experimental setup for measuring scattering diagrams is shown in Supporting Information, Figure REF .",
"The sample was illuminated using a linearly polarized Ti:Sapphire laser, tuned at a wavelength of $\\sim $ 800 nm.",
"The laser beam was weakly focused onto the sample using a 10$\\times $ objective of numerical aperture NA = 0.20.",
"The full-width-at-half-maximum (FHWM) of the focused beam spot was $\\sim $ 5 $\\mu $ m. The scattered light was collected using a 63$\\times $ oil-immersion collection objective, with a NA = 1.25.",
"The structures on the sample were positioned facing the oil-immersion objective, embedded in the index-matching oil ($n = 1.518$ ).",
"An imaging system, which consists of two lenses and two spatial filters, was used to image the back focal plane (BFP) of the collection objective with a charge-coupled device (CCD) camera.",
"The BFP and its image are also referred to here as the Fourier plane, because it shows angular distribution of the scattering (i.e., scattering diagram).",
"The two spatial filters have the following important functionalities.",
"The first filter is a micrometric metallic ball (diameter $\\sim $ 300 $\\mu $ m), glued on a glass substrate, and is used to stop the directly-transmitted light from reaching the CCD camera to avoid saturation.",
"It is a Fourier-plane filter, and ideally it should be placed at the BFP inside the collection objective.",
"Nevertheless, since the objective collimates the scattered light, the filter can be placed at the rear aperture of the objective and still produce the same filtering effect.",
"The second filter is a pinhole, positioned at the image plane (focal point of the first lens), and is used to stop all the unwanted scattering from the surroundings of the nanostructure (i.e., from impurities in the glass substrate and oil).",
"The images are captured with the CCD camera, located at the focal distance of the second lens, which matches with the Fourier plane.",
"Far-field spectroscopy.",
"Transmission spectroscopy was performed on a standard inverted optical microscope (Zeiss Axio Observer), equipped with a halogen light source, modified detection path, and fiber-coupled spectrometer (Ocean Optics QE Pro).",
"The light was collected using a $\\times $ 100 immersion-oil objective (Zeiss $\\alpha $ Plan-FLUAR, NA = 1.45) and the same index-matching oil ($n = 1.518$ ).",
"Schematically the setup is similar to the one for measuring scattering diagrams (Supporting Information, Figure REF ), allowing filtering both in Fourier and direct image planes.",
"We used an iris diaphragm for Fourier plane to limit detected NA to $\\sim $ 0.3 (in order to replicate measurements of transmitted plane wave), and another iris diaphragm was set as a filter in the image plane, corresponding to the area with a diameter of $\\sim $ 3 $\\mu $ m in a sample plane (that is, when image plane was recorded by a camera, the opening was $\\sim $ 3 times larger than the ring with the outer diameter of 1 $\\mu $ m).",
"The measured transmission spectra were normalized on the transmission spectrum through the glass substrate without any structure.",
"Acknowledgments The authors acknowledge financial support from the European Research Council (the PLAQNAP project, Grant No.",
"341054) and the University of Southern Denmark (SDU2020 funding), from scholarship 299967.",
"N.A.M.",
"is a VILLUM Investigator supported by Villum Fonden (Grant No.",
"16498).",
"C.E.G.-O and V.C.",
"acknowledge the technical assistance of Fabiola Armenta with the experimental setup.",
"V.C.",
"and C.E.G.-O.",
"acknowledge funding from CONACYT Basic Scientific Research Grants No.",
"250719 and No.",
"252621.",
"RM and AVL acknowledge the financial support from Villum Fonden ”DarkSILD project” (Grant No.",
"11116) as well as the support of the National Centre for Nano Fabrication and Characterization (DTU Nanolab) for fabrication of the structures.",
"A.B.E.",
"and B.N.C.",
"acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122, Project No.",
"390833453) and DFG Project CH179/34-1.",
"Numerical simulation was partially supported by the Russian Science Foundation (Grant No.",
"18-19-00684).",
"Supplementary Note 1: analytical expressions for multipole scattering diagrams The angular dependence of the scattered field in the far-field zone for a perfect sphere in an even dielectric environment is following [43]: $\\left\\lbrace \\begin{aligned}& {{E}_{\\theta }}\\left( \\theta ,\\varphi \\right)\\approx {{E}_{0}}\\frac{{{e}^{ikr}}}{-ikr}\\cos \\varphi \\sum \\limits _{n}{\\frac{2n+1}{n\\left( n+1 \\right)}}\\left( {{a}_{n}}\\frac{dP_{n}^{1}}{d\\theta }+{{b}_{n}}\\frac{P_{n}^{1}}{\\sin \\theta } \\right), \\\\& {{E}_{\\varphi }}\\left( \\theta ,\\varphi \\right)\\approx -{{E}_{0}}\\frac{{{e}^{ikr}}}{-ikr}\\sin \\varphi \\sum \\limits _{n}{\\frac{2n+1}{n\\left( n+1 \\right)}}\\left( {{a}_{n}}\\frac{P_{n}^{1}}{\\sin \\theta }+{{b}_{n}}\\frac{dP_{n}^{1}}{d\\theta } \\right), \\\\\\end{aligned} \\right.$ where $\\theta $ and $\\phi $ are the polar and azimuthal scattering angles, respectively, $E_0$ is the strength of the incident field, $k$ is the wavevector in the surrounding environment, and $n$ is the order of the multipole (e.g., $n$ = 1 defines dipole, $n$ = 2 defines quadrupole, etc.).",
"$a_n$ ($b_n$ ) are the complex Mie coefficients, corresponding to the electric (magnetic) multipoles.",
"$P_{n}^{1}$ represents the set of associated Legendre polynomials of order 1, which are convenient to rewrite using following angle-dependent functions [40]: ${{\\pi }_{n}}\\left( \\theta \\right)\\equiv \\frac{P_{n}^{1}}{\\sin \\theta },\\text{\\ \\ \\ \\ \\ \\ \\ }{{\\tau }_{n}}\\left( \\theta \\right)\\equiv \\frac{dP_{n}^{1}}{d\\theta }$ They can be computed iteratively using following relations: $\\left\\lbrace \\begin{aligned}& {{\\pi }_{n}}=\\frac{2n-1}{n-1}\\cos \\theta {{\\pi }_{n-1}}-\\frac{n}{n-1}{{\\pi }_{n-2}}, \\\\& {{\\tau }_{n}}=n\\cos \\theta {{\\pi }_{n}}-\\left( n+1 \\right){{\\pi }_{n-1}}, \\\\& {{\\pi }_{0}}=0,\\text{ }{{\\pi }_{1}}=1.",
"\\\\\\end{aligned} \\right.$ The scattering cross-section into a unit solid angle is proportional to the square of the $E$ -field: $\\sigma \\left( \\theta ,\\varphi \\right)\\propto {{\\left| {{E}_{\\theta }}\\left( \\theta ,\\varphi \\right) \\right|}^{2}}+{{\\left| {{E}_{\\varphi }}\\left( \\theta ,\\varphi \\right) \\right|}^{2}},$ therefore the scattering diagram from the given multipole of order n can be written as following: $\\left\\lbrace \\begin{aligned}& {{\\sigma }_{\\text{En}}}\\left( \\theta ,\\varphi \\right)\\propto {{\\cos }^{2}}\\varphi {{\\tau }_{n}}^{2}+{{\\sin }^{2}}\\varphi {{\\pi }_{n}}^{2}, \\\\& {{\\sigma }_{\\text{Mn}}}\\left( \\theta ,\\varphi \\right)\\propto {{\\sin }^{2}}\\varphi {{\\tau }_{n}}^{2}+{{\\cos }^{2}}\\varphi {{\\pi }_{n}}^{2}={{\\sigma }_{\\text{En}}}\\left( \\theta ,\\tfrac{1}{2}\\pi -\\varphi \\right).",
"\\\\\\end{aligned} \\right.$ One can see that these angle-dependent functions define the dependence of the scattering cross-section on the polar angle $\\theta $ within cross-sections at $\\phi = 0$ at $\\phi = \\pi /2$ .",
"The table below collects explicit expressions for angle-dependent functions, 3D illustration of the scattering diagram, and its corresponding sections at $\\phi = 0$ at $\\phi = \\pi /2$ for the first four orders of multipoles: Figure: NO_CAPTION Table S1: scattering diagrams of the first four multipole orders and their correspondent angle-dependent functions.",
"One can see that the principal scattering lobes for any multipole order $n > 1$ are defined by the angle-dependent function $\\tau _{n}$ .",
"Their position $\\theta _{\\rm max}$ can be easily found by taking the derivative of $\\tau _{n}$ : ${{\\left.",
"\\frac{d{{\\tau }_{n}}}{d\\theta } \\right|}_{\\theta =\\theta \\max }}=0$ Angles of non-trivial scattering lobes for octopole ($n$ = 3) and hexadecapole ($n$ = 4) are thus: $\\left\\lbrace \\begin{aligned}& {{\\theta }_{\\text{O,}\\max }}=\\arccos \\left( \\frac{1}{2}\\sqrt{\\frac{44}{45}} \\right)\\approx \\frac{\\pi }{3}, \\\\& {{\\theta }_{\\text{H,}\\max }}=\\arccos \\left( \\frac{1}{\\sqrt{2}}\\sqrt{\\frac{27}{28}} \\right)\\approx \\frac{\\pi }{4}.",
"\\\\\\end{aligned} \\right.$ Regarding the number of all scattering lobes, one can actually find two additional tiny scattering lobes for electric (magnetic) octopole and four for electric (magnetic) hexadecapole at $\\phi = \\pi /2$ ($\\phi = 0$ ) section, defined by the other angle-dependent function $\\pi _n$ .",
"Thus the number of scattering lobes would depend on their definition, and we believe it has no fundamental direct correlation with the multipole order.",
"The total scattering cross-section can be calculated as following [40]: ${{\\sigma }_{\\text{total}}}=\\frac{2\\pi }{{{k}^{2}}}\\sum \\limits _{n}{\\left( 2n+1 \\right)}\\left( {{\\left| {{a}_{n}} \\right|}^{2}}+{{\\left| {{b}_{n}} \\right|}^{2}} \\right),$ from which one can determine individual contributions from each multipole.",
"Assume that only magnetic dipole (MD) and electric octopole (EO) are present.",
"The asymmetry between forward ($\\theta = 0$ ) and backward scattering ($\\theta = \\pi $ ) can be calculated using Eqs.",
"REF and REF : $\\frac{{{\\sigma }_{\\text{forward}}}}{{{\\sigma }_{\\text{backward}}}}=\\frac{\\sigma \\left( \\theta =0,\\varphi \\right)}{\\sigma \\left( \\theta =\\pi ,\\varphi \\right)}={{\\left| \\frac{\\tfrac{7}{2}{{a}_{\\text{EO}}}+\\tfrac{3}{2}{{b}_{\\text{MD}}}}{\\tfrac{7}{2}{{a}_{\\text{EO}}}-\\tfrac{3}{2}{{b}_{\\text{MD}}}} \\right|}^{2}}={{\\left| \\frac{1+\\tfrac{3}{7}{{b}_{\\text{MD}}}/{{a}_{\\text{EO}}}}{1-\\tfrac{3}{7}{{b}_{\\text{MD}}}/{{a}_{\\text{EO}}}} \\right|}^{2}}$ From Eq.",
"REF it follows that $\\frac{\\left| {{b}_{\\text{MD}}} \\right|}{\\left| {{a}_{\\text{EO}}} \\right|}=\\sqrt{\\frac{7}{3}\\frac{{{\\sigma }_{\\text{MD}}}}{{{\\sigma }_{\\text{EO}}}}},$ If backward scattering cannot be completely canceled, then the maximum asymmetry in forward/backward scattering is: $\\max \\left\\lbrace \\frac{{{\\sigma }_{\\text{forward}}}}{{{\\sigma }_{\\text{backward}}}} \\right\\rbrace ={{\\left| \\frac{1+\\sqrt{\\tfrac{3}{7}{{{\\sigma }_{\\text{MD}}}}/{{{\\sigma }_{\\text{EO}}}}\\;}}{1-\\sqrt{\\tfrac{3}{7}{{{\\sigma }_{\\text{MD}}}}/{{{\\sigma }_{\\text{EO}}}}\\;}} \\right|}^{2}}$"
]
] | 2001.03489 |
[
[
"Arm-interarm gas abundance variations explored with MUSE: the role of\n spiral structure in the chemical enrichment of galaxies"
],
[
"Abstract Spiral arms are the most characteristic features of disc galaxies, easily distinguishable due to their association with ongoing star formation.",
"However, the role of spiral structure in the chemical evolution of galaxies is unclear.",
"Here we explore gas-phase abundance variations between arm and interarm regions for a sample of 45 spiral galaxies using high spatial resolution VLT/MUSE Integral Field Spectroscopy data.",
"We report the presence of more metal-rich HII regions in the spiral arms with respect to the corresponding interarm regions for a large subsample of galaxies ($45-65\\%$ depending on the adopted calibrator for the abundance derivation).",
"A small percentage of the sample is observed to display the opposite trend, that is, more metal-poor HII regions in the spiral arms compared to that of the interarms ($5-20\\%$ depending on the calibrator).",
"We investigate the dependence of the variations with three galaxy properties: the stellar mass, the presence of bars, and the flocculent/grand design appearance of spiral arms.",
"In all cases, we observe that the arm-interarm abundance differences are larger (positive) in more massive and grand-design galaxies.",
"This is confirmed by an analogous spaxel-wise analysis, which also shows a noticeable effect of the presence of galactic bars, with barred systems presenting larger (positive) arm-interarm abundance variations than unbarred systems.",
"The comparison of our results with new predictions from theoretical models exploring the nature of the spirals would highly impact on our knowledge on how these structures form and affect their host galaxies."
],
[
"Introduction",
"The study of the gas-phase chemical composition of spiral galaxies has proven to be a powerful tool to improve our knowledge on the evolution of these complex systems.",
"In particular, the analysis of Hii regions is of great importance, as it is through the birth and death of stars that galaxies chemically evolve.",
"The most studied trend in the oxygen abundance distribution of Hii regions in spiral galaxies is the well-known negative radial gradient , , , , , , .",
"Observational works on this topic have enabled constraints on chemical evolution models aimed at explaining galaxy formation and evolution , , , , , .",
"Nevertheless, by restricting the information to one dimension they were unable to properly describe the effect of two-dimensional (2D) structures, such as spiral arms or bars, on the chemical evolution of their host galaxies.",
"From a theoretical point of view, numerical simulations have recently revealed how the presence of spiral arms produces azimuthal differences in the chemical distribution , , .",
"One proposed mechanism responsible for these differences is the redistribution of metals induced by streaming motions of gas and stars along the spiral arms.",
"However, it has also been concluded that the passage of spiral density waves across galactic discs alone can cause azimuthal metallicity variations , , .",
"Observationally, one of the most straightforward methods to study the effect of the spiral structure on the chemical distribution of galaxies is via the analysis of arm-interarm abundance variations.",
"However, the first attempts to carry out such study found no significant differences in the gas metallicity between these two distinct regions , .",
"More recently, analysed a sample of 63 galaxies from the CALIFA survey and reported very subtle differences between the arm and interarm abundance gradients for two specific subgroups of galaxies: barred and flocculentFlocculent galaxies are defined as those displaying short, asymmetric, and patchy arms that fade over the gaseous disc .",
"See Sec.",
"REF for more information on this morphological classification of spiral arms.",
"systems.",
"also analysed four CALIFA spiral galaxies, showing a very small enhancement ($0.01-0.06$ dex) of the oxygen abundance in the spiral arms compared to the interarm region.",
"A drawback of those previous studies - that could account for the absence of strong evidence of arm-interarm abundance variations - is the lack of the spatial resolution of the data that might be needed to detect such differences.",
"In this regard, the advent of Integral Field Spectroscopy (IFS) instruments combining large field-of-views (FoV) and high spatial resolution has meant a revolution in the progress of 2D abundance distribution studies.",
"Since then, a growing number of works have been able to detect such elusive variations in individual galaxies: NGC 6754 (, see also ), NGC 1365 , HCG 91c , and NGC 2997 .",
"The magnitude of the reported variations differ from study to study (from 0.06 in NGC 2997 to 0.4 dex in NGC 1365), but in all cases more metal-rich gas has been observed associated with the spiral structure.",
"In spite of the improvement in the spatial resolution of the available data, there are still studies reporting an absence of arm-interarm oxygen abundance variations.",
"analysed MUSE data of NGC 628 and detected 391 Hii regions covering both arm and interarm regions, deriving a metallicity distribution independent of the environment.",
"However, the data did not map the total extent of the galaxy, but were restricted to a region of 12 kpc$^2$ , barely 10% of the total areaAssuming $R_{25}=5.25$ arcmin ..",
"It is therefore not clear that strong conclusions can be drawn when such a small percentage of the galaxy was analysed.",
"A first attempt to study azimuthal oxygen abundance variations in an homogeneous way for a number of galaxies has been recently performed by .",
"Analysing a sample of eight spiral galaxies from the PHANGS-MUSE project, the authors qualitatively find subtle azimuthal variations tenuously associated with the spiral pattern in half of the sample.",
"This spatial overlapping with the spiral structure was more clearly observed in some galaxies (such as NGC 1087 or NGC 1672, where enhanced abundances were detected along the eastern arm) than in others.",
"In general, they find stronger correlations with local physical conditions of the ISM (especially with the ionisation parameter) than with environmental parameters (arm/interarm masks or stellar mass surface density offset).",
"The authors conclude that the spiral arms can play an important role in mixing the ISM, in combination with a complex interplay between galaxy dynamics and enrichment patterns.",
"In order to shed light on the prevalence of the arm-interarm oxygen abundance variations, in this study we analyse high spatial resolution MUSE data for a large sample of 45 spiral galaxies.",
"To perform such a study, we compare the gas abundances of the spiral arms with those of the underlying disc (interarm region).",
"In order to assess which factors may enhance or diminish the emergence of arm-interarm differences, we examine the dependence of the variations with three galaxy properties: the stellar mass, the presence of bars, and the flocculent/grand design appearance of spiral arms.",
"Bars have been proposed as a key mechanism in the dynamical evolution of disc galaxies, for instance, by inducing gas flows , .",
"We already mentioned that the existence of streaming motions along the spiral arms could cause the rise of azimuthal abundance variations as proved by numerical simulations, making the presence of a bar a relevant factor worth investigating.",
"Based on our results, we discuss how the observed scenario could be explained according to the different theories on the nature of spiral structures.",
"The paper is organised as follows: Sec.",
"provides a description of the sample as well as the data used in this study.",
"In Sec.",
"we explain the methodology employed to detect the Hii regions, separate arm and interarm regions, and derive the corresponding oxygen abundances.",
"The results of the analysis are presented in Sec.",
", including the study of the arm-interarm abundance differences, their dependence on different properties of the galaxies (Sec.",
"REF ), and the comparison with a spaxel-wise analysis (Sec.",
"REF ).",
"The discussion and the main conclusions are given in Sec. .",
"Finally, a brief summary of the study is provided in Sec.",
"."
],
[
"Observations and galaxy sample",
"The analysed data belong to AMUSING++ (López-Cobá et al.",
"submitted), a compilation of nearby galaxies observed with the MUSE instrument , .",
"This collection comprises 534 galaxies from different MUSE projects in combination with archival data covering the redshift interval $0.0002 < z < 0.1$ .",
"The core of the compilation comes from the AMUSING survey , an ongoing project aimed at studying the environment of supernovae (SNe).",
"In this work we make use of the 450 AMUSING galaxies observed up to Period 101 (September 2018) for which all data reduction, quality assessment, and sample characterisation has been performed (currently the project comprises about 600 galaxies).",
"Observations for 362 of them constitute the long term observing campaign carried out as part of the AMUSING project, the remaining coming from the European Southern Observatory (ESO) archive with the only criterion of having hosted a SN.",
"AMUSING++ complements this AMUSING sample by adding selected galaxies observed by MUSE not hosting SNe (see López-Cobá et al.",
"submitted, for more details on the final AMUSING++ sample).",
"Despite being the analysed dataset the result of a compilation, and therefore not comprising a homogeneously selected and well-defined sample, in López-Cobá et al.",
"(submitted) the authors show that AMUSING++ includes galaxies which properties resemble those of a diameter-selected sample.",
"Not clear biases towards any particular morphological type, color, or magnitude are observed.",
"The MUSE instrument , is mounted on the Unit 4 telescope (UT4) at the Very Large Telescope (VLT) in Cerro Paranal Observatory.",
"In its Wide Field Mode, this integral-field spectrograph presents a FoV of approximately $1^{\\prime }\\times 1^{\\prime }$ and a pixel size of $0.2^{\\prime \\prime }$ , which limits the spatial resolution of the data to the atmospheric seeing during the observations.",
"Finally, MUSE covers a wavelength range from $\\rm 4750$ to $\\rm 9300$ Å, with a spectral sampling of $\\rm 1.25$ Å and a spectral resolution $\\lambda /\\Delta \\lambda \\sim 1800-3600$ (from the blue edge to the red end of the spectrum).",
"A detailed explanation of the AMUSING data reduction is provided in .",
"Briefly, we make use of version 1.2.1 of the MUSE reduction pipeline and the Reflex environment .",
"The sky subtraction is performed using the Zurich Atmosphere Purge package , employing offset pointings to blank sky positions, or blank sky regions within the science frames.",
"For the galaxies selected from the archive, the reduced cubes provided by ESO were used.",
"The effects of Galactic extinction were also corrected based on the reddening estimates from .",
"For this study, we select from the AMUSING mother sample a subset of 45 spiral galaxies.",
"To reach that number we first remove galaxies with low image quality (visually presenting low signal in the source and/or with seeing values above 2).",
"In addition, observations that do not cover the centre of the galactic discs (hampering the derivation of required parameters such as the inclination or position angle of the galaxy) are also discarded from the sample.",
"From a physical point of view, we select isolated spiral galaxies with morphological types between Sa and Sm (including barred galaxies) according to the HyperLeda extragalactic databasehttp://leda.univ-lyon1.fr .",
"For a proper separation of the arm and interarm regions, we restrict the sample to intermediate to low inclined galaxies ($i < 65 $ ).",
"Moreover, to perform a suitable characterisation of the abundance distribution of both areas, we discard those galaxies from the AMUSING sample that present a deprojected disc radius smaller than $\\sim 20 $ and/or a radial coverage of less than one effective disc radius.",
"In order to guarantee a feasible detection of the possible abundance variations, we only preserve galaxies for which a physical resolution higher than 1 kpc is achieved (determined from the seeing value of the observations).",
"In addition, we also reject galaxies whose spiral arms can not be properly traced because of their poor definition (see Sec.",
"REF for details).",
"These cases correspond to the most flocculent galaxies of the sample, whose fragmented and blurred arms are not easy to track, and galaxies with tightly wrapped ringlike arms, that are also very difficult to disentangle (see Sec.",
"REF ).",
"Finally, galaxies for which the Hii regions visually provide a poor spatial coverage of the disc (patchy and discontinuous) are also discarded.",
"The required selection criteria ensure a proper characterisation of both the arm and interarm oxygen abundance distributions in our sample.",
"However, due to these criteria the sample may not cover the full range of properties of the galaxy population in the Local Universe.",
"In this sense, we must bear in mind that this study represents the first attempt to extend the analysis of arm-interarm abundance variations to a statistically significant sample.",
"Having a complete sample was never the intention of the present work.",
"Upcoming surveys massively sampling the Local Universe with higher spatial resolutions will enable our results to be placed within a broader framework.",
"For the final sample of 45 galaxies, we present in Table REF a summary of the most important characteristics of the galaxies.",
"Information on the redshift, physical resolution (limited by the seeing corresponding to the observations of each object), morphological type, arm classification, and stellar mass is provided in the table.",
"Table: Fundamental properties of the galaxy sample."
],
[
"Measurement of emission lines with ",
"We make use of the fitting package FIT3D and Pipe3d , , an IFS analysis pipeline developed to characterise the properties of both the stellar population and the ionised gas.",
"Here we provide a brief outline of the procedure of fitting and subtracting the underlying stellar population and measuring the emission lines using FIT3D.",
"The entire scheme together with other algorithms of Pipe3d are extensively described in , .",
"Briefly, FIT3D fits each spectrum by a linear combination of simple stellar population templates after correcting for the appropriate systemic velocity and velocity dispersion, and also for the stellar dust extinction .",
"After the stellar component is subtracted, FIT3D measures the emission line fluxes by performing a multi-component fitting using a weighted moment analysis, as described in .",
"The measured line fluxes covered by MUSE include H$\\alpha $ , H$\\beta $ , $[{O}{iii} ]\\lambda 4959$ , $[{O}{iii} ]\\lambda 5007$ , $[{N}{ii} ]\\lambda 6548$ , $[{N}{ii} ]\\lambda 6584$ , $[{S}{ii} ]\\lambda 6717$ , and $[{S}{ii} ]\\lambda 6731$ , among others.",
"FIT3D provides the intensity, equivalent width (EW), systemic velocity, and velocity dispersion for each emission line of each spectrum.",
"For further analysis, emission line intensities are corrected for dust attenuation making use of the extinction law from , with $R_V = 3.1$ , and the H$\\alpha $ /H$\\beta $ Balmer decrement, considering the theoretical value for the unobscured H$\\alpha $ /H$\\beta $ ratio of 2.86, which assumes a case B recombination ($T_e = 10\\,000$ K, $n_e=100$ cm$^{-3}$ , )."
],
[
"Selection of Hii regions",
"We detect clumpy regions of ionised gas in each galaxy, candidates for Hii regions, based on its H$\\alpha $ intensity map using HIIexplorer .",
"A first description of the algorithm is provided there, but here we use an updated version modified for high resolution data like MUSE, that was described in .",
"Regions presenting $S/N < 3$ in H$\\alpha $ emission are rejected.",
"On average we detect $\\sim 200$ Hii region candidates per galaxy, thus a total of 8896 regions in the full sample.",
"Figure: BPT diagram of the Hii regions in NGC 3905.",
"The solid and dashed lines represent the and demarcation curves.",
"Star-forming regions are considered to be below the curve.From these candidates, we select true star-forming Hii regions using the diagnostic BPT diagram proposed by , based on the $[{N}{ii} ]\\lambda 6584$ /H$\\alpha $ and $[{O}{iii} ]\\lambda 5007$ /H$\\beta $ line ratios.",
"For this diagram we adopt the curve that separates regions ionised by OB stars (below the curve) from those associated with other sources of ionisation such as active galactic nuclei (AGNs) or shocks (above the curve).",
"An additional criterion on H$\\alpha $ EW (greater than 6 Å) is also assumed to ensure the exclusion of low-ionisation sources , and the presence of a significant percentage (at least $20\\%$ ) of young stars contributing to the emission of the Hii regions .",
"This procedure has proven to perform a good classification to describe real Hii regions, as shown by Espinosa et al.",
"(in prep.).",
"Figure REF shows the BPT diagram for one example galaxy in the sample, NGC 3905.",
"We only represent the selected regions that are finally associated with star formation, distinguishing between those that belong to the spiral arms (orange) and the interarm area (green, see Sec.",
"REF for details in the separation of the arm and interarm areas).",
"It is evident that both distributions spread differently across the BPT diagram, with the interarm regions presenting higher $\\rm [{O}{iii} ]/H\\beta $ line ratios than the arm regions (on average interarm values are 0.3 dex higher than arm values)."
],
[
"Separation of arm and interarm regions",
"The procedure followed here to separate the arm and interarm regions was used in for CALIFA data.",
"The outline of the spiral arms was based on previous and successful tracing of other morphological features such as dust lanes in galactic bars .",
"Figure: Left panel: Outline of the spiral arms on the gg-band image for two example galaxies.",
"Right panel: Colour map of the oxygen abundance residuals (derived by subtracting the radial profile) of the detected Hii regions.",
"Black solid lines enclose the area defined as arm region.",
"Dashed ellipses on both panels indicate the minimum and maximum radial limits (given by the minimum and maximum galactocentric distances of the arm region) to be considered for the interarm region.We first depict the spiral arms by visually tracing them on $g$ -band images of the galaxies reconstructed from the datacubes.",
"The marked points are then interpolated using a cubic spline (a numeric function that is piecewise-defined by a third-degree polynomial).",
"When discontinuous arms are considered, we individually outline all fragmented parts that are easily distinguishable.",
"The left panels of Fig.",
"REF show the spline fit of the detected spiral arms (dark grey solid lines) superimposed on the $g$ -band images of two example galaxies, NGC 3905 (top) and PGC 128348 (bottom).",
"Once the spiral structure is traced, we consider as arm Hii regions those separated from the closest point of the arms less than a certain distance that is visually chosen from the $g$ -band image.",
"The arm widths therefore range from 1 to 5, that correspond to $0.2-1.6$ kpc , .",
"The remaining Hii regions are associated with the interarm regions.",
"The right panels of Fig.",
"REF show the division of arm and interarm Hii regions, being the ones belonging to the spiral arms those contained inside the line-delimited areas.",
"The interarm Hii regions are those located outside the defined areas.",
"This method of tracing spiral arms restricts us to the most strong and prominent arms.",
"Weaker spiral arms might be missed, being the Hii regions belonging to them wrongly associated to the interarm regions.",
"This effect may dilute possible differences, and in this case the reported variations, if existing, should be considered as a lower limit to the real ones.",
"Nevertheless, it is expected that the most prominent and easily tracked arms correspond to those having the strongest impact on the chemical distribution, producing the largest and most clearly observable abundance differences."
],
[
"Derivation of oxygen abundances",
"The wavelength range covered by MUSE (which does not include the $[{O}{ii} ]\\lambda 3727$ emission line) prevents the use of several calibrators often adopted to measure oxygen abundances.",
"This is the case for those based on the R23 or ONS indicators , , .",
"Among the remaining available calibrators, the ones based on the O3N2 index, defined as $\\rm O3N2 = \\log ([{O}{iii} ]\\lambda 5007/H\\beta \\times H\\alpha /[{N}{ii} ]\\lambda 6584)$ , stand out because of its monotonic dependence on the abundance and the close distance in wavelength between the lines of both ratios, which makes the index barely affected by dust attenuation.",
"The O3N2 indicator was first introduced by [1] and later modified by .",
"In this study we derive the oxygen abundances of the Hii regions making use of this indicator in combination with a later calibration proposed by , hereafter M13.",
"By employing $T_e$ -based abundances of $\\sim 600$ Hii regions from the literature together with new measurements from the CALIFA survey, it constitutes one of the most accurate calibrations to date for the O3N2 index.",
"The improvement of this calibration is especially significant in the high-metallicity regime, where previous calibrators based on this index lack high quality observations , .",
"When measuring oxygen abundances using strong-line indicators, one has to keep in mind the systematic discrepancies arising between the different proposed diagnostics and calibrations , .",
"For this reason, in order to ensure that the results are not contingent on the adopted method to derive the abundances we also make use of the calibration, hereafter D13, that is based on the MAPPINGS IV code developed by the authors.",
"This calibration is based on a grid of photoionisation models, that cover a wide range of abundances and ionisation parameters typical of Hii regions in galaxies.",
"D13 can be used through a Python module implemented by the authors, known as pyqz, which is publicly availablehttp://dx.doi.org/10.4225/13/516366F6F24ED.",
"Figure: Probability distribution function (PDF) of the oxygen abundance residuals of the Hii regions for the two example galaxies shown in Fig. .",
"The red (grey) histogram correspond to the arm (interarm) distribution.",
"The vertical solid lines represent the mean value for each distribution.For the sake of clarity, below we only show the results based on the use of the M13 calibrator.",
"The results derived from the D13 diagnostic are provided in Appendix , showing that the main conclusions of the paper are independent of the adopted calibration, although slight differences arise.",
"These differences will be discussed in Sec. .",
"Figure: Distribution of average arm-interarm abundance variations for the galaxy sample derived from the analysis of the Hii regions.",
"Individual values together with the errors are shown in the bottom panel.",
"Red dots represent galaxies that are compatible with an absence of arm-interarm differences (vertical line centred at zero).As introduced in Sec., the principal behaviour observed in the oxygen abundance distribution of spiral galaxies is a negative radial gradient , , , , , , .",
"Apart from this radial decline, other trends have been observed very often in the inner and outer parts of the discs, namely a decrease towards the centre of the galaxies , , , , and a flattening in the outskirts .",
"Following the procedure described in , we determine the characteristic radial profile of each galaxy in the sample, and then remove it in order to highlight the arm-interarm abundance variations.",
"The so-called abundance residuals of the Hii regions ($\\rm \\delta \\,([O/H])$ ) are therefore derived by subtracting the corresponding values for their galactocentric distances according to this radial trend to the observed ones.",
"The abundances residuals range between $-0.25$ and $0.30$ dex, thus, clearly larger than the expected uncertainties for the considered calibratorThe derived abundances with M13 have a calibration error of $\\pm 0.08$ dex, and the typical errors associated with the pure propagation of the errors in the measured emission lines are about 0.02 dex..",
"Figure: Arm-interarm abundance variations as a function of: galaxy mass (left panel), appearance of the spiral arms (middle panel), and presence of bars (right panel).",
"Specific information for each panel is provided in the legend.",
"Arrows represent the mean value of the distribution for each considered subset of galaxies.",
"The result of the subtraction of both values is indicated in the top left corner of the panels and shown as black vertical lines.",
"Grey shaded histograms represent a bootstrapped distribution of the difference in the average arm-interarm variations between the two split subsamples for each analysed parameter.",
"A stellar mass cut of 10 11 M ⊙ 10^{11} \\, \\rm M_\\odot has been used.The right panels of Fig.",
"REF display the map of oxygen abundance residuals of the Hii regions detected in two example galaxies.",
"Distinguishing the regions that fall inside the line-delimited areas indicating the spiral arms from those located outside, we can derive separately the arm and interarm residual abundance distributions.",
"Their probability distribution functions (PDF) are represented in Fig.",
"REF for the same example galaxies as red and grey histograms, respectively.",
"For a fair comparison, we restrict the analysed interarm distribution to the Hii regions within the radial range covered by the spiral arms (defined by the dashed ellipses represented in Fig.",
"REF ).",
"Vertical lines of Fig.",
"REF represent the mean abundance residual of the arm ($\\rm \\overline{\\delta \\,([O/H])}_{\\,arm}$ ) and interarm ($\\rm \\overline{\\delta \\,([O/H])}_{\\,interarm}$ ) distributions.",
"For the first galaxy, we can see how the Hii regions of the spiral arms are on average more metal rich than those belonging to the interarm area.",
"The subtraction of the mean values of both distributions ($\\rm \\Delta \\,([O/H])_{HII} = \\overline{\\delta \\,([O/H])}_{\\,arm} - \\overline{\\delta \\,([O/H])}_{\\,interarm}$ ) yields a difference of approximately $0.040$ dex.",
"The second galaxy, in contrast, displays more metal-poor spiral arms compared with the rest of the disc ($\\rm \\Delta \\,([O/H])_{HII} = -0.033$ dex).",
"The distribution of average arm-interarm abundance variations $\\rm \\Delta \\,([O/H])_{HII}$ for the galaxy sample is shown in Fig.",
"REF .",
"The individual values are listed in Table REF and also represented with error bars in the bottom panel of the Figure.",
"We find that 15 galaxies of the sample ($\\sim 33\\,\\%$, red dots) are compatible within errors with presenting no significant variations between the arm and interarm abundance distributions.",
"In addition, 20 galaxies ($\\sim 45\\,\\%$ , grey squares) display positive arm-interarm abundance variations, that is, spiral arms that are more metal rich than the interarm region.",
"Finally, the remaining 10 galaxies ($\\sim 22\\,\\%$ , grey triangles) show negative arm-interarm abundance variations, meaning that spiral arms are more metal poor than the interarm area in these systems.",
"The magnitude of the reported differences is in all cases very small, with the values ranging from -0.06 to 0.05 dex.",
"Nevertheless, we should not forget that these differences are averaged over the whole arm and interarm areas.",
"Abundance variations of up to $0.2-0.3$ dex exit when individual Hii regions in both areas are considered."
],
[
"Dependence on galaxy properties",
"Figure REF exposes the diversity of behaviours displayed by the galaxy sample, with some objects exhibiting a metal rich spiral structure with respect to the interarm region, and other objects showing the opposite trend.",
"A homogeneous arm-interarm abundance distribution has also been revealed for some galaxies.",
"The large number of systems comprising the sample allows to explore how the emergence of arm-interarm abundance variations may depend on different galaxy properties.",
"Here we focus on three global parameters that may enable light to be shed on the origin of the spiral structure: the stellar mass, the appearance of the spiral arms (flocculent or grand design), and the presence of bars.",
"The values of these parameters for the whole galaxy sample are collected in Table REF .",
"The left panel of Fig.",
"REF shows the effect of the galaxy mass.",
"Stellar masses were derived as in by applying the mass-luminosity ratio of to the $g$ and $r$ apparent magnitudes recovered from the data (equations in were used to transform from SDSS magnitudes to the Johnson system).",
"Galaxies have been split into two subsets: higher- (blue dashed histogram) and lower- (red) mass systems, using a cut of $10^{11} \\, \\rm M_\\odot $ in order to have a comparable number of elements in both subsets.",
"On average, more massive galaxies present larger positive $\\rm \\Delta \\,([O/H])_{HII}$ values, that is, more metal rich spiral arms with respect to the interarm area.",
"The difference in the mean $\\rm \\Delta \\,([O/H])_{HII}$ values between higher-mass (blue arrow) and lower-mass (red arrow) systems is $0.016$ dex (vertical black line).",
"Performing a bootstrapping scheme of the data (technique based on random sampling with replacement) 100 times (grey shaded histogram), we obtain that this difference is always systematically positive, concluding that the found dependence on the galaxy mass is not driven by extreme cases.",
"Furthermore, we perform a two-sample Kolmogorov-Smirnov test (K-S test) to check if the differences found in the results between higher- and lower-mass galaxies are statistically significant.",
"The resulting $P$ -value is $0.1\\%$ , well below the significance level of 5 per cent, supporting the result that lower- and higher-mass systems have a different distribution of arm-interarm abundance variations.",
"The second analysed parameter is the appearance of the spiral structure.",
"In this way, galaxies are divided into flocculent and grand design systems according to the symmetry and continuity of the spiral pattern , .",
"Flocculent galaxies present small and patchy spiral arms while grand designs are characterised by the presence of long, symmetric and continuous arms.",
"The classification was carried out based on a visual inspection of $g$ -band images recovered from the data, resulting in 11 flocculent galaxies and 32 grand designs (two galaxies were ambiguous and therefore not considered).",
"The middle panel of Fig.",
"REF shows the distribution of average arm-interarm abundance variations for flocculent (yellow dashed histogram) and grand design (green) systems.",
"Although the covered range for both distributions is similar, on average the grand design galaxies display larger positive arm-interarm abundance variations (i.e.",
"more metal rich spiral arms compared to the interarm area) than the flocculent ones.",
"The difference in the mean $\\rm \\Delta \\,([O/H])_{HII}$ values is $0.015$ dex (vertical black line).",
"Again, the performed bootstrapping yields a distribution of values always above 0.0 dex, indicating that the arm-interarm abundance variations are systematically larger (positive) in grand design galaxies than in flocculent ones.",
"This is confirmed by the $P$ -value of $3\\%$ obtained from the K-S test.",
"Finally, we investigate the role of the galaxy bar on the emergence and magnitude of the arm-interarm abundance variations, the outcome of which is shown in the right panel of Fig.",
"REF .",
"Galaxies are separated in barred (purple dashed histogram) and unbarred (brown) systems according to the information provided in HyperLeda.",
"One galaxy, for which this information is not available in the database, has been excluded.",
"In this case, the distributions of average arm-interarm abundance variations for barred and unbarred galaxies are very similar, with a $\\rm \\overline{\\Delta \\,([O/H])}_{HII}$ value of $0.005$ dex.",
"The performed bootstrapping, with a distribution of values compatible with zero, and the K-S test, with a $P$ -value of $21\\%$ , confirm this result.",
"In summary, our results seem to indicate that the chemical enrichment associated with the spiral arms is clearly correlated with the galaxy mass and the appearance of the spiral structure.",
"More massive and grand design galaxies show spiral arms that are more metal rich than the interarm area compared with less massive and flocculent objects.",
"This enrichment of the spiral arms seems to weaken until reversing, with some cases in which the spiral structure presents less metal rich gas than the interarm regions, preferentially observed in lower-mass and flocculent galaxies.",
"The presence of a bar does not seem to play any role in this scenario, with no significant differences in the arm-interarm abundance distribution reported between barred and unbarred galaxies."
],
[
"Comparison with a spaxel-by-spaxel approach",
"A drawback of selecting Hii regions in order to characterise the gas abundance distribution is the limitation in the statistics and the coverage of the arm and/or the interarm areas.",
"The advent of IFS techniques offers the opportunity to overcome these limitations by tracing the distribution of ionised gas using all the available information (spectra) across the entire galaxies extent.",
"However, this approach suffers from other problems, such as possible dilution effects and an imprecise decontamination of the underlying diffuse nebular emission , , .",
"In order to reinforce the results derived analysing individual Hii regions, we carry out here an alternative analysis using a spaxel by spaxel approach.",
"In this way, to derive the arm and interarm abundance distributions we follow the same procedure described in Sec.",
"for the clumpy regions detected with HIIexplorer.",
"However, in this case we base the analysis on all the spaxels from the datacubes that fulfil the criteria defined in Sec.",
"REF to be associated with star formation.",
"The high spatial resolution of the data entails an obstacle to carry out a spaxel by spaxel analysis.",
"Their physical resolution may allow us to resolve the ionised structure of the Hii regions in some galaxies, depending on the atmospheric seeing during the observations and the redshift of the object .",
"This represents an issue since the empirical calibrators used to derive the oxygen abundances cannot be applied when resolving different areas of Hii regions , .",
"To overcome this difficulty, we degrade the data to reach a physical resolution below the order of the typical size of a “giant” extragalactic Hii region .",
"For this, we perform a simple spatial binning scheme (squared bins) in order to have a final spatial unit of $\\rm \\sim 400 \\,pc$.",
"However, when the original physical resolution of the datacubes is very high, we would have to add a lot of pixels to reach the final resolution of $\\rm \\sim 400 \\,pc$, loosing significant spatial information when degrading the data.",
"For this reason, we exclude from this analysis all galaxies that present an original physical resolution of the datacubes ($< 150$ pc).",
"In addition, galaxies for which the analysed star-forming spaxels visually provide a poor spatial coverage of the disc (patchy and discontinuous) are also not considered (ESO 246-G021, NGC 0835, and NGC 0881).",
"The final sample for the spaxel by spaxel analysis comprises 33 galaxies.",
"Figure REF shows the spaxel-wise residual oxygen abundance distribution of NGC 3905, for which the same map based on the individual Hii regions is displayed in the top-right panel of Fig.",
"REF .",
"In this map it can be clearly observed how the spiral arms present higher abundance residuals (orange-yellow colours) than the interarm region (blue-purple).",
"Deriving the PDFs of the arm and interarm abundance distributions in a similar way as for the Hii regions, we find that NGC 3905 has an average arm-interarm abundance variation $\\rm \\Delta \\,([O/H])_{sp}=0.03$ (compared to the 0.04 value obtained from the Hii region analysis).",
"For the whole galaxy sample, we identify 14 galaxies with positive arm-interarm abundance variations ($43\\%$ ), and 7 galaxies with negative variations ($21\\%$ ).",
"The remaining 12 galaxies ($36\\%$ ) are compatible within errors with an absence of significant arm-interarm differences.",
"The $\\rm \\Delta \\,([O/H])_{sp}$ values for the galaxies are provided in Table REF .",
"We note that although the $\\rm \\Delta \\,([O/H])$ values slightly change between the analysis of the Hii regions and the spaxel-by-spaxel approach, the qualitative behaviour regarding the presence of positive or negative arm-interarm variations (or the compatibility with an absence of them) is the same (taking into account the error intervals) for all galaxies except for a very small percentage of the sample (ESO 506–G004, IC 1320, NGC 1285, and NGC 5339).",
"Figure: Colour map of the oxygen abundance residuals (derived by subtracting the radial profile) of the star-forming spaxels in NGC 3905.",
"See caption of the right panels of Fig.",
"for more details.Figure: Spaxel-wise arm-interarm abundance variations as a function of: galaxy mass (left panel), appearance of the spiral arms (middle panel), and presence of bars (right panel).",
"See caption of Fig.",
"for more details.Finally, we investigate again the dependence of the arm-interarm abundance variations with the same three previously studied galaxy properties: the stellar mass, the appearance of the spiral arms, and the presence of bars.",
"The outcome of this test is shown in Fig.",
"REF .",
"Analogously to when analysing the individual Hii regions, a correlation with the galaxy mass (left panel) appears, with more massive galaxies displaying more metal rich spiral arms compared to the interarm area.",
"The difference in the mean $\\rm \\Delta \\,([O/H])_{sp}$ values between higher-mass (blue arrow) and lower-mass (red arrow) systems is $0.021$ dex, a bit larger that the value obtained in the Hii region analysis.",
"The significance of this difference is supported by the performed bootstrapping (positive distribution of values) and the K-S test ($P$ -value of $0.5\\%$ ).",
"Similarly, when examining the effect of the appearance of the spiral structure, the spaxel by spaxel analysis reinforces the results from the Hii region one: grand design galaxies (green histogram) tend to present larger positive arm-interarm abundance variations than flocculent systems (yellow, $\\rm \\overline{\\Delta \\,([O/H])}_{sp} = 0.019$ dex).",
"The bootstrapping confirms this trend, although in this case the $P$ -value of the K-S test is above the significance level ($15\\%$ ).",
"It can not be discarded that the low number of flocculent systems (11) may affect the accuracy of the K-S test.",
"Lastly, regarding the role of the galactic bar in shaping the arm-interarm abundance variations, the spaxel wise analysis yields larger differences between barred (purple) and unbarred (brown) galaxies ($\\rm \\overline{\\Delta \\,([O/H])}_{sp} = 0.014$ dex).",
"The outcome of the performed bootstrapping, with a distribution of values always above zero, and the K-S test, with a $P$ -value of $3\\%$ , confirm the statistical significance of this result.",
"In summary, we have tried to improve the number statistics and the coverage of the arm and interarm areas by following a spaxel by spaxel approach for a smaller galaxy sample (33 of the 45 galaxies comprising the original sample).",
"This analysis reinforces the results obtained with the classical approach of selecting individual Hii regions regarding the role of the galaxy mass and the appearance of the spiral structure in the emergence and magnitude of the arm-interarm abundance variations.",
"In addition, despite the weak effect previously observed for the presence of a galactic bar, in this case the analysis shows significant arm-interarm abundance variations between barred and unbarred systems, with barred galaxies displaying spiral arms that are more metal rich than the interarm area compared with unbarred objects."
],
[
"Discussion",
"In the past, the chemical distribution of the ISM in spiral galaxies was considered to be highly homogeneous (, see also , ).",
"However, the advent of new IFS instruments combining high spatial resolution and large field-of-views has recently brought to light the presence of azimuthal variations on the gas oxygen abundance distribution in different galaxies , , , , .",
"All these works point to a connection between the observed chemical inhomogeneities and the spiral structure of the galaxy.",
"Although the above studies agree that the gas in the spiral arms is more metal-rich than the rest of the disc (interarm region), the magnitude of the reported variations diverges.",
"The fact that the analysed data are collected using different instruments and that the employed methodologies vary, hampers a proper comparison of the arm-interarm abundance differences among the explored galaxies, and therefore, hinders the assessment of the physical scenario behind the observed trends.",
"In this study we take advantage of the availability of high spatial resolution MUSE data for a large sample of galaxies within the AMUSING project .",
"For 45 spiral galaxies (that remain from the initial sample after the selection criteria outlined in Sec.",
"), we compare the gas abundance distribution of the spiral arms with that of the interarm region in order to evaluate (with larger statistics) whether the spiral structure is the main driver of the abundance variations.",
"We report the presence of more metal-rich gas in the spiral arms with respect to the interarm region for a subsample of galaxies ($\\sim 40-45\\%$ ), confirming previous studies.",
"Surprisingly, a small percentage of the sample ($\\sim 20\\%$ ) shows the opposite trend, that is, spiral arms that are more metal poor than the interarm area.",
"The remaining $\\sim 35-40\\%$ of the galaxies do not display any clear enrichment pattern associated with the spiral arms, in agreement with , that only find arm-interarm abundance variations in half of their sample.",
"However, Fig.",
"REF shows that the PDF of the abundance variations clearly differs from the normal distribution centred at zero, which is confirmed by performing a Lilliefors test that yields a $P$ -value of $3\\%$ .",
"Despite this, the amplitude of the stated abundance variations are very small (up to 0.05 dex) when compared to previous studies, that report variations as large as 0.4 dex .",
"However, we note that these differences are averaged over the whole arm and interarm areas, whereas in most of previous studies the provided values represent the maxima of the detected variations.",
", following a similar methodology to separate arm and interarm areas, report abundance differences in NGC 2997 of 0.02 dex when the same calibrator is used, totally compatible with the range of values obtained here.",
"The obtained arm-interarm abundance variations rely on the defined width for the spiral arms.",
"For this study, based on a thorough visual inspection of the $g$ -band images of the galaxies, we measured arm widths ranging from $1-5$ (see Sec.",
"REF for details).",
"In order to check how the choice of this value may affect the results, we perform several tests using other values of the arm width in the analysis.",
"When modifying the lower limit of 1 to 2, similar results are observed.",
"However, as we increase the arm width the observed differences begin to disappear.",
"This result is expected; by increasing the arm width, more interarm Hii regions are erroneously included in the spiral arms, and the arm-interarm abundance differences are diluted.",
"Because of the nature of the AMUSING project, the observations used in this study present a wide variety of characteristics.",
"In particular, the spatial resolution of the galaxies in the sample is very diverse, spanning from approximately 70 to 870 pc (see Table REF ).",
"addressed the issue of how the spatial resolution could affect the detectability of arm-interarm abundance differences.",
"They concluded that the magnitudes of the variations decrease with worsening spatial resolution.",
"However, this effect started to be noticeable with resolutions above 1 kpc, which is the upper limit selected for this study.",
"Nevertheless, we represent in Fig.",
"REF the $\\rm \\Delta \\,([O/H])$ values of the galaxies in the sample as a function of the spatial resolution.",
"No clear trends are observed between both parameters, supported by a Pearson correlation coefficient of $r=0.27$ and a p-value for testing non-correlation of $8\\%$ (i.e.",
"above the significance level).",
"This test suggests that the variety of detected arm-interarm abundance variations is not driven by differences in the spatial resolution of the data.",
"Figure: Average arm-interarm abundance variations for the galaxy sample as a function of the spatial resolution.",
"Red triangles represent the mean values in bins of 200 pc, with the error bars indicating the standard deviation.The large sample of galaxies analysed in this work enables the study of the dependency of the arm-interarm abundance variations with physical parameters of the galaxies.",
"This analysis could help to identify which mechanisms may enhance or diminish the emergence of such variations.",
"Here we focus in three galaxy properties: the stellar mass, the appearance of the spiral structure, and the presence of bars, which might provide key information about the nature of the spiral pattern.",
"We observe that the first two factors seem to affect the chemical distribution of the galaxies, with more massive and grand design systems presenting larger positive arm-interarm abundance variations than their lower-mass and flocculent counterparts.",
"This result is found when analysing the individual Hii regions and is reinforced by a second analysis following a spaxel-wise approach for a subsample of 33 galaxies.",
"Regarding the role of the bar, a very weak enrichment of the spiral arms is observed in its presence when individual Hii regions are analysed, being significantly stronger in the spaxel-wise analysis.",
"Further analysis based on a larger sample may help to evaluate whether the improvement in the number statistics and spatial coverage of the disc in the second approach is behind this detection, otherwise elusive (Hii region analysis).",
"In order to assess if the three studied galaxy properties act independently in shaping the gas abundance distribution we check whether there is any correlation between these parameters or not.",
"Regarding the influence of bars, we find no clear correlation between the presence of a bar and any of the other analysed attributes (i.e.",
"the stellar mass of the galaxy or the flocculent/grand design appearance of the spiral structure).",
"This suggests that galactic bars themselves do seem to influence the chemical distribution of the host galaxies, by favouring the enrichment of the gas associated with the spiral arms.",
"As regards the two remaining factors, the larger positive differences found in grand design and higher-mass galaxies might be related, since both properties seem to be connected.",
", analysing a sample of 46 galaxies from the S$^4$ G project, observed a tendency in which early types (generally more massive) have grand design spirals, while the late types have flocculent ones.",
"In our case, this behaviour is not so clear, but we find that higher-mass galaxies tend to present grand design spiral arms (at lower masses we find both flocculent and grand design arms), and a flocculent pattern is preferentially found in lower-mass galaxies (whereas grand design arms are equally found in lower- and higher-mass galaxies).",
"We would like to note that although the sample has been divided in lower- and higher- mass galaxies in order to investigate the effect of the galaxy mass on the results, the used cut is $10^{11}$ M$_\\odot $ (to have a comparable number of elements in both subsets).",
"In general the sample comprises quite massive systems with values of log M$_\\odot $ above $9.8$ (see Table REF ), which could influence the lack of correlation with the presence of a bar or the weak one observed with the appearance of the spiral pattern.",
"A larger sample with a better coverage of the parameter space, including the low mass range, would be needed in order to corroborate these tendencies and disentangle the effect of both the galaxy mass and the appearance of the spiral structure in the chemical distribution of the galaxies.",
"The dependence of the found arm-interarm abundance variations with global galaxy properties is not new.",
"analysed the radial abundance gradient in a sample of 63 galaxies from the CALIFA survey differentiating between arm and interarm regions.",
"The authors reported very subtle differences in the observed gradients when segregating the sample in different subgroups, with the ionised gas in the inter-arm regions exhibiting a shallower gradient with a lower zero-point value and a larger dispersion in the oxygen abundances compared to that of the spiral arms.",
"These differences displayed by the two areas were only observed for barred and flocculent galaxies.",
"This is in agreement with our current results that larger (positive) arm-interarm abundance variations exist in barred galaxies with respect to unbarred systems, but does not agree with the larger (positive) variations observed in this work for grand design galaxies compared to flocculent systems.",
"However, the different approaches followed in both studies (in they study radial abundance gradients whereas here we directly compare the distribution of abundance values) does not allow us to further investigate the origin of the discrepancies.",
"More recently, studied local spatial variations of the gas oxygen abundance distribution (as a proxy of the gas metallicity Z$_g$ ) using a large sample of $\\sim 700$ galaxies from the MaNGA survey , based on data of much lower spatial resolution than in this study.",
"Significant chemical inhomogeneities up to 0.2 dex were revealed across the discs, that were also found to correlate with local variations of star formation rate.",
"The authors showed that the slope of the SFR-Z$_g$ correlation depended on the average gas-phase metallicity of the galaxy and its stellar mass.",
"More metal-poor (and low mass) galaxies displayed the lowest slopes (i.e.",
"the strongest SFR-Z$_g$ anticorrelations), reversing the relation for more metal-rich (and high mass) systems (at $\\rm \\sim 10^{10.5}-10^{11} M_\\odot $ ).",
"Since the spiral arms of galaxies are usually associated with regions of enhanced star formation rate , , the positive correlation found between SFR and Z$_g$ for massive systems is somehow analogous to the positive arm-interarm abundance variations found in this study.",
"In the same way, the anticorrelation displayed by low mass galaxies is comparable to the negative arm-interarm abundance variations also revealed by our results.",
"Although the number of systems displaying positive correlations (and equivalently, positive arm-interarm abundance variations) was lower in than in this study, it can be explained by the dominance of low mass galaxies in their sample.",
"The SFR-Z$_g$ correlation was suggested to be motivated by external gas accretion.",
"Indeed, it could be the case that in low mass galaxies the spiral arms are channels of pristine gas (i.e.",
"metal-poor) coming from the intergalactic medium, whereas in high mass systems the spiral arms are dominated by localised metal recycling by preexisting gas.",
"The arm and interarm abundance distributions compared in this study are determined making use of the empirical calibration proposed by for the O3N2 indicator.",
"In addition, the analysis is also reproduced using an alternative calibrator described in based on photoionisation models (see Appendix ).",
"Both analyses lead to equivalent results despite their origin being quite distinct, which reflects the robustness of the results independently of the adopted method to derive the oxygen abundances.",
"The dependence of the arm-interarm abundance variations with the explored galaxy properties maintain.",
"The only significant difference that arises from the use of these two calibrators is the number of galaxies displaying spiral arms that are more metal poor than the interarm area.",
"Whereas the percentage of galaxies presenting negative $\\rm \\Delta \\,([O/H])$ values is as high as $20\\%$ for the M13 calibrator, this number decreases to barely $5\\%$ in the case of D13.",
"This could indicates that the two calibrators are affected in a different way by their relations with the ionisation parameter , that is, the shape of the ionising spectrum.",
"Thus, the differences or lack of them could be tracing changes not entirely related with the oxygen abundance distribution.",
"Further analysis would be necessary in order to assess the impact of the ionisation parameter on the results.",
"Thus, although we can not confirm the existence of galaxies with metal-poor spiral arms (compared to the interarm region), the opposite behaviour is strongly supported.",
"From this study it is clear that the abundance distribution in spiral galaxies is not completely homogeneous and that the spiral structure plays a significant role in the local chemical enrichment of these systems, which somehow is also affected by the galaxy mass, the presence of a bar, and the appearance of the spiral pattern.",
"A first plausible explanation for the distinct oxygen abundance distributions shown by the ionised gas in the spiral arms and the underlying disc could suggest the presence of Hii regions of a different nature.",
", and later , reported the existence of certain Hii regions that distinguish themselves from the `classical' ones due to a stronger low-ionisation forbidden emission.",
"Although first associated with the centres of galaxies, they were later found at any galactocentric distance .",
"The involvement of other sources of ionisation such as shocks were proposed to explain the origin of these Hii regions, although other stellar processes such as ageing were also suggested to produce the same effects .",
"Independently of their origin, this type of Hii regions is less easily detected in spiral arms due to its low number compared with that of classical Hii regions and the crowding of the later in this area.",
"For this reason, they could be responsible for the arm-interarm abundance variations found in this study.",
"However, these Hii regions have been found to present enhanced $[{N}{ii} ]\\lambda 6584$ /H$\\alpha $ ratios and lower $[{O}{iii} ]\\lambda 5007$ /H$\\beta $ values, and to occupy the region in the BPT diagram between the and demarcation lines .",
"As we can see in Fig.",
"REF , this is not the case for the Hii regions linked to the interarm area in NGC 3905, which exhibit higher $[{O}{iii} ]\\lambda 5007$ /H$\\beta $ ratios and cover the same range of $[{N}{ii} ]\\lambda 6584$ /H$\\alpha $ values than the Hii regions belonging to the spiral arms.",
"They could still be responsible for the negative arm-interarm abundance variations found in a small subsample of galaxies, but they are insufficient to explain the whole picture.",
"Besides this possibility, the most plausible explanation for the detected arm-interarm abundance variations is that the physical processes behind these variations are connected to the dynamics of the gas and the spiral pattern.",
"Recent 2D chemical evolution models that incorporate the role of the spiral pattern have shown how spiral density waves associated with this structure can be responsible of producing azimuthal abundance variations , .",
"The azimuthal trend would be the result of an enhanced star formation rate linked to an increase in the gas surface density due to the passage of the density wave.",
"This effect would be strengthen by a raise in the probability of cloud-cloud collisions because of the produced shocks when the gas enters the arm .",
"However, the predicted abundance variations decrease significantly with time, especially for the SWS and SWR models presented in and that of , the three including the rotation of both the disc and spiral pattern.",
"In these cases the mixing of the gas is higher due to rotation, diluting very quickly the azimuthal differences.",
"As a consequence, the fact that we are able to observe clear arm-interarm abundance variations implies that the spiral arms had to be formed recently (less than 1-2 Gyr before the observations).",
"Another alternative is that the production of the spiral pattern is a recurrent process along the evolution of the disc and we are observing the effects of the last created density wave.",
"In this framework, the different reported amplitudes of the arm-interarm abundance variations could be the result of different time intervals from the formation of the spiral arms.",
"Except for the above mentioned chemical evolution models, there are few theoretical works centred on the effect of spiral arms on the metallicity distribution of galaxies.",
"Moreover, the existing studies are focused on the stellar metallicity rather than the behaviour of the gas.",
", analysing N-body simulations, have shown the existence of significant azimuthal variations in the metallicity distribution of old stars as a consequence of the effect of radial migration along spiral arms induced by a bar over a pre-existing radial metallicity gradient.",
"Also based on N-body simulations, found evidences of radial migration induced by both a galactic bar and the spiral structure producing variations in the stellar metallicity distribution with respect to the radial gradient .",
", using high-resolution cosmological simulations, exposed an azimuthal variation in the stellar metallicity driven by streaming motions of star particles along the leading and trailing sides of the spiral arms.",
"These radial flows produce an overdensity of metal-rich stars on the trailing edge of the spiral and metal-poor stars on the leading edge.",
"Regarding the effect of the spiral arms on the gas metallicity distribution, compared their observations on NGC 6754 with an N-body simulation showing how streaming motions of gas along spiral arms could also produce the same trends reported by on the gas oxygen abundance distribution.",
"The analysed simulations described spiral arm morphological features that were transient and rotated at a similar speed as the gas at every radius , proposing a particular scenario for spiral structure formation of NGC 6754.",
"On the contrary, stated that the action of radial flows alone could not explain the high abundance variations observed in NGC 1365, and proposed another scenario within the density wave paradigm based on a simple chemical evolution model.",
"According to this scenario, the arm-interarm abundance variations observed in the galaxy were caused by self-enrichment undergone by gas when crossing the interarm region followed by a mixing-induced dilution by the passage of spiral density waves.",
"Based on the conclusions reached by previous works, we find that the complementarity of the analysis of the chemical distribution presented here with a future analysis on the gas velocity distribution for the same galaxy sample could be key in order to investigate the existence of radial flows in these galaxies and their connection with the observed arm-interarm abundance distributions.",
"Furthermore, the analysis on the velocity pattern would enable the derivation of the corotation radius location, in order to confirm theoretical predictions finding stronger abundance variations close by these regions .",
"Both aims will be addressed in a forthcoming paper.",
"In addition, more theoretical works focused on the response of the gas abundance distribution to the presence of the spiral structure would be crucial in order to investigate the effect of spiral arms of different nature .",
"Of particular interest would be the predictions on the behaviour of the arm-interarm abundance variations as a function of the galaxy properties analysed in this study in order to distinguish between different spiral arm models.",
"In this regard, simulations covering galaxies with different values of these properties (in particular, low mass and flocculent galaxies) could enable the finding of systems with more metal-poor spiral arms, allowing us, among other things, to confirm or discard the existence of negative arm-interarm abundance variations."
],
[
"Summary",
"In this work we compare the arm and interarm oxygen abundance distributions using high resolution IFS MUSE data for a sample of 45 spiral galaxies in order to assess the role of the spiral structure on the chemical enrichment of galaxies.",
"We follow two complementary methodologies: one based on the analysis of the individual Hii regions, and another one using the information from all the spaxels associated with star formation.",
"Whereas the first approach is more successful dealing with dilution effects and the decontamination of the underlying diffuse ionsed gas, the latter allows us to improve the statistics and the spatial coverage of the explored regions.",
"Regarding the derivation of the oxygen abundances, we make use of two independent methods: an empirical calibrator based on the O3N2 indicator , and a theoretical calibrator based on photoionisation models .",
"Comparing the abundance distribution of the arm and interarm regions of the galaxies, we find more metal-rich Hii regions in the spiral arms with respect to the interarm area for a large subsample of galaxies ($45\\%$ using the M13 calibration, $65\\%$ for D13).",
"This is in agreement with previous studies on individual galaxies , , , .",
"Particularly relevant is the compatibility found between the range of values obtained here (up to $0.08$ dex) and the one derived by for NGC 2997 following a similar methodology.",
"In addition, surprisingly, we observe the opposite trend in a small percentage of the sample, that is, more metal-poor Hii regions in the spiral arms compared to the interarm region.",
"This finding is highly dependent on the used calibrator, with $\\sim 20\\%$ galaxies exhibiting this behaviour for M13, and barely $5\\%$ for D13, and therefore, further analysis is required to confirm it.",
"We investigate the dependence of the arm-interarm abundance variations with three galaxy properties: the stellar mass, the flocculent/grand design appearance of the spiral structure, and the presence of galactic bars.",
"Following the two described approaches, we observe that the arm-interarm abundance differences are larger (positive) in more massive and grand-design galaxies than in low-mass and flocculent systems.",
"In addition, the spaxel-wise analysis yields also significant differences when analysing the effect of bars, with barred galaxies presenting larger (positive) abundance variations.",
"These trends are found irrespective of the used calibrator.",
"The results suggest that the detected abundance variations are connected to the dynamics of the gas and the spiral structure.",
"Further analysis on the gas velocity distribution could be useful in order to link these variations with the presence of radial flows, if existing.",
"In addition, predictions from theoretical works on the response of the gas abundance distribution to the action of the spiral structure, as well as how the arm-interarm variations would depend on the analysed galaxy properties, are important to interpret the observations and distinguish between different spiral arm models."
],
[
"Acknowledgements",
"We would like to thank the anonymous referee for comments which helped to improve the paper.",
"We are grateful to Merce Romero and Mercedes Mollá for their valuable comments.",
"LSM acknowledges financial support from the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU) via grant AYA2016-79724-C4-2-P. SFS is grateful for the support of the CONACYT grants CB-285080 and FC-2016-01-1916, and funding from the PAPIIT-DGAPA-IA101217 (UNAM) project.",
"TRL acknowledges financial support through the grants (AEI/FEDER, UE) AYA2017-89076-P, AYA2016-77237-C3-1-P and AYA2015-63810-P, as well as by the MCIU, through the State Budget and by the Consejería de Economía, Industria, Comercio y Conocimiento of the Canary Islands Autonomous Community, through the Regional Budget.",
"TRL is supported by a MCIU Juan de la Cierva - Formación grant (FJCI-2016-30342).",
"LG was funded by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No.",
"839090.",
"This study was based on observations made with ESO Telescopes at the Paranal Observatory (programmes 60.A-9317(A), 60.A-9392(A), 095.B-0532(A), 095.D-0172(A), 095.D-0091(B), 096.B-0309(A), 096.B-0951(A), 096.D-0263(A), 096.D-0296(A), 097.B-0640(A), 097.D-0408(A), 098.D-0115(A), 099.D-0022(A), 100.D-0341(A), and 0101.D-0748(B)).",
"This research makes use of python (http://www.python.org), of Matplotlib , a suite of open-source python modules that provides a framework for creating scientific plots, and Astropy, a community-developed core Python package for Astronomy ."
]
] | 2001.03450 |
[
[
"A Deep Learning Approach Towards Prediction of Faults in Wind Turbines"
],
[
"Abstract With the rising costs of conventional sources of energy, the world is moving towards sustainable energy sources including wind energy.",
"Wind turbines consist of several electrical and mechanical components and experience an enormous amount of irregular loads, making their operational behaviour at times inconsistent.",
"Operations and Maintenance (O&M) is a key factor in monitoring such inconsistent behaviour of the turbines in order to predict and prevent any incipient faults which may occur in the near future.",
"Machine learning has been applied to the domain of wind energy over the last decade for analysing, diagnosing and predicting wind turbine faults.",
"In particular, we follow the idea of modelling a turbine's performance as a power curve where any power outputs that fall off the curve can be seen as performance errors.",
"Existing work using this idea has used data from a turbine's Supervisory Control & Acquisition (SCADA) system to filter and analyse fault & alarm data using regression techniques.",
"In contrast to previous work, we explore how deep learning can be applied to fault prediction from open access meteorological data only."
],
[
"Introduction",
"Northern Lights Deep Learning Workshop, Tromso, Norway, January 2019.",
"With the rising costs of conventional sources of energy, the world is moving towards sustainable energy sources including wind energy.",
"Wind turbines consist of several electrical and mechanical components and experience an enormous amount of irregular loads, making their operational behaviour at times inconsistent.",
"Operations and Maintenance (O&M) is a key factor in monitoring such inconsistent behaviour of the turbines in order to predict and prevent any incipient faults which may occur in the near future.",
"Machine learning has been applied to the domain of wind energy over the last decade for analysing, diagnosing and predicting wind turbine faults.",
"In particular, we follow the idea of modelling a turbine's performance as a power curve where any power outputs that fall off the curve can be seen as performance errors.",
"Existing work using this idea [1] has used data from a turbine's Supervisory Control & Acquisition (SCADA) system to filter and analyse fault & alarm data using regression techniques.",
"In [3], the authors showed that traditional supervised learning techniques, e.g.",
"support vector regression, are able to make reliable predictions from a power curve.",
"In this study, we investigate the applicability of deep learning techniques to turbine fault prediction from power curves.",
"Deep learning [2] is relatively new in its application to wind energy.",
"We use the open access NREL Western Wind Datasethttps://www.nrel.gov/grid/western-wind-data.html for the year 2012 to first predict the wind power output from turbines and secondly identify operational faults based on data points that fall off the optimal performance curve.",
"In contrast to previous work, we explore how deep learning can be applied to fault prediction from open access meteorological data only.",
"Figure: Power Curve for the Wind Turbine where any instance in Region 1 or Region 3 signifies a fault.",
"Actual power curve is modelled based on the labelled data (black) while the predicted power (yellow) is modelled using a neural network"
],
[
"Modelling of Power Curve",
"A power curve is an integral part of wind turbines' O&M, showing the relation between the wind speed and the turbine's power output.",
"A turbine is said to be operating normally when the wind speed is above the cut-in speed—which is the minimum wind speed at which the turbine generates power, or when the wind speed is below the cut-out speed—which is the maximum wind speed, beyond which turbine should stop operating to prevent failures.",
"The ideal turbine operates at the rated speed—which is the speed at which turbine generates its rated power.",
"Whenever a turbine enters regions 1 and 3 of the power curve (i.e.",
"below cut-in wind speed or above cut-out), it can be interpreted as an impending fault in the turbine.",
"See Figure REF for an illustration.",
"In the absence of openly available SCADA data on turbine faults, we use the 29,736 data samples in the NREL database to model faults based on the power curve and meteorological data only.",
"Features include timestamps (month, day, hour, min), wind speed (m/s), air temperature (Kelvin), air pressure (Pascals), wind direction (Degrees) and density at hub height (Kg/$ m^3 $ ).",
"We use a 70%-30% split of the data set of training and testing and find that a Medium Gaussian Kernel based Support Vector Regression (SVR) model achieves a minimal square error (MSE) of 0.098063 using 5-fold cross-validation.",
"This is a much better MSE obtained than [3], in which SVR with a radial basis function Kernel yields an MSE of 0.6696.",
"Figure REF shows the resulting power curve including predicted faults.",
"It can clearly be enunciated from the graph that the actual and predicted power is quite close, and the actual power curve resembles the predicted power curve in most of the context.",
"The deviations from the curve, which are demonstrated by the curve showing extreme asymptotic spikes pertains to the turbine moving over to above cut-out speed (Region 3) or below cut-in speed (Region 1)."
],
[
"Prediction of Faults",
"We compare various neural network (NN) models for identifying and predicting faults in the NREL dataset.",
"We treat fault situations as those where the turbine operates in Region 1 or 3 of the power curve.",
"Accordingly, the entire dataset is annotated with binary labels—1 for a fault and 0 for normal operation.",
"The dataset is split into training (70%), validation (15%) and test (15%) sets and we use MSE as an optimisation criterion and early stopping.",
"Table REF shows a comparison of neural networks, alongside performance parameters.",
"We compared a feedforward NN, recurrent NN, convolutional NN, sparse autoencoder and dynamic time series Non Linear Autoregressive (NAR) NN.",
"Results show that the recurrent neural net performs best (by 10%) but needs longest to train (by 9%) compared to sparse autoencoder (2nd best).",
"Table: Various Neural Network Models used and their performance evaluation"
],
[
"Conclusion and Future Work",
"As is evident from existing literature most work on O&M of wind turbines is based on traditional machine learning algorithms but has not yet explored deep learning.",
"We present a feasibility study demonstrating that deep learning is promising for application in the wind energy domain, outperforming traditional regression models.",
"In addition, we have explored the possibility of predicting impeding turbine faults from meteorological data — thus circumventing the problem that machine fault data is often treated as commercially sensitive and therefore not readily openly available.",
"In future work we seek to obtain such data and confirm our results, paving the way for predicting incipient faults in the electrical and mechanical components of wind turbines."
],
[
"Acknowledgement",
"We are thankful to the University of Hull's High Performance Computing facility for providing us access to MATLAB on Viper."
]
] | 2001.03612 |
[
[
"A Josephson phase battery"
],
[
"Abstract A battery is a classical apparatus which converts a chemical reaction into a persistent voltage bias able to power electronic circuits.",
"Similarly, a phase battery is a quantum equipment which provides a persistent phase bias to the wave function of a quantum circuit.",
"It represents a key element for quantum technologies based on quantum coherence.",
"Unlike the voltage batteries, a phase battery has not been implemented so far, mainly because of the natural rigidity of the quantum phase that, in typical quantum circuits, is imposed by the parity and time-reversal symmetry constrains.",
"Here we report on the first experimental realization of a phase battery in a hybrid superconducting circuit.",
"It consists of an n-doped InAs nanowire with unpaired-spin surface states and proximitized by Al superconducting leads.",
"We find that the ferromagnetic polarization of the unpaired-spin states is efficiently converted into a persistent phase bias $\\varphi_0$ across the wire, leading to the anomalous Josephson effect.",
"By applying an external in-plane magnetic field a continuous tuning of $\\varphi_0$ is achieved.",
"This allows the charging and discharging of the quantum phase battery and reveals the symmetries of the anomalous Josephson effect predicted by our theoretical model.",
"Our results demonstrate how the combined action of spin-orbit coupling and exchange interaction breaks the phase rigidity of the system inducing a strong coupling between charge, spin and superconducting phase.",
"This interplay opens avenues for topological quantum technologies, superconducting circuitry and advanced schemes of circuit quantum electrodynamics.}"
],
[
"Acknowledgement",
"The work of E.S.",
"was supported by a Marie Curie Individual Fellowship (MSCA-IFEF-ST No.660532-SuperMag).",
"E.S., N.L and F.G acknowledge partial financial support from the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant No.",
"615187- COMANCHE.",
"E.S., A.I., O.D., N.L, F.S.B.",
"and F.G were partially supported by EU's Horizon 2020 research and innovation program under Grant Agreement No.",
"800923 (SUPERTED).",
"L.S and V. Z. acknowledge partial support by the SuperTop QuantERA network and the FET Open And QC.",
"I.V.T, C.S.F., and F.S.B., acknowledge financial support by the Spanish Ministerio de Ciencia, Innovacion y Universidades through the Projects No.",
"FIS2014-55987-P, FIS2016-79464-P and No.",
"FIS2017-82804-P and by the grant “Grupos Consolidados UPV/EHU del Gobierno Vasco” (Grant No.",
"IT1249-19).",
"A.B.",
"acknowledges the CNR-CONICET cooperation program “Energy conversion in quantum nanoscale hybrid devices,” the SNS-WIS joint laboratory QUANTRA, funded by the Italian Ministry of Foreign Affairs and International Cooperation and the Royal Society through the international exchanges between the United Kingdom and Italy (Grant No.",
"IEC R2192166).",
"E.S.",
"A.I.",
"and O.D.",
"performed the experiment and analyzed the data.",
"R.C.,C.S.F.,C.G.,I.V.T., A.B.",
"and F.S.B.",
"provided theoretical support.",
"M.R., N.L.",
"and O.D.",
"fabricated the phase battery on the InAs nanowires grown by V.Z.",
"and L.S.. E.S.",
"conceived the experiment together with F.G. that supervised the project.",
"E.S., A.I., I.V.T.and F.S.B.",
"wrote the manuscript with feedback from all authors."
],
[
"Device fabrication",
"Hybrid proximity DC SQUIDs devices were fabricated starting from gold catalyzed $n$ -doped InAs nanowires with typical length of ${1.5}{}$ and a diameter of $\\sim $ ${85}{}$ grown by chemical beam epitaxy gomescontrolling2015.",
"The $n$ -doping was obtained with Se wallentindoping2011 and the metalorganic precursors for the nanowire growth was trimethylindium (TMIn), tertiarybutylarsine (TBAs) and ditertiarybutylselenide (DTSe), with line pressures of 0.6, 1.5 and 0.3 Torr respectively.",
"Nanowires were drop-casted onto a substrate consisting of ${300}{}$ thick SiO$_2$ on $p$ -doped Si.",
"Afterwards, a ${280}{}$ -thick layer of positive-tone Poly(methyl methacrylate) (PMMA) electron beam resist was spun onto the substrate.",
"The devices were then manually aligned to the randomly distributed InAs nanowires and patterned by means of standard electron beam lithography (EBL) followed by electron beam evaporation (EBE) of superconducting Ti/Al (${5/100}{}$ ) electrodes.",
"Low-resistance ohmic contacts between the superconducting leads and the InAs nanowires were promoted by exposing the InAs nanowire contact areas to a highly diluted ammonium polysolfide (NH$_4$ )S$_x$ solution, which selectively removes the InAs native oxide and passivates the surface, prior to EBE.",
"The fabrication process was finalized by dissolving the PMMA layer in acetone.",
"From transport characterization on similar wires and normal metal electrodes ioriovectorial2019, we estimate a typical electron concentration $n \\simeq {2e18}{^{-3}}$ and mobility $\\mu \\simeq {1200}{^2/}$ .",
"The corresponding Fermi velocity $v_F$ , mean free path $l_e$ and diffusion coefficient $D = v_F l_e/3$ , are evaluated to be $v_F \\simeq {2e6}{/}$ , $l_e \\simeq {30}{}$ and $D \\simeq {200}{^2/s}$ ."
],
[
"Kondo resistance $R(T)$",
"To quantify the amount of unpaired spins in our system the temperature dependence of the normal-state resistance $R(T)$ has been studied in the range 10300.",
"The data exhibits an upturn at $T\\sim {80}{}$ (see Fig.",
"REF ) suggesting a Kondo scattering mechanisms between the free electrons and the unpaired spins in the weak-links.",
"Since the InAs nanowires were synthesized without incorporating any magnetic impurity (to the best of our knowledge, Se doping cannot provide by itself any magnetism), we conjecture that unpaired spins are originated from oxides states at the nanowire surface, in analogy with what is observed in metallic nanowires rogachevmagnetic-field2006,sapkotaobservations2018.",
"Indeed, the $R(T)$ of a diluted magnetic alloy follows the universal non-monotonic relation kondoresistance1964 $R(T) = R_0 + R_{el-ph}(T) + R_K(T),$ where $R_0$ is the residual resistance while $R_{el-ph}(T)$ and $R_K(T)$ are the contribution given respectively by the electron-phonon and the Kondo scattering.",
"The temperature dependence of the former can be expressed according to the Bloch-Gruneisen model as PhysRevB.74.035426 $R_{el-ph}(T) = a\\left(\\frac{T}{\\theta _D}\\right)^5 \\int _0^{\\theta _D/T} \\frac{x^5}{(e^{x}-1)(1-e^{-x})}\\text{d}x.$ For the Kondo contribution many analytical approximations are available according to the range of temperature investigated.",
"In the full range of temperature the exact solution exist from the numerical renormalization group theory (NRG).",
"In the following we use an empirical fitting function derived as an analytical approximation of the NRG given by goldhaber-gordonkondo1998,Parks1370, malletscaling2006,costikondo2009 $R_K^{NRG} = R_K(0) \\left( \\frac{T_K^{\\prime 2}}{T^2+T_K^{\\prime 2}} \\right)^s,$ with $T_K^{\\prime }$ related to the actual Kondo temperature $T_K$ by $T_K^{\\prime }=T_K/(2^{1/s}-1)^{1/2}$ .",
"Note that Eq.",
"REF is defined such that $R_K(T_K)=R_K(0)/2$ and the parameter $s$ is fixed to $s=0.22$ as expected for a spin $1/2$ impurity.",
"In Fig.",
"REF we show the fit of the experimental data with Eq.",
"REF , from which we extract a Kondo temperature $T_K = 58 \\pm 5$ K, a residual magnetic impurity resistance $R_K(0)=13.5 \\pm 0.3~\\Omega $ , a coefficient $a=87 \\pm 6~\\Omega $ and a Debye temperature $\\theta _D = 551 \\pm 31$ K. From $R_K(0)$ is possible to estimate the density of unpaired spin, that form the Hamann expression of the residual Kondo resistance in the unitary limit is given by hamannnew1967 $R_K(0) = \\frac{L}{A} \\frac{4\\pi c \\hbar }{n k_F e^2},$ with $L\\sim {80}{}$ junction lenghts, $A \\sim \\pi r^2$ with $r \\sim {45}{}$ nanowire cross-sectional area, $k_F$ Fermi wavevector and $n$ electron carrier density, from which we estimate the density of magnetic impurities $c \\simeq {1.36e17}{^{-3} }$ .",
"This corresponds to a concentration of $\\sim $ 4 ppm ($= c /n_{InAs}$ , with the InAs atomic density $n_{InAs} = {3.59e22}{^{-3}}$ ) of unpaired spins in the InAs nanowire."
],
[
"First “magnetization” curve",
"The persistent hysteretic loops of the $\\varphi _0$ -shift, shown in Fig.",
"1d-f of the main text, are consistent with the presence of a ferromagnetic background of unpaired spin.",
"To support this hypothesis, we show in Fig.",
"REF a and b the first magnetization curve of this spin ensemble measured in the same device.",
"Initially, the magnetization of the sample is lifted by warming the system above 3 K. Then, the SQUID voltage drop $\\Delta V (\\Phi )$ is measured in the absence of the in-plane magnetic field $B_y$ which is gradually turned on thus polarizing the unpaired spins.",
"The resulting $\\Delta V (\\Phi )$ shows no shifts at low $B_y$ while, only above ${5}{}$ a clear shift is generated.",
"The resulting $\\varphi _{int}$ extracted by fitting $\\Delta V (\\Phi )$ is shown in the violet curve of Fig.",
"REF c. By reversing $B_y$ the $\\varphi _{int}$ then evolves with the typical hysteretic curve of a ferromagnetic system (blue and red curves in Fig.",
"REF c)."
],
[
"SQUID with anomalous Josephson junctions",
"The critical current of a SQUID interferometer can be evaluated from the CPR of the two JJs forming the interferometer.",
"Using a sinusoidal CPR, the currents through the two junctions can be written as $i_1 = I_{c} \\sin \\left[(\\varphi _C-\\varphi _L)+\\varphi _0^{(1)}\\right] \\qquad i_2 = I_{c} \\sin \\left[(\\varphi _C-\\varphi _R)+\\varphi _0^{(2)}\\right],$ where $\\varphi _L,\\varphi _C,\\varphi _R$ are the left,central and right superconducting phases and $I_{c}$ is the critical current of each JJ.",
"The supercurrent of the SQUID is the sum of the two contributions ($I_s = i_1 + i_2,$ ) and, with the constraint on the superconducting phases of the flux quantization $(\\varphi _L -\\varphi _C) + (\\varphi _C - \\varphi _R) +2\\pi \\frac{\\Phi }{\\Phi _0}&= 2\\pi ~\\text{(mod n)},$ it has the form $I_s &= 2I_C \\sin (\\delta _0) \\cos \\left[\\frac{1}{2}\\left(2\\pi \\frac{\\Phi }{\\Phi _0} +\\varphi _{tot}\\right)\\right],$ Figure: Comparison between SQUID geometries.",
"The SQUID geometry employed in this work a allows a simple readout of the anomalous phase ϕ tot \\varphi _{tot} generated by a uniform magnetic field B in B_{in} since the opposite direction of the supercurrents in the two JJs.",
"This would not be possible in a conventional geometry as the one shown in b.where $\\delta _0 =\\varphi _C- \\tfrac{\\varphi _L+\\varphi _R}{2} +\\tfrac{(\\varphi _0^{(2)}+\\varphi _0^{(1)})}{2}$ and $\\varphi _{tot}=\\varphi _0^{(2)}-\\varphi _0^{(1)}$ is the total anomalous phase built in the interferometer.",
"With the geometry depicted in Fig.",
"REF a, the two junctions experience the same in-plane magnetic field orientation but the supercurrents flow in opposite directions resulting in $\\varphi _0^{(1)}=-\\varphi _0^{(2)}=\\varphi _0$ and $\\varphi _{tot}=2\\varphi _0$ .",
"The stable state configuration of the SQUID is achieved by minimizing the total Josephson free energy obtained at $\\delta _0=\\pi /2$ , and then the maximum sustainable supercurrent results to be $I_{S}(\\Phi ) = 2I_C\\left|\\cos \\left(\\pi \\frac{\\Phi }{\\Phi _0} +\\frac{\\varphi _{tot}}{2}\\right)\\right|.$ It follows that in absence of magnetic flux, the maximum supercurrent is reduced by a factor $\\sim |\\cos (\\varphi _{tot}/2)|$ compared to the non-anomalous case as a consequence of the anomalous supercurrent already present in the interferometer.",
"In a more conventional geometry as the one showed in Fig.",
"REF b, the anomalous phases acquired by the two junctions would be the same $\\varphi _0^{(1)}=\\varphi _0^{(2)}=\\varphi _0$ , making impossible its detection in the phase-to-current readout employed in the present work."
],
[
"Lateral $\\varphi _0$ -junction",
"The origin of the anomalous phase $\\varphi _0$ is the singlet-triplet conversion mediated by the spin-orbit coupling (SOC), which in the normal state corresponds to the charge-spin conversion bergerettheory2015.",
"The calculations of the anomalous Josephson current in $\\varphi _0$ -junctions have been done for ideal planar S-N-S junctions, in which the superconducting electrodes and the normal region with SOC are separated by sharp boundaries bergerettheory2015,konschelletheory2015,buzdindirect2008, where the singlet-triplet coupling takes place only at N region.",
"As shown in Ref.",
"bergerettheory2015, this assumption leads to a monotonically increase of the anomalous phase $\\varphi _0$ as a function of the applied magnetic field, which contrasts with curves extracted from our experiment (see Fig.",
"3a in the main text).",
"It is however clear that our experimental setup (Fig.",
"1 in the main text) differs from an ideal S-N-S junction.",
"Indeed, in each junction, the superconducting leads are covering part of the wires over distances larger than the coherence length.",
"This means that the SOC, and hence the spin-charge conversion, is also finite in the portion of the wire covered by the superconductor.",
"As we show in this section, this feature is essential to understand the experimental findings; in particular, the dependence to $\\varphi _0$ from the external magnetic field.",
"In this calculation, we focus on the dependence of $\\varphi _0$ on the $y$ direction of the field, i.e., $B_{in}$ at $\\theta = \\pi /2$ (see Figs.",
"3a and b in the main text).",
"Figure: Schematic view of the S-N-S junctions.",
"At |x|>L/2|x|>L/2, the InAs nanowire (red region) is partially covered by the Al superconducting leads (dark blue regions).",
"The gray region corresponds to the substrate.",
"This schematic view corresponds to the zz-xx plane of Fig.",
"3b (top panel) in the main text.To be specific, we consider the junction sketched in Fig.",
"REF .",
"We assume an infinite diffusive quasi-one dimensional nanowire along the $x$ -axis, which is partially covered by two semi-infinite Al superconducting leads at $x<L/2$ and $x>L/2$ .",
"We assume, for simplicity, that the proximity effect is weak and that the wire is diffusive.",
"In such a case, the condensate function, which determines the Josephson current, obeys the linearized Usadel equation, which results in two coupled differential equations for the singlet and triplet components as shown in Ref. bergerettheory2015.",
"Because the wire lies on a substrate plane, the system has an uniaxial asymmetry in the $z$ direction perpendicular to the substrate (see Fig.",
"REF ).",
"In the presence of SOC, this allows for a gradient singlet-triplet coupling generated by a differential operator of the form $C^a_k\\partial _k\\sim (\\hat{z}\\times \\nabla )^a$ , which converts a scalar (the singlet) into a pseudovector (the triplet) and vice-versa konschelletheory2015,bergerettheory2015.",
"We consider the case when the external field is applied in the $y$ direction, and hence, the superconducting condensate function has the form $f=f_s+if_t{\\rm sgn}\\omega \\sigma ^y$ , where $f_{s,t}$ are the singlet and triplet components and $\\omega $ the Matsubara frequency.",
"The linearized Usadel equation reads: $\\begin{split}&\\frac{D}{2}\\nabla ^2 f_s - |{\\omega }| f_s + \\left( h - iD \\kappa _{\\rm {sc}} \\partial _x \\right) f_t = 0 \\; , \\\\&\\frac{D}{2}\\nabla ^2 f_t - |{\\omega }| f_t - \\left( h - 2i D\\kappa _{\\rm {sc}} \\partial _x \\right) f_s = 0 \\; .",
"\\\\\\end{split}$ Here $D$ is the diffusion coefficient and $h={\\mu _{\\rm {B}} g_{\\rm {s}}} B_{\\rm {in}} / 2$ is the Zeeman field.",
"The last term in both equations describes the spin-charge conversion due to the SOC.",
"It is proportional to the effective inverse length $\\kappa _{\\rm {sc}}$ and the spatial variation of the condensate in the direction of the wire axis.",
"The form of this term is determined by the uniaxial anisotropy of the setup in combination with the fact that we assume that the field is applied only in $y$ direction.",
"Equation (REF ) is written for the full 3D geometry.",
"To obtain an effective 1D Usadel equation, we integrate Eq.",
"(REF ) over the wire cross-section and use boundary conditions imposed on the condensate function at the surface of the wire.",
"In the part of the wire which is covered by the superconductor, the interface between the wire and the superconductor is described by the linearized Kupryianov-Lukichev boundary condition: $\\left.\\partial _x f_s \\right|_{\\text{InAs/Al}}=\\gamma f_{\\rm {BCS}}e^{i\\phi },$ where $\\gamma $ is a parameter describing the InAs/Al interface, $f_{\\rm {BCS}} = \\Delta / \\sqrt{\\omega ^2 + \\Delta ^2}$ is the BCS bulk anomalous Green's function in the superconducting leads, and $\\phi $ is the phase of the corresponding lead.",
"In the uncovered parts of the wire, we impose a zero current flow which corresponds to $\\left.\\partial _x f_{s/t}\\right|_{\\text{InAs/vac.",
"}}=0$ .",
"The integration of Eq.",
"(REF ) over the cross-section of the wire results in two coupled equations for the singlet and triplet components: $\\begin{split}&\\partial _x^2 f_s - \\kappa _{\\omega }^2 f_s + \\left( \\kappa _{h}^2 - 2i \\kappa _{\\rm {sc}} \\partial _x \\right) f_t = S(x) \\; , \\\\&\\partial _x^2 f_t - \\kappa _{\\omega }^2 f_t - \\left( \\kappa _{h}^2 - 2i \\kappa _{\\rm {sc}} \\partial _x \\right) f_s = 0 \\; , \\\\\\end{split}$ with $\\begin{split}S(x) = \\gamma f_{\\rm {BCS}} \\left[ \\Theta \\left( x - \\frac{L}{2} \\right) e^{i\\frac{\\varphi }{2}} + \\Theta \\left( - x - \\frac{L}{2} \\right) e^{-i\\frac{\\varphi }{2}} \\right] \\; ,\\end{split}$ $\\kappa _{\\omega }^2 = \\frac{2|\\omega |}{D}$ , $\\kappa _h^2 = \\frac{2h}{\\hbar D}$ , and $\\varphi $ is the phase difference between these two Al leads.",
"After a cumbersome but straightforward procedure, we solve Eq.",
"(REF ) for continue and finite $f_{s,t}$ .",
"From the knowledge of the singlet and triplet components one determines the Josephson current as follows bergerettheory2015: $\\begin{split}&j(x) = \\frac{\\pi \\sigma _{\\rm {D}} T}{e} \\sum _{\\omega } \\operatorname{\\mathbb {I}m} \\lbrace f_s^* \\partial _x f_s - f_t^* \\partial _x f_t - i \\kappa _{\\rm {sc}} \\left( f_s^* f_t + f_s f_t^* \\right) \\rbrace \\; .\\end{split}$ The resulting current can be written as $j = I_c \\sin (\\varphi + \\varphi _0)$ , with the anomalous-phase given by: $\\begin{split}\\varphi _0 &= \\arctan \\left\\lbrace \\frac{ \\sum _{\\omega }\\operatorname{\\mathbb {I}m}\\left\\lbrace f_{\\rm {BCS}}^2 e^{-q L} \\frac{\\sinh (\\kappa _{\\rm {sc}} L) (q^{2} + \\kappa _{\\rm {sc}}^2) + 2 q\\kappa _{\\rm {sc}}\\cosh (\\kappa _{\\rm {sc}} L)}{q (q^{2} - \\kappa _{\\rm {sc}}^2)^2} \\right\\rbrace }{ \\sum _{\\omega }\\operatorname{\\mathbb {R}e}\\left\\lbrace f_{\\rm {BCS}}^2 e^{-q^* L} \\frac{\\cosh (\\kappa _{\\rm {sc}} L) (q^{*2} + \\kappa _{\\rm {sc}}^2) + 2 q^*\\kappa _{\\rm {sc}}\\sinh (\\kappa _{\\rm {sc}} L)}{q^* (q^{*2} - \\kappa _{\\rm {sc}}^2)^2} \\right\\rbrace } \\right\\rbrace \\; , \\\\\\end{split}$ where $q^2 = \\kappa _{\\omega }^2 + \\kappa _{\\rm {sc}}^2 - i \\kappa _{h}^2$ .",
"In order to compare with the experimental data, we assume a Rashba-like SOC and use the expression derived in Ref.",
"bergerettheory2015 for the spin-charge coupling parameter, namely $\\kappa _{\\rm {sc}}=2\\tau \\alpha ^3m^{*2}/\\hbar ^5$ .",
"By using typical values for the parameters of a InAs/Al system: $\\xi _0 \\simeq {100}{}$ , $\\Delta \\simeq {150}{}$ , $m^* = 0.023~m_e$ , $T\\simeq {25}{}$ , $g_s \\simeq 12$ , and $\\alpha \\simeq {0.24}{}$ , we find the $\\varphi _0(B_{in})$ dependence corresponding to the one shown in Fig.",
"3a of the main text.",
"We see that our model provides a good qualitative explanation of the two main observed features.",
"Namely, the linear increase of $\\varphi _0$ for small fields and a kind of saturation at $\\varphi _0 \\approx \\pm 0.5 \\pi $ .",
"In Fig.",
"REF , we show different $\\varphi _0 (B_{in})$ curves obtained from our general expression (REF ).",
"Whereas for small fields the experimental slope (dashed grey line) can be obtained from different values of the parameters, the behaviour of $\\varphi _0$ at larger fields depends strongly on these parameters.",
"Figure: Theoretical model of the extrinsic anomalous phase.",
"The dependence of the extrinsic anomalous-phase on the magnetic field applied in yy direction (B in B_{in} at θ=π/2\\theta = \\pi /2) for a) different temperatures, with α≃0.24\\alpha \\simeq {0.24}{} and for b) different values of α\\alpha , with T≃10T\\simeq {10}{}.",
"In a), the solid blue line coincides with the one shown in Fig.",
"3a of the main text, with T≃25T\\simeq {25}{} and g s ≃12g_s \\simeq 12.",
"For the dashed orange line, we choose T≃10T\\simeq {10}{} and g s ≃5g_s \\simeq 5, and for the dashed-dotted line, T≃5T\\simeq {5}{} and g s ≃2g_s \\simeq 2.",
"In b), the solid blue line corresponds to α≃0.1\\alpha \\simeq {0.1}{} and g s ≃37g_s \\simeq 37, the dashed orange line to α≃0.18\\alpha \\simeq {0.18}{} and g s ≃7g_s \\simeq 7, and the dashed-dotted green line to α≃0.4\\alpha \\simeq {0.4}{} and g s ≃3g_s \\simeq 3.",
"In both a) and b), the dashed grey line corresponds to the measured slope at low fields.Indeed, it is important to emphasize that the saturation value at $\\varphi _0\\approx \\pm 0.5 \\pi $ is not an universal property of the phase-battery.",
"This value depends on the intrinsic properties of the system.",
"In particular, larger values of the SOI $\\alpha $ leads to larger values of $\\varphi _{\\rm {ex}}$ at values of the field larger than those accessed in the experiment.",
"This is shown in Fig.",
"(REF )b, where we plot the $\\varphi _0 (B_{in})$ dependence for different values of $\\alpha $ , with $T \\simeq {10}{}$ .",
"As in Fig.",
"(REF )a, we change the $g_s$ value to maintain the experimental slope in the low-field region.",
"The linear behavior for the low-field region is shared by all the $\\varphi _0 (B_{in})$ curves, as shown in Fig.",
"(REF ).",
"In this regime, we can thus find the slope value by linearizing Eq.",
"(REF ): $\\varphi _{0} \\simeq C_1 B_{in} + O(B_{in}^3) ,$ with $C_1 \\simeq {0.035}{\\pi }$ , which is in agreement with the value extracted from the experiment."
],
[
"Trivial mechanisms to induce phase shifts",
"Trivial hypotheses, alternative to the anomalous $\\varphi _0$ effect, have been also considered for the for the generation of a hysteretic phase shift: trapped magnetic fluxes and Abrikosov vortices.",
"Trapped magnetic fluxes can be observed in superconducting loops with a non negligible ring inductance $L$ and, more precisely, for a screening parameter $\\beta _L =\\frac{ 2\\pi LI_c}{\\Phi _0} \\gtrsim 1$ , with $I_C$ being the critical current of a single junction clarkesquid2004.",
"This indeed can lead to a hysteretic behavior due to the presence of a circulating current in the ring.",
"For our interferometer we estimated $\\beta _L \\lesssim 10^{-2}$ ($I_C \\sim {300}{}$ and $L \\sim {10}{}$ dambrosionormal2015) that is very unlikely to induce any magnetic hysteresis.",
"Still, if circulating currents are present, hysteretic jumps should be sharp, periodic and visible even at low $B_z$ .",
"The absence of any hysteretic behavior at low magnetic field is further confirmed by the continuous interference patterns shown in Figs.",
"1e and 1f.",
"Abrikosov vortices, also known as fluxons, can be often induced in type-II superconductors – like the thin Al film used in our SQUID devices – when an out-of-plane magnetic field is applied.",
"To avoid vortex intrusion into the ring surface, which might induce a parasitic phase shift, we limit our out-of-plane component to $|B_z| < {0.8}{}$ , which guarantees the absence of any fluxon.",
"Indeed, upon the application of a larger field $B_z \\gtrsim (3-4)~\\text{mT}$ , also in our case abrupt phase shifts appear with a density that increases by increasing $B_z$ , as shown in Fig.",
"REF .",
"Figure: Fluxons induced phase shifts at high B z B_z.",
"Voltage drop ΔV(B z )\\Delta V(B_z) across the SQUID for I sd =1I_{sd} = {1}{} versus applied magnetic field B z B_z up to 8 recorded at T=100T={100}{}.",
"The grey area indicated in the plot (corresponding to 0≤B z ≤0.80\\le B_z\\le {0.8}{}) is the one used to track and evaluate the induced phase shift in our interferometer.",
"For B z ≳3B_z \\gtrsim {3}{} abrupt jumps in the phase start to appear due to trapped fluxons piercing the SQUID area.",
"Inset: back (gray) and forth (red) traces at high B z B_z in the same conditions as before show an hysteretic behavior which is expected for fluxons pick-up.This is what is expected for fluxons pinning in the Al, i.e., stochastic and abrupt events providing a discrete jump of the phase PhysRevLett.104.227003.",
"Notice also the hysteretic behavior expected for fluxon inclusion, which is underlined in the inset of the figure showing a local back and forth measurement.",
"With respect to the in-plane magnetic fields, the thickness of the Al film (thinner than the superconducting coherence length) ensures the complete penetration of the magnetic field, and thereby the absence of generated fluxons.",
"This is consistent with the lack of any stochastic shift upon the application of $B_{in}$ ."
],
[
"Supplementary device measured",
"In this section we repeated the same magnetic characterization of the Josephson phase battery shown in Fig.",
"2 and 3 of the main text, performed on a different device to demonstrate the high reproducibility of the effect, apart sample-specific details.",
"Notice that the behaviour of $\\varphi _{tot}$ and $\\varphi _{int}$ (Fig.",
"REF ) is qualitatively similar, but with a smaller total phase shift of $\\sim 0.4 \\pi $ stemming for a weaker exchange interaction induced by the unpaired-spin.",
"Moreover the angle dependence of $\\varphi _{ex}(\\theta )$ shown in Fig.",
"REF is in very good agreement with the evolution observed in Fig.",
"2 and expected from the model presented in Section .",
"Figure: Charging loops of the Josephson phase battery (second device).a, Voltage drop ΔV(Φ)\\Delta V(\\Phi ) at constant current bias I=1I={1}{} versus in-plane magnetic field B y B_{y} applied orthogonal to the nanowire axis .",
"At large |B y ||B_{y}|, the amplitude of ΔV(Φ)\\Delta V(\\Phi ) is lowered due to the suppression of superconductivity inside the wire.",
"Each trace is vertically offset for clarity.b, Selected traces ΔV(Φ)\\Delta V(\\Phi ) extracted from a for different B y B_{y}.",
"Data are vertically offset for clarity.c, Extracted phase shift ϕ tot \\varphi _{tot} from the curves in a.d, Color plot of the persistent voltage drop ΔV(Φ)\\Delta V(\\Phi ) measured at B y =0B_{y}=0 after the magnetic field was swept to the values shown on the yy-axis.e, Selected traces ΔV(Φ)\\Delta V(\\Phi ) corresponding to the cuts in d.f, Intrinsic phase shift ϕ int \\varphi _{int} extracted from d. ϕ int \\varphi _{int} stems from the ferromagnetic polarization magnetic impurities.",
"All data were recorded at 30{30}{} of bath temperature.Figure: Vectorial symmetry of the anomalous phase ϕ 0 \\varphi _0 (second device).a, Dependence of the extrinsic anomalous phase ϕ ex \\varphi _{ex} on B y B_y.b, Evolution of the anomalous phase ϕ ex \\varphi _{ex} on θ\\theta and B in B_{in}.c, dϕ ex /dB in d \\varphi _{ex}/d B_{in} versus θ\\theta together with a sinusoidal fit (red curve) from Eq. ().",
"The slope has been evaluated by a linear fit of the data in b for |B in |<10|B_{in}|< {10}{}.",
"The error bar is the RMS of the fit.All the data were recorded at 30{30}{} of bath temperature."
],
[
"Low magnification SEM image of the device",
"style bibliography.bib"
]
] | 2001.03393 |
[
[
"Laguerre-Gaussian wave propagation in parabolic media"
],
[
"Abstract We report a new set of Laguerre-Gaussian wave-packets that propagate with periodical self-focusing and finite beam width in weakly guiding inhomogeneous media.",
"These wave-packets are solutions to the paraxial form of the wave equation for a medium with parabolic refractive index.",
"The beam width is defined as a solution of the Ermakov equation associated to the harmonic oscillator, so its amplitude is modulated by the strength of the medium inhomogeneity.",
"The conventional Laguerre-Gaussian modes, available for homogenous media, are recovered as a particular case."
],
[
"Introduction",
"The study of optical beams having complex structures is a subject of intense activity in current times, mainly because the properties of structured light open new possibilities for the manipulation of individual atoms and small molecules [1], [2].",
"This subject represents a feedback pathway between theory and experiment: theoretical advances suggest new experiments while significant experimental results require either new theoretical models or improvements in our understanding of the behavior of light.",
"Remarkably, after realizing that azimuthally phased beams carry angular momentum [3], it was understood that the concept of photon angular momentum is not limited to spin [4], but it may include either extrinsic or intrinsic orbital angular momentum [5] (for a recent discussion on the matter see e.g.",
"[6]).",
"However, it is important to emphasize that, although spin and orbital angular momentum behave quite similar in some instances, “orbital angular momentum has its own distinctive properties and its own distinctive optical components” [7].",
"Such a subtlety is fundamental in the investigation of light-matter interactions [8], [9], [10].",
"In this context it is notable that the wavefront structure of the Laguerre-Gaussian beams allows the production of force fields that have no counterpart in conventional optical beams [11], [12].",
"From the practical point of view, it has been found that Hermite-Gaussian beams with no orbital angular momentum can be transformed into Laguerre-Gaussian beams carrying orbital angular momentum [13], [14], [15].",
"Thus, one can use either cylindrical lenses [13] or Fork diffractive gratings [14], [15] to produce Laguerre-Gaussian beams in the laboratory (sophisticated spatial light modulators can be used instead).",
"Nevertheless, the propagation of Laguerre-Gaussian beams in free (homogeneous) space implies that the corresponding beam width diverges as the propagation variable increases, which may tie down the usefulness of such beams.",
"In this paper we address the problem of finding Laguerre-Gauss wave-packets with finite beam width along all the propagation axis.",
"With this aim we solve the paraxial form of the wave equation for a weakly guiding inhomogeneous medium, the refractive index of which is quadratic (parabolic) in the coordinates transverse to the propagation.",
"Our method is based on the approach introduced in [16], where a Gaussian wave-packet is used to solve the Schrödinger equation for time-dependent and nonlinear Hamiltonian operators via complex Riccati equations.",
"The main point in [16] is that the width of the packet is defined as a solution of the Ermakov equation [17] associated to the one-dimensional oscillator.",
"Such approach was already applied to study the propagation of waves in non-homogeneous media [18], [19], [20], were the close relationship between the paraxial wave equation and the Schrödinger equation is successfully exploited to construct Hermite-Gaussian wave-packets for quadratic refractive index optical media.",
"In the present case the beam width is an oscillatory solution of the Ermakov equation that depends on the propagation variable and such that its amplitude is modulated by the strength of the medium inhomogeneity.",
"The Laguerre-Gaussian wave-packets reported here correspond to non-dispersive beams of finite transverse optical power (localized beams) that propagate with periodical self-focusing profile in the medium.",
"Our approach generalizes the methods to define finite beam widths already reported by other authors [21], [22], [23], [24], [25], [26], and confirm that the distinctive angular momentum properties of the Laguerre-Gaussian modes are better prepared and exploited if the related beam propagates in parabolic media.",
"The generalities to construct the above described Laguerre-Gaussian wave-packets are outlined in Section , where it is also shown that the conventional Laguerre-Gaussian modes arise after turning-off the inhomogeneity of the medium, just as a particular case.",
"In Section we summarize our results and provide some directions for future work."
],
[
"Paraxial wave equation for parabolic media",
"Consider the $z$ -propagation of waves through a weakly guiding inhomogeneous medium, the refractive index of which is quadratic in the transverse coordinates (the $xy$ -plane).",
"Using polar coordinates to write the position vector transverse to beam propagation as $\\rho = (\\rho , \\theta )$ , the refractive index we are dealing with is of the form $n^2(\\rho ) = n_0^2 \\left(1 - \\Omega ^2 \\rho ^2\\right), \\quad \\Omega ^2 \\rho ^2 \\ll 1.$ Here, $n_0$ stands for the refractive index at the optical axis and $\\Omega \\ge 0 $ is a parameter that characterizes the focusing properties of the medium.",
"The corresponding paraxial wave equation is given by $-\\frac{1}{2 k_0^2 n_0} \\nabla _{\\perp }^2 U + \\frac{n_0}{2}\\Omega ^2 \\rho ^2 U = \\frac{i}{k_0} \\frac{\\partial U}{\\partial z}.$ The solutions $U = U(\\rho ,z)$ of (REF ) describe the transversal amplitude of the electric field in the medium.",
"Hereafter $\\nabla _{\\perp }^2$ stands for the transversal component of the Laplacian operator and $k_0$ is the wave number in free space."
],
[
"Lowest-order Gaussian mode",
"Following [16], [18], [19], as a fundamental nonstationary solution of the paraxial wave equation (REF ), we propose the Gaussian wave-packet $U(\\rho ,z) = N(z) e^{iS(z)\\rho ^2}.$ The straightforward calculation shows that the normalization factor $N(z)$ and the coefficient $S(z)$ are respectively given by $N(z) = \\frac{N_0}{w(z)} e^{-i \\chi (z)}, \\quad S(z) = \\frac{k_0n_0}{2} \\frac{d}{dz} \\ln w(z)+ \\frac{i}{w^2(z)},$ where $N_0$ is a normalization constant, $w(z)$ is a solution of the Ermakov equation for the one-dimensional harmonic oscillator $\\frac{d^2w}{dz^2} + \\Omega ^2 w = \\frac{4}{k_0^2n_0^2 w^3},$ and $\\chi (z) = \\frac{2}{k_0n_0} \\int ^z \\frac{1}{ w^2(x)} dx.$ It is a matter of substitution to verify that after the identification $\\frac{1}{R(z)} = \\frac{d}{dz} \\ln w(z),$ we can rewrite Eq.",
"(REF ) in the `canonical' form of the lowest-order Gaussian mode $U(\\rho ,z) = \\frac{N_0}{w(z)} \\exp \\left( \\frac{- \\rho ^2}{w^2(z)} \\right)\\exp \\left( i \\left[\\frac{k_0n_0\\rho ^2}{2R(z)} - \\chi (z)\\right]\\right).$ In analogy with the homogenous case [28], [27], we see that $N_0$ refers to the maximum electric field strength, $w(z)$ corresponds to the beam width, $R(z)$ to the radius of curvature, and $\\chi (z)$ to the Gouy phase.",
"Notice however that the main difference between the wave-packet (REF ) and the Gaussian mode of the homogeneous case [28], [27] is the $z$ -dependence of the beam width.",
"Indeed, we have already mentioned that $w(z)$ in (REF ) is a solution of the Ermakov equation (REF ), so we follow [29] to get $w(z) = w_0\\left[\\cos ^2 \\left[\\Omega (z-z_0)\\right] + \\frac{1}{\\left(\\Omega z_R\\right)^2} \\sin ^2 \\left[\\Omega (z-z_0)\\right]\\right]^{1/2},$ with $z_0$ an integration constant, $w_0 = w(z_0)$ , and $z_R = \\tfrac{1}{2} k_0n_0w_0^2$ .",
"Figure: R(z)R(z)For $\\Omega > 0$ the amplitude of the beam width (REF ) oscillates with period $\\pi /\\Omega $ between $w_0$ and $w_0/(\\Omega z_R)^2$ .",
"The latter values are respectively reached at the points $z= \\left( n+\\tfrac{1}{2} \\right) \\frac{\\pi }{\\Omega } + z_0$ and $z= n \\frac{\\pi }{\\Omega } + z_0$ , with $n=0,1,2,\\ldots $ Thus, in contraposition to the homogeneous case where $w(z)$ diverges for large values of $z$ (see Section REF ), the beam width introduced in (REF ) is finite over all the $z$ -axis.",
"Moreover, the maximum amplitude reached by $w(z)$ can be adjusted by varying $\\Omega $ , see Figure REF (a).",
"Now let us write explicitly the expression for the radius of curvature.",
"From (REF ) and (REF ), one obtains $R(z) =\\frac{z_R^2 \\Omega ^2 \\cot [\\Omega (z-z_0)] + \\tan [\\Omega (z-z_0)]}{ \\Omega ( 1 - \\Omega ^2 z_R^2)}.$ The latter expression diverges at the critical points of $w(z)$ , see Figure REF (b).",
"Thus, the wavefront of the Gaussian wave-packet (REF ) is plane at either the beam waist (minimum beam width) or the beam `hip' (maximum beam width).",
"On the other hand, the oscillatory profile of the beam width (REF ) is inherited to the wave-packet (REF ).",
"Indeed, the field intensity $\\vert U(\\rho ,z)\\vert ^2$ describes a spot centered at $\\rho =0$ that increases its diameter till a maximum value as the wave-packet propagates along the $z$ -axis.",
"Then, the spot starts to reduce its diameter up to recover its initial configuration.",
"This stretching and squeezing phenomenon is repeated over and over along the propagation axis, see Figure REF .",
"Such behavior corresponds to the self-focusing of the beam, which is a consequence of the parabolic profile of the refractive index (REF ).",
"The number of stretching and squeezing of the beam width in a given interval $z \\in (a,b) \\subset \\mathbb {R}$ is determined by the parameter $\\Omega $ , in complete agreement with the periodical profile of the beam width (REF ).",
"Figure: Field intensity |U(ρ,z)| 2 \\vert U(\\rho ,z)\\vert ^2 of the Gaussian wave-packet () for z 0 =0z_0=0 and k 0 =n 0 =w 0 =1k_0=n_0=w_0=1.",
"The upper and lower rows correspond to Ω=0.5\\Omega = 0.5 and Ω=1\\Omega =1, respectively.",
"From left-to-right the columns show the transversal plane where the beam width reaches the first of its maxima, the longitudinal plane y=0y=0, and the propagation of the wave-packet along the zz-axis.",
"Compare with Figure ."
],
[
"Recovering the results for homogeneous media",
"The description of the propagation in homogeneous media is obtained at the limit $\\Omega \\rightarrow 0^+$ .",
"For instance, at such a limit the beam width acquires the well known form [27], [28]: $w(z)_{\\Omega \\rightarrow 0^+} = w_{\\operatorname{hom}}(z) = w_0 \\sqrt{1 + \\left(\\frac{z-z_0}{z_R}\\right)^2}.$ In this case $w_0 = w_{\\operatorname{hom}}(z_0)$ corresponds to the beam waist.",
"In turn, $z=z_0$ defines the focal plane and $z_R$ stands for the distance from such plane to the position at which the spot of the beam doubles its size.",
"On the other hand, for large values of $z$ , Eq.",
"(REF ) can be approximated as $w_{\\operatorname{hom}}(z) \\sim \\frac{w_0}{z_R} (z-z_0)$ .",
"Thus, the beam in homogeneous media diverges as $\\vert z \\vert \\rightarrow \\infty $ , see Figure REF (a), and describes a cone of half-angle defined by the beam angular divergency $\\theta _0 \\sim \\frac{w_0}{z_R}$ (it is subtended by the gray-dashed line with respect to the horizontal axis in the figure).",
"The corresponding radius of curvature $R_{\\operatorname{hom}}(z)$ is depicted in Figure REF (b), where it is compared with the radius of curvature of a spherical wavefront produced by a point source located at the center of the beam waist (gray-dotted line in the figure), and with the function $R(z)$ defined in (REF ) for $\\Omega =0.3$ (magenta curve in the figure).",
"Notice that $R_{\\operatorname{hom}}(z)$ has only a singular point, located at $\\rho =0$ , which means that the wavefront of the corresponding wave-packet is plane at the beam waist only, so no self-focusing is predicted for homogeneous media, as expected.",
"Figure: R hom (z)R_{\\operatorname{hom}}(z)"
],
[
"Laguerre-Gaussian wave-packets",
"We wonder whether there are other wave-packet solutions to the paraxial wave equation associated with a parabolic refractive index.",
"The answer is positive (see, e.g.",
"and [18]) and it may be shown that, using cylindric coordinates, a particularly useful set of solutions can be cast in the form $U(\\rho ,z) = \\frac{\\mathcal {N}}{w(z)}\\exp \\left( i \\left[\\frac{k_0n_0\\rho ^2}{2R(z)} - \\beta \\chi (z)\\right]\\right) \\Phi (r(\\rho ,z),\\theta ),$ where $\\mathcal {N}$ and $\\beta $ are constants to be determined, $\\theta $ stands for the polar coordinate in the transversal plane, and $r(\\rho ,z) = \\sqrt{2} \\frac{\\rho }{w(z)}$ is the corresponding radial coordinate.",
"The straightforward calculation shows that the function $\\Phi (r,\\theta )$ satisfies the differential equation $- \\nabla _{\\perp }^2\\Phi + r^2 \\Phi = 2 \\beta \\Phi ,$ which resembles the stationary Schrödinger equation for a two dimensional oscillator in the radial variable $r$ .",
"The resemblance is complete if one considers that square integrability in the Hilbert space corresponds to finite transverse optical power $P_0$ for localized beams in the $(r,\\theta )$ -plane.",
"That is, we demand the field intensity of the wave-packets (REF ) to satisfy the condition $P_0 = \\int _0^{2\\pi } \\int _0^\\infty \\left|U_\\ell ^p(\\rho ,z)\\right|^2 \\rho d\\rho d\\theta = 1.$ Figure: Field intensity |U ℓ p (ρ,z)| 2 \\vert U_{\\ell }^p (\\rho , z) \\vert ^2 of the Laguerre-Gaussian wave-packet () for p=0p=0 and ℓ=1\\ell =1, with the same parameters and distribution as the panel shown in Figure .The conventional approach used to face stationary problems in quantum mechanics yields the eigenvalues $\\beta _\\ell ^p = \\vert \\ell \\vert + 2p + 1, \\quad \\ell \\in \\mathbb {Z}, \\quad p=0,1,2,\\ldots ,$ together with the eigenfunctions $\\Phi _{\\ell }^p (r,\\theta ) = \\left(\\frac{\\sqrt{2}\\rho }{w(z)}\\right)^{\\vert \\ell \\vert }L_p^{(\\vert \\ell \\vert )} \\left(\\frac{2\\rho ^2}{w^2(z)}\\right) \\exp \\left(-\\frac{\\rho ^2}{w^2(z)} + i \\ell \\theta \\right),$ and the constant $\\mathcal {N}_{\\ell }^p = (-1)^p \\sqrt{\\frac{2 \\Gamma (p + 1)}{\\pi \\Gamma (\\vert \\ell \\vert + p + 1)}}.$ The introduction of (REF )-(REF ) into (REF ) gives the expressions $U_{\\ell }^p(\\rho ,z)$ we were looking for.",
"Hereafter they are referred to as Laguerre-Gaussian (LG) wave-packets.",
"The results discussed in Section REF for the nonstationary Gaussian wave-packets are recovered from the above formulae after making $p=0$ and $\\ell =0$ , which respectively means the lowest radial parameter and null orbital angular momentum.",
"For $p=0$ and $\\ell \\ne 0$ the field intensity $\\vert U_{\\ell }^p (\\rho , z) \\vert ^2$ describes a ring centered at $\\rho =0$ , see Figure REF .",
"In general, for $\\ell \\ne 0$ and any allowed value of $p$ , the field intensity of the LG wave-packets (REF ) exhibits a well known ring-shaped distribution.",
"The size of the rings depends on $\\ell $ while $p$ defines the number of nodes in the radial coordinate, see Figure REF and compare with Figs.",
"REF and REF .",
"Thus, controling the values of $p$ , $\\ell $ , and $\\Omega $ , one can manipulate the profile as well as the collapse-revival (self-focusing) properties of the LG wave-packets (REF ).",
"Figure: Field intensity |U ℓ p (ρ,z)| 2 \\vert U_{\\ell }^p (\\rho , z) \\vert ^2 of the Laguerre-Gaussian wave-packet () for p=1p=1 and ℓ=8\\ell =8, with the same parameters and distribution as the panel shown in Figure ."
],
[
"Recovering the results for guided Laguerre-Gaussian modes",
"If the parameters $w_0$ and $\\Omega $ are related as $k_0n_0 \\Omega = 2/w_0 ^2$ , then $\\Omega z_R = 1$ and, according to (REF ), the beam width becomes a constant $w(z)=w_0$ , see [20].",
"In such a case the radius of curvature $R(z)$ diverges and the Gouy phase turns into a linear function of $z$ .",
"Thus, the wavefront of the LG wave-packets is plane.",
"The corresponding field amplitudes, known as guided LG modes, take the form $\\begin{array}{rl}U_\\ell ^p(\\rho ,z) & = (-1)^p \\sqrt{\\frac{2 \\Gamma (p + 1)}{\\pi w_0^2\\Gamma (\\vert \\ell \\vert + p + 1)}} \\left(\\frac{\\sqrt{2}\\rho }{w_0}\\right)^{\\vert \\ell \\vert }L_p^{(\\vert \\ell \\vert )} \\left(\\frac{2\\rho ^2}{w^2_0}\\right) \\\\[2ex]& \\qquad \\times \\exp \\left( -\\frac{\\rho ^2}{w^2_0} -\\beta _\\ell ^p \\Omega (z-z_0) + i \\ell \\theta \\right),\\end{array}$ and are stationary eigenmodes of the Schrödinger-like operator $H= -\\frac{1}{2 k_0^2 n_0} \\nabla _{\\perp }^2 + \\frac{n_0}{2}\\Omega ^2 \\rho ^2,$ with the propagation constants $\\varepsilon _\\ell ^p = \\frac{\\Omega }{k_0} \\beta _\\ell ^p \\equiv \\frac{\\Omega }{k_0}(\\vert \\ell \\vert + 2p + 1), \\quad \\ell \\in \\mathbb {Z}, \\quad p=0,1,2,\\ldots $"
],
[
"Discussion of results",
"We have shown that the beam width of the Laguerre-Gaussian wave-packets is finite and periodic along all the propagation axis if it is a solution of the Ermakov equation associated with the one-dimensional harmonic oscillator.",
"The amplitude of the beam width can be modulated by the strength of the medium inhomogeneity.",
"The wave-packets so constructed have finite transverse optical power and propagate with periodical self-focusing in the medium.",
"The conventional Laguerre-Gaussian modes are recovered as a particular case, after turning-off the inhomogeneity.",
"Since the orbital angular momentum is a consequence of the mode structure of a given beam, one would guess that such a property is also present in single photons [6].",
"The statement seems to be strengthened by recalling that idealized plane waves with only transverse fields do not carry angular momentum, no matter their degree of polarization [30].",
"A clue may be found in the phenomenon of parametric-down conversion [31], [32], where the quantum state of down-converted photon pairs is entangled in at least one of their physical variables.",
"As entanglement is a fingerprint of the quantum world, the production of photon pairs entangled in their orbital angular momentum states [33] bets on a quantum nature of structured light.",
"It is then interesting to formulate an approach based on operators in Hilbert spaces to describe the propagation of structured light beams in diverse media.",
"Some initial steps on the matter were given in [34].",
"The complete operator-like description for the propagation of Hermite-Gaussian wave-packets in parabolic media has been provided in [18], [19], [20], where the Lie group formalism was addressed to construct generalized coherent states as linear superpositions of Hermite-Gaussian modes.",
"The same approach can be applied to the Laguerre-Gaussian wave-packets reported in this paper.",
"Work in this direction is in progress."
],
[
"Acknowledgment",
"This research was funded by Consejo Nacional de Ciencia y Tecnología (Mexico), grant number A1-S-24569, and by Instituto Politécnico Nacional (Mexico), project SIP20195981."
]
] | 2001.03456 |
[
[
"Identification and nonlinearity compensation of hysteresis using NARX\n models"
],
[
"Abstract This paper deals with two problems: the identification and compensation of hysteresis nonlinearity in dynamical systems using nonlinear polynomial autoregressive models with exogenous inputs (NARX).",
"First, based on gray-box identification techniques, some constraints on the structure and parameters of NARX models are proposed to ensure that the identified models display a key-feature of hysteresis.",
"In addition, a more general framework is developed to explain how hysteresis occurs in such models.",
"Second, two strategies to design hysteresis compensators are presented.",
"In one strategy the compensation law is obtained through simple algebraic manipulations performed on the identified models.",
"It has been found that the compensators based on gray-box models outperform the cases with models identified using black-box techniques.",
"In the second strategy, the compensation law is directly identified from the data.",
"Both numerical and experimental results are presented to illustrate the efficiency of the proposed procedures."
],
[
"Introduction",
"Hysteresis is a nonlinear behavior that is present in several systems and devices.",
"It is commonly related to the phenomena of ferromagnetism, plasticity, and friction, among others [1].",
"Some examples include mechanical, electronic and biomedical systems, as well as sensors and actuators such as magneto-rheological dampers, piezoelectric actuators and pneumatic control valves [2], [3], [4].",
"An intrinsic feature of such systems is the memory effect, meaning that their output depends on the history of the corresponding input.",
"In addition to the memory effect, the literature provides different definitions and conditions to distinguish such systems and characterize the hysteretic behavior.",
"In some cases, the occurrence of hysteresis has been associated with the existence of several equilibrium points whenever these systems are subject to a constant [5] or time-varying [6] input signal.",
"Additionally, hysteresis has also been defined as a hard nonlinearity that depends on the magnitude and rate of the input signal.",
"These aspects can pose various performance limitations if not properly taken into account during the control design [7], [3].",
"Hence, a common goal is to attenuate the hysteretic behavior of the system [8], [9], [10].",
"In many approaches, the compensation of hysteresis starts with obtaining a suitable model.",
"In the literature, several hysteresis models have been proposed based on phenomenological, black-box and gray-box modeling approaches.",
"In the realm of models based on first principles, important contributions have been made based on differential equations and operators [11], such as the Bouc-Wen model [12], the Duhem model [13], the Preisach model [14] and the Prandtl-Ishlinskii operator [15].",
"These models have been widely used to predict the hysteresis behavior due to their ability to describe a variety of hysteresis loops that resemble the proprieties of a wide class of real nonlinear hysteretic systems [16].",
"Besides, such models are known to be challenging for system identification techniques [17].",
"Apart from the computational effort required in the identification of phenomenological models, their application in the design of compensators is somewhat limited due to their structural complexity [4], [11].",
"Black-box modeling does not rely on prior knowledge about the system [18], [19], [20].",
"Unfortunately, relevant features that should be present in a model to reproduce hysteresis and an appropriate structure for designing compensators are not ensured by black-box techniques.",
"Hence, the search for models that have specific features, that are accurate and that have a suitable structure for designing compensators remains an open problem.",
"Models obtained using gray-box techniques can be tailored to reproduce specific relevant features [21].",
"In this context, nonlinear autoregressive with exogenous inputs (NARX) models are considered a convenient choice due to their ability to predict a wide class of nonlinear behaviors [22], [23].",
"Another interesting feature is the structural flexibility of such models.",
"Therefore, enforcing constraints on the model structure (e.g., in order to make it suitable for designing compensators) does not drastically affect its predictive ability [24].",
"Despite these promising aspects, the literature on this approach for modeling and compensating of hysteretic systems is scarce [25], [26], [27], [28], [6], [29].",
"In this sense, an important step in modeling the hysteresis nonlinearity was advanced in [6], in which sufficient conditions are presented for NARX models to display a hysteresis loop when subject to a certain class of input signals.",
"The concept of a bounding structure ${\\cal H}$ formed by sets of stable equilibria and its implication on the existence of the hysteresis loop in the identified models has also introduced in [6].",
"However, for more general cases, this concept and conditions need to be adapted.",
"For instance, the conditions proposed in [6] are not sufficient to ensure the existence of several equilibrium points at steady-state [30], [5].",
"Also, the concept of bounding structure is limited to cases in which the sets of equilibria that form this structure are stable.",
"Recently, a NARX model was identified for an experimental electronic circuit with hysteresis [29].",
"In this paper, we will propose ways to overcome some shortcomings pointed out in the aforementioned references and a more general framework will be put forward to explain how hysteresis takes place in identified models.",
"The main contributions of this work are: the proposition of a specific parameter constraint that ensures reproducing a key-feature of hysteresis through identified NARX models.",
"A framework is put forward to explain how the hysteresis loop results from an interplay of attracting and repelling regions in the input-output plane.",
"Moreover, some structural specifications are enforced during the identification procedure in such a way that the identified NARX model can be effectively used to mitigate the hysteresis nonlinearity.",
"Hence, two model-based compensation strategies are introduced.",
"In the first the compensation law is obtained through simple algebraic manipulations performed on the identified models.",
"It has been found that the compensators based on gray-box models outperform the cases with models identified using black-box techniques.",
"In the second strategy, the compensation law is directly identified from the data.",
"This work is organized as follows.",
"Section presents the background.",
"A constraint to ensure hysteresis in the identified models and a framework for understanding how the hysteresis loop is formed are provided in section .",
"Based on NARX models, two strategies to design compensators are detailed in section .",
"The numerical and experimental results for the model identification and the compensator design are, respectively, given in sections and .",
"Section presents the concluding remarks."
],
[
"Background",
"A NARX model can be represented as [22]: $y_k = \\tilde{F}\\big (y_{k-1},\\cdots ,y_{k-n_{y}}, \\,u_{k-\\tau _{\\rm d}},\\cdots ,u_{k-n_u}\\big ),$ where $y_k \\in \\mathbb {R}$ is the output at instant $k \\in \\mathbb {N}$ , $u_k \\in \\mathbb {R}$ is the input, $n_y$ and $n_u$ are the maximum lags for the output and input, respectively, $\\tau _{\\rm d} \\in \\mathbb {N}^{+}$ is the pure time delay, and $\\tilde{F}(\\cdot )$ is a nonlinear function of the lagged inputs and outputs.",
"This work considers a linear-in-the-parameters extended model set [31] of the NARX model (REF ) with the addition of specific functions, such as absolute value, trigonometric, and sign function.",
"The goal is to choose functions that allow the models to predict systems whose nonlinearities cannot be well approximated using only regressors based on monomials of lagged input and output values.",
"For instance, [31] recommends the addition of absolute value and sine functions as candidate regressors for the identification of a damped and forced nonlinear oscillator.",
"In the case of the identification of systems with hysteresis, [6] shows that including the regressor given by sign of the first difference of the input, i.e.",
"${\\rm sign}(u_k-u_{k-1})$ , in addition to polynomial terms is a sufficient condition to reproduce hysteresis.",
"Therefore, in this work the models are of the general type: $y_k & = & F^{\\ell }\\big (y_{k-1},\\cdots ,y_{k-n_{y}}, \\,u_{k-1},\\cdots ,u_{k-n_u}, \\nonumber \\\\& ~ &\\hspace{11.38092pt} \\phi _{1,\\,k-1}, \\,\\phi _{2,\\,k-1} \\big ),$ where $\\phi _{1,\\,k}{=}u_k-u_{k-1}$ , $\\phi _{2,\\,k} {=} {\\rm sign}(\\phi _{1,\\,k})$ , and $F^{\\ell }(\\cdot )$ is a polynomial function of the regressor variables up to degree $\\ell \\in \\mathbb {N}^{+}$ .",
"Evaluating model (REF ) along a data set of length $N$ , the resulting set of equations can be expressed in matrix form as: ${y}=\\Psi \\hat{{\\theta }}+{\\xi },$ where ${y} \\triangleq [y_k\\,\\, y_{k-1}\\,\\cdots \\, y_{k+1-N}]^T \\in \\mathbb {R}^N$ is the vector of output measurements, $\\Psi \\triangleq [\\psi _{k-1}^T;\\, \\cdots ;\\, \\psi ^T_{k-N}] \\in \\mathbb {R}^{N\\times n_{\\theta }}$ is the matrix composed by measurements of the regressors vector $\\psi _{k-1} \\in \\mathbb {R}^{n_{\\theta }}$ which contains linear and nonlinear combinations of the variables that compose $F^{\\ell }(\\cdot )$ in (REF ) weighted by the parameter vector $\\hat{{\\theta }} \\in \\mathbb {R}^{n_{\\theta }}$ , ${\\xi } \\triangleq [\\xi _k\\,\\, \\xi _{k-1}\\,\\cdots \\, \\xi _{k+1-N}]^T \\in \\mathbb {R}^{N}$ is the residual vector and $T$ indicates the transpose.",
"The unconstrained least squares batch estimator is given by $\\hat{{\\theta }}_{\\rm LS}=(\\Psi ^T\\Psi )^{-1}\\Psi ^T{y}.$ Assume the set of equality constraints on the parameter vector written as ${c}=S{\\theta }$ , where ${c} \\in \\mathbb {R}^{n_c}$ and $S \\in \\mathbb {R}^{n_c\\times n_{\\theta }}$ are known constants.",
"Then, the constrained least squares estimation problem is given by $\\hat{{\\theta }}_{\\rm CLS} = \\underset{{\\theta }\\,:\\,{c}=S{\\theta }}{\\arg \\min } \\big [{\\xi }^T{\\xi }\\big ],$ whose solution is [32]: $\\hat{{\\theta }}_{\\rm CLS} {=} \\hat{{\\theta }}_{\\rm LS}{-}(\\Psi ^T\\Psi )^{-1}S^T[S(\\Psi ^T\\Psi )^{-1}S^T]^{-1}(S\\hat{{\\theta }}_{\\rm LS}{-}{c}).",
"$ For the model structure selection, we use the error reduction ratio (ERR) [33] together with Akaike's information criterion (AIC) [34].",
"Other approaches that have proved to be useful in more demanding contexts are found in [35], [36], [37], [38], [39]."
],
[
"Identification of Systems with Hysteresis",
"The scientific community has been investigating which relevant features must be present in a model to reproduce hysteresis.",
"Some of these features are: a characteristic loop behavior displayed on the input-output plane [30], several stable equilibrium points [5], and multi-valued mapping [26].",
"However, which and how these features can be used in the identification procedure remains an open problem.",
"Here, a constraint is proposed to ensure a key-feature of hysteresis.",
"Also, it is shown how the hysteresis loop can be seen as an interplay of attracting and repelling regions in certain models.",
"Then, the resulting models will be used to design compensators.",
"First, a property of hysteresis based only on the main aspect discussed in [6], [30], [5] is presented.",
"In the sequel, this property is used to obtain constraints on the structure and parameters of the model.",
"Property 1 An identified model of hysteresis, under a constant input, has two or more real non-diverging equilibria.",
"$\\square $ In [6], Property REF was attained by ensuring that the model had at least one equilibrium point under loading-unloading inputs, with different values for loading and unloading.",
"Thus, in (REF ) $\\phi _{1,\\,k}{=}u_k-u_{k-1}$ and $\\phi _{2,\\,k} {=} {\\rm sign}(\\phi _{1,\\,k})$ , with $\\phi _{2,\\,k} {=}1$ for loading, and $\\phi _{2,\\,k} {=}-1$ for unloading.",
"Hence, hysteresis is a nonlinear behavior that appears in both the static response and the dynamics.",
"In some works, this nonlinearity is classified as quasi-static because the analyses are performed when the system is excited by a periodic signal that is very slow compared to the system dynamics [40].",
"Based on a static analysis of NARX models (REF ), we will show which constraints need to be considered in the identification procedure in order for Property REF to be satisfied.",
"Thereafter, a quasi-static analysis will be used to describe how hysteresis happens in these models and an illustrative example will be presented."
],
[
"Static analysis",
"By means of static analysis it is possible to determine the fixed points of a model.",
"Assumption 1 In order to comply with Property REF , considering the recommendation of the literature, the identified models should not have the following regressors: (i) $y^{p}_{k-\\tau _y}$ , $y^{p}_{k-\\tau _y}\\phi _{1,\\,k-\\tau _u}^m$ and $y^{p}_{k-\\tau _y}\\phi _{2,\\,k-\\tau _u}^m$ for $p{>}1,~\\forall m$ [41], (ii) ${\\rm sign}(u_{k-\\tau _u}-u_{k-\\tau _u-1})^m=\\phi ^m_{2,\\,k-\\tau _u}$ for $m>1$ [6], as will be shown in this paper, the following regressors can also be removed (iii) $y^{p}_{k-\\tau _y}u^m_{k-\\tau _u},~\\forall p,\\,m$ , where $\\tau _y$ and $\\tau _u$ are any time lags.",
"$\\square $ The steady-state analysis of a model that complies with Assumption REF is done by taking $y_k=\\bar{y},\\,\\forall k$ , $u_k=\\bar{u},\\,\\forall k$ and, consequently, $\\phi _{1,\\,k}=u_k-u_{k-1}=0$ $\\phi _{2,\\,k}={\\rm sign}(\\phi _{1,\\,k})=0,\\,\\forall k$ , thus yielding $\\bar{y}=\\Sigma _y\\bar{y}$ , where $\\Sigma _y$ is the sum of all parameters of all linear output regressors.",
"For $\\Sigma _y \\ne 1$ the model has a single fixed point at $\\bar{y}=0$ with stability domain given by: $-1 < \\qquad \\theta _1 \\qquad < 1.$ If $|\\theta _1| < 1$ ($|\\theta _1| > 1$ ), then $\\bar{y}=0$ is only non-diverging (diverging) equilibrium and, as a result, Property REF is not satisfied.",
"In order to solve this problem, we start be reviewing the following definition.",
"Definition 1 (Continuum of equilibrium points [5]).",
"A model has a continuum of equilibrium points if for any constant value of the input its corresponding output in steady-state is an equilibrium solution.",
"$\\square $ Based on Definition REF and the problem aforementioned, the following lemma is stated.",
"Lemma 1 Given that Assumption REF holds, if $\\Sigma _y=1$ is verified, then the identified model has a continuum of equilibrium points at steady-state.",
"$\\square $ Proof.",
"The steady-state analysis of a model that satisfies Assumption REF and Lemma REF yields $\\bar{y}=\\bar{y}$ which is trivially true for any value $\\bar{y}$ .",
"Hence, the model has a continuum of equilibrium points and Property REF is satisfied.",
"$\\square $"
],
[
"Quasi-static analysis",
"The core idea of the framework proposed in [6] to identify models with a hysteresis loop is to build a bounding structure ${\\cal H}$ made of sets of equilibria and to ensure that one set is stable during loading and the other one, during unloading.",
"Such a scenario is effective, but it does not help to understand models with more complicated structures and with both attracting and repelling regions in the $u \\times y$ plane.",
"This section aims at enlarging the scenario developed in [6].",
"In quasi-static analysis, it is assumed that the input $u_k$ is a loading-unloading signal that is much slower than the system dynamics to the point that, at a given time $k$ , the system will be in a certain attracting region, avoiding any possible repelling regions.",
"Also, such regions depend on $u_k$ , $\\phi _{1,\\,k}$ and $\\phi _{2,\\,k}$ .",
"More specifically, there will be two sets of regions, one for loading and another for unloading.",
"In quasi-static analysis, we assume that $y_k\\approx y_{k-j}=\\tilde{y},~j=1,\\,2,\\ldots ,\\,n_y$ , such that (REF ) is given by $\\tilde{y} & \\approx & F^{\\ell }\\big (\\tilde{y}, \\,u_{k-1},\\cdots ,u_{k-n_u}, \\, \\phi _{1,\\,k-1}, \\,\\phi _{2,\\,k-1} \\big ),$ which can be usually solved for $\\tilde{y}$ , especially if higher powers of the output are not in $F^{\\ell }(\\cdot )$ [41].",
"This is achieved in practice by removing such group of terms from the set of candidates.",
"If the model has no inputs, then $\\tilde{y}$ coincides with the fixed points.",
"Alternatively, if the inputs are all constant, then $\\tilde{y}$ is a family of fixed points that depends on the set of constant inputs.",
"Given the slow input, if $\\tilde{y}$ is in an attractive region, then the model output moves towards $\\tilde{y}$ .",
"In what follows, $\\tilde{y}_{\\rm L}^{\\rm a}$ and $\\tilde{y}_{\\rm U}^{\\rm a}$ are, respectively, the solutions to (REF ) in attracting regions under loading and unloading.",
"Likewise, $\\tilde{y}_{\\rm L}^{\\rm r}$ and $\\tilde{y}_{\\rm U}^{\\rm r}$ are their counterparts in repelling regions.",
"The conditions for $\\tilde{y}$ to be attracting is $\\left| {\\rm eig} \\left( \\frac{\\partial F^{\\ell }({y},\\,u_{k-1},\\,\\phi _{1,\\,k-1},\\, \\phi _{2,\\,k-1})}{\\partial {y}} \\right) \\right|<1 ,$ where ${y}=[y_{k-1}\\, \\ldots y_{k-n_y}]^T$ .",
"This procedure resembles that of determining the stability of fixed points.",
"Here the Jacobian matrix is not evaluated at fixed points.",
"Hence we do not speak in terms of stable and unstable fixed points.",
"To illustrate how this helps to understand the formation of a hysteresis loop, consider the schematic representation in figure REF .",
"The input is a loading-unloading signal such that $u_{\\rm min} \\le u_k \\le u_{\\rm max},~\\forall k$ .",
"The sets $\\tilde{y}_{\\rm L}^{\\rm a}$ , $\\tilde{y}_{\\rm U}^{\\rm a}$ , $\\tilde{y}_{\\rm L}^{\\rm r}$ and $\\tilde{y}_{\\rm U}^{\\rm r}$ are shown.",
"Consider the point A, which takes place under loading.",
"Given that the system is under the direct influence of $\\tilde{y}_{\\rm L}^{\\rm r}$ , which is responsible for pushing upwards (see vertical component $y_{\\rm A}$ ), and it is the loading regime, there is a horizontal component $u_{\\rm A}$ (related to the input) that points to the right.",
"The resulting effect is to pull the system along the loop in the NE direction.",
"The same can be said for point B; however, at that point the vertical component is the result of the attracting action of $\\tilde{y}_{\\rm L}^{\\rm a}$ .",
"A similar analysis can be readily done for the unloading regime, given by points D and E. At the turning points C and F, $\\phi _{2,\\,k}$ switches from 1 to -1 and from -1 to 1, respectively.",
"Hence the analysis also switches from using $\\tilde{y}_{\\rm L}^{\\rm a}$ and $\\tilde{y}_{\\rm L}^{\\rm r}$ , to using $\\tilde{y}_{\\rm U}^{\\rm a}$ and $\\tilde{y}_{\\rm U}^{\\rm r}$ .",
"This analysis will be useful in section to understand the formation of hysteresis loops in identified models.",
"Figure: Schematic representation of hysteresis loop in the u×yu \\times y plane.Attracting sets are shown in black continuous lines, whereas the repellingsets are indicated in red dash-dot.",
"The hysteresis loop is indicated by dotted lines.As a final remark, it is important to point out that the assumption that the set $\\tilde{y}$ comes in two disjoint parts, either for loading or unloading, is a consequence of the solution of (REF ) being rational instead of polynomial.",
"This is useful to analyse models with more general model structures.",
"The following example illustrates the application of this analysis and show which constraints should be considered to comply with Property REF .",
"Example 1 Consider the following NARX model that complies with Assumption REF : $y_k&{=}\\theta _1y_{k-1}+\\theta _2\\phi _{2,\\,k-1}{+}\\theta _3\\phi _{1,\\,k-1}u_{k-1} \\nonumber \\\\&{+}\\theta _4\\phi _{2,\\,k-1}\\phi _{1,\\,k-1}y_{k-1}{+}\\theta _5\\phi _{1,\\,k-1}.$ In this case, the constraint $\\theta _1=1$ will be used such that, according to Lemma REF , the resulting model will have a continuum of equilibrium points.",
"This can be achieved using estimator (REF ) with $c=1$ and $S=[1\\,\\, 0\\,\\, 0\\,\\, 0\\,\\, 0]$ .",
"For a more complicated model structure, the constraint in Lemma REF is still in the form $1=S{\\theta }$ (REF ) but with $S$ having more that one element equal to one, e.g.",
"as shown in [42] to obtain NARX models able to reproduce dead-zone and in [43] for a quadratic nonlinearity.",
"The quasi-static analysis of model (REF ) is performed following the steps provided in section REF .",
"So rewriting this model as (REF ), we have $\\tilde{y} & \\approx & \\theta _1\\tilde{y}{+}\\theta _2{\\phi }_{2,k-1}{+}\\theta _3{\\phi }_{1,k-1}u_{k-1} \\nonumber \\\\& ~ & +\\theta _4{\\phi }_{2,k-1}{\\phi }_{1,k-1}\\tilde{y} {+}\\theta _5{\\phi }_{1,k-1}, \\nonumber $ which can be described by $\\tilde{y}({u},{\\phi }_{1},{\\phi }_{2}) {=}\\left\\lbrace \\begin{array}{ll}{\\frac{\\theta _2+\\theta _3{\\phi }_{1}{u}+\\theta _5{\\phi }_{1}}{1-\\theta _1-\\theta _4{\\phi }_{1}}, {\\rm for} \\,\\,{\\phi }_{2}{=}\\,\\,\\,1; \\\\\\frac{-\\theta _2+\\theta _3{\\phi }_{1}{u}+\\theta _5{\\phi }_{1}}{1-\\theta _1+\\theta _4{\\phi }_{1}}, {\\rm for} \\,\\,{\\phi }_{2}{=}-1,\\\\}\\end{array}\\right.where the time indices have been omitted for simplicity.",
"Therefore, the solutiongiven at the top in (\\ref {190919a}) represents the set \\tilde{y}_{\\rm L}, while the bottomis the set \\tilde{y}_{\\rm U}.$ To define whether the solutions to (REF ) are in the attracting or repelling regions, (REF ) should be computed for model (REF ) as $-1<& \\theta _1 + \\theta _4{\\phi }_{2,k-1}{\\phi }_{1,k-1} &<1, \\nonumber \\\\\\frac{-1-\\theta _1}{\\theta _4\\phi _{2,k-1}}<& {\\phi }_{1,k-1} &< \\frac{1-\\theta _1}{\\theta _4\\phi _{2,k-1}}.$ Since it is assumed that the known input $u_k$ is a non-zero loading-unloading signal, then the conditions (REF ) to ensure that the solutions (REF ) are in attracting regions can be readily verified numerically.",
"In sections and , the same analyses will be performed for the identified models.",
"$\\square $"
],
[
"Compensator Design",
"The proposed strategies to design compensators based on NARX models are detailed in this section, starting with some preliminary assumptions.",
"In this paper, a key point is to investigate if hysteresis in the models estimated according to section have any impact on the regulation performance."
],
[
"Preliminaries",
"Given a nonlinear system $\\mathcal {S}$ , the first step is to obtain hysteretic models for $\\mathcal {S}$ ; see figure REF (a).",
"To achieve that, two procedures will be followed.",
"The first one aims at identifying a model $\\mathcal {M}$ based on the direct relationship between $u$ and $y$ , whose simulation yields $\\hat{y}_k$ , according to section REF .",
"The second procedure is based on the identification of the inverse relationship, in which case a model $\\breve{\\mathcal {M}}$ is obtained to yield $\\hat{u}_k$ , as illustrated in figure REF (a) and following section REF .",
"In the second step, the identified model is used to design a compensator $\\mathcal {C}$ that yields the compensation signal $m_k$ for a given reference $r_k$ ; see figure REF (b).",
"Figure: Compensator design based on identified NARX models.",
"(a) Model identification, and (b) compensator design based on identified models.In this paper,the following additional assumptions are made for NARX models (REF ).",
"Remark 1 For compensation design, $y_k$ can be replaced by $r_k$ , and $u_k$ by $m_k$ , respectively, in the models $\\mathcal {M}$ and $\\breve{\\mathcal {M}}$ .",
"$\\square $ The motivation behind this is that $y_k$ should ideally be equal to $r_k$ under compensation, that is, when $m_k$ is used as an input to the dynamical system.",
"In what follows, the main idea is to use an identified model to determine the compensation input $m_{k-\\tau _{\\rm d}+1}$ ."
],
[
"Model-Based Compensation",
"The aim here is to specify a general model structure for $\\mathcal {M}$ in order to find $m_k$ analytically from this model.",
"To achieve that, the following assumptions are needed.",
"Assumption 2 Assume that: (i) the only regressor involving $u_{k-\\tau _{\\rm d}}$ is linear; (ii) $n_u>\\tau _{\\rm d}$ ; (iii) the compensation signal $m_k$ is known up to time $k-\\tau _{\\rm d}$ ; and (iv) the reference signal $r_k$ is known up to time $k+1$ .",
"$\\square $ Assumption REF imposes conditions on the selection of the model structure.",
"Note that (i) ensures that $u_{k-\\tau _{\\rm d}}$ can be isolated in the identified models; (ii) allows that input terms with a delay longer than $\\tau _{\\rm d}$ to be regressors in the identification procedure; and the other constraints guarantee that the control action can be computed from known values.",
"Therefore, the model $\\mathcal {M}$ is rewritten as $A(q)y_k {=} B(q)u_{k}{+}f\\big (y_{k-1},{\\cdots },y_{k-n_{y}},u_{k-\\tau _{\\rm d}-1},{\\cdots },u_{k-n_u}\\big )\\!,$ where $q^{-1}$ is the backward time-shift operator such that $q^{-1}u_k{=}u_{k-1}$ , and the linear regressors are grouped in $A(q)y_k$ and $B(q)u_{k}$ with $A(q) &=& 1 - a_1q^{-1} - a_2q^{-2} - \\cdots - a_{n_{y}}q^{-n_{y}}, \\\\B(q) &=& b_{\\tau _{\\rm d}}q^{-\\tau _{\\rm d}} + \\underbrace{b_{\\tau _{\\rm d}+1}q^{-\\tau _{\\rm d}-1} + \\cdots + b_{n_{u}}q^{-n_{u}}}_{B^*(q)}, $ and $f(\\cdot )$ includes all the nonlinear terms and possibly the constant term of the NARX model (REF ).",
"Using (), model (REF ) can be rewritten as $A(q)y_k & = & b_{\\tau _{\\rm d}} u_{k-\\tau _{\\rm d}}{+}B^*(q)u_{k}{+}f\\big (y_{k-1}, \\cdots ,y_{k-n_{y}},\\nonumber \\\\& ~& \\hspace{56.9055pt} u_{k-\\tau _{\\rm d}-1}, \\cdots , u_{k-n_u} \\big ).$ From Remark REF , we have $A(q)r_{k{+}1} & = & b_{\\tau _{\\rm d}} m_{k-\\tau _{\\rm d}+1} {+}B^*(q)m_{k{+}1}{+}f\\big (r_{k},\\cdots ,\\nonumber \\\\& ~& \\hspace{14.22636pt} r_{k-n_{y}{+}1}, m_{k-\\tau _{\\rm d}},\\cdots ,m_{k-n_u{+}1}\\big ),$ which, for convenience, has been written an instant of time ahead, i.e.",
"$k \\rightarrow k+1$ .",
"From the Assumption REF , the compensation input can be obtained from (REF ) as $m_{k-\\tau _{\\rm d}+1} & = & \\frac{1}{b_{\\tau _{\\rm d}} } \\Big [A(q)r_{k{+}1}-B^*(q)m_{k{+}1}-f\\big (r_k,\\cdots ,\\nonumber \\\\& ~& \\hspace{14.22636pt} r_{k-n_y{+}1}, m_{k-\\tau _{\\rm d}},\\cdots ,m_{k-n_u{+}1}\\big )\\Big ].$ To illustrate the application of this strategy, assume that the constraints discussed in section and Assumption REF are verified in the identification procedure.",
"Example 2 Consider the NARX model described by $y_k&{=}\\theta _1y_{k-1}{+}\\theta _2{\\phi }_{2,\\,k-2}{+}\\theta _3{\\phi }_{1,\\,k-2}u_{k-2} \\nonumber \\\\&{+}\\theta _4{\\phi }_{2,\\,k-2}{\\phi }_{1,\\,k-2}y_{k-1}{+}\\theta _5{\\phi }_{1,\\,k-1}.$ Since $\\phi _{1,\\,k}{=}u_k-u_{k-1}$ and $\\phi _{2,\\,k} {=} {\\rm sign}(\\phi _{1,\\,k})$ , we have $y_k&{=}\\theta _1y_{k-1}{+}\\theta _2{\\rm sign}(u_{k-2}-u_{k-3}){+}\\theta _3[u_{k-2}-u_{k-3}]u_{k-2} \\nonumber \\\\&{+}\\theta _4{\\rm sign}(u_{k-2}{-}u_{k-3})[u_{k-2}{-}u_{k-3}]y_{k-1}{+}\\theta _5[u_{k-1}{-}u_{k-2}], \\nonumber $ which is in the form (REF ) and, therefore, $A(q)y_k {=} B(q)u_{k}{+}f\\big (y_{k-1},u_{k-2},u_{k-3},{\\rm sign}(u_{k-2}-u_{k-3})\\big ),$ where $A(q) &=& 1 - \\theta _1q^{-1}, \\\\B(q) &=& \\theta _5q^{-1} - \\theta _5q^{-2}, \\\\f(\\cdot ) &=& \\theta _2{\\rm sign}(u_{k-2}-u_{k-3}){+}\\theta _3[u_{k-2}-u_{k-3}]u_{k-2} \\nonumber \\\\&~&{+}\\theta _4{\\rm sign}(u_{k-2}-u_{k-3})[u_{k-2}-u_{k-3}]y_{k-1}.$ From Remark REF , the model (REF ) is recast as $A(q)r_{k{+}1} {=} \\theta _5m_{k}{-}&\\theta _5m_{k-1}{+}f\\big (r_{k},m_{k-1},m_{k-2},\\nonumber \\\\&{\\rm sign}(m_{k-1}-m_{k-2})\\big ),$ and $m_k & = &\\frac{1}{\\theta _5} \\Big [A(q)r_{k{+}1}{+}\\theta _5m_{k-1} \\nonumber \\\\& ~& \\hspace{5.69046pt} -f\\big (r_k,m_{k-1},m_{k-2},{\\rm sign}(m_{k-1}{-}m_{k-2})\\big )\\Big ], \\nonumber \\\\& = &\\frac{1}{\\theta _5}\\Big [r_{k{+}1}-\\theta _1r_{k}{+}\\theta _5m_{k-1} \\nonumber \\\\& ~& \\hspace{5.69046pt}-\\theta _2{\\rm sign}(m_{k-1}{-}m_{k-2})-\\theta _3[m_{k-1}{-}m_{k-2}]m_{k-1} \\nonumber \\\\& ~& \\hspace{5.69046pt}-\\theta _4{\\rm sign}(m_{k-1}{-}m_{k-2})[m_{k-1}{-}m_{k-2}]r_{k} \\Big ],$ which is computed due to Assumption REF .",
"$\\square $"
],
[
"Compensation Based on Compensator Identification",
"Here, the strategy is to identify NARX models $\\breve{\\mathcal {M}}$ that are able to describe the inverse relationship between the input $u$ and output $y$ signals of the system $\\mathcal {S}$ .",
"The advantage of this strategy is that the compensator $\\mathcal {C}$ is obtained directly from $\\breve{\\mathcal {M}}$ , according to the variable solutions presented in Remark REF .",
"However, some issues related to the identification procedure of these models need to be addressed.",
"For nomenclature simplicity, in this section, we assume that $\\tau _{\\rm d}=1$ .",
"For the inverse model $\\breve{\\mathcal {M}}$ , the output $\\hat{u}_k$ depends on $y_k$ .",
"Hence in order to avoid the lack of causality, $y_k$ should be delayed by $\\tau _{\\rm s}$ time steps with respect to $u_k$ , yielding [44]: $\\hat{u}_k {=} \\breve{F}\\big (\\hat{u}_{k-1},\\cdots ,\\hat{u}_{k-n_{u}},y_{k-1{+}\\tau _{\\rm s}},\\cdots ,y_{k-n_y+\\tau _{\\rm s}}\\big ), $ where $\\breve{F}(\\cdot )$ is the inverse nonlinear function and $\\hat{u}_k \\in \\mathbb {R}$ and $y_k \\in \\mathbb {R}$ are related as shown in figure REF (a).",
"It should be noted that $\\tau _{\\rm s} \\ge \\tau _{\\rm d}+1$ , where usually the equality is preferred.",
"Similar ways to avoid noncausal models can be found in the literature [45], [29].",
"Assumption 3 Assume that: (i) there is at least one regressor of the output $(y_k)^j$ for $j \\ge 1$ ; (ii) the compensation signal $m_k$ is known up to time $k-1$ ; and (iii) the reference signal $r_k$ is known up to time $k-1+\\tau _{\\rm s}$ .",
"$\\square $ Assumption REF should be imposed during the structure selection of the inverse model $\\breve{\\mathcal {M}}$ .",
"Note that (i) ensures that there is at least one input signal $y_k$ in the identified models; (ii) and (iii) ensure that the compensation input $m_k$ to be computed at time $k$ is the only unknown variable.",
"Given Assumption REF and Remark REF , the compensation signal $m_k$ can be obtained directly from $\\breve{\\mathcal {M}}$ as $m_k {=} \\breve{F}\\big (m_{k-1},\\cdots ,m_{k-n_{u}},r_{k-1{+}\\tau _{\\rm s}},\\cdots ,r_{k-n_y+\\tau _{\\rm s}} \\big ).$"
],
[
"Numerical Results",
"This section identifies models to predict the behavior of a hysteretic system from simulated data, and evaluates the performance of these models in predicting dynamics and compensating the nonlinearity of the simulated system."
],
[
"Identification of a bench test system",
"Consider the piezoelectric actuator with hysteretic nonlinearity modeled by the Bouc-Wen model [12] and whose mathematical model is given by [45] $\\left\\lbrace \\begin{array}{ll}{\\dot{h}(t) = ~A\\dot{u}(t) - \\beta |\\dot{u}(t)|h(t) - \\gamma \\dot{u}(t)|h(t)|,\\\\y(t) =~d_{\\rm p} u(t) - h(t),\\\\}\\end{array}\\right.where y(t) is the displacement, u(t) is the voltage applied to the actuator,d_{\\rm p}\\hspace{-1.42271pt}=\\hspace{-1.42271pt}1.6\\, {\\rm \\frac{\\mu m}{V}} is thepiezoelectric coefficient, h(t) is the hysteretic nonlinear term andA\\hspace{-1.42271pt}=\\hspace{-1.42271pt}0.9\\, {\\rm \\frac{\\mu m}{V}},\\beta \\hspace{-1.42271pt}=\\hspace{-1.42271pt}0.008\\, {\\rm V^{-1}} and\\gamma \\hspace{-1.42271pt}=\\hspace{-1.42271pt}0.008\\, {\\rm V^{-1}} are parametersthat determine the shape and scale of the hysteresis loop.$ Model (REF ) was integrated numerically using a fourth-order Runge-Kutta method with integration step $\\delta t=0.001\\,{\\rm s}$ .",
"The excitation signal was generated by low-pass filtering a white Gaussian noise [6].",
"In this work, a fifth-order low-pass Butterworth filter with a cutoff frequency of 1 Hz was used; see figure REF (a).",
"The sampling time is set to $T_{\\rm s}=\\delta t=0.001\\,{\\rm s}$ and a frequency of 1 Hz is chosen to validate the identified models [45].",
"The data sets are $50\\,{\\rm s}$ long $(N=50000)$ .",
"The constraints defined in section are here considered to build NARX models for system (REF ).",
"Input and output signals are shown in figure REF .",
"In addition to the monomial regressors in $u_k$ and $y_k$ , the following regressors are also used: $\\phi _{1,\\,k}=u_k-u_{k-1}$ and the sign of this first difference $\\phi _{2,\\,k} = {\\rm sign}(\\phi _{1,\\,k})$ .",
"The maximum nonlinear degree for regressors is cubic, $\\ell =3$ , and the maximum delays are $n_y=n_u=1$ .",
"This choice is based on the fact that discrete models of hysteresis that have only unit delayed regressors typically perform well and result in models with simple structures, which are advantageous for model-based control [6], [29].",
"The model structure is selected using the ERR criterion to rank the regressors according to importance and the AIC determine the final number of model terms.",
"In this case, the standard least squares solution is used for parameter estimation.",
"Also, as proposed in section , to obtain models that describe some features of hysteresis, the constrained parameter estimation was used in order to comply with the condition established in Property REF .",
"Figure: Signals used to identify system ().",
"(a) excitation, and (b) simulated output.In this example we take the following metaparameters: $n_u=2$ which is the smallest value that complies with Assumption REF -(ii); while $n_y$ and $\\ell $ maintain the values which were determined above.",
"Using the data shown in figure REF and considering Assumption REF , the following model structure is obtained $y_k&=\\theta _1y_{k-1}+\\theta _2\\phi _{1,\\,k-1} +\\theta _3\\phi _{2,\\,k-2}\\phi _{1,\\,k-2}u_{k-2} \\nonumber \\\\&+\\theta _4\\phi _{2,\\,k-2}\\phi _{1,\\,k-2}y_{k-1}+\\theta _5\\phi _{1,\\,k-2}u^2_{k-2} \\nonumber \\\\&+\\theta _6\\phi _{1,\\,k-2}u_{k-2}y_{k-1},$ where $\\theta _i$ for $i=1,\\cdots ,6$ are estimated by least squares.",
"In steady-state, we have $y_k=\\bar{y},\\,\\forall k$ , $\\phi _{1,\\,k}=0,\\,\\forall k$ ; hence, the resulting expression is $\\bar{y}=\\theta _1\\bar{y}$ .",
"Therefore, based on Lemma REF and Example REF , for model (REF ) to fulfill Property REF , the constraint $\\Sigma _y=\\theta _1=1$ should be imposed.",
"This can be done using (REF ) with the constraint written as: $c = 1;\\qquad S {=} [1 ~~ 0 ~~ 0 ~~ 0 ~~ 0 ~~ 0].$ Hence, the parameter values estimated by the constrained least squares estimator (REF ) are shown in Table REF .",
"Table: Model parameters obtained with() and ().Next, a quasi-static analysis of the identified model is performed as discussed in section REF and illustrated in Example REF .",
"First, we write for (REF ) the corresponding to (REF ) as $\\tilde{y} & \\approx & \\theta _1\\tilde{y}+\\theta _2{\\phi }_{1,k-1} +\\theta _3{\\phi }_{2,k-2}{\\phi }_{1,k-2}u_{k-2} \\nonumber \\\\& ~ & {+}\\theta _4{\\phi }_{2,k-2}{\\phi }_{1,k-2}\\tilde{y}\\!+\\!\\theta _5{\\phi }_{1,k-2}u_{k-2}^2\\!+\\!\\theta _6{\\phi }_{1,k-2}u_{k-2}\\tilde{y}, \\nonumber $ yielding $\\tilde{y}({u},{\\phi }_{1},{\\phi }_{2}) {=} \\left\\lbrace \\begin{array}{ll}{\\frac{\\theta _2{\\phi }_{1}+\\theta _3{\\phi }_{1}{u}+\\theta _5{\\phi }_{1}{u}^2}{1-\\theta _1-\\theta _4{\\phi }_{1}-\\theta _6{\\phi }_{1}{u}}, {\\rm for} \\,\\,{\\phi }_{2}{=}\\,\\,\\,1; \\\\\\frac{\\theta _2{\\phi }_{1}-\\theta _3{\\phi }_{1}{u}+\\theta _5{\\phi }_{1}{u}^2}{1-\\theta _1+\\theta _4{\\phi }_{1}-\\theta _6{\\phi }_{1}{u}}, {\\rm for} \\,\\,{\\phi }_{2}{=}-1,}\\end{array}\\right.where the time indices have been omitted for brevity.$ The top expression in (REF ) gives the set $\\tilde{y}_{\\rm L}$ , while the bottom one, $\\tilde{y}_{\\rm U}$ .",
"Computing the derivative of (REF ) with respect to $y_{k-1}$ and using (REF ), we obtain $-1< \\quad \\theta _1 + \\theta _4\\phi _{1,k-2}{\\phi }_{2,k-2}+\\theta _6{\\phi }_{1,k-2}u_{k-2} \\quad <1, \\nonumber \\\\\\frac{-1-\\theta _1-\\theta _4\\phi _{1,k-2}{\\phi }_{2,k-2}}{\\theta _6\\phi _{1,k-2}}< u_{k-2} < \\frac{1-\\theta _1-\\theta _4\\phi _{1,k-2}{\\phi }_{2,k-2}}{\\theta _6\\phi _{1,k-2}}.$ Taking ${\\phi }_{2,k-2}{=}1$ or ${\\phi }_{2,k-2}{=}-1$ , the conditions for attracting regions under load or unloading, respectively, are obtained.",
"Considering the parameter values presented in Table REF and a loading-unloading input signal, the points (REF ) and their attraction conditions (REF ) are computed numerically and shown in figure REF .",
"Figure REF should be compared to figure REF , whose main elements are analogous.",
"Hence, in this way it is possible to see how model (REF ) is able to describe the hysteresis nonlinearity.",
"Figure: Results of quasi-static analysis for model () with input u k =70sin(2πk)Vu_k{=}70\\sin (2\\pi k)\\,{\\rm V}.",
"The hysteresis loop indicated with (⋯\\cdots ) is a result of the interaction of(—) attracting (y ˜ L a \\tilde{y}_{\\rm L}^{\\rm a}, y ˜ U a \\tilde{y}_{\\rm U}^{\\rm a}) and(-·\\,\\cdot \\,-) repelling (y ˜ L r \\tilde{y}_{\\rm L}^{\\rm r}, y ˜ U r \\tilde{y}_{\\rm U}^{\\rm r}) sets.",
"() indicates the orientation of the hysteresis loop.Model (REF ) is simulated with a loading-unloading input (see left side of figure REF ) and, in cases where the input becomes constant, either during loading or unloading (see right side of figure REF ), the system remains at the corresponding point of the hysteresis loop.",
"This is a direct consequence of using Lemma REF .",
"This feature is not generally present in identified models found in the literature.",
"Figure: Free-run simulation of model ().",
"This figure is arranged in columns, which have: (a) sinusoidal input of voltage u k =40sin(2πk)Vu_k{=}40\\sin (2\\pi k)\\,{\\rm V} and in (b) the case where this input becomes constant during a loading (•\\bullet ) and unloading (⧫\\blacklozenge ) regime with the final value of 16.8V16.8\\,{\\rm V}, its temporal responses are shown in (c) and (d) while the hysteresis loops are in (e) and (f), respectively.",
"(—) represents the measured data and (- -) is the estimated output of the model.",
"The full records have N=50000N=50000 data points.The mean absolute percentage error (MAPE) ${\\rm MAPE} = \\frac{100 \\sum _{k=1}^{N} |y_k-\\hat{y}_k|}{N|\\max ({y}) - \\min ({y})|}, $ computed for the case in figure REF (c), is shown in Table REF ."
],
[
"Estimating $\\breve{\\mathcal {M}}$ .",
"The identified model that complies with Assumptions REF and REF is given by $\\hat{u}_k&=\\theta _1\\hat{u}_{k-1}+\\theta _2\\breve{\\phi }_{1,\\,k-1} +\\theta _3\\breve{\\phi }_{2,\\,k-1}\\breve{\\phi }_{1,\\,k-1}\\hat{u}_{k-1} \\nonumber \\\\&+\\theta _4\\breve{\\phi }_{2,\\,k-1}\\breve{\\phi }_{1,\\,k-1}y_{k-1}+\\theta _5\\breve{\\phi }_{2,\\,k-1}y_{k-1}\\hat{u}_{k-1} \\nonumber \\\\&+\\theta _6\\breve{\\phi }_{2,\\,k-1}y^2_{k-1},$ where $\\breve{\\phi }_{1,\\,k}=y_k-y_{k-1}$ , $\\breve{\\phi }_{2,\\,k} = {\\rm sign}(\\breve{\\phi }_{1,\\,k})$ , $\\hat{u}_k$ is the estimated input (model output), and $y_k$ is the output of system (REF ) (model input).",
"Note that the regressors of (REF ) and of (REF ) are different.",
"In both cases, the regressors are automatically chosen from the pool of candidates using the ERR criterion.",
"Nevertheless, also for (REF ), the steady-state analysis yields $\\bar{\\hat{u}}{=}\\theta _1\\bar{\\hat{u}}$ , which is similar to the result found for model (REF ).",
"Proceeding as before, the constrained least squares estimated parameters are shown in Table REF .",
"Consider now the quasi-static analysis of model (REF ).",
"The formation of the hysteresis loop for this model is shown in figure REF .",
"Interestingly, the ability of model (REF ) to describe hysteresis is also present in model (REF ).",
"The main difference between them is the orientation of the hysteresis loop, as discussed in [46] and illustrated in figures REF and REF .",
"Figure: Results of quasi-static analysis for model ().",
"For meaning of linepatterns refer to captions of figure and of figure .Table REF shows the prediction performance of models (REF ) and (REF ) are similar.",
"In addition, Table REF reports the prediction performance of a black-box NARX polynomial model.",
"The results obtained for (REF ) are similar to those shown in figure REF and are omitted for brevity.",
"Table: Performance of the modeling step.",
"Simulation results.Next, the models identified in the previous section are used to design compensators using the procedure illustrated in figure REF (b)."
],
[
"Design of the compensation input signals.",
"Applying the steps described in section REF to model (REF ), the following compensation signal is obtained $m_k{=}\\frac{1}{\\theta _2}&\\Big [r_{k{+}1}-\\theta _1r_{k}{+}\\theta _2m_{k-1} \\nonumber \\\\&{-}[\\theta _3m_{k-1}{+}\\theta _4r_{k}]{\\rm sign}(m_{k-1}{-}m_{k-2})[m_{k-1}{-}m_{k-2}] \\nonumber \\\\&{-}[\\theta _5m^2_{k-1}{+}\\theta _6m_{k-1}r_{k}][m_{k-1}-m_{k-2}]\\Big ].$ Similarly, using the method described in section REF to identify the inverse model (REF ), the following compensation signal is obtained $\\breve{m}_k&=\\theta _1\\breve{m}_{k-1}{+}\\theta _2[r_{k{+}1}-r_{k}] \\nonumber \\\\&+[\\theta _3\\breve{m}_{k-1}+\\theta _4r_{k{+}1}]{\\rm sign}(r_{k{+}1}-r_{k})[r_{k{+}1}-r_{k}]\\nonumber \\\\&+[\\theta _5r_{k{+}1}\\breve{m}_{k-1}+\\theta _6r^2_{k{+}1}]{\\rm sign}(r_{k{+}1}-r_{k}).$ Since the parameters of compensators (REF ) and (REF ) have been estimated (Table REF ), and Assumptions REF and REF are satisfied, the compensation inputs $m_k$ and $\\breve{m}_k$ can be computed."
],
[
"Evaluating the performance of the compensation.",
"The results are summarized in figure REF (b).",
"From the hysteresis loops shown in figure REF (c), it is clear that the compensators enforced a practically linear relation between the reference and the output.",
"This would greatly facilitate the design and increase the performance of a feedback controller.",
"Figure: Hysteresis compensation for the piezoelectric actuator ().",
"(a) Compensation inputs, (b) temporal responses and in (c) hysteresis loops.",
"(- -) results obtained with compensator ()(⋯\\cdots ) results with compensator (),(-·\\,\\cdot \\,-) system output without compensation, and (—) displacementreference r=40sin(2πt)μr=40\\sin (2\\pi t)\\,\\mu m.The accuracy achieved by each compensator was quantified by the MAPE index (REF ).",
"In order to quantify the compensation effort, the normalized sum of the absolute variation of the input (NSAVI) ${\\rm NSAVI}=\\sum _{k=1}^{N-1} \\frac{\\big |m_{k+1}-m_k\\big |}{\\big |r_{k+1}-r_k\\big |},$ is calculated.",
"These indices are shown in Table REF .",
"Table: Performance of the compensation step.",
"Simulation results.The results shown in figure REF and Table REF indicate that the compensators may provide a significant improvement in the tracking performance of system (REF ).",
"The tracking error was reduced by about $93\\%$ at the cost of a $14\\%$ increase in the compensation effort.",
"Although the compensator strategies yield similar results, the design strategy of section REF yielded results with lower compensation effort and tracking error.",
"To further characterize the performance of the proposed designs, the influence of the sampling time $T_{\\rm s}$ is also investigated.",
"In figure REF , it can be seen that the model accuracy somewhat deteriorates as $T_{\\rm s}$ is increased.",
"It should be noted that even the largest values of $T_{\\rm s}$ in figure REF are still comfortably small in terms of the sampling theorem.",
"However, since one of the regressors is the first difference of the input, then the identification of systems with hysteresis seems to be particularly sensitive to the sampling time [47].",
"Another conclusion that can be drawn from figure REF is that, for both design strategies, the compensation performance is correlated to the model accuracy, and that the strategy in section REF (figure REF (a)) is somewhat less sensitive to such accuracy.",
"Figure: MAPE index () computed for the models and compensators described, respectively, by equations (a) () and (); (b) () and ().",
"(∘\\circ ) model and (•\\bullet ) tracking accuracies.",
"(▴\\blacktriangle ) accuracy of uncompensated system.Finally, the same analysis was carried out for situations with different shapes of the hysteresis loop varying $\\beta $ in the range $0.004 \\le \\beta \\le 0.1$ with increments of $\\Delta =0.002$ (see figure REF ).",
"The results are quite similar to the ones described so far and are not shown.",
"Figure: Bouc-Wen hysteresis loops within the investigated range."
],
[
"Experimental Results",
"Both identification and compensation strategies are now applied to an experimental pneumatic control valve.",
"This type of actuator is widely used in industrial processes, for which control performance can degrade significantly due to valve problems caused by nonlinearities [48] such as friction [49], [50], dead-zone, dead-band and hysteresis [2].",
"Hence, in this section we aim at compensation hysteresis using the developed techniques.",
"The measured output is the stem position of the pneumatic valve and the input is a signal that, after passing V/I and I/P conversion, becomes a pressure signal applied to the valve.",
"The sampling time is $T_{\\rm s}=0.01\\,{\\rm s}$ .",
"For model identification, the input is set as white noise low-pass filtered at $0.1\\,{\\rm Hz}$ .",
"For model validation, the input is a sinusoid with frequency $0.1\\,{\\rm Hz}$ .",
"Both data sets are $200\\,{\\rm s}$ long ($N=20000$ ).",
"The identification of the direct $\\mathcal {M}$ and inverse $\\breve{\\mathcal {M}}$ models was performed as in section .",
"The pool of candidate terms is generated with $\\ell =3$ , $n_{y}=1$ and $n_{u}=2$ .",
"The model parameters are estimated using (REF ) in order to comply with Lemma REF .",
"The estimated model $\\mathcal {M}$ is $y_k&=y_{k-1}-19.76\\phi _{1,\\,k-2}+19.32\\phi _{1,\\,k-1}\\nonumber \\\\&{+}9.44\\phi _{2,\\,k{-}2}\\phi _{1,\\,k{-}2}u_{k{-}2}{-}12.61\\phi _{2,\\,k{-}2}\\phi _{1,\\,k{-}2}y_{k{-}1},$ and the inverse model $\\breve{\\mathcal {M}}$ is $\\hat{u}_k&=\\hat{u}_{k{-}1}+86.67\\breve{\\phi }_{1,\\,k{-}1}-85.02\\breve{\\phi }_{1,\\,k{-}2}{-}0.98\\breve{\\phi }_{1,\\,k{-}1}y_{k{-}2} \\nonumber \\\\&+1.72\\breve{\\phi }_{2,\\,k{-}2}\\breve{\\phi }_{1,\\,k{-}2}y_{k{-}2}{-}1.13\\breve{\\phi }_{2,\\,k{-}2}\\breve{\\phi }_{1,\\,k{-}2}\\hat{u}_{k{-}1},$ which was estimated from a smoothed version of $y_k$ obtained by quadratic regression.",
"This is done only to estimate $\\breve{\\mathcal {M}}$ to avoid the error-in-the-variables problem, since $y_k$ serves as the input for $\\breve{\\mathcal {M}}$ .",
"Each model performance is given in figure REF and Table REF .",
"Figure: Left column refers to model () and right column to model ().",
"(a) input u k =0.56sin(0.2πk)+3Vu_k{=}0.56\\sin (0.2\\pi k)+3\\,{\\rm V} and(c) the corresponding measured output (—) yy and (- -) model() free-run simulation; (b) smoothed version of yy in (c);(d) the corresponding output which is u k u_k in (a) and (- -) model() free-run simulation.",
"(e) and (f) show the same data as (c) and (d),respectively.Table: Performance of the modeling step.",
"Experimental results.Models (REF ) and (REF ) are used to implement the strategies described in sections REF and REF , thus yielding, respectively, the compensation inputs $m_k{=}\\frac{1}{19.32}&\\!\\Big [r_{k{+}1}{-}r_{k}{+}19.32m_{k{-}1}{+}19.76[m_{k{-}1}{-}m_{k{-}2}] \\nonumber \\\\&-9.44{\\rm sign}(m_{k-1}-m_{k{-}2})[m_{k{-}1}-m_{k{-}2}]m_{k{-}1} \\nonumber \\\\&{+}12.61{\\rm sign}(m_{k{-}1}{-}m_{k{-}2})[m_{k{-}1}{-}m_{k{-}2}]r_{k}\\Big ]\\!,$ and $\\breve{m}_k&=\\breve{m}_{k{-}1}{+}86.67[r_{k{+}1}-r_{k}]{-}85.02[r_{k}-r_{k{-}1}] \\nonumber \\\\&-0.98[r_{k{+}1}-r_{k}]r_{k}+1.72{\\rm sign}(r_{k}-r_{k{-}1})[r_{k}-r_{k{-}1}]r_{k}\\nonumber \\\\&-1.13{\\rm sign}(r_{k}-r_{k{-}1})[r_{k}-r_{k{-}1}]\\breve{m}_{k-1}.$ The results of the experiment are shown in figure REF and assessed in Table REF .",
"Note that both approaches significantly reduce the tracking error.",
"Based on both numerical and experimental results, it seems that the performance of the compensators is directly related to the accuracy of the identified model; see Table REF and Table REF .",
"Hence, as before, these results suggest that the compensation effort tends to be lower and more effective whenever the identified models are more accurate.",
"The compensation produced by (REF ) is smoother than the one obtained with (REF ); see figure REF (a).",
"This occurs because, for the compensator (REF ), the argument of the sign function depends on the difference of the reference signal, while, for the compensator (REF ), it depends on the difference of the autoregressive variable which usually produces stronger oscillations and sudden changes; see figure REF (a), e.g.",
"in the range of $51{-}53\\,{\\rm s}$ .",
"As a result, larger compensation effort is required as quantified by NSAVI (REF ) in Table REF .",
"Figure: Hysteresis compensation for the pneumatic valve.",
"(a) Compensation inputs, (b) and (c) its temporal responses and in (d) and (e) the hysteresis loops.",
"(- -) illustrates the results obtained with compensator (),(⋯\\cdots ) refers to the results by using compensator (),(-·\\,\\cdot \\,-) the system output without compensation, and (—) the referencer=0.56sin(0.2πt)+3Vr{=}0.56\\sin (0.2\\pi t){+}3\\,{\\rm V}.Table: Performance of the compensation step.",
"Experimental results."
],
[
"Conclusions",
"This work addressed the problems of identification and compensation of hysteretic systems.",
"In the context of system identification, the contribution is twofold.",
"First, we build models with regressors that use the sign function of the first difference of the input, as proposed by [6], and present an additional condition in order to guarantee a continuum of equilibrium points at steady-state, which is an important ingredient for hysteresis [30], [5].",
"To this aim, a particular constraint on the parameters is presented in Lemma REF .",
"As a consequence, the identified models are able to describe both dynamical and static features of the hysteresis nonlinearity.",
"Second, following a quasi-static analysis of these models, a schematic framework is proposed to explain how the hysteresis loop occurs on the input-output plane (see Figure REF ).",
"In the context of hysteresis compensation, this paper introduces two strategies to design compensators.",
"An important aspect of such procedures is that they show how to restrict the pool of candidate regressors aiming at solving the compensation problem.",
"Such strategies are not limited to hysteresis and can be extended to other nonlinearities.",
"The effectiveness of the compensation schemes is illustrated by means of numerical and experimental tests.",
"For the strategy described in Section REF , the compensation law is obtained from the identified model by simple algebraic manipulations.",
"In the case of the strategy introduced in Section REF , the compensators are identified directly from the data.",
"This, however, leads to models that are noncausal by nature.",
"Guidelines to circumvent this problem are provided.",
"The compensators designed by both strategies can be readily employed in online compensation schemes.",
"As a general remark, we observed in all our examples that the quality of the achieved compensation is correlated with the accuracy of the identified model (compare Table REF with Table REF and Table REF with Table REF ).",
"Finally, we noticed that the identified models have a discontinuity due to the sign function used in some regressors.",
"When the model has many such terms, it sometimes happens that the compensation signal presents abrupt transitions.",
"The use of smoother functions in place of the sign function, in order to alleviate this problem, will be investigated in the future."
],
[
"Acknowledgments",
"The authors would like to thank Arthur N Montanari for the insightful discussions.",
"PEOGBA, BOST and LAA gratefully acknowledge financial support from CNPq (Grants Nos.",
"142194/2017-4, 310848/2017-2 and 302079/2011-4) and FAPEMIG (TEC-1217/98)."
]
] | 2001.03604 |
[
[
"Measurement-induced transitions of the entanglement scaling law in\n ultracold gases with controllable dissipation"
],
[
"Abstract Recent studies of quantum circuit models have theoretically shown that frequent measurements induce a transition in a quantum many-body system, which is characterized by the change of the scaling law of the entanglement entropy from a volume law to an area law.",
"In order to propose a way for experimentally observing this measurement-induced transition, we present numerical analyses using matrix-product states on quench dynamics of a dissipative Bose-Hubbard model with controllable two-body losses, which has been realized in recent experiments with ultracold atoms.",
"We find that when the strength of dissipation increases, there occurs a measurement-induced transition from volume-law scaling to area-law scaling with a logarithmic correction in a region of relatively small dissipation.",
"We also find that the strong dissipation leads to a revival of the volume-law scaling due to a continuous quantum Zeno effect.",
"We show that dynamics starting with the area-law states exhibits the breaking of ergodicity, which can be used in experiments for distinguishing them from the volume-law states."
],
[
"Introduction",
"In a generic quantum many-body system, a pure state is thermalized via long-time evolution, i.e., its expectation values of local observables are very close to those given by a statistical (micro-) canonical ensemble [1], [2], [3].",
"The entanglement entropy of such a thermal pure state obeys volume-law scaling, corresponding to the fact that the entropy of a thermal density matrix is extensive [4], [5], [6].",
"Recent advances in understanding and controlling coherent quantum many-body dynamics have revealed a few exceptional systems which do not show such thermalization.",
"First, in integrable systems, such as the Lieb-Liniger model and the one-dimensional (1D) Ising model with a transverse field, many integrals of motion prevent a pure state from relaxation towards a thermal state [7], [8], [1], [9].",
"Second, in many-body localized (MBL) systems, disordered potentials forbid ballistic propagation of quantum information such that the entanglement entropy grows only logarithmically with time [10], [11], [2], [3].",
"Recent theoretical studies of quantum circuit models have proposed another class of exceptional systems [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].",
"In these studies, random unitary dynamics with probabilistic measurements have been investigated.",
"It has been shown that when the probability of measurements increases, the scaling law of the entanglement entropy exhibits a transition from a volume law to an area law at a certain critical point.",
"Since the volume-law scaling is a necessary condition for a pure state to be thermal, the emergence of the area-law scaling means that many measurements prevent a state after long-time evolution from the thermalization.",
"Despite the intensive interest in this measurement-induced transition (MIT), its experimental observation is still lacking.",
"In order to observe the MIT, one needs an experimental system with long coherence time and high controllability of measurements.",
"Ultracold gases have served as an ideal platform for analyzing long-time coherent dynamics of many-body systems thanks to their long thermalization time and isolation from the environment.",
"Indeed, coherent quantum dynamics of integrable systems [7] and MBL systems [22] has been observed in this platform for the first time.",
"Recent experiments have successfully introduced controllable dissipation to ultracold-gas systems to create and manipulate quantum many-body states [23], [24], [25], [26], [27].",
"Since the introduced dissipation corresponds to a continuous quantum measurement, which can be interpreted as probabilistic measurements in terms of quantum trajectory representation of open quantum systems, we expect that it may be utilized for causing the MIT.",
"In this paper, we propose a specific protocol to realize the MIT with use of ultracold gases in optical lattices.",
"By means of the quantum trajectory method implemented with matrix product states (MPS) [28], [29], [30], we analyze the 1D Bose-Hubbard model with two-body losses, which can be widely controlled in experiment by the strength of a photoassociation (PA) laser [26].",
"We find that this system exhibits a MIT from volume-law scaling to area-law scaling with a logarithmic correction (ALSLC) when the strength of the two-body losses increases in a weakly dissipative regime.",
"Moreover, we find another MIT in a strongly dissipative regime.",
"The latter transition can be attributed to a continuous quantum Zeno effect (QZE) and has not been reported in previous literature studying quantum circuit models.",
"We show that the experimentally accessible momentum distribution reflects the changes of the scaling laws.",
"We also analyze dynamics after release of the particles to an empty space in order to show that the states with ALSLC can be distinguished from the volume-law states by observing the strong suppression of particle transport, which can be recognized as the tendency towards the ergodicity breaking.",
"The rest of the paper is organized as follows.",
"In Sec.",
", we define the master equation describing ultracold bosons with a PA laser in a 1D optical lattice, and introduce the quantum trajectory method for analyzing the master equation.",
"We also define “entanglement entropy” used in this study in the section.",
"In Sec.",
", we show that there exist two MIT in this system and that the MIT has the reentrant structure.",
"In Sec.",
", we discuss how to detect the MIT in ultracold-gas experiments.",
"In Sec.",
", we summarize the results."
],
[
"Model and methods",
"We consider ultracold bosons confined in an optical lattice.",
"We assume that the lattice potential in the transverse (yz) directions is so deep that the hopping in these direction is forbidden, i.e., the system is 1D.",
"We also assume that the lattice potential in the longitudinal (x) direction is deep enough for the tight-binding approximation to be valid.",
"The two-body losses can be introduced by exposing the system to a PA laser [26], which couples a local two-atom state to a molecular state with a very short lifetime.",
"In this system, the time-evolution of a density matrix $\\hat{\\rho }(t)$ can be effectively described by the master equation in Lindblad form [26], [31], [32] $\\frac{\\mathrm {d}}{\\mathrm {d} t} \\hat{\\rho }(t) &= -\\frac{\\mathrm {i}}{\\hbar }[\\hat{H}, \\hat{\\rho }(t)] + \\hat{L}[\\hat{\\rho }(t)]$ with the 1D Bose-Hubbard Hamiltonian $\\hat{H} = -J\\sum ^{M-1}_{i=1} (\\hat{b}^\\dagger _i \\hat{b}_{i+1} + \\mathrm {H.c.}) + \\frac{U}{2} \\sum ^M_{i=1} \\hat{n}_i(\\hat{n}_i - 1),$ and the Lindblad superoperator for two-body atom losses $\\hat{L}[\\hat{\\rho }] = - \\frac{\\gamma }{2} \\sum _i (\\hat{b}^\\dagger _i \\hat{b}^\\dagger _i \\hat{b}_i \\hat{b}_i \\hat{\\rho } + \\hat{\\rho } \\hat{b}^\\dagger _i \\hat{b}^\\dagger _i \\hat{b}_i \\hat{b}_i - 2\\hat{b}_i\\hat{b}_i\\hat{\\rho }\\hat{b}^\\dagger _i\\hat{b}^\\dagger _i).$ Here, $J$ is the hopping amplitude, $M$ is the number of lattice sites, $\\hat{b}^\\dagger _i$ ($\\hat{b}_i$ ) creates (annihilates) a boson at site $i$ , $U$ is the on-site Hubbard interaction, $\\hat{n}_i = \\hat{b}^\\dagger _i \\hat{b}_i$ , and $\\gamma $ is the strength of the two-body inelastic collision which can be controlled by the intensity of the PA laser.",
"We denote the number of remaining particles in the system as $N$ , i.e., $N = \\sum _i \\mathinner {\\langle {\\hat{n}_i}\\rangle }$ .",
"At initial time $t=0$ , we assume that the system is a Mott insulating state at unit filling, i.e., $\\mathinner {|{\\psi _0}\\rangle } = \\prod _i \\hat{b}^\\dagger _i\\mathinner {|{0}\\rangle }$ and thus $\\hat{\\rho }(0) = \\mathinner {|{\\psi _0}\\rangle }\\mathinner {\\langle {\\psi _0}|}$ , where $\\mathinner {|{0}\\rangle }$ denotes the vacuum state.",
"Solving the master equation (REF ) requires a very high numerical cost in general because the number of coefficients in the density matrix is the square of the dimension of the Hilbert space.",
"To circumvent this difficulty, we use quantum trajectory techniques which treat pure states in the density matrix [29], [28] instead of treating the density matrix directly.",
"Following the quantum trajectory techniques, we calculate the time-evolved state $\\mathinner {|{\\psi (t)}\\rangle } = \\mathrm {e}^{-\\mathrm {i}\\frac{t}{\\hbar }\\hat{H}_\\mathrm {eff}}\\mathinner {|{\\psi _0}\\rangle }$ with the effective non-hermitian Hamiltonian $\\hat{H}_\\mathrm {eff} = \\hat{H} - \\mathrm {i}\\frac{\\hbar \\gamma }{2} \\sum _i \\hat{b}^\\dagger _i \\hat{b}^\\dagger _i \\hat{b}_i \\hat{b}_i.$ As time $t$ increases, the norm of the time-evolved state $\\mathinner {|{\\psi (t)}\\rangle }$ decreases because of the non-hermitian part of the effective Hamiltonian $\\hat{H}_\\mathrm {nh}$ .",
"When the squared norm of the time-evolved state becomes lower than a random number generated from the uniform distribution $(0, 1)$ , we calculate a probability $p_i \\propto \\mathinner {\\langle {\\psi (t)|\\hat{b}^\\dagger _i\\hat{b}^\\dagger _i\\hat{b}_i\\hat{b}_i|\\psi (t)}\\rangle }$ and choose one index $j$ according to the probability $p_i$ .",
"Then, the jump operator $\\hat{b}_j \\hat{b}_j$ is applied to $\\mathinner {|{\\psi (t)}\\rangle }$ and the state is normalized.",
"This stochastic process emulates the open dynamics described by the master equation in Lindblad form, and the expectation values are obtained by the sample average $\\begin{aligned}\\mathinner {\\langle {\\hat{O}(t)}\\rangle } &= \\mathrm {Tr}[\\hat{O}\\hat{\\rho }(t)] \\\\&\\simeq \\frac{1}{K}\\sum ^K_{l=1}\\frac{\\mathinner {\\langle {\\psi _l(t)|\\hat{O}|\\psi _l(t)}\\rangle }}{\\mathinner {\\langle {\\psi _l(t)|\\psi _l(t)}\\rangle }},\\end{aligned}$ where $\\mathinner {|{\\psi _l(t)}\\rangle }$ is the $l$ -th sample of the stochastic process and $K$ is the number of samples.",
"Notice that the application of the jump operator and the subsequent normalization correspond to a quantum measurement.",
"In the sense that a series of the measurement events stemming from the dissipation occur probabilistically according to the random number and the probability distribution $p_i$ , the dissipation can be interpreted as probabilistic measurements.",
"For numerically efficient calculations in 1D, we represent a state $\\mathinner {|{\\psi (t)}\\rangle }$ with MPS and perform the time evolution by means of the time-evolving block decimation algorithm [33], [34], [35], [36] using the optimized Forest-Ruth-like decomposition [37].",
"The truncation error is set to be less than $10^{-8}$ , and the time step $\\Delta t$ is adaptively changed after each jump operation as $\\Delta t = \\min \\lbrace -\\log (0.9)\\hbar /\\mathinner {\\langle {\\psi (t)|\\mathrm {i}\\hat{H}_\\mathrm {nh}|\\psi (t)}\\rangle }, \\Delta t_{\\max }\\rbrace $ in order to avoid a rapid decrease in the norm of wavefunction.",
"Here, $\\Delta t_{\\max }$ is the upper bound of the time step that we set to be $0.05 \\hbar /J$ ($0.02 \\hbar /J$ ) for small to intermediate $\\hbar \\gamma /J$ (large $\\hbar \\gamma / J \\ge 100 $ ).",
"It should be cautioned that we have to define what we call “entanglement entropy” in this study because the ordinary entanglement entropy is defined only for pure states $\\mathinner {|{\\phi }\\rangle }$ on a system biparted into subsystems $A$ and $B$ as $S_A(t) = -\\mathrm {Tr}\\hat{\\rho }_A(t) \\ln \\hat{\\rho }_A(t),$ where $\\hat{\\rho }_A$ is a reduced density matrix defined as $\\hat{\\rho }_A(t) = -\\mathrm {Tr}_B\\mathinner {|{\\phi (t)}\\rangle }\\mathinner {\\langle {\\phi (t)}|}.$ Here, $\\mathrm {Tr}_B$ means a partial trace over the subsystem $B$ .",
"In this study, as well as other studies investigating the MIT, the statistical average of the entanglement entropy of $\\mathinner {|{\\psi (t)}\\rangle }/\\sqrt{\\mathinner {\\langle {\\psi (t)|\\psi (t)}\\rangle }}$ is called “entanglement entropy” and the size dependence of the “entanglement entropy” is discussed.",
"In other words, what we discuss is typical behaviors of the entanglement entropy of relevant states in a density matrix $\\hat{\\rho }(t)$ .",
"An equal bipartition does not always give the maximal entanglement entropy in the presence of the two-body loss.",
"Therefore, we define the average of the maximal bipartite entanglement entropy $S_{\\max }(t) = \\left\\langle {\\max _A S_A(t)}\\right\\rangle ,$ where $\\max _A$ means the biparted subsystem $A$ that gives the maximal entanglement entropy.",
"In this study, we discuss the scaling law of the “entanglement entropy” based on $S_{\\max }(t)$ .",
"Hereafter, we call $S_{\\max }(t)$ as entanglement entropy for simplicity.",
"It is worth noting that the MIT in the Bose-Hubbard model (REF ) with local projective measurements has been studied in Ref. [21].",
"In contrast to the previous study, here we incorporate the specific form of controllable dissipation that has been experimentally realized and show an observable suited for characterizing the transitions."
],
[
"Measurement-induced transitions in the dissipative Bose-Hubbard models",
"Figure REF shows the time-evolution of the entanglement entropy for different values of $\\hbar \\gamma / J$ and $M$ at $U/J = 5.0$ .",
"By comparing the case of $(\\hbar \\gamma / J, M) = (0.5, 24)$ with that of $(\\hbar \\gamma / J, M) = (5.0, 24)$ , we see that the dissipation suppresses the growth of the entanglement entropy.",
"Thanks to this suppression, when $\\hbar \\gamma / J = 5.0$ , we can compute long-time dynamics of a relatively large system, say $M=256$ .",
"The general tendency of the entanglement entropy in the presence of the two-body losses is that it rapidly grows in a short time regime and gradually decreases due to the two-body losses after taking a maximal value.",
"We show below that the maximal entanglement entropy during the time evolution at $\\hbar \\gamma / J = 5.0$ obeys ALSLC.",
"In Fig.",
"REF (b), we see that a steady-value region, where $S_{\\max }(t)$ takes almost the same value as the maximal value, develops when the system size increases (see, e.g., the region $15 \\lesssim tJ/\\hbar \\lesssim 30$ in the case of $(\\hbar \\gamma / J, M) = (5.0, 128)$ ).",
"The presence of the steady-value region allows us to identify the states with ALSLC analyzed in the present work as those in the realm of the MIT [12], [13], [14].",
"Figure: System size MM dependence of the maximal entanglement entropy max t S max (t)\\max _t S_{\\max }(t) (a) in the linear scale for MM and (b) in the logarithmic scale for MM at U/J=5.0U/J = 5.0.",
"The blue solid, orange dashed, green dashed-dotted, and red dotted lines correspond to ℏγ/J=0.5\\hbar \\gamma / J = 0.5, 5.05.0, 50.050.0, and 500.0500.0, respectively.",
"Error bars indicate 1σ1\\sigma uncertainty.",
"Figure REF shows the maximal values of the entanglement entropy, $\\max _t S_{\\max }(t)$ , during the time evolution as a function of the system size $M$ for $\\hbar \\gamma / J = 0.5$ , $5.0$ , $50.0$ , and $500.0$ at $U/J = 5.0$ .",
"When the dissipation is as small as $\\hbar \\gamma / J = 0.5$ or is as large as $\\hbar \\gamma / J = 500.0$ , the entanglement entropy grows linearly with $M$ within the system size that we can numerically compute ($M \\simeq 24$ ), i.e., it follows the volume-law scaling.",
"On the contrary, in an intermediate dissipation regime, including $\\hbar \\gamma / J = 5.0$ and $50.0$ , the entanglement entropy grows logarithmically with $M$ , i.e., it follows ALSLC.",
"We call the scaling with the logarithmic correction as area law in the sense that the correction grows more slowly with the system size than algebraic growth, i.e., it is not extensive.",
"This observation means that when the strength of the dissipation increases, the system exhibits a transition from a volume-law state to an ALSLC state at a relatively small value of dissipation and the other transition to another volume-law state at a relatively large value.",
"In short, in the present system the MIT has the reentrant structure.",
"The presence of the volume-law state at the small dissipation, $\\hbar \\gamma / J =0.5$ , implies that there is a finite critical value ${(\\hbar \\gamma /J)}_\\mathrm {c}$ for the MIT likewise the cases of random unitary circuits.",
"On the other hand, that at the large dissipation, $\\hbar \\gamma / J = 500.0$ , can be interpreted as a consequence of the continuous QZE.",
"More specifically, the strong two-body losses suppress double occupation at each site such that the particles in the system behave as hardcore bosons [38].",
"Hence, after several loss events at an early time range, which create a considerable number of holes, the measurement events rarely happen so that the holes spread ballistically to lead to the volume-law entanglement.",
"Figure: Time evolutions of the number of remaining particles per site for several system sizes with ℏγ/J=0.5\\hbar \\gamma / J = 0.5 and ℏγ/J=5.0\\hbar \\gamma / J = 5.0.",
"The Hubbard interaction U/JU/J is set to 5.0.",
"Linear fits are obtained from the data for t>10ℏ/Jt > 10 \\hbar / J (t>5.0ℏ/Jt > 5.0 \\hbar / J) in the largest MM for ℏγ/J=0.5\\hbar \\gamma / J = 0.5 (ℏγ/J=5.0\\hbar \\gamma / J = 5.0) case.",
"Error bars indicate 1σ\\sigma uncertainty.",
"Figure: The system size MM dependencies of the number of remaining particles NN when the entanglement entropy takes the maximal value for (a) ℏγ/J=0.5\\hbar \\gamma / J = 0.5 and (b) ℏγ/J=5.0\\hbar \\gamma / J = 5.0.",
"Error bars indicate 1σ\\sigma uncertainty.",
"Since the number of remaining particles $N$ continues to decrease in the Lindblad dynamics, one might suspect that the transitions of the scaling law are results of the decrease of particles.",
"Figure REF represents the time evolution of the average density $N/M$ in the dynamics shown in Fig.",
"REF .",
"For both $\\hbar \\gamma / J = 0.5$ and $5.0$ cases, the density decreases algebraically in long-time dynamics and its dependence on the system size is almost absent.",
"The exponents of the algebraic decreases estimated from the linear fits are $-0.65$ for $\\hbar \\gamma / J = 0.5$ and $-0.66$ for $\\hbar \\gamma / J = 5.0$ .",
"These exponents are almost the same before and after the scaling law transition.",
"Furthermore, as shown in Fig.",
"REF , the number of remaining particles when the entanglement entropy takes the maximal value increases almost linearly as the system size increases for both $\\hbar \\gamma / J = 0.5$ and $5.0$ cases.",
"Therefore, the scaling of $N$ is not so different before and after the transitions of the scaling laws, and thus the transitions cannot be understood as the result of the decrease of particles.",
"Figure: System size MM dependencies of the maximal entanglement entropy max t S max (t)\\max _t S_{\\max }(t) (a) with U/J=1.0U/J = 1.0 and (b) with U/J=10.0U/J = 10.0 for several strengths of the dissipation γ\\gamma .",
"Error bars indicate 1σ\\sigma uncertainty.The reentrant structure we found is present in a broad range of $U/J$ .",
"Figure REF represents the system size dependencies of the maximal value of the entanglement entropy for $U/J = 1.0$ and $U/J = 10.0$ cases.",
"One can see the reentrant structure for both cases.",
"For the $U/J = 1.0$ case, even with a small dissipation $\\hbar \\gamma / J = 0.5$ , the scaling law of the entanglement entropy is ALSLC in contrast to the $U/J = 5.0$ case.",
"This can be attributed to the fact that the double occupancy rate increases compared to the $U/J = 5.0$ case and thus the probability of measurement is effectively increased."
],
[
"How to experimentally detect measurement-induced transitions",
"In closed systems, a kind of the entanglement entropy, namely the 2nd order Rény entropy, has been observed in experiments with ultracold gases in optical lattices by preparing a copy of the target system and measuring interference between the target and the copy [39], [40].",
"However, in open systems with dissipation, it is hard to use the same protocol because the copy cannot perfectly mimic measurement events which happen in a stochastic manner.",
"Hence, it is imperative to point out alternative experimental observables that can distinguish the ALSLC states from the volume-law states."
],
[
"Momentum distribution",
"In this subsection, we show that the momentum distribution $\\mathinner {\\langle {\\hat{n}_k}\\rangle } = \\frac{1}{M}\\sum _{ij}\\mathinner {\\langle {b^\\dagger _i b_j}\\rangle }\\mathrm {e}^{\\mathrm {i} k(i-j)},$ which is a standard observable in ultracold-gas experiments, reflects the scaling law of the entanglement entropy.",
"Here, we set the lattice spacing to unity.",
"We set $U/J$ to $5.0$ in this subsection.",
"Figure: Normalized momentum distributions for several values of the dissipation strength γ\\gamma at the time that gives max t S max (t)\\max _t S_{\\max }(t).",
"The blue solid, orange dashed, green dashed-dotted, and red dotted lines correspond to ℏγ/J=0.5\\hbar \\gamma / J = 0.5, 5.05.0, 50.050.0, and 500.0500.0, respectively.",
"The system size MM is set to 20 in order to investigate a vast range of γ\\gamma .",
"Error bars indicate 1σ1\\sigma uncertainty.",
"Figure REF shows the normalized momentum distributions for $\\hbar \\gamma / J = 0.5$ , $5.0$ , $50.0$ , and $500.0$ at the time that gives $\\max _t S_{\\max }(t)$ (See Sec.",
"for the definition).",
"The system size is set to $M = 20$ in order to compute states with the volume-law entanglement.",
"In each of the three different regions of the dissipation strength, $\\mathinner {\\langle {\\hat{n}_k}\\rangle }/N$ exhibits a distinct signal.",
"Here, $N$ is the total number of remaining particles in the system.",
"In the case of the small dissipation, $\\hbar \\gamma / J = 0.5$ , there exists a single peak at $k=0$ .",
"In the intermediate region, including $\\hbar \\gamma / J = 5.0$ and $50.0$ , the dips at $|k| = \\pi /2$ are developed.",
"In the case of the strong dissipation, $\\hbar \\gamma / J = 500.0$ , the distribution is almost flat.",
"In order to characterize the signals more quantitatively, we show in Fig.",
"REF the visibility $\\mathinner {\\langle {\\hat{n}_\\pi }\\rangle }/\\mathinner {\\langle {\\hat{n}_{\\pi /2}}\\rangle }$ as a function of $\\hbar \\gamma / J$ .",
"Figure: Visibility 〈n ^ π 〉/〈n ^ π/2 〉\\mathinner {\\langle {\\hat{n}_\\pi }\\rangle } / \\mathinner {\\langle {\\hat{n}_{\\pi /2}}\\rangle } as a function of the dissipation strength γ\\gamma .",
"Although it is in practice impossible to precisely determine the critical points with our matrix product states method, we have checked that the states in the region 2≤ℏγ/J≤502 \\le \\hbar \\gamma / J \\le 50 safely obey ALSLC.",
"Error bars indicate 1σ1\\sigma uncertainty.",
"Since the visibility becomes considerably large in the intermediate region, where the states with ALSLC emerge, it can be used for distinguishing the states with ALSLC from the volume-law states.",
"Notice that the visibility at $M=20$ shown in Fig.",
"REF does not exhibit any singular behaviors across the transition points because the system size is too small.",
"Figure: (a) A possible loss process in the early time dynamics.",
"Atoms with finite group velocity tend to be lost.",
"(b) Momentum distributions before (blue solid line) and after (orange dashed line) a loss event in the early time dynamics.",
"(c) Difference of momentum distributions before and after the loss event.",
"To obtain the momentum distributions, we set M=64M=64.",
"The emergence of the dip structure in the intermediate region can be understood as a quantum Zeno effect in the momentum space.",
"At $t=0$ , there is no doubly-occupied site as depicted in Fig.",
"REF (a).",
"This means that in order for the loss events to happen, particles have to move with finite group velocity.",
"In other words, the loss event is more probable for faster particles.",
"Since the group velocity is the largest at $|k|=\\pi /2$ in the single-particle band of the one-dimensional Bose-Hubbard model, which is $-2J\\cos k$ , the particles with $|k|=\\pi /2$ is the most likely to be lost.",
"In Figs.",
"REF (b) and (c), we compare the momentum distribution right before and after a loss event during early-time dynamics and see that the momentum distribution of the lost two particles is indeed peaked at $|k| = \\pi /2$ .",
"As a consequence of series of such loss events, $\\mathinner {\\langle {\\hat{n}_k}\\rangle }/N$ forms the dips at $|k| = \\pi / 2$ .",
"After the formation of the dip structure, the stronger dissipation for faster particles suppresses the redistribution of the particles towards states around $|k| = \\pi /2$ ."
],
[
"Strong suppression of particle transport",
"The momentum distribution reflects the transition of the scaling law of the entanglement entropy, and is relatively easily accessible in ultracold-gas experiments.",
"However, the relation between the momentum distribution and the entanglement entropy seems unclear.",
"To resolve this difficulty, we propose a more direct signature of the entanglement transitions.",
"For this purpose, we borrow an idea from the experimental confirmations of the MBL states [22], [41], which has utilized the breaking of ergodicity as an indicator of the area-law states.",
"In an area-law state, a part of a system does not possess an extensive entanglement entropy and thus cannot act as a thermal bath for the rest of the system [42].",
"The absence of the thermal bath results in the ergodicity breaking that is manifested, e.g., by the spatial imbalance of particles [22], [41].",
"However, the true equilibrium state of this system is the vacuum state $\\mathinner {|{0}\\rangle }$ regardless of the scaling laws because the number of remaining particles continues to decrease.",
"Therefore, what one can observe is only dynamics towards a transient spatially imbalanced state.",
"We expect that particle transport reflects a tendency toward the spatially imbalanced non-ergodic state.",
"In order to confirm this expectation, we simulate the following dynamics: In a 2$M$ -site system, we prepare an initial state $\\mathinner {|{\\psi _0}\\rangle } = \\prod ^M_{i=1} \\hat{b}^\\dagger _i\\mathinner {|{0}\\rangle }$ in the left half of the system and set a high barrier potential $100J\\sum ^{2M}_{i=M+1} \\hat{n}_i$ which prevents the particles from coming into the right half.",
"We set $U/J$ to $5.0$ .",
"After $t=t_\\mathrm {rel}$ , we turn off the barrier potential and release the particles to the right half of the system.",
"$t_\\mathrm {rel}$ is chosen to be within the time region where $S_{\\max }(t)$ takes almost a steady value.",
"We characterize particle transport by comparing the number of particles in the right half, i.e., $N_\\mathrm {r} = \\sum ^{2M}_{i=M+1}\\mathinner {\\langle {\\hat{n}_i}\\rangle }$ with the half of the number of remaining particles $N/2$ .",
"Both $N_\\mathrm {r}$ and $N/2$ can be measured in experiments [41].",
"Figure: Time evolution of the number of particles in the right half of the system N r N_\\mathrm {r} (blue solid line) and the half of the remaining number of particles N/2N/2 (orange dashed line) for (a) (ℏγ/J,2M,t rel J/ℏ)=(5.0,128,10)(\\hbar \\gamma / J, 2M, t_\\mathrm {rel} J / \\hbar ) = (5.0, 128, 10) and (b) (ℏγ/J,2M,t rel J/ℏ)=(0.5,40,5)(\\hbar \\gamma / J, 2M, t_\\mathrm {rel} J / \\hbar ) = (0.5, 40, 5).Figure: Spatial distribution of particles at t=t rel +ℏM/Jt = t_\\mathrm {rel}+\\hbar M / J in the dynamics of (a) Fig.",
"(a) and (b) Fig.",
"(b).Figure REF (a) represents the time evolution of $N_\\mathrm {r}$ and $N/2$ for $(\\hbar \\gamma / J, 2M, t_\\mathrm {rel}J/\\hbar ) = (5.0, 128, 10)$ , where the state at $t=t_\\mathrm {rel}$ is an ALSLC state.",
"For the broad region $60 \\lesssim t J / \\hbar \\lesssim 100$ , there is a visible difference between $N/2$ and a converged $N_\\mathrm {r}$ .",
"The difference means that the delocalization of the particles is suppressed due to the dissipation, thus signaling the tendency toward a state without ergodicity.",
"By contrast, for $(\\hbar \\gamma / J, 2M, t_\\mathrm {rel}J/\\hbar ) = (5.0, 128, 10)$ , where the state at $t=t_\\mathrm {rel}$ is a volume-law state, $N_\\mathrm {r}$ exceeds $N/2$ before the convergence as seen in Fig.",
"REF (b).",
"This overshoot behavior implies ballistic transport of the particles, which is consistent with the fact that in a volume-law state quantum information ballistically propagates.",
"The difference of tendencies is also visible as the inhomogeneity of the spatial distribution of particles, left-leaning or right-leaning, as shown in Fig.",
"REF .",
"In another volume-law region, say $\\hbar \\gamma / J = 500.0$ , the spatial distribution of particles and the time evolution of $N_\\mathrm {r}$ and $N/2$ shown in Fig.",
"REF are similar to those of $\\hbar \\gamma / J = 0.5$ in terms of the ballistic propagation and the right-leaning spatial distribution of particles.",
"Thus, the strong suppression of particle transport clearly distinguishes the scaling law of the entanglement entropy as expected.",
"For a fair comparison, we also present the dynamical behaviors of $\\hbar \\gamma / J = 5.0$ case with system size $2M=40$ in Fig.",
"REF .",
"Even in the small system, the absence of the particle excess and the left-leaning spatial distribution of particles, which characterize the localization, are also visible.",
"Figure: (a) Time evolution of the number of particles in the right half of the system N r N_\\mathrm {r} (blue solid line) and the half of the remaining number of particles N/2N/2 (orange dashed line) for (ℏγ/J,2M,t rel J/ℏ)=(500.0,40,30.0)(\\hbar \\gamma / J, 2M, t_\\mathrm {rel}J/\\hbar ) = (500.0, 40, 30.0).",
"(b) Spatial distribution of particles at t=t rel +ℏM/Jt = t_\\mathrm {rel} + \\hbar M / J.Figure: (a) Time evolution of the number of particles in the right half of the system N r N_\\mathrm {r} (blue solid line) and the half of the remaining number of particles N/2N/2 (orange dashed line) for (ℏγ/J,2M,t rel J/ℏ)=(5.0,40,5.0)(\\hbar \\gamma / J, 2M, t_\\mathrm {rel}J/\\hbar ) = (5.0, 40, 5.0).",
"(b) Spatial distribution of particles at t=t rel +ℏM/Jt = t_\\mathrm {rel} + \\hbar M / J.Figure: The imbalance PP versus the dissipation strength ℏγ/J\\hbar \\gamma / J at t obs =t rel +ℏM/Jt_\\mathrm {obs} = t_\\mathrm {rel} + \\hbar M / J.For the simulation with 2M=402M = 40, we use t rel J/ℏ=5.0t_\\mathrm {rel}J/\\hbar = 5.0 for ℏγ/J≤5.0\\hbar \\gamma / J \\le 5.0 and set t rel J/ℏt_\\mathrm {rel}J/\\hbar to 6.0, 10.0, 15.0, 20.0, and 30.0 for ℏγ/J=10.0,20.0,50.0,100.0\\hbar \\gamma / J = 10.0, 20.0, 50.0, 100.0, and 500.0, respectively.",
"For 2M=642M=64 (128) case, we use t rel J/ℏ=10.0t_\\mathrm {rel}J/\\hbar = 10.0 for ℏγ/J≤20.0\\hbar \\gamma / J \\le 20.0 (10.0) and t rel J/ℏ=20.0t_\\mathrm {rel}J/\\hbar = 20.0 for ℏγ/J=50.0\\hbar \\gamma / J = 50.0 (20.0).",
"In order to quantify how much particle transport is suppressed, we calculate the imbalance between $N_\\mathrm {r}$ and $N/2$ defined as $P = \\frac{N/2 - N_\\mathrm {r}}{N/2 + N_\\mathrm {r}}$ at $t_\\mathrm {obs} = t_\\mathrm {rel} + \\hbar M / J$ .",
"$t_\\mathrm {obs} - t_\\mathrm {rel}$ corresponds to a rough estimate of the time scale in which the particles released at $t=t_\\mathrm {rel}$ reaches the right edge of the system.",
"If $P$ significantly exceeds zero, the state at $t=t_\\mathrm {rel}$ is an ALSLC state.",
"If $P \\le 0$ , that is a volume-law state.",
"Otherwise, the state lies in an intermediate regime.",
"Figure REF represents $P$ versus $\\hbar \\gamma / J$ for $2M = 40$ , 64, and 128.",
"The distinction between the volume-law and ALSLC states made from $P$ is consistent with the scaling law shown in Fig.",
"REF .",
"We also see that the imbalance becomes more visible as the system size increases.",
"Although it is quite difficult to access the volume-law region in the case of the larger systems ($2M=64$ and 128) with numerical simulations, it is expected that $P \\le 0$ with the ballistic particle transport regardless of the system size.",
"In short, the imbalance $P$ serves as an indicator of whether the initial state of the release dynamics is the ALSLC or the volume-law state, which can be observed in experiments."
],
[
"Summary",
"We proposed the measurement-induced transitions (MITs), which have been theoretically found in recent studies of quantum circuit models [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], can be experimentally observed by using ultracold bosons in optical lattices with controllable dissipation.",
"We employed a quasi-exact numerical method to investigate effects of dissipation on quench dynamics of the one-dimensional Bose-Hubbard model with a two-body loss term.",
"By computing the maximal entanglement entropy of the system during the time evolution, we found two MITs.",
"Specifically, when the strength of the dissipation increases, the scaling of the entanglement changes from a volume law to an area law with a logarithmic correction, and again to the volume law.",
"We showed that the momentum distribution, a standard observable in ultracold-gas experiments, reflects the change of the scaling laws.",
"We also suggested that the strong suppression of particle transport in the dynamics after release of the particles to an empty space are more direct observable signatures in experiments for distinguishing the area-law states from the volume-law states.",
"We could not locate precisely the critical points for the two MITs because it was impossible to efficiently describe the volume-law states of the dissipative Bose-Hubbard model with currently available numerical techniques.",
"Since in experiments with ultracold gases the tractable system size is not limited by the volume-law entanglement, the determination of the critical points will be a meaningful target of quantum simulations.",
"We thank S. Nakajima, Y. Takahashi, Y. Takasu, and T. Tomita for fruitful discussions.",
"The MPS calculations in this work are performed with ITensor library, http://itensor.org.",
"This work was financially supported by KAKENHI from Japan Society for the Promotion of Science: Grant No.",
"18K03492 and No.",
"18H05228, by CREST, JST No.",
"JPMJCR1673, and by MEXT Q-LEAP Grant No.",
"JPMXS0118069021."
]
] | 2001.03400 |
[
[
"Inductive Document Network Embedding with Topic-Word Attention"
],
[
"Abstract Document network embedding aims at learning representations for a structured text corpus i.e.",
"when documents are linked to each other.",
"Recent algorithms extend network embedding approaches by incorporating the text content associated with the nodes in their formulations.",
"In most cases, it is hard to interpret the learned representations.",
"Moreover, little importance is given to the generalization to new documents that are not observed within the network.",
"In this paper, we propose an interpretable and inductive document network embedding method.",
"We introduce a novel mechanism, the Topic-Word Attention (TWA), that generates document representations based on the interplay between word and topic representations.",
"We train these word and topic vectors through our general model, Inductive Document Network Embedding (IDNE), by leveraging the connections in the document network.",
"Quantitative evaluations show that our approach achieves state-of-the-art performance on various networks and we qualitatively show that our model produces meaningful and interpretable representations of the words, topics and documents."
],
[
"Introduction",
"Document networks, e.g.",
"social media, question-and-answer websites, the scientific literature, are ubiquitous.",
"Because these networks keep growing larger and larger, navigating efficiently through them becomes increasingly difficult.",
"Modern information retrieval systems rely on machine learning algorithms to support users.",
"The performance of these systems heavily depends on the quality of the document representations.",
"Learning good features for documents is still challenging, in particular when they are structured in a network.",
"Recent methods learn the representations in an unsupervised manner by combining structural and textual information.",
"Text-Associated DeepWalk (TADW) [28] incorporates text features into the low-rank factorization of a matrix describing the network.",
"Graph2Gauss [2] learns a deep encoder, guided by the network, that maps the nodes' attributes to embeddings.",
"GVNR-t [3] factorizes a random walk based matrix of node co-occurrences and integrates word vectors of the documents in its formulation.",
"CANE [25] introduces a mutual attention mechanism that builds representations of a document contextually to each of its direct neighbors in the network.",
"Apart from Graph2gauss, these methods are not intended to generate representations for documents with no connection to other documents and thus cannot induce a posteriori representations for new documents.",
"Moreover, they provide little to no possibility to interpret the learned representations.",
"CANE is a notable exception since its attention mechanism produces interpretable weights that highlight the words explaining the links between documents.",
"Nevertheless, it lacks the ability to explain the representations for each document independently.",
"In this paper, we describe and evaluate an inductive and interpretable method that learns word, topic and document representations in a single vector space, based on a new attention mechanism.",
"Our contributions are the following: we present a novel attention mechanism, Topic-Word Attention (TWA), that produces representations of a text where latent topic vectors attend to the word vectors of a document; we explain how to train the parameters of TWA by leveraging the links of the network.",
"Our method, Inductive Document Network Embedding (), is able to produce representations for previously unseen documents, without network information; we quantitatively assess the performance of on several networks and show that our method performs better than recent methods in various settings, including when new documents, not part of the network, are inductively represented by the algorithms.",
"To our knowledge, we are the first to evaluate this kind of inductive setting in the context of document network embedding; we qualitatively show that our model learns meaningful word and topic vectors and produces interpretable document representations.",
"The rest of the paper is organized as follows.",
"In Section we survey related works.",
"We present in details our attention mechanism and show how to train it on networks of documents in Section .",
"Next, in Section , we present a thorough experimental study, where we assess the performance of our model following the usual evaluation protocol on node classification and further evaluating its capacity of inducting representations for text documents with no connection to the network.",
"In Section , we study the ability of our method to provide interpretable representations.",
"Lastly, we conclude this paper and provide future directions in Section .",
"The code for our model, the datasets and the evaluation procedure are made publicly available https://github.com/brochier/idne."
],
[
"Related Work",
"Network embedding (NE) provides an efficient approach to represent nodes in a low dimensional vector space, suitable for solving various machine learning tasks.",
"Recent techniques extend NE for document networks, showing that text and graph information can be combined to improve the resolution of classification and prediction tasks.",
"In this section, we first cover important works in document NE and then relate recent advances in attention mechanisms."
],
[
"Document Network Embedding",
"DeepWalk [22] and node2vec [9] are the most well-known NE algorithms.",
"They train dense embedding vectors by predicting nodes co-occurrences through random walks by adapting the Skip-Gram model initially designed for word embedding [19].",
"VERSE [24] propose an efficient algorithm that can handle any type of similarity over the nodes.",
"Text-Associated DeepWalk (TADW) [28] extends DeepWalk to deal with textual attributes.",
"Yang et al.",
"prove, following the work in [17], that Skip-Gram with hierarchical softmax can be equivalently formulated as a matrix factorization problem.",
"TADW then consists in constraining the factorization problem with a pre-computed representation of the documents $T$ by using Latent Semantic Analysis (LSA) [6].",
"The task is to optimize the objective: $ \\mathrm {argmin}_{W,H} ||M - W^{\\intercal } HT||^2_F .$ where $M=(A+A^2)/2$ is a normalized second-order adjacency matrix of the network, $W$ is a matrix of one-hot node embeddings and $H$ a feature transformation matrix.",
"Final document embeddings are the concatenation of $W$ and $HT$ .",
"Graph2Gauss (G2G) [2] is an approach that embeds each node as a Gaussian distribution instead of a vector.",
"The algorithm is trained by passing node attributes through a non-linear transformation via a deep neural network (encoder).",
"GVNR-t [3] is a matrix factorization approach for document network embedding, inspired by GloVe [21], that simultaneously learns word, node and document representations.",
"In practice, the following least-square objective is optimized: $ \\underset{U,W}{\\mathrm {argmin}} \\sum _{i=1}^{n_d} \\sum _{j=1}^{n_d} \\big (u_i \\cdot \\frac{\\delta _j ~ W}{|\\delta _j|_1} - \\log (1 + x_{ij})\\big )^2 .$ where $x_{ij}$ is the number of co-occurrences of nodes $i$ and $j$ , $u_i$ is a one-hot encoding of node $i$ and $\\frac{\\delta _j ~ W}{|\\delta _j|_1}$ is the average of the word embeddings of document $j$ .",
"Context-Aware Network Embedding (CANE) [25] consists in a mutual attention mechanism trained on a document network.",
"It learns several embeddings for a document according to its different contextual documents, represented by its neighbors in the network.",
"The attention mechanism selects meaningful features from text information in pairs of documents that explain their relatedness in the graph.",
"A similar approach is presented in [4] where the links between pairs of documents are predicted by computing the mutual contribution of their word embeddings.",
"In this work, we aim at constructing representations of documents that reflect their connections in a network.",
"A key motivation behind our approach is to be able to predict a document's neighborhood given only its textual content.",
"This allows our model to inductively produce embeddings for new documents for which no existing link is known.",
"To that extend, Graph2Gauss is a similar approach.",
"On the contrary, TADW and GVNR-t are not primarily designed for this purpose as they both learn one-hot embeddings for each node in the document network.",
"Note that if some methods like GraphSage [10], SDNE [27] and GAE [13] also enable induction on new nodes, they cannot deal with nodes that have no known connection.",
"Also, our approach differs from CANE since this latter needs the neighbors of a document to generate its representation.",
"learns to produce a single interpretable vector for each document in the network.",
"In the next section, we review recent works in attention mechanisms for natural language processing (NLP) that inspired the conception of our method."
],
[
"Attention Mechanism",
"An attention mechanism uses a contextual representation to highlight or hide some parts of input data.",
"Attention is an essential element of state-of-the-art neural machine translation (NMT) algorithms [18] by providing a powerful way to capture dependencies between words.",
"The Transformer [26] introduces a formalism of attention mechanisms for NMT.",
"Given a query vector $q$ , a set of key vectors $K$ and a set of value vectors $V$ , an attention vector is produced with the following formula: $ v_a = \\omega (qK^T)V .$ $qK^T$ measures the similarity between the query and each key $k$ of $K$ .",
"$\\omega $ is a normalization function such that all attention weights are positive and sum to 1.",
"$v_a$ is then the weighted sum of the values $V$ according to the attention weights.",
"Multiple attention vectors can be generated by using a set of queries $Q$ .",
"In CANE, as for various NLP tasks [7], an attention mechanism generates attention weights that represent the strengths of relation between pairs of input words.",
"However, in this paper, we do not seek to learn dependencies between pairs of words, but rather between words and some global topics.",
"In this direction, the Set Transformer [16] constitutes a computationally efficient attention mechanism where the queries are replaced with a fixed-size set of learnable global inducing points.",
"This model is originally not intended for NLP tasks, therefore we will explore the capacity of such inducing points to play the role of topic representations when applied to textual data.",
"Even if we introduce the concept of topic vectors, the aim of this work is not to propose another topic model [5], [23].",
"We hypothesize that the introduction of global topic vectors in an attention mechanism can (1) lead to useful representations of documents for different tasks and (2) bring an interpretable sight on the patterns learned by the model.",
"Interpretability can help both machine learning practitioners to better refine their models and end users to understand automated recommendations."
],
[
"Method",
"We are interested in finding low dimensional vector space representations of a set of $n_d$ documents organized in a network, described by a document-term matrix $ X \\in \\mathbb {N}^{n_d \\times n_w}$ and an adjacency matrix $A \\in \\mathbb {N}^{n_d \\times n_d}$ , where $n_w$ stands for the number of words in our vocabulary.",
"The method we propose, Inductive Document Network Embedding (), learns to represent the words and topics underlying the corpus in a single vector space.",
"The document representations are computed by combining words and topics through an attention mechanism.",
"In the following, we first describe how to derive the document vectors from known word and topic vectors through a novel attention mechanism, the Topic-Word Attention (TWA).",
"Next, we show how to estimate the word and topic vectors, guided by the links connecting the documents of the network."
],
[
"Representing Documents with Topic-Aware Attention",
"We assume a $p$ -dimensional vector space in which both words and topics are represented.",
"We note $W \\in \\mathbb {R}^{n_w \\times p}$ the matrix that contain the $n_w$ word embedding vectors and $T \\in \\mathbb {R}^{n_t \\times p}$ the matrix of $n_t$ topic vectors.",
"Figure REF shows the matrix computation of the attention weights.",
"Given a document $i$ and its bag-of-word encoding $X_i \\in \\mathbb {N^+}^{n_w}$ , we measure the attention weights between topics and words, $Z^i \\in \\mathbb {R}^{n_t \\times n_w}$ , as follows: $Z^i = \\text{g}\\big (TW^{\\intercal }\\text{diag}(X_i)\\big ).$ The activation function $g$ must satisfy two requirements: (1) all the weights are non-negative and (2) columns of $Z^i$ sum to one.",
"The intuition behind the first requirement is that enforcing non-negativity should lead to sparse and interpretable topics.",
"The second requirement transforms the raw weights into word-wise relative attention weights, which can be read as probabilities similarly to what is done in neural topic models [23].",
"An obvious choice would be column-wise softmax, however, we empirically find that ReLU followed by a column-wise normalization performs best."
],
[
"Document Representation",
"Given $Z^i$ , we are able to calculate topic-specific representations of the document $i$ .",
"From the perspective of topic $k$ , the $p$ -dimensional representation of document $i$ is: $ D^i_k = \\frac{Z^i_k \\text{diag}(X_i)W}{|X_i|_1}.$ Similarly to Equation REF , each topic vector, akin to a query, attends to the word vectors that play the role of keys to generate $Z^i$ .",
"The topic-specific representations are then the weighted sum of the values, also played by the word vectors.",
"The final document vector is obtained by simple summation of all the topic-specific representations, which leads to $d^i = \\sum _k{D^i_k}$ .",
"Scaling by $\\frac{1}{|X_i|_1}$ in Equation REF ensures that the document vectors have the same order of magnitude as the word vectors.",
"Figure: Matrix computation of the attention weights.",
"Here W i W^i is the compact view of diag(X i )W\\text{diag}(X_i)W where zero-columns are removed since they do not impact on the result.",
"n w i n_{w^i} denotes the number of distinct words in document ii.",
"Each element z jk z_{jk} of Z i Z^i is the column-normalized rectified scalar product between the topic vector t j t_{j} and the word embedding w k i w^i_{k} and represents the strength of association between the topic jj and the word kk in document ii.",
"The final document representation is then the sum of the topic-specific representations D i =Z i W i |X i | 1 D^i = \\frac{Z^i W^{i}}{|X_i|_1}.Since the corpus is organized in a network, we propose to estimate the parameters, $W$ and $T$ , by leveraging the links between the documents.",
"We posit that the representations of documents connected by a short path in the network should be more similar in the vector space than those that are far apart.",
"Thus, we learn $W$ and $T$ in a supervised manner, through the training of a discriminative model.",
"Let $\\Delta \\in \\mathbb {\\lbrace }0,1\\rbrace ^{n_d \\times n_d}$ be a binary matrix, so that $\\delta _{ij} = 1$ if document $j$ is reachable from document $i$ and $\\delta _{ij} = 0$ otherwise.",
"We model the probability of a pair of documents to be connected, given their representations, in terms of the sigmoid of the dot-product of $d_i$ and $d_j$ : $P(Y = 1 | d_i, d_j ; W, T) = \\sigma (d_i \\cdot d_j).$ Assuming the document representations are i.i.d, we can express the log-likelihood of $\\Delta $ given $W$ and $T$ : $ \\ell (W,T) &= \\sum _{i=1}^{n_d} \\sum _{j=1}^{n_d} \\log P(Y = \\delta _{ij} | d_i, d_j ; W,T) \\nonumber \\\\&= \\sum _{i=1}^{n_d} \\sum _{j=1}^{n_d} \\delta _{ij} \\log \\sigma (d_i \\cdot d_j) + (1 - \\delta _{ij}) \\log \\sigma (-d_i \\cdot d_j).$ Through the maximization of this log-likelihood via a first-order optimization technique, we back-propagate the gradient and thus learn the word and topic vectors that lead to the document representations that best reconstruct $\\Delta $ ."
],
[
"Quantitative Evaluation",
"Common tasks in document network embedding are classification and link prediction.",
"We assess the quality of the representations learned with for these tasks in two different settings: (1) a traditional setting where all links and documents are observed and (2) an inductive setting where only a fraction of the links and documents is observed during training.",
"The first setting corresponds to a scenario where the goal is to propagate labels associated with a small portion of the documents.",
"The second represents a scenario where we want to predict labels and links for new documents that have no network information, once the algorithm is already trained.",
"This is common setting in real world applications.",
"As an example, when a new user asks a new question on a Q&A website, we would like to suggest tags for its question and to recommend potential similar questions.",
"In this case, the only information available to the algorithm is the textual content of the question."
],
[
"Experimental Setup",
"We detail here the setup we use to train ."
],
[
"Computing the $\\Delta $ matrix.",
"We consider paths of length up to 2 and compute the $\\Delta $ matrix in the following manner: $\\delta _{ij} = {\\left\\lbrace \\begin{array}{ll}1 &\\text{~if~} (A + A^2)_{ij} > 0,\\\\0 &\\text{~otherwise}.\\end{array}\\right.",
"}$ This means that two documents are considered close in the network if they are direct neighbors or share at least one neighbor.",
"Note that this matrix is the binarized version of the matrix TADW factorizes."
],
[
"Optimizing the log-likelihood.",
"We perform mini-batch SGD with the ADAM [12] update rule.",
"Because most document networks are sparse, rather than uniformly sampling entries of $\\Delta $ , we sample 5000 balanced mini-batches in order to favor convergence.",
"We sample 16 positive examples ($\\delta _{ij} = 1$ ) and 16 negative ones ($\\delta _{ij} = 0$ ) per mini-bacth.",
"Positive pairs of documents are drawn according to the number of paths of length 1 or 2 linking them.",
"Negative samples are uniformly drawn.",
"The impact of the number of steps is detailed in Section REF ."
],
[
"Networks",
"We consider 4 networks of documents of various nature: A well-known scientific citation network extracted from Cora https://linqs.soe.ucsc.edu/data.",
"Each document is an article labelled with a conference.",
"New York Times (NYT) titles of articles from January 2007.",
"Articles are linked according to common tags (e.g.",
"business, arts, technology) and are labeled with the section they appear in (e.g.",
"opinion, news).",
"This network is particularly dense and documents have a short length.",
"Two networks of the Q&A website Stack Exchange (SE) https://archive.org/details/stackexchange from June 2019, namely gaming.stackexchange.com and travel.stackexchange.com.",
"We only keep questions with at least 10 user votes and that have at least one answer with 10 user votes or more.",
"We build the network by linking questions with their answers and by linking questions and answers of the same user.",
"The labels are the tags associated with each question.",
"Table: General properties of the studied networks."
],
[
"Tasks and Evaluation Metrics",
"For each network, we consider a traditional classification tasks, an inductive classification task and an inductive link prediction task.",
"the traditional task refers to a setting where the model is trained on the entire network and the learned representations are used as features for a one-vs-all linear classifier with a training set of labelled documents ranging from 2% to 10% for multi-class networks and from 10% to 50% for multi-label networks.",
"the inductive tasks refer to a setting where 10% of the documents are removed from the network and the model is trained on the resulting sub-network.",
"For the classification task, a linear classifier is trained with the representations and the labels of the observed documents.",
"Representations for hidden documents are then generated in an inductive manner, using their textual content only.",
"Classifications and link predictions are then performed on these induced representations.",
"To classify the learned representations, we use the LIBLINEAR [8] logistic regression [14] algorithm and we cross validate the regularization parameter for each dataset and each model.",
"Every experiment is repeated 10 times and we report the micro average of the area under the ROC curve (AUC).",
"The AUC uses the probabilities of the logistic regression for all classes and evaluates the quality of the resulting ranking given the true labels.",
"This metric is thus suitable for information retrieval tasks where we want to penalize wrong predictions depending on their ranks.",
"For link prediction, we rank pairs of documents according to the cosine similarity between their representations."
],
[
"Compared Representations",
"For all document networks, we process the documents by tokenizing text into words, discarding punctuation, stop words and words that appear less than 5 times or in more than 25% of the documents.",
"We create document-term matrices that are used as input for 6 algorithms.",
"Our baselines are representative of the different approaches for document NE.",
"TADW and GVNR-t are based on matrix factorization whereas CANE and G2G are deep learning models.",
"For each of them, we used the implementations of the authors: LSA: we use a 256-dimensional SVD decomposition of the tf-idf vectors as a text-only baseline; TADW: we follow the guidelines of the original paper by using 20 iterations and a penalty term $\\lambda =0.2$ .",
"For induction, we generate a document vector by computing the textual component $HT$ in Equation REF ; Graph2gauss (G2G): we make sure the loss function converges before the maximum number of iterations; GVNR-t: we use $\\gamma =10$ random walks of length $t=40$ , a sliding window of size $l=5$ and a threshold $x_{\\text{min}}=5$ with 1 iteration.",
"For induction, we compute $ \\frac{\\delta _j ~ W}{|\\delta _j|_1}$ in Equation REF ; CANE: we use the same parameters as in the original paper; : we run all experiments with $n_t=32$ topic vectors.",
"The effect of $n_t$ is discussed in Section REF ."
],
[
"Results Analysis",
"Tables REF and REF detail the AUC scores on the traditional classification task.",
"We report the results for CANE only for Cora since the algorithm did not terminate within 10 hours for the other networks.",
"In comparison, our method takes about 5 minutes to run on each network on a regular laptop.",
"The classifier performs well on the representations we learned, achieving similar or better results than the baseline algorithms on Cora, Gaming and Travel Stack Exchange.",
"However, regarding the New York Times network, GVNR-t and TADW have a slight advantage.",
"Because of its high density, the links in this network are little informative which may explain the relative good scores of the LSA representations.",
"We hypothesize that (1) TADW benefits from its input LSA features and that (2) GVNR-t benefits both from its random walk based matrix of node co-occurrences [20], which captures more precisely the proximities of the nodes in such dense network, and from the short length of the documents making the word embedding averaging efficient [15], [1].",
"Table REF shows the AUC scores in the inductive settings.",
"For link prediction performs best on three networks, showing its capacity to learn meaningful word and topic representations according to the network structure.",
"For classification, LSA and GVNR-t achieve the best results while reaches similar but slightly lower scores on all datasets.",
"On the contrary, TADW and Graph2gauss show weaknesses on NYT and Gaming SE.",
"In summary, shows constant performances across all settings where other methods lack of robustness against the type of network or the type of task.",
"A surprising result is the good scores of GVNR-t for inductive classification which we didn't expect given that its textual component only is used for this setting.",
"However, for the traditional classification, GVNR-t has difficulties to handle networks wih longer documents.",
"does not suffer the same problem because TWA carefully select discriminative words before averaging them.",
"In Section , we further show that learns meaningful representations of words and topics and builds interpretable document representations."
],
[
"Impact of the Number of Topics and Convergence Speed",
"Figure REF shows the impact of the number of topic vectors $n_t$ and of the number of steps (mini-batches) on the AUC scores obtained in traditional classification with Cora.",
"Note that we observe a similar behavior on the other networks.",
"We see that the scores improve from 1 to 16 topics and tend to stagnate for upper values.",
"In a similar manner, performances improve up to 5000 iterations after which no increase is observed.",
"Table: Micro AUC scores on Cora and NYTTable: Micro AUC scores on Stack Exchange NetworksTable: Micro AUC scores for inductive classification and inductive link predictionFigure: Impact of the number of topics and of the number of steps on the traditional classification task on Cora with ."
],
[
"Qualitative Evaluation",
"We first show in Section REF that is capable of learning meaningful word and topic vectors.",
"Then, we provide visualizations of documents that highlight the ability of the topic-word attention to reveal topics of interest.",
"For all experiments, we set the number of topics to $n_t = 6$ ."
],
[
"Word and topic vectors",
"Table REF shows the closest words to each topic, computed as the dot product between their respective vectors, learned on Cora.",
"Word and topic vectors are trained to predict the proximity of the nodes in a network, meaningless words are thus always dissimilar to the topic vectors, since they do not help to predict a link.",
"This can be verified by observing the words that have the largest and the smallest norms, also reported in Table REF .",
"Even though the topics are learned in an unsupervised manner, we notice that, when we set the number of topics close to the number of classes, each topic seems to capture the semantics of one particular class.",
"Table: Topics with their closest words produced by on Cora and words whose vector L 2 L_2 norms are the largest (resp.",
"the smallest) reported in parenthesis.",
"The labels in this dataset are: Case Based, Genetic Algorithms, Neural Networks, Probabilistic Methods, Reinforcement Learning, Rule Learning and Theory."
],
[
"Topic Attention Weights Visualization",
"To further highlight the ability of our model to bring interpretability, we show in Figure REF the topics that most likely generated the words of a document according to TWA.",
"The document is the abstract of this paper whose weights are inductively calculated with previously trained on Cora.",
"We compute its attention weights $Z^i$ and associate each word $k$ to the maximum value of its column $Z^i_k$ .",
"We then colorize and underline each word associated to the two most represented topics in the document, if its weight is higher than $\\frac{1}{2}$ .",
"We see that the major topic (green and single underline), that accounts for 32% of the weights, deals with the type of data, here document networks.",
"The second topic (blue and double underline), which represents 18% of the weights, relates to text modeling, with words like interpretable and topics.",
"Figure: Topics provided by in the abstract of this very paper trained on Cora."
],
[
"Discussion and Future Work",
"In this paper, we presented , an inductive document network embedding algorithm that learns word and latent topic representations via TWA, a topic-word attention mechanism able to produce interpretable document representations.",
"We showed that IDNE performs state-of-the-art results on various network in different settings.",
"Moreover, we showed that our attention mechanism provides an efficient way of interpreting the learned representations.",
"In future work, we would like to study the effect of the sampling of the documents on the learned topics.",
"In particular, the matrix $\\Delta $ could capture other types of similarities between documents such as SimRank [11] which measures structural relatedness between nodes instead of proximities.",
"This could reveal complementary topics underlying a document network and could provide interpretable explanations of the roles played by documents in networks."
]
] | 2001.03369 |
[
[
"Review of Sterile Neutrino Experiments"
],
[
"Abstract There are $\\sim3\\sigma$ or more evidence of eV-scale sterile neutrinos from several different measurements.",
"Many dedicated experiments are (being) created and (will) take data to confirm or refute the eV-scale sterile neutrinos.",
"In this talk, a mini review is presented on current experimental efforts and status on sterile neutrino search, especially using reactor and accelerator neutrinos."
],
[
"Introduction",
"The first experimental evidence of eV-scale sterile neutrino is from LSND (Liquid Scintillator Neutrino Detector) experiment in 2001 by observing 3.8$\\sigma $ excess of $\\overline{\\nu }_{\\rm {e}}$ events from a $\\overline{\\nu }_e$ appearance channel at 30 m baseline from 30 MeV $\\overline{\\nu }_{\\mu }$ beam produced by muon decay at rest coming from 800 MeV proton beam on water target in Los Alamos National Lab.",
"The main goal of MiniBooNE experiment at Fermilab was to confirm or refute the LSND result by observing $\\nu _{\\rm {e}}$ ($\\overline{\\nu }_{\\rm {e}}$ ) appearance at 500 m baseline from 500 MeV $\\nu _{\\mu }$ ($\\overline{\\nu }_{\\mu }$ ) beam produced by pion decay in flight coming from 8 GeV proton beam on Beryllium target.",
"Note that MiniBooNE has a similar $L/E$ as LSND but different neutrino energy, signal, background and systematics.",
"In 2009, MiniBooNE observed 3$\\sigma $ excess of events in low ($<$ 475 MeV) energy region and this is consistent with the LSND excess but could not identify the excess is signal or unknown background.",
"MicroBooNE was designed to identify the origin of the excess in MiniBooNE and currently has been taking data for few years.",
"Another evidence of eV-scale sterile neutrino was observed in 2011 by radio-chemical solar neutrino experiments, GALLEX and SAGE, from 2.6$\\sigma $ deficit of $\\nu _{\\rm {e}}$ events from $^{51}$ Cr and $^{37}$ Ar radioactive sources to calibrate their detectors.",
"Very short baseline (VSBL) reactor neutrino experiments in the 1990's also showed $\\sim 6\\%$ deficit ($\\sim $ 3$\\sigma $ ) of $\\overline{\\nu }_{\\rm {e}}$ events when compared to an updated calculation of reactor neutrino flux in 2011 by Mention et al [1].",
"This deficit is so called RAA (Reactor Anti-neutrino Anomaly) and stimulated follow-up experiments even though there is a rather large uncertainty (2.4%) of predicted reactor neutrino flux from Huber [2] and Muller [3] model.",
"The RAA best fit value with 3+1 $\\nu $ scheme is $\\Delta m^{2}_{41} = 2.4$ eV$^2$ and sin$^22\\theta _{14} =$ 0.14.",
"Recent VSBL reactor neutrino experiments are further motivated by the \"5 MeV excess\" observed by RENO, Daya Bay and Double Chooz in 2012 to identify the origin of the 5 MeV excess.",
"In these VSBL experiments, background subtraction is done by subtracting reactor-OFF from reactor-ON data assuming background is time invariant, and data analyses are performed in a model-independent way.",
"In the following sections, more details on current reactor and accelerator neutrinos experiments to search for an eV-scale sterile neutrinos are described.",
"CEvNS experiments such as RED100, MINER, and CONUS etc, can also search for light sterile neutrinos as well as BEST, source experiment in BAksan, and KATRIN at Karlsruhe, but they are not discussed here due to limited space."
],
[
"Reactor $\\nu $ Experiments for Sterile {{formula:bf402657-ef34-40a0-af5b-3729c22a27f4}} Search",
"There are two types of detectors in VSBL reactor neutrino detectors.",
"One is liquid scintillator (LS) and the other is plastic scintillator (PS) detector.",
"LS detectors can be either homogeneous or segmented while PS detectors are all segmented.",
"Segmented detectors are better to reduce background and to identify the vertex position.",
"To further reduce background and to improve detection efficiency, gadolinium (Gd) or lithium ($^{6}$ Li) are often loaded in one way or another in these detectors.",
"NEOS, Neutrino-4, STEREO, and PROSPECT are LS detectors and all are segmented and use research reactors except NEOS.",
"DANSS and SoLid are PS detectors and use commercial and research reactors, respectively.",
"NuLat and CHANDLER are mobile $\\nu $ labs with PS detectors for nuclear monitoring at the surface and currently under R&D and preparing a full detector, respectively.",
"Table REF compares some important differences of these VSBL experiments.",
"In the following subsections, more details on current VSBL experiments are discussed.",
"Table: Comparison of the current VSBL experiments."
],
[
"NEOS",
"NEOS is a $\\sim 1m^3$ LS detector doped with 0.5% Gd and located in a tendon gallery of the 5$^{\\rm {th}}$ reactor (2.8 GW$_{\\rm {th}}$ ) of the Hanbit nuclear power plant which is also used by RENO.",
"The distance from the 5$^{\\rm {th}}$ reactor core to the NEOS detector is 24 m and overburden is about 20 m.w.e.",
"Total 38 8 inch PMTs are mounted on two side buffers and the energy resolution is measured to be about 5% at 1 MeV.",
"In phase-I, NEOS has taken 180 (46) days reactor-ON (-OFF) data from Aug. 2015 to May 2016 with $\\sim $ 2000 IBDs/day where signal to background ratio is about 22 thanks to 10% Ultima-Gold-F added to GdLS for good PSD (pulse shape discrimination).",
"The 5 MeV excess was clearly observed in this NEOS-I data set when compared with the Huber-Miller model.",
"According to the shape analysis by comparing the prompt energy spectra of NEOS to Daya Bay, an oscillation pattern was seen but this was still within systematic uncertainty, and therefore only an exclusion region was set.",
"The RAA best fit was disfavored by NEOS-I at 4.3$\\sigma $ [4].",
"NEOS phase-II has started taking data in September 2018 after refurbishing (new GdLS and muon veto system) its phase-I detector.",
"Main goals of NEOS-II are “rate and shape” analysis for sterile neutrino search and spectral decomposition of $^{235}$ U and $^{239}$ Pu for a full fuel cycle of 500 days data, important for understanding origin of the 5 MeV excess."
],
[
"NEUTRINO-4",
"Neutrino-4 is a segmented 1.8 m$^{3}$ LS detector doped with 0.1% Gd and the GdLS is equally filled in optically isolated 10$\\times $ 5 rectangular cells (W: 22.5 cm x H: 85 cm L: 22.5 cm) up to 70 cm height.",
"Each of total 50 PMTs is mounted on the top of each cell and energy resolution is measured as 16% at 1 MeV.",
"Neutrino-4 detects neutrinos from SM-3 research reactor (100 MW$_{\\rm {th}}$ ) in Russia with baseline from 6 to 12 m under a shallow overburden (3$\\sim $ 5 m.w.e.).",
"Neutrino-4 took 480 (278) days of reactor-ON (-OFF) data [5] where signal to background ratio is about 0.54 due to no PSD capability.",
"Oscillation pattern in 3+1 $\\nu $ scheme was observed with the best fit value of $\\Delta m^{2}_{41} \\approx 7$ eV$^2$ and sin$^22\\theta _{14} \\approx $ 0.4 at 3$\\sigma $ .",
"However, this contradicts the absolute flux measurements at a longer baseline by Daya Bay and RENO due to much bigger deficit by Neutrino-4 at the averaged oscillation.",
"Neutrino-4 plans to upgrade its detector to Neutrino-6 where new GdLS is filled with higher concentration of Gd and PSD capability.",
"Expected sensitivity of Neutrino-6 is about 3 times better than current result of Neutrino-4."
],
[
"STEREO",
"STEREO is a segmented 2.4 m$^{3}$ LS detector doped with 0.2% Gd and the GdLS target is equally filled in 6 rectangular cells (W: 89.2cm, H: 91.8 cm, L:36.9 cm) optically isolated.",
"The 6 cells of GdLS target are surrounded by two short LS cells in front and end and two long LS cells in both sides.",
"Four 8 inch PMTs are mounted on top of each target cell and each short LS cell and energy resolution is measured as 9% at 1 MeV.",
"For each long LS cell, eight 8 inch PMTs are mounted on top of it.",
"STEREO detects neutrinos from ILL research reactor (58 MW$_{\\rm {th}}$ ) in France with baseline from 9 to 11 m under an overburden ($\\sim $ 15 m.w.e.).",
"STEREO took 179 (235) days of reactor-ON (-OFF) data with $\\sim $ 400 IBDs/day where signal to background ratio is about 0.9.",
"No excess around 5 MeV but deficit above 6 MeV was observed [6].",
"No oscillation pattern was also observed and the RAA best fit value was rejected at $>$ 99.9%CL."
],
[
"PROSPECT",
"PROSPECT is a segmented 4 ton LS detector with 0.1% $^6$ Li doping and the $^6$ LiLS is equally filled in 154 rectangular cells (14.5 cm x 14.5 cm x 117.6 cm per cell) optically isolated.",
"A 5 inch PMT is attached to each side, apart by 117.6 cm length, of a cell and energy resolution is measured as 4.5% at 1 MeV.",
"PROSPECT detects neutrinos from HFIR research reactor (80 MW$_{\\rm {th}}$ ) in USA with baseline from 7 to 9 m under a very shallow overburden ($<$ 1 m.w.e.).",
"PROSPECT took 33 (23) days of reactor-ON (-OFF) data with 771 IBDs/day where signal to background ratio is about 2.2 (1.32) for accidental (correlated) background thanks to good PSD power.",
"No clear excess around 5 MeV was observed and no oscillation pattern was also observed due to large statistical uncertainty, and the RAA best fit value was rejected at 2.2$\\sigma $ [7].",
"PROSPECT is currently analyzing additional data (June-October 2018) with twice more IBD event statistics than the data used in current results."
],
[
"DANSS",
"DANSS is a segmented 1 m$^3$ PS detector consisting of 2,500 strips (W: 4 cm, H: 1 cm, L: 1 m per strip) of PS embedded with 3 wavelength shifting (WLS) fibers per strip coated with reflective material with 0.35% Gd loading.",
"Each of total 2,500 SiPMs is attached to one end of each strip, and additionally 50 PMTs are attached to 50 bundles of 50 strips resulting in $\\sim $ 28% energy resolution at 1 MeV.",
"DANSS detects neutrinos from WWER-1000 reactor (3.1 GW$_{\\rm {th}}$ ) in Russia with baseline from 10.7 to 12.7 m under an overburden of 50 m.w.e.",
"DANSS has been taking data since October 2016 with $\\sim $ 5,000 IBDs/day (signal/background $\\approx $ 33) at 10.7 m, and result with 193 days of reactor-ON data showed no excess around 5 MeV.",
"When the prompt energy distribution at bottom position was compared to that of top position, small oscillation pattern was observed but did not match with the RAA best fit value which was rejected at 5$\\sigma $ [8].",
"More recent preliminary result from DANSS [9] with about 2.5 times increased statistics showed some excess around 5 MeV region but still weak (1.8$\\sigma $ ) evidence of 3+1 $\\nu $ oscillation."
],
[
"SoLid",
"SoLid [10] is a highly segmented 1.6 ton PS detector consisting of about 13,000 cubes (5x5x5 cm$^3$ per cube) of PS embedded with parallel and vertical optical fibers per cube covered with LiF:ZnS(Ag) sheets.",
"Readout is done by total about 32,000 SiPMs operating at 5$^{O}$ C and energy resolution is measured as $\\sim $ 14% at 1 MeV.",
"It detects neutrinos from BR2 research reactor (50-80 MW$_{\\rm {th}}$ ) in Belgium with baseline from 6 to 9 m under an overburden of 10 m.w.e.",
"About 300 IBDs/day with $\\sim $ 30% efficiency is expected with a good signal to background ratio of three to one.",
"Data has been taken for more than a year so far and the first physics result is expected to come out soon."
],
[
"Daya Bay, RENO and JUNO",
"Short baseline (SBL) reactor neutrino experiments, Daya Bay [11] and RENO [12], also searched for eV-scale sterile neutrinos assuming 3+1 $\\nu $ scheme using 621 and 1,500 days data, respectively, with a Far-to-Near ratio method in their data analysis.",
"Figure REF [13] shows NEOS, DANSS, and Daya Bay exclusion regions together with JUNO sensitivity.",
"Figure: Exclusion limits and sensitivity for eV-scale sterile neutrino searches by NEOS, DANSS, Daya Bay and JUNO.",
"Adapted from ."
],
[
"MiniBooNE",
"MiniBooNE is a 800 ton mineral oil Cherenkov detector which detects neutrinos at a 500 m baseline from pion decay in flight from the Booster neutrino beam (E$_{\\nu } = $ 500 MeV) line at Fermilab.",
"By taking additional 6.38$\\times $ 10$^{20}$ POT $\\nu $ data since 2015, so far MiniBooNE has taken total 3$\\times $ 10$^{21}$ POT with almost same amount of $\\nu $ and $\\overline{\\nu }$ for more than 15 years.",
"Using this data MiniBooNE observed excess of $\\nu _{\\rm {e}}$ and $\\overline{\\nu }_{\\rm {e}}$ events, 460.5 $\\pm $ 95.8, corresponding to 4.8$\\sigma $ , and this result is consistent with LSND excess (see the left plot in Fig.",
"REF ) [14].",
"Figure: Left: Appearance/prediction vs. L/E shows LSND and MiniBooNE results agree well.Right: Allowed regions by LSND and MiniBooNE mostly overlap but OPERA and KARMEN2 exclude some of the allowed regions.",
"Adapted from .The best fit sin$^22\\theta $ value for active to sterile neutrino oscillation is close to 1 and this is too big to explain non-observation of sterile neutrinos from OPERA and a combined result of MINOS and MINOS+ that gives best limit on sin$^22\\theta _{24}$ for $\\Delta m^2 <$ 0.3 eV$^2$ [15].",
"The right plot in Fig.",
"REF shows the allowed (excluded) regions by MiniBooNE and LSND (OPERA and KARMEN2) [14].",
"Additionally, the origin of the excess (signal vs. unknown background) events must be identified by MicroBooNE which can distinguish electron-like and gamma-like events.",
"In addition to the MicroBooNE, the two other detectors in Fermilab SBN program [16] using the Booster neutrino beam, ICARUS and ND using the same technology as MicroBooNE, are expected to take data in 2020 or 2021.",
"Therefore, the long time mystery of the LSND result would be expected to be resolved soon."
],
[
"JSNS$^{2}$",
"JSNS$^2$ is a 50 ton GdLS (0.2% Gd) detector in a cylindrical vessel surrounded by gamma catcher region filled with LS surrounded by buffer region filled with mineral oil where 150 10 inch PMTs are mounted on the wall resulting in $\\sim $ 15% energy resolution at 1 MeV.",
"The outermost region is a muon veto region filled with water.",
"JSNS$^2$ is located in the 3$^{\\rm {rd}}$ floor of MLF building, JPARC, wehere 3 GeV proton beam is interacted in a mercury target.",
"Unlike MiniBooNE, JSNS$^2$ uses a similar base line (24 m) and the same neutrino source (E$_{\\nu } = $ 30 MeV) from muon decay at rest as LSND.",
"JSNS$^2$ is expected to take data in 2020 and with 3 years of data it can exclude LSND result at $\\Delta m^2 > 2$ MeV$^2$ region but some phase space is still allowed at $\\Delta m^2 < 2$ MeV$^2$ ."
],
[
"Acknowledgments",
"This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Ministry of Science and ICT (MSIT) (No.",
"2017R1A2B4012757 and IBS-R016-D1-2019-b01)."
]
] | 2001.03349 |
[
[
"Near-horizon geodesics of high-spin black holes"
],
[
"Abstract We provide an exhaustive and illustrated classification of timelike and null geodesics in the near-horizon region of near-extremal Kerr black holes.",
"The classification of polar motion extends to Kerr black holes of arbitrary spin.",
"The classification of radial motion leads to a simple parametrization of the separatrix between bound and unbound motion.",
"Furthermore, we prove that each timelike or null geodesic is related via conformal transformations and discrete symmetries to spherical orbits and we provide the explicit mappings.",
"We detail the high-spin behavior of both the innermost stable and the innermost bound spherical orbits."
],
[
"Introduction and summary",
"The study of timelike and null geodesics of the Kerr metric has a long history which is still ongoing [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].",
"There are at least two motivations for studying Kerr geodesics.",
"First, null geodesics (together with the modeling of light sources) underpin the field of black hole imaging [3], [22], [23], [9], [24], [25], [26], [27], which has recently become an observational science [28].",
"Second, timelike geodesics provide the zeroth-order motion of binary systems in the perturbative small-mass-ratio expansion, which leads in the adiabatic approximation to the leading-order gravitational waveforms of extreme-mass-ratio inspirals (EMRIs) [29], [30], [31], [32].",
"In the extremely high-spin limit, conformal $SL(2,\\mathbb {R})$ symmetry appears in the near-horizon limit [33], [34], [35], [36].",
"Previous studies of geodesic motion in the high-spin near-horizon limit were restricted either to equatorial orbits [37], [38], [39], [40], [41], [17], [42], to specific orbits [43], or to parametrically generic geodesics [18] that discard relevant measure-zero sets in parameter space such as the separatrix between bound and unbound motion.",
"The first objective of this paper is to derive the complete classification of geodesics in the high-spin near-horizon Kerr region.",
"While our treatment of polar motion applies to arbitrary geodesics in Kerr, the radial motion is only studied in the high-spin limit and in the near-horizon region.",
"For the ease of readability, we provide extended tables and figures summarizing our classification.",
"In their range of applicability, all the results of Kapec and Lupsasca [18] agree with ours.",
"We will be attentive to defining an intuitive nomenclature for each geodesic class.",
"It was previously shown in Refs.",
"[43], [16] that any null orbit that enters or leaves the near-horizon region has a polar motion bounded by the minimal angle $\\cos ^2\\theta _{\\text{\\text{min}}} = 2 \\sqrt{3}-3$ ($47^\\circ \\lessapprox \\theta \\lessapprox 133^\\circ $ ), which corresponds to the polar inclination of the velocity-of-light surface in the near-horizon and high-spin limit.",
"As an outcome of our analysis, we confirm this property and further prove that it holds for any timelike geodesic as well.",
"The polar motion is more restricted for the innermost bound spherical orbits (IBSOs): $\\cos ^2\\theta _{\\text{\\text{min}}} = 1/3$ ($55^\\circ \\lessapprox \\theta \\lessapprox 125^\\circ $ ), as already shown in Refs.",
"[42], [21] and even more restricted for the innermost stable spherical orbits (ISSOs): $\\cos ^2\\theta _{\\text{\\text{min}}} = 3-2 \\sqrt{2}$ ($65^\\circ \\lessapprox \\theta \\lessapprox 115^\\circ $ ), as independently shown in Ref.",
"[21] at the time of finalizing this draft.",
"We also prove that the separatrix between bound and unbound motion consists of all orbits with an angular momentum equal to that of the ISSO.",
"Conformal symmetry in the near-horizon high-spin Kerr geometry leads to potentially observable signatures if such high-spin black holes are realized in nature.",
"The behavior of null geodesics on the image of an extremely spinning Kerr black hole leads to the NHEKline [3], [44] and to specific polarization whorls [45].",
"Gravitational waveforms on adiabatic inspirals lead to exponentially decaying tails at fixed oscillation frequencies with amplitudes suppressed as $(1-J^2/M^4)^{1/6}$ [37], [46], while plunging trajectories lead to impact-dependent polynomial quasinormal ringing with a power ranging from inverse time to square root of inverse time [17].",
"With the aim of deriving further consequences of conformal symmetry, we provide in this paper the action of conformal symmetry on arbitrary timelike and null geodesics in near-horizon high-spin Kerr.",
"It was shown in Ref.",
"[17] that conformal symmetry together with a discrete symmetry leads to equivalence classes of equatorial timelike geodesics with circular orbits as distinguished representatives.",
"This allows us to simplify the computation of Teukolsky waveforms by applying conformal transformations to the seed circular waveform [38], [17].",
"In this paper, we show that in whole generality conformal symmetry and discrete symmetries lead to equivalence classes which each admit spherical orbits as distinguished representatives.",
"For orbits with an angular momentum lower than the ISSO, which we denote as subcritical orbits, no spherical orbit exists but a “complex spherical orbit” exists that generates the equivalence class.",
"Such a complex spherical orbit can be used as a seed and complexified conformal transformations allow to reach all real subcritical geodesics.",
"We provide as ancillary files the following four Mathematica sheets: the explicit polar integrals describing geodesics in Kerr and in (near-)NHEK, the explicit radial integrals describing geodesics in (near-)NHEK, the list of conformal maps between (near-) NHEK radial geodesics and finally the code allowing to draw the Penrose diagram of NHEK geodesics.",
"The rest of the paper is organized as follows.",
"In Sec.",
"we provide the complete classification of polar geodesic motion in Kerr.",
"In Sec.",
", we present the complete classification of timelike and null geodesics in the near-horizon region of a high-spin Kerr black hole.",
"We discuss the properties of spherical geodesics in Sec.",
"and we derive the conformal mappings between near-horizon geodesics in high-spin Kerr in Sec.",
".",
"Several appendixes contain the explicit form of all prograde near-horizon geodesics as well as relevant integrals.",
"A comparison with the radial geodesic motion in high-spin Kerr obtained in [18] is performed in Appendix ."
],
[
"Polar geodesic motion in Kerr",
"After a short summary of some essential features of Kerr geodesic motion, we will provide a complete classification of polar geodesic motion in Kerr.",
"Our results complete the inspiring work of Ref.",
"[18] by treating all marginal cases (i.e., measure-zero sets) that were discarded such as $Q=0$ and $\\ell = 0$ (for the notation, see below).",
"We will also provide extensive tables and figures that summarize the classification and allow us to visually grasp all main features.",
"We will be attentive to defining an intuitive nomenclature for each geodesic class.",
"We will also derive manifestly real and positive explicit analytical forms for each polar motion with $\\ell \\ne 0$ .",
"We do not provide the exhaustive real analytical forms for $\\ell = 0$ by lack of astrophysical interest.",
"In Boyer-Lindquist coordinates $\\hat{x}^\\mu = ({\\hat{t}},{\\hat{r}},\\theta ,{\\hat{\\varphi }})$ , the Kerr metric is $s^2=-\\frac{\\Delta }{\\Sigma }({\\hat{t}}-a\\sin ^2\\theta {\\hat{\\varphi }})^2+\\Sigma (\\frac{{\\hat{r}}^2}{\\Delta }+\\theta ^2)+\\frac{\\sin ^2\\theta }{\\Sigma }(({\\hat{r}}^2+a^2){\\hat{\\varphi }}-a{\\hat{t}})^2$ with $\\Delta ({\\hat{r}})&\\triangleq {\\hat{r}}^2-2M{\\hat{r}}+a^2,\\qquad \\Sigma ({\\hat{r}},\\theta )\\triangleq {\\hat{r}}^2+a^2\\cos ^2\\theta .$ Assuming the validity of the cosmic censorship conjecture, the angular momentum per unit mass $a$ of the hole is bounded by its mass $M$ : ${a}\\le M$ .",
"The Kerr metric admits a Killing-Yano tensor $J_{\\mu \\nu }=J_{[\\mu \\nu ]}$ [47], [48], [49], [50] and a derived Killing-Stäckel tensor $K_{\\mu \\nu }=K_{(\\mu \\nu )}$ [51], where (see also Ref.",
"[18]) $\\begin{split}K_{\\mu \\nu } &= -J_\\mu ^{\\;\\; \\lambda }J_{\\lambda \\nu } , \\\\\\frac{1}{2}J_{\\mu \\nu } \\hat{x}^\\mu \\wedge \\hat{x}^\\nu &= a \\cos \\theta \\hat{r} \\wedge (\\hat{t} - a \\sin ^2\\theta \\hat{\\phi })+\\hat{r} \\sin \\theta \\theta \\wedge \\left( (\\hat{r}^2+a^2) \\hat{\\phi }- a \\hat{t}\\right) .\\end{split}$ It obeys $K^{\\mu \\nu }=2 \\Sigma \\ell ^{(\\mu } n^{\\nu )}+\\hat{r}^2 g^{\\mu \\nu }$ , where the two principal null directions of Kerr are given by $ \\ell ^\\mu \\partial _\\mu =\\Delta ^{-1}( (\\hat{r}^2+a^2)\\partial _{\\hat{t}}+a \\partial _{\\hat{\\phi }})$ and $ n^\\mu \\partial _\\mu =(2 \\Sigma )^{-1}( (\\hat{r}^2+a^2)\\partial _{\\hat{t}}-\\Delta \\partial _{\\hat{r}} +a \\partial _{\\hat{\\phi }} )$ ."
],
[
"Geodesic equations",
"As anyone can check, geodesic motion of a test pointwise particle in the Kerr geometry can be described by the following set of four first-order differential equations: $\\Sigma \\,{{\\hat{t}}}{\\tau }&=a \\ell -a^2{\\hat{E}}\\sin ^2\\theta +({\\hat{r}}^2+a^2)\\frac{P({\\hat{r}})}{\\Delta ({\\hat{r}})},\\\\\\Sigma \\,{{\\hat{r}}}{\\tau }&=\\pm _{\\hat{r}}\\sqrt{\\hat{R}({\\hat{r}})},\\\\\\Sigma \\,{\\cos \\theta }{\\tau }&=\\pm _\\theta \\sqrt{\\Theta (\\cos ^2 \\theta )},\\\\\\Sigma \\,{{\\hat{\\varphi }}}{\\tau }&=-a{\\hat{E}}+ \\ell \\csc ^2\\theta +a\\frac{P({\\hat{r}})}{\\Delta ({\\hat{r}})}$ where $\\mu $ is the mass, $\\tau $ is either $\\mu $ times the proper time for timelike geodesics or an affine parameter for massless particles (i.e.",
"$\\mu = 0$ ).",
"The momentum is denoted as $p^\\mu = \\frac{d\\hat{x}^\\mu }{d\\tau }$ .",
"The geodesic is characterized by its energy $\\hat{E} = -p_{\\hat{t}}$ , its angular momentum $\\ell =p_{\\hat{\\phi }}$ and its non-negative Carter constant $k= K_{\\mu \\nu }p^\\mu p^\\nu = p_\\theta ^2 + a^2 \\mu ^2 \\cos ^2\\theta + (p_{\\hat{\\phi }} \\csc \\theta + p_{\\hat{t}} a \\sin \\theta )^2 \\ge 0$ or, equivalently, its Carter constant $Q = k- ( \\ell - a \\hat{E})^2 $ [52], [53].",
"We define $P({\\hat{r}})&\\triangleq {\\hat{E}}({\\hat{r}}^2+a^2)-a \\ell ,\\\\\\hat{R}({\\hat{r}})&\\triangleq P^2({\\hat{r}})-\\Delta ({\\hat{r}})(\\mu ^2{\\hat{r}}^2+k),\\\\\\Theta (\\cos ^2\\theta )&\\triangleq Q \\sin ^2 \\theta + a^2({\\hat{E}}^2-\\mu ^2)\\cos ^2\\theta \\sin ^2 \\theta - \\ell ^2 \\cos ^2\\theta .$ In a given Kerr geometry $(M,a)$ , a geodesic is fully characterized by the quadruplet of parameters $(\\mu \\ge 0,{\\hat{E}}\\ge 0, \\ell \\in \\mathbb {R},Q \\ge -(\\ell - a \\hat{E})^2$ ), its initial spacetime position and two signs, $s_r^i \\equiv \\pm _{\\hat{r}}|_{\\tau = \\tau _i}$ , $s_\\theta ^i \\equiv \\pm _{\\theta }|_{\\tau = \\tau _i}$ , that correspond to the signs of the radial and polar velocity at the initial time $\\tau _i$ .",
"The motion can be integrated using Mino time defined as $\\textrm {d}\\lambda = d\\tau /\\Sigma $ thanks to the property $\\lambda = \\frac{\\hat{r}}{\\pm _{\\hat{r}} \\sqrt{\\hat{R}({\\hat{r}})} } = \\frac{\\cos \\theta }{\\pm _\\theta \\sqrt{\\Theta (\\cos ^2\\theta )}}.$ We consider a geodesic path linking the initial event $({\\hat{t}}_i,{\\hat{r}}_i,\\theta _i,{\\hat{\\varphi }}_i)$ at Mino time $\\lambda _i$ and the final event $({\\hat{t}}_f,{\\hat{r}}_f,\\theta _f,{\\hat{\\varphi }}_f)$ at Mino time $\\lambda _f$ .",
"The equations of motion can be formally integrated as $\\hat{t}(\\lambda _f)-\\hat{t}(\\lambda _i) &=& a(\\ell - a \\hat{E})(\\lambda _f-\\lambda _i) +a^2 \\hat{E}\\, (\\hat{T}_{\\theta }(\\lambda _f)-\\hat{T}_{\\theta }(\\lambda _i) )+\\hat{T}_{\\hat{r}}(\\lambda _f)-\\hat{T}_{\\hat{r}}(\\lambda _i) ,\\nonumber \\\\{\\hat{\\varphi }}(\\lambda _f) -{\\hat{\\varphi }}(\\lambda _i) &=& (\\ell - a \\hat{E}) (\\lambda _f-\\lambda _i) + \\ell (\\hat{\\Phi }_\\theta (\\lambda _f)-\\hat{\\Phi }_\\theta (\\lambda _i)) + a (\\hat{\\Phi }_{\\hat{r}}(\\lambda _f)-\\hat{\\Phi }_{\\hat{r}}(\\lambda _i)) \\nonumber $ where $\\lambda &=& \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{\\hat{r}}{\\pm _{\\hat{r}} \\sqrt{\\hat{R}({\\hat{r}})} } = \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{\\cos \\theta }{\\pm _\\theta \\sqrt{\\Theta (\\cos ^2\\theta )}},\\\\\\hat{T}_\\theta (\\lambda ) &\\triangleq & \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{\\cos ^2 \\theta \\,\\cos \\theta }{\\pm _\\theta \\sqrt{\\Theta (\\cos ^2 \\theta )}}, \\qquad \\hat{\\Phi }_\\theta (\\lambda ) \\triangleq \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{ \\cos \\theta }{\\pm _\\theta \\sqrt{\\Theta (\\cos ^2\\theta )}}(\\csc ^2 \\theta -1), \\\\\\hat{T}_{\\hat{r}}(\\lambda ) &\\triangleq & \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{(\\hat{r}^2 + a^2)P(\\hat{r}) \\,\\hat{r}}{\\pm _{\\hat{r}} \\sqrt{\\hat{R}({\\hat{r}})} \\Delta ({\\hat{r}})},\\qquad \\hat{\\Phi }_{\\hat{r}}(\\lambda ) \\triangleq \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{P(\\hat{r}) \\,\\hat{r}}{\\pm _{\\hat{r}} \\sqrt{\\hat{R}({\\hat{r}})} \\Delta ({\\hat{r}})}.$ The notation $\\scriptsize \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-10.0pt}\\int $ indicates that the signs $\\pm _r$ and $\\pm _\\theta $ are flipped each time a zero of $\\hat{R}$ and $\\Theta $ , respectively, is encountered.",
"Since the signs $\\pm _r$ , $\\pm _\\theta $ are identical to the signs of $r$ and $\\theta $ , respectively, the integral (REF ) is monotonic around each turning point, as it should be in order to define an increasing Mino time $\\lambda $ along the geodesic.",
"It is the same notation used in Ref.",
"[18].",
"Note that $\\hat{T}_\\theta ,~\\hat{\\Phi }_\\theta $ are normalized to be vanishing for equatorial motion.",
"The initial signs $s_r^i \\equiv \\pm _{\\hat{r}}|_{\\lambda = \\lambda _i}$ , $s_\\theta ^i \\equiv \\pm _{\\theta }|_{\\lambda = \\lambda _i}$ , as well as the initial spacetime position, are fixed as a part of the specification of the orbit.",
"If we denote by $w(\\lambda ),m(\\lambda )$ the number of turning points in the radial and polar motion, respectively, at Mino time $\\lambda $ , then as the velocity changes sign at each turning point, $\\pm _r = s_r^i (-1)^w, \\qquad \\pm _\\theta = s_\\theta ^i (-1)^m.$"
],
[
"Taxonomy of polar geodesic motion in Kerr",
"This section aims to describe in full detail the classification of the polar geodesic motion around a Kerr black hole.",
"We will complete the classification given in the recent analysis [18] to all geodesics including in particular $\\ell = 0$Note the conversion of notations $Q_{\\text{here}}=Q_{\\text{there}}$ , ${\\hat{E}}_{\\text{here}} = \\omega _{\\text{there}}$ , $G_\\theta ^{(\\text{there})}=\\lambda ^{\\text{here}}$ , $G_\\phi ^{(\\text{there})}=\\hat{\\Phi }^{(\\text{here})}_\\theta $ , $G_t^{(\\text{there})}=\\hat{T}^{(\\text{here})}_\\theta $ .",
"In addition, we name Type A as Pendular and Type B as Vortical.. Our taxonomy is depicted in Figs.",
"REF and REF .",
"The phenomenology of the polar behavior is principally governed by the sign of $Q$ .",
"The details of the classification are summarized in Tables REF and REF .",
"Due to mirror symmetry between the two hemispheres, the natural variable to describe the polar motion is $z\\triangleq \\cos ^2\\theta $ .",
"Let us define $\\epsilon _0(\\hat{E},\\mu ) \\triangleq a^2 (\\hat{E}^2 - \\mu ^2), $ and $z_\\pm \\triangleq \\Delta _\\theta \\pm \\,\\text{sign}(\\epsilon _0) \\sqrt{\\Delta _\\theta ^2+\\frac{Q}{\\epsilon _0}},\\qquad \\Delta _\\theta \\triangleq \\frac{1}{2}(1-\\frac{Q+\\ell ^2}{\\epsilon _0}),\\qquad z_0\\triangleq \\frac{Q}{Q+\\ell ^2}.$ The classification is determined by the roots of the polar potential ().",
"Assuming $a \\ne 0$ , we can rewrite it as $\\Theta (z)=-\\ell ^2 z+(Q+\\epsilon _0 z)(1-z)= \\left\\lbrace \\begin{array}{ll}\\epsilon _0(z_+-z)(z-z_-), & \\epsilon _0\\ne 0 ;\\\\(Q+\\ell ^2)(z_0-z), & \\epsilon _0=0.\\end{array}\\right.$ Our definition of the roots $z_\\pm $ implies the ordering $z_-<z_+$ (and respectively $z_+<z_-$ ) for $\\epsilon _0>0$ (respectively $\\epsilon _0<0$ ).",
"This is a convenient convention because in both cases the maximal angle will be related to $z_+$ .",
"The positivity of the polar potential implies that the poles $z=1$ ($\\theta =0,\\pi $ ) can only be reached if $\\ell =0$ .",
"Note that when a geodesic crosses a pole, its $\\hat{\\varphi }$ coordinates discontinuously jump by $\\pi $ .",
"The invariance of the polar geodesic equation under $(\\hat{E},\\ell )\\rightarrow (-\\hat{E},-\\ell )$ allows us to reduce the analysis to prograde $\\ell \\ge 0$ orbits.",
"We distinguish the orbits with angular momentum $\\ell \\ne 0$ and without, $\\ell = 0$ : I. Nonvanishing angular momentum $\\ell \\ne 0$ .",
"We must consider the following cases: $-(\\ell -a\\hat{E})^2\\le Q<0$ can only occur if $\\epsilon _0>0$ , otherwise leading to $\\Theta <0$ .",
"For $\\epsilon _0>0$ , the motion is vortical; i.e., it takes place only in one of the two hemispheres without crossing the equatorial plane and is bounded by $0<z_-\\le z\\le z_+<1.$ This vortical motion can only occur provided $\\ell ^2\\le (\\sqrt{\\epsilon _0}-\\sqrt{-Q})^2$ .",
"$Q>0$ leads to motion crossing the equator and symmetric with respect to it, bounded by $0 &\\le z \\le z_+<1\\qquad (\\epsilon _0\\ne 0),\\\\0 & \\le z \\le z_0<1\\qquad (\\epsilon _0=0).$ We will refer to such a motion as pendular; $Q=0$ allows us to write $\\Theta (z)=\\epsilon _0\\, z\\, (1-\\frac{\\ell ^2}{\\epsilon _0}-z).$ If $\\epsilon _0\\le 0$ , the positivity of the polar potential enforces the motion to be equatorial.",
"For $\\epsilon _0\\ge 0 $ , equatorial motion exists at $z=0$ .",
"For $\\epsilon _0\\ge 0 $ and $\\ell ^2 \\le \\epsilon _0$ , another motion exists bounded by $0 < z \\le 1-\\frac{\\ell ^2}{\\epsilon _0}<1,$ which is a marginal case separating the pendular and vortical regimes; the motion then admits only one turning point and asymptotes to the equator both at future and at past times.",
"Since we could not find a terminology for such a motion in the literature, we propose to call it equator-attractiveThis neologism accurately reflects the fact that the motion is polar and that the equator is an attractor.",
"The terminology “homoclinic” is already used in the literature to refer to radial motion..",
"In the special case where $z=0$ at the initial time, the motion remains $z=0$ at all times: it is equatorial.",
"II.",
"Vanishing angular momentum $\\ell =0$ .",
"The polar potential reduces to $\\Theta (z)=\\left\\lbrace \\begin{array}{ll}\\epsilon _0(\\frac{Q}{\\epsilon _0}+z)(1-z), & \\epsilon _0\\ne 0\\\\Q(1-z), & \\epsilon _0=0.\\end{array}\\right.$ We distinguish the following cases: $\\epsilon _0=0$ leads to motion over the whole polar range $0\\le z \\le 1$ for $Q>0$ ; we called it polar motion.",
"The only turning point is located at $z=1$ .",
"For $Q=0$ , the potential vanishes identically and the polar angle remains constant; we call it azimuthal motion; for $Q<0$ the potential is positive only if the motion takes place along the black hole axis $z=1$ ; we call it axial motion.",
"$\\epsilon _0>0$ leads to a polar motion $0\\le z\\le 1$ for $Q>0$ .",
"For $Q=0$ and $z=0$ , the motion is equatorial.",
"For $Q=0$ and $z\\ne 0$ , $z=0$ is an asymptotic attractor of the motion which only takes place in one of the hemispheres.",
"It is therefore a special case of equator-attractive motion where the turning point is at the pole $z=1$ .",
"For $Q<0$ , the motion is either vortical ($0<-\\frac{Q}{\\epsilon _0}\\le z \\le 1$ ) for $-\\epsilon _0<Q<0$ or axial with $z=1$ for $Q\\le -\\epsilon _0<0$ .",
"$\\epsilon _0<0$ leads to a polar motion $0\\le z\\le 1$ for $Q\\ge -\\epsilon _0>0$ and to a pendular one ($0\\le z \\le -\\frac{Q}{\\epsilon _0}<1$ ) for $0<Q<-\\epsilon _0$ .",
"For $Q=0$ , the motion is either equatorial or axial for the potential to be positive.",
"For $Q<0$ , the motion also has to take place along the axis.",
"Let us finally notice that, for any value of $\\epsilon _0$ and $Q\\ge -(a{\\hat{E}})^2$ , an axial motion is always possible.",
"Table: Polar taxonomy of Kerr geodesics with ℓ≥0\\ell \\ge 0.",
"The orbits with ℓ<0\\ell <0 are obtained from ℓ>0\\ell >0 by flipping the signs of both E ^\\hat{E} and ℓ\\ell .",
"Table: Polar taxonomy of Kerr geodesics with ℓ=0\\ell = 0.Figure: Polar taxonomy of ℓ=0\\ell = 0 Kerr geodesics.",
"In addition to the possible motions depicted in the figure, an axial motion is always possible for any value of ϵ 0 \\epsilon _0 and Q≥-(aE ^) 2 Q\\ge -(a{\\hat{E}})^2."
],
[
"Solution to the polar integrals",
"After having classified the different types of motion allowed, we will provide manifestly real and positive explicit solutions in terms of elliptic integrals for each type of polar motion with $\\ell \\ne 0$ in line with the recent analysis [18].",
"All such integrals will turn out to agree with Ref.",
"[18], but our presentation will be slightly simpler.",
"The solution to the polar integrals (REF ) and () can be organized in terms of the categories of polar motion with $\\ell \\ne 0$ : Table: NO_CAPTIONEach type of motion yields to a specific decomposition of the line integrals $\\scriptsize \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-10.0pt}\\int $ in terms of basic integrals.",
"In order to simplify the notations, we drop the “$f$ \" indices labeling the final event and define $h \\equiv \\,\\text{sign}( \\cos \\theta )$ , $\\theta _a \\triangleq \\arccos \\sqrt{z_a}$ ($a=+,-,0$ ), as well as the initial and final signs $\\eta _i$ , $\\eta $ : $\\eta _i \\triangleq - s_\\theta ^i \\,\\text{sign}(\\cos \\theta _i) ,\\qquad \\eta \\triangleq - (-1)^m s_\\theta ^i \\,\\text{sign}(\\cos \\theta ).$ We are now ready to perform the explicit decomposition: Pendular motion.",
"We have $0< z_+ \\le 1$ , and $\\theta $ therefore belongs to the interval $\\theta _+ \\le \\theta \\le \\pi - \\theta _+$ .",
"The polar integral can be written (see Ref.",
"[18]) $\\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int _{\\cos \\theta _i}^{\\cos \\theta } &= 2 m \\left| \\int _{0}^{\\cos \\theta _+} \\right| -\\eta \\left| \\int _0^{\\cos \\theta }\\right|+\\eta _i \\left| \\int _0^{\\cos \\theta _i} \\right|,\\qquad \\epsilon _0\\ne 0,\\\\\\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int _{\\cos \\theta _i}^{\\cos \\theta } &= 2 m \\left| \\int _{0}^{\\cos \\theta _0} \\right| -\\eta \\left| \\int _0^{\\cos \\theta }\\right|+\\eta _i \\left| \\int _0^{\\cos \\theta _i} \\right|,\\qquad \\epsilon _0= 0.$ It is useful to note that our definitions of the roots imply $\\epsilon _0\\,z_-<0,\\qquad \\epsilon _0(z-z_-)>0,\\qquad \\frac{z_+}{z_-}\\le 1.$ Vortical motion.",
"We have $\\epsilon _0>0$ and $0 < z_-\\le \\cos ^2\\theta \\le z_+ < 1$ .",
"The motion therefore never reaches the equator.",
"The sign of $\\cos \\theta $ is constant and determines whether the motion takes place in the northern or the southern hemisphere.",
"Without loss of generality, let us focus on the northern hemisphere: $0 \\le \\theta _+\\le \\theta \\le \\theta _- < \\frac{\\pi }{2}$ ; we denote again as $m$ the number of turning points at Mino time $\\lambda $ .",
"The polar integral can be written (see Ref.",
"[18] and Appendix A of Ref.",
"[19]): $\\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int _{\\cos \\theta _i}^{\\cos \\theta } = \\left( m -\\eta _i \\frac{1-(-1)^m}{2} \\right) \\left| \\int _{\\cos \\theta _-}^{\\cos \\theta _+} \\right| -\\eta \\left| \\int _{\\cos \\theta _-}^{\\cos \\theta } \\right|+ \\eta _i \\left| \\int _{\\cos \\theta _-}^{\\cos \\theta _i} \\right| .$ Equator-attractive motion.",
"This is a limit case of the vortical motion reached in the limit $z_-\\rightarrow 0$ , $z_+\\rightarrow 2\\Delta _\\theta $ .",
"As detailed in Ref.",
"[18], the turning point $z_-=0$ corresponds to a nonintegrable singularity of the polar integrals and the motion exhibits consequently at most one turning point at $z_+=2\\Delta _\\theta $ , leading to the line-integral decomposition $\\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int _{\\cos \\theta _i}^{\\cos \\theta } = \\eta {\\int _{\\cos \\theta _+}^{\\cos \\theta }}-\\eta _i{\\int _{\\cos \\theta _+}^{\\cos \\theta _i}}.$ In all cases but the equator-attractive case, the polar motion is periodic.",
"Denoting by $\\Lambda _\\theta $ its period, one can easily give an explicit formula for the number of turning points $m$ as a function of the Mino time: $m(\\lambda )=\\left\\lbrace \\begin{array}{ll}\\rule {0pt}{6pt}\\left\\lfloor \\frac{2}{\\Lambda _\\theta }(\\lambda -\\lambda _i^\\theta )+\\frac{1}{2}\\right\\rfloor , & Q>0\\\\\\rule {0pt}{16pt}\\left\\lfloor \\frac{2}{\\Lambda _\\theta }(\\lambda -\\lambda _i^\\theta )\\right\\rfloor + \\left\\lfloor \\frac{2}{\\Lambda _\\theta }(\\lambda _i^\\theta -\\lambda _i)\\right\\rfloor + \\frac{3-s^i_\\theta }{2}, & Q<0\\end{array}\\right.$ with $\\lambda _i^\\theta \\triangleq \\lambda _i-s^i_\\theta \\int _{0}^{\\cos \\theta _i}\\frac{\\cos \\theta }{\\sqrt{\\Theta (\\cos ^2 \\theta )}}$ and where the floor function is defined as $\\left\\lfloor x \\right\\rfloor \\triangleq \\max {n\\in \\mathbb {Z}|n\\le x}$ .",
"For the equator-attractive case, one has simply $m(\\lambda )=\\theta (\\lambda -\\lambda _i^\\theta )$ where $\\theta $ is here the Heaviside step function.",
"The integrals introduced above are solved explicitly in Appendix .",
"For each case, the corresponding solutions are detailed below and schematically depicted in Fig.",
"REF .",
"Figure: Angular taxonomy of ℓ≠0\\ell \\ne 0 Kerr geodesics.",
"The angular behavior is depicted in spherical coordinates on the unit sphere: the polar angle is θ(λ)\\theta (\\lambda ), and the azimuthal angle is the purely angular part of the Kerr azimuthal angle (ℓ-aE ^)(λ-λ i )+ℓΦ ^ θ (λ)(\\ell - a \\hat{E}) (\\lambda -\\lambda _i) + \\ell \\hat{\\Phi }_\\theta (\\lambda )."
],
[
"Pendular$(\\hat{E},Q)$ motion.",
"The motion exhibits a positive Carter constant $Q$ and can occur for any $\\epsilon _0\\ne 0$ ; our definition of the roots $z_\\pm $ allows us to treat simultaneously the two cases $\\epsilon _0<0$ and $\\epsilon _0>0$ , which is a simplification with respect to the analysis carried out in Ref.",
"[18].",
"The period of the polar motion (comprising two turning points) in Mino time is given by $\\Lambda _\\theta = 4 \\int _0^{\\cos \\theta _+} \\frac{\\text{d} \\cos \\theta }{\\sqrt{\\Theta (\\cos ^2\\theta )}}\\triangleq 4\\hat{I}^{(0)}(\\sqrt{z_+}) =\\frac{4}{\\sqrt{-\\epsilon _0z_-}}K(\\frac{z_+}{z_-}).$ Using the basic integrals of Appendix , one can write (REF ) as $\\lambda -\\lambda _i&=\\frac{1}{\\sqrt{-\\epsilon _0z_-}}\\left[2mK(\\frac{z_+}{z_-})+s^i_\\theta (-1)^mF(\\Psi ^+(\\cos \\theta ),\\frac{z_+}{z_-})\\right.\\nonumber \\\\&~\\left.-s_\\theta ^iF(\\Psi ^+(\\cos \\theta _i),\\frac{z_+}{z_-})\\right]$ where we define $\\Psi ^+(x)\\triangleq \\arcsin (\\frac{x}{\\sqrt{z_+}})$ .",
"Using (REF ), one can invert (REF ) as $\\cos \\theta =s^i_\\theta (-1)^m\\sqrt{z_+}\\,\\text{sn}\\,(\\sqrt{-\\epsilon _0 z_-}(\\lambda -\\lambda _i^\\theta )-2mK(\\frac{z_+}{z_-}),\\frac{z_+}{z_-})$ where we introduce $\\lambda _i^\\theta &\\triangleq \\lambda _i-\\frac{s^i_\\theta }{\\sqrt{-\\epsilon _0z_-}}F(\\Psi ^+(\\cos \\theta _i),\\frac{z_+}{z_-}).$ This expression matches with Eq.",
"(38) of Ref.",
"[12].",
"Using the periodicity property (REF ) of the elliptic sine, we can further simplify it to $\\cos \\theta (\\lambda )=s^i_\\theta \\sqrt{z_+}\\,\\text{sn}\\,( \\sqrt{-\\epsilon _0 z_-}(\\lambda -\\lambda _i^\\theta ), \\frac{z_+}{z_-}).$ It consistently obeys $\\cos \\theta (\\lambda _i)=\\cos \\theta _i$ and $\\,\\text{sign}(\\cos \\theta ^{\\prime }(\\lambda _i))=s_\\theta ^i$ .",
"This formula agrees with (53) of Ref.",
"[18] but it is written in a simpler form.",
"We also obtain $\\hat{T}_\\theta &=\\frac{-2z_+}{\\sqrt{-\\epsilon _0 z_-}} \\left[ 2 m E^{\\prime } (z_+,\\frac{z_+}{z_-}) +( \\pm _\\theta ) E^{\\prime }(z_+,\\Psi ^+(\\cos \\theta ),\\frac{z_+}{z_-})\\right.\\nonumber \\\\&\\left.~- s_\\theta ^i E^{\\prime }(z_+, \\Psi ^+(\\cos \\theta _i),\\frac{z_+}{z_-} )\\right],$ $\\hat{\\Phi }_\\theta &= \\frac{1}{\\sqrt{-\\epsilon _0 z_-}} \\left[ 2 m \\Pi (z_+,\\frac{z_+}{z_-}) +( \\pm _\\theta ) \\Pi (z_+,\\Psi ^+(\\cos \\theta ),\\frac{z_+}{z_-})\\right.\\nonumber \\\\&~\\left.- s_\\theta ^i \\Pi (z_+,\\Psi ^+(\\cos \\theta _i) ,\\frac{z_+}{z_-})\\right]-(\\lambda -\\lambda _i).$ where $\\lambda -\\lambda _i$ is given by (REF ).",
"All quantities involved are manifestly real.",
"These final expressions agree with Ref.",
"[18]."
],
[
"Pendular${}_\\circ (Q)$ motion.",
"We now consider the critical case $|\\hat{E}| = \\mu $ .",
"The period of the polar motion is $\\Lambda _\\theta =4\\hat{I}^{(0)}(\\sqrt{z_0})=2\\pi \\sqrt{\\frac{z_0}{Q}}.$ In this critical case, (REF ) leads to $\\lambda -\\lambda _i=\\sqrt{\\frac{z_0}{Q}}[m\\pi +s^i_\\theta (-1)^m\\arcsin {\\frac{\\cos \\theta }{\\sqrt{z_0}}}-s^i_\\theta \\arcsin {\\frac{\\cos \\theta _i}{\\sqrt{z_0}}}],$ which can be simply inverted as $\\cos \\theta =s_\\theta ^i\\sqrt{z_0}\\,\\sin (\\sqrt{\\frac{Q}{z_0}}(\\lambda -\\lambda _i^\\theta )),\\qquad \\lambda _i^\\theta \\triangleq \\lambda _i-\\sqrt{\\frac{z_0}{Q}}\\arcsin {\\frac{\\cos \\theta _i}{\\sqrt{z_0}}}.$ The other polar integrals are $\\hat{T}_\\theta &=\\frac{1}{2}{z_0(\\lambda -\\lambda _i)-\\sqrt{\\frac{z_0}{Q}}[(\\pm _\\theta )\\cos \\theta \\sqrt{z_0-\\cos ^2\\theta }-s^i_\\theta \\cos \\theta _i\\sqrt{z_0-\\cos ^2\\theta _i}]},\\\\\\hat{\\Phi }_\\theta &=\\sqrt{\\frac{z_0}{Q(1-z_0)}}\\left[m\\pi +(\\pm _\\theta )\\arcsin (\\sqrt{\\frac{1-z_0}{z_0}}\\cot \\theta )-s^i_\\theta \\arcsin (\\sqrt{\\frac{1-z_0}{z_0}}\\cot \\theta _i)\\right]\\nonumber \\\\&~-(\\lambda -\\lambda _i).$"
],
[
"Vortical$(\\hat{E},Q)$ motion.",
"The period in Mino time is given by $\\Lambda _\\theta = 2 \\left| \\int _{\\cos \\theta _-}^{\\cos \\theta _+} \\frac{\\text{d} \\cos \\theta }{\\sqrt{\\Theta (\\cos ^2\\theta )}} \\right|=\\frac{2}{\\sqrt{\\epsilon _0 z_+}} K(1-\\frac{z_-}{z_+}).$ Using the basic integrals of Appendix , one has $\\lambda -\\lambda _i&=\\frac{1}{\\sqrt{\\epsilon _0z_+}}\\left[(m-h s^i_\\theta \\frac{1-(-1)^m}{2})K(\\tilde{m})-s^i_\\theta (-1)^mF(\\Psi ^-(\\cos \\theta ),\\tilde{m})\\right.\\nonumber \\\\&~\\left.+s^i_\\theta F(\\Psi ^-(\\cos \\theta _i),\\tilde{m})\\right]$ where $\\tilde{m}\\triangleq 1-\\frac{z_-}{ z_+},\\qquad \\Psi ^-(x)=\\arcsin {\\sqrt{\\frac{z_+-x^2}{z_+-z_-}}}.$ Using the inversion formula (REF ) and the periodicity property (), we obtain $\\cos \\theta =h\\sqrt{z_+}\\,\\text{dn}\\,(\\sqrt{\\epsilon _0z_+}(\\lambda -\\lambda _\\theta ^i),\\tilde{m})$ with $\\lambda _i^\\theta \\triangleq \\lambda _i+\\frac{s^i_\\theta h}{\\sqrt{\\epsilon _0z_+}}F(\\Psi ^-(\\cos \\theta _i),\\tilde{m}).$ Again, one has $\\cos \\theta (\\lambda _i)=\\cos \\theta _i$ and $\\,\\text{sign}(\\cos \\theta ^{\\prime }(\\lambda _i))=s_\\theta ^i$ .",
"The two other polar integrals are $\\hat{T}_\\theta &=\\sqrt{\\frac{z_+}{\\epsilon _0}}\\left[(m-h s^i_\\theta \\frac{1-(-1)^m}{2})E(\\tilde{m})-(\\pm _\\theta )E(\\Psi ^-(\\cos \\theta ),\\tilde{m})\\right.\\nonumber \\\\&~\\left.+s^i_\\theta E(\\Psi ^-(\\cos \\theta _i),\\tilde{m})\\right],\\\\\\hat{\\Phi }_\\theta &=\\frac{1}{(1-z_+)\\sqrt{\\epsilon _0z_+}}\\left[(m-h s^i_\\theta \\frac{1-(-1)^m}{2})\\Pi (\\frac{z_--z_+}{1-z_+},\\tilde{m})\\right.\\nonumber \\\\&~\\left.-(\\pm _\\theta )\\Pi (\\frac{z_--z_+}{1-z_+},\\Psi ^-(\\cos \\theta ),\\tilde{m})+s^i_\\theta \\Pi (\\frac{z_--z_+}{1-z_+},\\Psi ^-(\\cos \\theta _i),\\tilde{m})\\right]\\nonumber \\\\& ~-(\\lambda -\\lambda _i)$ in agreement with the results of Ref.",
"[18]."
],
[
"Equator-attractive$(\\hat{E})$ motion.",
"This is the only polar motion which is not periodic.",
"One has $\\lambda -\\lambda _i=\\frac{h}{\\sqrt{\\epsilon _0z_+}}[-(\\pm _\\theta )\\,\\text{arctanh}\\sqrt{1-\\frac{\\cos ^2\\theta }{z_+}}+s^i_\\theta \\,\\text{arctanh}\\sqrt{1-\\frac{\\cos ^2\\theta _i}{z_+}}]$ leading to $\\cos \\theta =h\\sqrt{z_+}\\,\\text{sech}(\\sqrt{\\epsilon _0z_+}(\\lambda -\\lambda _i^\\theta )),\\qquad \\lambda _i^\\theta \\triangleq \\lambda _i+\\frac{s^i_\\theta h}{\\sqrt{\\epsilon _0z_+}}\\text{arctanh}\\sqrt{1-\\frac{\\cos ^2\\theta _i}{z_+}}.$ The polar integrals are $\\hat{T}_\\theta &=\\frac{h}{\\sqrt{\\epsilon _0}}[-(\\pm _\\theta )\\sqrt{z_+-\\cos ^2\\theta }+s^i_\\theta \\sqrt{z_+-\\cos ^2\\theta _i}],\\\\\\hat{\\Phi }_\\theta &=\\frac{h}{\\sqrt{\\epsilon _0(1-z_+)}}[-(\\pm _\\theta )\\arctan \\sqrt{\\frac{z_+-\\cos ^2\\theta }{1-z_+}}+s^i_\\theta \\arctan \\sqrt{\\frac{z_+-\\cos ^2\\theta _i}{1-z_+}}].$ This agrees with the results of Ref.",
"[18]."
],
[
"Near-horizon geodesics in the high-spin limit",
"In this section, we derive a complete classification of timelike and null geodesic trajectories lying in the near-horizon region of a quasiextremal Kerr black hole, the so-called near-horizon extremal Kerr (NHEK) region.",
"We will provide explicit manifestly real analytic expressions for all geodesic trajectories.",
"We will present the classification in terms of the geodesic energy, angular momentum, and Carter constant $Q$ .",
"We will also illustrate each radial motion in NHEK with a Penrose diagram.",
"Partial classifications were performed in Refs.",
"[17] and [18].",
"In Ref.",
"[17], equatorial timelike prograde incoming (i.e.",
"that originate from the Kerr exterior geometry) geodesics were classified.",
"Such geodesics reach the spatial boundary of the near-horizon region at infinite past proper time and therefore physically reach the asymptotically flat Kerr region once the near-horizon is glued back to the exterior Kerr region.",
"It turns out that bounded geodesics in the near-horizon Kerr region also arise in the study of gravitational waves since they correspond to the end point of the transition motion [54].",
"Timelike outgoing geodesics originating from the white hole horizon and reaching the near-horizon boundary are also relevant for particle emission within the near-horizon region [44].",
"In addition, null outgoing geodesics are relevant for black hole imaging around high-spin black holes [27].",
"The generic nonequatorial geodesics were obtained in Ref.",
"[18].",
"In particular, real forms were obtained for each angular integral involved in geodesic motion.",
"However, zero-measure sets of parameters were discarded.",
"These zero-measure sets include in particular the separatrix between bounded and unbounded radial motion which plays a key role in EMRIs.",
"In the following, we do not make any assumption on the geodesic parameters.",
"We will treat both timelike and null geodesics, prograde or retrograde, and with any boundary conditions.",
"Without loss of generality, we will consider future-directed orbits.",
"Past-directed geodesics can be obtained from future-directed geodesics using the $\\mathbb {Z}_2$ map: $T \\rightarrow -T,\\quad \\Phi \\rightarrow -\\Phi ,\\quad E\\rightarrow -E,\\quad \\ell \\rightarrow -\\ell ,$ which will play an important role in Sec.",
".",
"We will denote it as the $\\uparrow \\!\\downarrow $ -flip.",
"We will compare our classification with [18] in Appendix ."
],
[
"Near-horizon extremal Kerr (NHEK)",
"We now consider the near-extremal limit $\\lambda \\triangleq \\sqrt{1-\\frac{a^2}{M^2}}\\rightarrow 0$ combined with the near-horizon limit ${\\hat{r}}\\rightarrow {\\hat{r}}_+=M(1+\\lambda )$ .",
"Suitable coordinates in the near-horizon region are defined as (for a review, see Ref.",
"[55]) $T&=\\frac{{\\hat{t}}}{2M}\\lambda ^{2/3},\\qquad R =\\frac{{\\hat{r}}-{\\hat{r}}_+}{M \\lambda ^{2/3}},\\qquad \\Phi ={\\hat{\\varphi }}-\\frac{{\\hat{t}}}{2M} .$ Plugging (REF ) into the Kerr metric (REF ) and expanding the result in powers of $\\lambda $ gives the NHEK spacetime in Poincaré coordinates: $s^2=2M^2\\Gamma (\\theta )(-R^2T^2+\\frac{R^2}{R^2}+\\theta ^2+\\Lambda ^2(\\theta )(\\Phi +R\\,T)^2)+\\mathcal {O}(\\lambda ^{2/3})$ where $\\Gamma (\\theta )&\\triangleq \\frac{1+\\cos ^2\\theta }{2},\\qquad \\Lambda (\\theta ) \\triangleq \\frac{2\\sin \\theta }{1+\\cos ^2\\theta }.$ The NHEK geometry admits a $SL(2,\\mathbb {R}) \\times U(1)$ symmetry generated by $\\partial _\\Phi $ and $H_0 = T \\partial _T - R \\partial _R,\\qquad H_+ = \\partial _T,\\qquad H_- = (T^2 + \\frac{1}{R^2})\\partial _T -2 T R \\partial _R - \\frac{2}{R}\\partial _\\Phi .$ The Killing tensor $K_{\\mu \\nu }$ (REF ) becomes reducible [43], [56] and can be expressed as $K^{\\mu \\nu } = M^2 g^{\\mu \\nu } + \\mathcal {C}^{\\mu \\nu } + (\\partial _\\Phi )^\\mu (\\partial _\\Phi )^\\nu $ where the $SL(2,\\mathbb {R})$ Casimir is given by $\\mathcal {C}^{\\mu \\nu } \\partial _\\mu \\partial _\\nu = -H_0 H_0 + \\frac{1}{2} (H_+ H_- + H_- H_+).$ We are interested in the Kerr geodesics that exist in the near-extremal limit within the NHEK geometry at leading order in $\\lambda $ .",
"The NHEK angular momentum $\\ell $ and Carter constant $Q$ are identical to their values defined in Boyer-Lindquist coordinates.",
"The NHEK energy is related to the Boyer-Lindquist energy $\\hat{E}$ as ${\\hat{E}}&=\\frac{\\ell }{2M}+\\frac{\\lambda ^{2/3}}{2M}E. $ From now on, we will consider the leading high-spin limit; i.e.",
"we will neglect all $\\mathcal {O}(\\lambda ^{2/3})$ corrections in (REF )."
],
[
"Geodesics",
"In the NHEK geometry, Mino time is defined as $\\lambda \\triangleq \\int ^\\tau \\frac{\\tau ^{\\prime }}{2M^2\\Gamma (\\theta (\\tau ^{\\prime }))}$ , and the geodesic equations of motion simplify to ${T}{\\lambda }&=\\frac{E}{R^2}+\\frac{\\ell }{R},\\\\{R}{\\lambda }&=\\pm _R\\sqrt{v_R(R)},\\\\{\\cos \\theta }{\\lambda }&=\\pm _\\theta \\sqrt{v_\\theta (\\cos ^2\\theta )},\\\\{\\Phi }{\\lambda }&=\\frac{\\ell }{\\Lambda ^2}-\\frac{E}{R}- \\ell ,$ with $v_R(R)&\\triangleq E^2+2E \\ell R+\\frac{R^2}{4}(3\\ell ^2-4(Q+\\mu ^2 M^2)),\\\\v_\\theta (\\cos ^2\\theta )&\\triangleq Q \\sin ^2\\theta +\\cos ^2\\theta \\sin ^2\\theta (\\frac{\\ell ^2}{4}-M^2\\mu ^2 )-\\ell ^2\\cos ^2\\theta .$ The limitation that $E$ remains real and finite implies from (REF ) that we are only considering orbits with energy close to the extremal value $\\ell /(2M)$ .",
"The ISSO angular momentum at extremality is denoted as [box=]align * = EISSOext = 23 M2 2 + Q where the energy of the ISSO will be defined in (REF ).",
"It will play a key role in the following.",
"On the equatorial plane ($Q=0$ ), the definition reduces to $\\ell _*$ used in Refs.",
"[17], [57].",
"We can also write down more simply $v_R(R)= E^2+2E \\ell R-\\mathcal {C} R^2$ where $\\mathcal {C}$ is the conserved quantity obtained from the $SL(2,\\mathbb {R})$ Casimir, $\\mathcal {C} \\triangleq \\mathcal {C}^{\\mu \\nu } P_\\mu P_\\nu = Q - \\frac{3}{4}\\ell ^2 + \\mu ^2 M^2 = \\frac{3}{4} (\\ell _*^2 -\\ell ^2).$ We also have $v_R(R)=\\left\\lbrace \\begin{array}{ll}-\\mathcal {C} (R-R_+)(R-R_-), & \\mathcal {C}\\ne 0 ;\\\\2E\\ell (R-R_0), & \\mathcal {C}=0 ,\\end{array}\\right.$ with $R_\\pm \\triangleq \\frac{E}{\\mathcal {C}} \\ell \\pm \\frac{\\vert E \\vert }{\\vert \\mathcal {C}\\vert } \\sqrt{\\ell ^2+\\mathcal {C}},\\qquad R_0\\triangleq -\\frac{E}{2\\ell } .$ The non-negative Carter constant $k$ is $k = Q+\\frac{\\ell ^2}{4} > 0$ , which implies that $\\mathcal {C} > -\\ell ^2$ and that $R_- < R_+$ with $R_\\pm $ both real.",
"These equations all agree with Ref.",
"[18].",
"Similarly, defining $z\\triangleq \\cos ^2\\theta $ , one can rewrite the polar potential as $v_\\theta (z)= -\\ell ^2 z+(Q+\\mathcal {C}_\\circ z)(1-z)=\\left\\lbrace \\begin{array}{ll}(Q+\\ell ^2)(z_0-z) & \\text{ for } \\mathcal {C}_\\circ =0\\\\\\mathcal {C}_\\circ (z_+-z)(z-z_-) & \\text{ for } \\mathcal {C}_\\circ \\ne 0\\end{array}\\right.$ where $\\mathcal {C}_\\circ $ is defined through the critical value of the angular momentum $\\ell _\\circ $ : $\\mathcal {C}_\\circ \\triangleq \\frac{\\ell ^2-\\ell ^2_\\circ }{4}, \\qquad \\ell _\\circ \\triangleq 2M\\mu .$ The roots of the polar potential are given by $z_0\\triangleq \\frac{Q}{Q+\\ell ^2}\\,\\qquad z_\\pm \\triangleq \\Delta _\\theta \\pm \\text{sign}(\\mathcal {C}_\\circ ) \\sqrt{\\Delta _\\theta ^2+\\frac{Q}{\\mathcal {C}_\\circ }},\\qquad \\Delta _\\theta \\triangleq \\frac{1}{2}(1-\\frac{Q+\\ell ^2}{\\mathcal {C}_\\circ }).$ Future orientation of the geodesic is equivalent to $dT/d\\lambda > 0$ or $E+\\ell R > 0.$ Future-oriented geodesics with $\\ell = 0$ have $E >0$ .",
"For $\\ell \\ne 0$ , we define the critical radius as in [18]: $R_c = -\\frac{E}{\\ell }.",
"$ Future-orientation of the orbit requires $R < R_c \\quad \\text{for} \\quad \\ell < 0, \\quad \\text{and} \\quad R > R_c \\quad \\text{for} \\quad \\ell > 0.$"
],
[
"Solving the equations of motion",
"Using the same reasoning as for the Kerr geometry, the formal solutions to the geodesic equations are given by $\\lambda _f -\\lambda _i&= T^{(0)}_R(R_f)-T^{(0)}_R(R_i) = \\lambda _\\theta (\\theta _f)-\\lambda _\\theta (\\theta _i) , \\\\T(\\lambda _f) -T(\\lambda _i)&= E ( T^{(2)}_R(R(\\lambda _f))-T^{(2)}_R(R(\\lambda _i))) \\nonumber \\\\&~+ \\ell ( T^{(1)}_R(R(\\lambda _f))-T^{(1)}_R(R(\\lambda _i))), \\\\\\Phi (\\lambda _f)-\\Phi (\\lambda _i) &=-\\frac{3}{4} \\ell (\\lambda _f-\\lambda _i) - E ( T^{(1)}_R(R(\\lambda _f))-T^{(1)}_R(R(\\lambda _i))) \\nonumber \\\\&~+ \\ell \\big ( \\Phi _\\theta (\\theta (\\lambda _f))-\\Phi _\\theta (\\theta (\\lambda _i))\\big )$ where the three radial integrals are $T^{(i)}_R(\\lambda ) & \\triangleq & \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{ R}{\\pm _RR^i\\sqrt{E^2+2E \\ell R-\\mathcal {C} R^2}} ,\\qquad i = 0,1,2$ and the two polar integrals are $\\lambda _\\theta (\\lambda ) & \\triangleq & \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{ \\cos \\theta }{\\pm _\\theta \\sqrt{v_\\theta (\\theta )}}, \\\\\\Phi _\\theta (\\lambda ) & \\triangleq & \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{ \\cos \\theta }{\\pm _\\theta \\sqrt{v_\\theta (\\theta )}}\\left( \\frac{1}{\\Lambda ^2(\\theta )} - \\frac{1}{4}\\right).$ The notation was explained previously.",
"We defined $\\Phi _\\theta $ such that it is zero for equatorial orbits (since $\\Lambda (\\pi /2)=2$ ).",
"After integration, the equation (REF ) can be inverted to give $R(\\lambda )$ and $\\theta (\\lambda )$ .",
"We need to solve these five integrals as a function of the geodesic parameters $\\mu ,E,\\ell ,Q,s_\\theta ^i,s_R^i,T_i,R_i,\\theta _i,\\Phi _i$ ."
],
[
"Polar behavior.",
"The results derived in Sec.",
"REF in the context of general Kerr still hold in the near-horizon high-spin limit which is obtained by the scaling limit $\\lambda \\rightarrow 0$ taken in the near-horizon coordinates (REF ).",
"We anticipate that the results also hold in the distinct near-NHEK limit $\\lambda \\rightarrow 0$ taken in the near-horizon coordinates (REF ).",
"Due to the high-spin limit, the following substitution can be made: $&a \\mapsto M, \\qquad \\hat{E} \\mapsto \\frac{\\ell }{2M},\\qquad \\epsilon _0 \\mapsto \\mathcal {C}_\\circ \\triangleq \\frac{\\ell ^2-\\ell _\\circ ^2}{4}, \\\\&\\Theta (z) \\mapsto v_\\theta (z),\\qquad \\hat{\\Phi }_\\theta -\\frac{1}{4}\\hat{T}_\\theta \\mapsto \\Phi _\\theta .$ Notice that the dependence on $\\hat{E}$ of $\\epsilon _0$ has been changed into a dependence in $\\ell $ , the Kerr energy being the same at zeroth order on $\\lambda $ for all trajectories.",
"Therefore, the quadratic term of the polar potential vanishes at the critical value $\\ell _\\circ $ of the angular momentum $\\ell $ .",
"One of the most striking features of the near-horizon polar motion is that $Q$ is non-negative as a consequence of the reality of polar motion, as noticed in Ref.",
"[18]: Proposition 1 $\\forall z\\in [0,1] :v_\\theta (z)\\ge 0\\Rightarrow Q\\ge 0.$ Proof.",
"This property is a consequence of the dependence on $Q$ of $\\mathcal {C}$ defined in (REF ).",
"Indeed, using the fact that $z=\\cos ^2\\theta \\in [0,1]$ one can write $Q = \\mathcal {C} + \\frac{3}{4}\\ell ^2 -M^2 \\mu ^2 \\ge \\mathcal {C} + (1-\\Lambda ^{-2})\\ell ^2 -M^2 \\mu ^2 \\ge v_\\theta (z) \\ge 0.$ A direct consequence is that the near-horizon polar motion cannot be vortical and is consequently either equatorial, pendular, polar or axial.",
"We note that the condition $\\epsilon _0\\ge \\ell ^2$ is never obeyed in the near-horizon case after using the definition (REF ), $a=M$ and $\\hat{E}=\\frac{\\ell }{2M}$ .",
"The equator-attractive class is therefore discarded.",
"The resulting polar classes are listed in Table REF and the phase space is represented in Fig.REF ."
],
[
"Radial behavior.",
"The radial behavior for generic inclined orbits can be solved using the equatorial results [17] thanks to the following observation: Proposition 2 The radial integrals $T_R^{(i)}(R)$ $(i=0,1,2)$ only depend upon the NHEK energy $E$ and angular momentum $\\ell $ while all the dependence upon the mass $\\mu $ and Carter constant $Q$ is through $\\ell _* = \\frac{2}{\\sqrt{3}}\\sqrt{M^2 \\mu ^2 +Q}$ .",
"This simple observation has far-reaching consequences.",
"For any timelike geodesic with $Q \\ne 0$ , one could directly reuse the classification established in Ref.",
"[17], modulo the substitution $\\frac{2}{\\sqrt{3}}M \\mu \\rightarrow \\ell _*$ in every expression encountered.",
"Moreover, null geodesics with $\\mu = 0$ have $Q \\ge 0$ from Proposition REF .",
"We can therefore reuse the classification established in Ref.",
"[17] to classify null geodesics modulo the substitution $M \\mu \\rightarrow Q$ in every expression encountered.",
"Overall, all radial integrals can be described in closed form for all cases by keeping the dependence upon $\\ell _*$ or, equivalently, upon the Casimir invariant $\\mathcal {C}$ .",
"Since the equatorial taxonomy of Ref.",
"[17] did not consider bounded orbits and only considered $\\ell > 0$ , we will expand the taxonomy to the general case.",
"The general classification can be achieved by studying the roots of $v_R$ and the range of $R$ where $v_R \\ge 0$ .",
"We only consider orbits outside the horizon, $R>0$ .",
"There are three broad categories depending on the angular momentum: the supercritical case $\\vert \\ell \\vert > \\ell _*$ or equivalently $\\mathcal {C} < 0$ , the critical case $\\vert \\ell \\vert = \\ell _*$ or equivalently $\\mathcal {C} = 0$ and the subcritical case $0 \\le \\vert \\ell \\vert < \\ell _*$ or $\\mathcal {C} > 0$ .",
"The relative position of the critical radius (REF ) with respect to the roots of $v_R$ may restrict the allowed classes of future-oriented orbits.",
"As a result of (REF ), subcritical $\\ell ^2 <\\ell ^2_*$ orbits have either $R_+ < R_c$ for $\\ell < 0$ or $R_c < 0$ for $\\ell > 0$ , and all orbits are future oriented.",
"Critical orbits $\\ell ^2 =\\ell ^2_*$ have either $R_c < 0$ for $\\ell = \\ell _*$ or $R_c > R_0$ for $\\ell = -\\ell _*$ .",
"This restricts the classes of orbits.",
"Supercritical orbits $\\ell ^2 > \\ell ^2_*$ with $E,\\ell >0$ are future directed.",
"Supercritical orbits with $E>0$ , $\\ell <0$ admit $R_- < R_c < R_+$ , and only bounded orbits with $R \\le R_-$ are admissible.",
"Finally, supercritical orbits with $E<0$ and $\\ell > 0$ obey $R \\ge R_+ > R_c $ and are therefore deflecting.",
"After a simple analysis, we reach the following taxonomy, displayed in Table REF and in Fig.",
"REF .",
"In comparison with Ref.",
"[17], the classes Outward$(E,\\ell )$ , Outward$_*(E)$ , Bounded$_>(E,$ $\\ell )$ , Bounded$^-_*(E)$ , and Bounded$_<(E,\\ell )$ are new, while all other classes with $\\ell > 0$ appeared in Ref.",
"[17].",
"The class $\\text{Osculating}(E,\\ell )$ is now better called $\\text{Def}\\text{lecting}(E,\\ell )$ .",
"The classes with $\\ell = \\pm \\ell _*$ will be denoted with a subscript $_*$ .",
"The Spherical$_*$ orbit with $\\ell = \\ell _*$ is also the prograde ISSO.",
"For $\\ell \\ge 0$ , the conformal diagrams corresponding to those orbits are depicted in Fig.",
"REF and their explicit forms are given in Appendix .",
"Past-oriented geodesics (not depicted) are obtained from a central symmetry around the origin $E=\\ell =0$ as a result of the $\\uparrow \\!\\downarrow $ -flip (REF ).",
"Table: Polar taxonomy of near-horizon geodesics with ℓ≥0\\ell \\ge 0.",
"The orbits with ℓ<0\\ell <0 are obtained from ℓ>0\\ell >0 by flipping the sign of ℓ\\ell with the rest unchanged.Figure: Polar taxonomy of near-horizon geodesics.",
"For clarity, the scale is not respected on the horizontal axis.",
"The dashed blue curves ℓ 2 =ℓ * 2 \\ell ^2=\\ell ^2_* represent the position of the spherical orbits in parameter space.",
"This figure contrasts with Figure ."
],
[
"Metric and geodesic equations",
"We completed the study of the inclined geodesic motion in NHEK spacetime, which describes in particular the motion of test particles near the ISSO and the separatrix at leading order in the high-spin limit, as we will further expand upon in Sec.",
".",
"We now turn to the study of such a motion in a closer vicinity of the black hole event horizon.",
"We introduce the so-called near-NHEK coordinates ($t,r,\\theta ,\\phi $ ), related to Boyer-Lindquist coordinates through the relations $t&=\\frac{{\\hat{t}}}{2M\\kappa }\\lambda ,\\qquad r=\\frac{\\kappa }{M}({\\hat{r}}-{\\hat{r}}_+)\\lambda ^{-1},\\qquad \\phi ={\\hat{\\varphi }}-\\frac{{\\hat{t}}}{2M}.$ At leading order in $\\lambda $ , the metric becomes $s^2=2M^2\\Gamma (\\theta )(-r(r+2\\kappa )t^2+\\frac{r^2}{r(r+2\\kappa )}+\\theta ^2+\\Lambda ^2(\\theta )(\\phi +(r+\\kappa )t)^2)+\\mathcal {O}(\\lambda ).$ Though the metric explicitly depends upon $\\kappa $ , no physical quantity depends upon $\\kappa $ , since it is introduced from a coordinate transformation.",
"The corresponding geodesic equations of motion are now given by ${t}{\\lambda }&=\\frac{e+\\ell (r+\\kappa )}{r(r+2\\kappa )},\\\\{\\phi }{\\lambda }&=-\\frac{e(r+\\kappa )+\\ell \\kappa ^2}{r(r+2\\kappa )}+\\ell (\\frac{1}{\\Lambda ^2}-1),\\\\{\\theta }{\\lambda }&=\\pm _\\theta \\sqrt{v_\\theta (\\cos ^2 \\theta )},\\\\{r}{\\lambda }&=\\pm _r\\sqrt{v_{r;\\kappa }(r)}$ where the radial potential can be written as $v_{r;\\kappa }(r)&\\triangleq (e+\\ell \\kappa )^2+2e \\ell r+\\frac{3}{4}(\\ell ^2-\\ell ^2_*)r(r+2\\kappa )$ and the angular potential is still as given in ().",
"Although $r$ is a meaningful radial coordinate (because of the horizon location at $r=0$ ), it is convenient to introduce the shifted radial variable $R\\triangleq r+\\kappa $ to get more elegant expressions.",
"The coordinate $R$ is also used in NHEK, but the context allows us to distinguish them.",
"The generators of $SL(2,\\mathbb {R})\\times U(1)$ are $\\partial _\\phi $ and $H_0=\\frac{1}{\\kappa }\\partial _t,\\qquad H_\\pm =\\frac{\\text{exp}({\\mp \\kappa t})}{\\sqrt{R^2-\\kappa ^2}}[\\frac{R}{\\kappa }\\partial _t\\pm (R^2-\\kappa ^2)\\partial _R-\\kappa \\partial _\\phi ].$ The $SL(2,\\mathbb {R})$ Casimir $\\mathcal {C}^{\\mu \\nu }\\partial _\\mu \\partial _\\nu $ takes the form (REF ), where the vectors are now given by (REF ).",
"The radial potential can be recast as $v_{R;\\kappa }(R)=\\left\\lbrace \\begin{array}{ll}-\\mathcal {C}(R-R_+)(R-R_-), & \\mathcal {C}\\ne 0 ;\\\\2e \\ell _* (R-R_0), & \\mathcal {C}=0\\end{array}\\right.$ where $R_\\pm \\triangleq \\frac{e \\ell }{\\mathcal {C}} \\pm \\frac{\\sqrt{(\\mathcal {C}+\\ell ^2)(e^2+\\kappa ^2\\mathcal {C})}}{|\\mathcal {C}|},\\qquad R_0\\triangleq -\\frac{e^2+\\kappa ^2\\ell _*^2}{2e\\ell _*}.$ The near-NHEK energy $e$ is related to Boyer-Lindquist energy by $\\hat{E}=\\frac{\\ell }{2M}+\\frac{\\lambda }{2M\\kappa }e.$ The (near-)NHEK and Boyer-Lindquist angular momenta and Carter constants $Q$ are equal.",
"We define again the critical radius $R_c = -\\frac{e}{\\ell }.$ and the future orientation of the orbit again requires (REF )."
],
[
"Solutions to the equations of motion",
"The (near-)NHEK geodesic equations being very similar, this section will only briefly point out the similarities and the differences between the two cases.",
"The formal solutions to the geodesic equations are $\\lambda _f-\\lambda _i&=t^{(0)}_{R;\\kappa }(R_f)-t^{(0)}_{R;\\kappa }(R_i)=\\lambda _\\theta (\\theta _f)-\\lambda _\\theta (\\theta _i),\\\\t(\\lambda _f)-t(\\lambda _i)&=e(t^{(2)}_{R;\\kappa }(R(\\lambda _f))-t^{(2)}_{R;\\kappa }(R(\\lambda _i)))\\nonumber \\\\&~+\\ell (t^{(1)}_{R;\\kappa }(R(\\lambda _f))-t^{(1)}_{R;\\kappa }(R(\\lambda _i))),\\\\\\phi (\\lambda _f)-\\phi (\\lambda _i)&=-\\frac{3}{4}\\ell (\\lambda _f-\\lambda _i)-e (t^{(1)}_{R;\\kappa }(R(\\lambda _f))-t^{(1)}_{R;\\kappa }(R(\\lambda _i)))\\nonumber \\\\&~-\\kappa ^2\\ell ( t^{(2)}_{R;\\kappa }(R(\\lambda _f))- t^{(2)}_{R;\\kappa }(R(\\lambda _i)))\\nonumber \\\\&~+\\ell (\\Phi _\\theta (\\theta (\\lambda _f))-\\Phi _\\theta (\\theta (\\lambda _i))) $ where the polar integrals are the same as in NHEK (see above) and the radial ones are defined by $t^{(0)}_{R;\\kappa }(\\lambda )&\\triangleq \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{R}{\\pm _R\\sqrt{v_{R;\\kappa }(R)}},\\\\t^{(i)}_{R;\\kappa }(\\lambda )&\\triangleq \\rotatebox {-90}{\\XMLaddatt {origin}{c}\\backslash }\\hspace{-14.0pt}\\int \\frac{R}{\\pm _R\\sqrt{v_{R;\\kappa }(R)}}\\frac{R^{2-i}}{R^2-\\kappa ^2},\\qquad i=1,2.$ Notice that NHEK geodesics equations can be recovered from near-NHEK ones by taking the formal limit $\\kappa \\rightarrow 0$ ; the normalization of the radial integrals has been chosen to satisfy $\\lim _{\\kappa \\rightarrow 0} t^{(i)}_{R;\\kappa }(R)=T^{(i)}(R)$ ($i=0,1,2$ ) as defined in (REF ).",
"Therefore, the formal solutions to NHEK geodesic equations can also be recovered by taking the limit $\\kappa \\rightarrow 0$ in (REF ) and ()."
],
[
"Radial behavior.",
"The only difference between NHEK and near-NHEK geodesic solutions lies in the terms involving the radial coordinate.",
"The proposition stating the equivalence relation between the equatorial and inclined radial parts of the geodesic motion takes the same form as in NHEK: Proposition 3 For a given normalization $\\kappa $ , the radial integrals $t^{(i)}_{R;\\kappa }(R)$ $(i=0,1,2)$ only depend upon the near-NHEK energy $e$ and angular momentum $\\ell $ while all the dependence upon the mass $\\mu $ and Carter constant $Q$ is through $\\ell _* = \\frac{2}{\\sqrt{3}}\\sqrt{M^2 \\mu ^2 +Q}$ .",
"As in NHEK, the radial taxonomy of Ref.",
"[17] is easily extended to bounded, outward and/or retrograde orbits by studying the roots and the sign of $v_{R;\\kappa }(R)$ .",
"This leads to the classification displayed in Table REF and Figure REF .",
"Table: Taxonomy of future-directed geodesics in near-NHEK.The future-orientation condition (REF ) implies $e> -\\kappa \\ell $ for each orbit that reaches the horizon at $R=\\kappa $ .",
"In the case $\\ell > \\ell _*$ and $e < 0$ , the condition $e \\le -\\kappa \\sqrt{-\\mathcal {C}}$ implies $e+\\kappa \\ell \\ge 0$ and therefore the parabola does not intersect the line.",
"Past-oriented geodesics (not depicted here) are obtained from a central symmetry around the origin $e=\\ell =0$ as a result of the $\\uparrow \\!\\downarrow $ -flip (REF ).",
"The explicit expressions of all near-NHEK geodesics are listed in Appendix .",
"Figure: Radial taxonomy of geodesics in near-NHEK.",
"For equatorial geodesics, ℓ * =2 3Mμ\\ell _*=\\frac{2}{\\sqrt{3}}M \\mu , while for orbits with inclination, ℓ * =2 3M 2 μ 2 +Q\\ell _* = \\frac{2}{\\sqrt{3}} \\sqrt{M^2\\mu ^2 + Q}."
],
[
"High-spin features of geodesic motion",
"Let us now discuss a few generic and universal features of near-horizon geodesic motion holding in the high-spin case."
],
[
"Radial motion.",
"A first straightforward conclusion one can derive from the analysis of the near-horizon radial geodesic motion is that Proposition 4 All radially unbounded NHEK or near-NHEK geodesics are prograde and either critical or supercritical; i.e., they satisfy $\\ell \\ge \\ell _*$ .",
"This feature of the near-horizon radial motion is directly visible in Figs.",
"REF and REF and leads to remarkable consequences concerning the polar behavior of such trajectories that we will derive in the following section.",
"The separatrix between bound and unbound motion is clearly visible in Figs.",
"REF and REF .",
"It consists of the geodesic classes Plunging$_*(E)$ and Outward$_*(E)$ for NHEK and the geodesic classes Plunging$_*(e)$ , Outward$_*(e)$ , and Bounded$_*(e)$ for near-NHEK that each lie at the critical angular momentum line $\\ell = \\ell _*$ ."
],
[
"Polar motion.",
"The polar motion of both NHEK and near-NHEK trajectories is bounded in an interval around the equator, $\\theta _{\\text{\\text{min}}}\\le \\theta \\le \\pi - \\theta _{\\text{\\text{min}}}$ , where $\\cos \\theta _{\\text{\\text{min}}}=\\sqrt{z_+}$ or $\\cos \\theta _{\\text{\\text{min}}}=\\sqrt{z_0}$ .",
"The maximal polar angle is determined for $\\ell ^2 \\ne \\ell ^2_\\circ = 4 M^2 \\mu ^2$ as $z_+(\\ell ,Q ) =\\frac{3\\ell ^2+4(Q+M^2\\mu ^2)-\\sqrt{9\\ell ^4+16(M^2\\mu ^2-Q)^2+8\\ell ^2(3M^2\\mu ^2+5Q)}}{2(4M^2\\mu ^2-\\ell ^2)}$ and for $\\ell = \\pm \\ell _\\circ $ as $z_0(Q) = \\lim _{\\ell \\rightarrow \\pm 2 M \\mu } z_+ = \\frac{Q}{Q+4 M^2 \\mu ^2}.$ Remember that $Q \\ge 0$ by consistency of polar motion.",
"The asymptotic values are $\\lim _{\\scriptsize {\\begin{array}{l} Q\\rightarrow 0 \\\\ \\ell \\text{ fixed} \\end{array}}}z_+(\\ell ,Q)&=0, \\qquad \\qquad \\qquad \\qquad \\qquad \\;\\;\\;\\;\\;\\;\\;\\; \\lim _{\\scriptsize {\\begin{array}{l} Q\\rightarrow \\infty \\\\ \\ell \\text{ fixed} \\end{array}}}z_+(\\ell ,Q)=1,\\\\\\lim _{\\scriptsize {\\begin{array}{l} \\ell \\rightarrow 0 \\\\ Q \\text{ fixed} \\end{array}}}z_+(\\ell ,Q)&=\\left\\lbrace \\begin{array}{cl}\\frac{Q}{M^2\\mu ^2} &\\text{ if }Q<M^2\\mu ^2\\\\1 & \\text{ if }Q\\ge M^2\\mu ^2\\end{array}\\right.",
",\\qquad \\lim _{\\scriptsize {\\begin{array}{l} \\ell \\rightarrow \\infty \\\\ Q \\text{ fixed} \\end{array}}}z_+(\\ell ,Q) = 0.$ For fixed $\\ell $ , $z_+$ is a monotonic function of $Q$ , and reciprocally $z_+$ is monotonic in $\\ell $ at fixed $Q$ .",
"The pendular oscillation around the equatorial plane will explore a larger range of $\\theta $ when $\\theta _\\text{\\text{min}}$ is smallest or $z_+$ closer to 1, which occurs either for small $\\ell $ and $Q \\ge M^2 \\mu ^2$ or large $Q$ .",
"Now, one can check that for critical or supercritical angular momentum $\\ell ^2 \\ge \\ell ^2_*(Q)$ , one has $z_+ < 2 \\sqrt{3}-3$ for $\\ell \\ne \\ell _\\circ (Q)$ and $z_0 < 2 \\sqrt{3}-3$ for $\\ell ^2 = \\ell ^2_\\circ $ .",
"The special angle $\\theta _{\\text{VLS}} \\triangleq \\arccos {\\sqrt{2\\sqrt{3}-3}} \\approx 47^\\circ $ is in fact the velocity-of-light surface in the NHEK geometry (REF ) (or near-NHEK geometry) defined as the polar angle such that $\\partial _T$ is null.",
"It obeys $\\Lambda (\\theta _{\\text{VLS}}) = 1$ .",
"The polar region closer to either the north or south poles admits a timelike Killing vector, namely $\\partial _T$ .",
"On the contrary, the polar region around the equator $\\theta \\in ]\\theta _{\\text{VLS}},\\pi -\\theta _{\\text{VLS}}[$ does not admit a timelike Killing vector.",
"The velocity-of-light surface separates these two polar regions.",
"We have therefore proven the following property: Proposition 5 All critical or supercritical orbits $\\ell ^2 \\ge \\ell ^2_*$ in (near-)NHEK geometry lie in the polar region $\\theta \\in ]\\theta _{\\text{VLS}},\\pi -\\theta _{\\text{VLS}}[$ where there is no timelike Killing vector.",
"This applies in particular to all spherical orbits.",
"The subcritical orbits $\\ell ^2 < \\ell ^2_*$ can explore all polar regions of the (near)-NHEK geometry.",
"As a consequence of Propositions REF and REF , we have Proposition 6 All radially unbounded geodesics in (near-)NHEK geometry lie in the polar region $\\theta \\in ]\\theta _{\\text{VLS}},\\pi -\\theta _{\\text{VLS}}[$ bounded by the velocity-of-light surface.",
"In particular, for null geodesics, this feature provides the “NHEKline” in the imaging of light sources around a nearly extreme Kerr black hole [3], [44].",
"In [43], [16], Proposition REF was proven for null geodesics.",
"Here, we show that it is a generic property of all timelike geodesics as well."
],
[
"Spherical geodesics",
"The spherical (near-)NHEK geodesics take a distinguished role among all geodesics.",
"First, a subclass of spherical geodesics in NHEK and near-NHEK constitute the innermost stable spherical orbits (ISSOs) and the innermost spherical bound orbits (ISBOs) in the high-spin limit, respectively.",
"Our first motivation is to fully characterize the ISSO, in order to generalize the analysis of the inspiral/merger transition performed around the equatorial plane in the high-spin limit [54], [58] to inclined orbits.",
"Second, as noticed in Ref.",
"[17], the equatorial NHEK (resp.",
"near-NHEK) orbits are the simplest representatives for each equivalence class of prograde incoming critical (respectively, supercritical) equatorial orbits under $SL(2,\\mathbb {R}) \\times U(1) \\times \\mathbb {Z}_2$ symmetry.",
"We will show in Sec.",
"that the spherical (near-)NHEK orbits are the simplest representatives for each equivalence class of arbitrary timelike (near-)NHEK geodesics under $SL(2,\\mathbb {R}) \\times U(1) \\times (\\mathbb {Z}_2)^3$ symmetry without any restriction.",
"These two reasons justify the comprehensive study of the spherical geodesics."
],
[
"Innermost stable spherical orbits",
"The ISSOs are defined as the last stable spherical orbits of Kerr.",
"They are defined from the solutions to $\\hat{R}(\\hat{r}) =\\partial _{\\hat{r}} \\hat{R}(\\hat{r})=\\partial _{\\hat{r}}\\partial _{\\hat{r}}\\hat{R}(\\hat{r}) = 0$ where $\\hat{R}$ is defined in (REF ).",
"They admit a constant radius ${\\hat{r}}$ and a fixed $\\hat{E}$ and $\\hat{\\ell }$ , which can be obtained as solutions of polynomial equations which we will not give explicitly.",
"There are two branches at positive $\\hat{E}$ corresponding to prograde ($\\ell \\ge 0$ ) and retrograde orbits ($\\ell < 0$ ).",
"For the Schwarzschild black hole, the parameters on the two branches of the ISSO are $\\hat{r}_{\\text{ISSO}}=6M,\\qquad \\frac{\\hat{E}_{\\text{ISSO}}}{\\mu }= \\frac{2\\sqrt{2}}{3},\\qquad \\frac{\\ell _{\\text{ISSO}}}{\\mu M} = \\pm \\sqrt{12 - \\frac{Q}{M^2\\mu ^2}},$ which implies the bound $Q \\le 12 M^2 \\mu ^2$ .",
"For arbitary spin, the innermost stable circular orbit (ISCO) is defined as the prograde ISSO equatorial orbit, i.e.",
"restricted to $Q=0$ ($\\theta = \\frac{\\pi }{2}$ ).",
"The parameters are [59] $\\frac{\\hat{E}_{\\text{ISCO}}}{M \\mu } = \\frac{1-2 /\\tilde{r}_{\\text{ISCO}}-\\tilde{a}/\\tilde{r}_{\\text{ISCO}}^{3/2}}{\\sqrt{1-3/\\tilde{r}_{\\text{ISCO}}-2\\tilde{a} / \\tilde{r}_{\\text{ISCO}}^{3/2}}},\\qquad \\frac{\\ell _{\\text{ISCO}}}{\\mu M} = \\frac{2}{\\sqrt{3 \\tilde{r}_{\\text{ISCO}}}} (3 \\sqrt{\\tilde{r}_{\\text{ISCO}}} +2\\tilde{a}),$ where $\\tilde{a} = a/M$ and $\\tilde{r}_{\\text{ISCO}} &\\triangleq & \\frac{\\hat{r}_{\\text{ISCO}}}{M} =3+Z_2-\\sqrt{(3-Z_1)(3+Z_1+2Z_2)}, \\\\Z_1 & \\triangleq & 1+(1-\\tilde{a}^2)^{1/3}[(1+\\tilde{a})^{1/3}+(1-\\tilde{a})^{1/3}],\\qquad Z_2 \\triangleq \\sqrt{3\\tilde{a}^2+(Z_1)^2}.$"
],
[
"Minimal polar angle.",
"In the generic case $\\ell \\ne 0$ , the polar motion is pendular – i.e., oscillating around the equator in the interval $[\\theta _\\text{\\text{min}},\\pi -\\theta _\\text{\\text{min}}]$ .",
"The minimal angle as a function of the spin $a$ and ISCO radius $\\hat{r}_{\\text{ISSO}}$ can simply be found by solving numerically the three equations (REF ) that define the ISSO together with the condition that there is a polar turning point, $\\Theta (\\cos \\theta _{\\text{\\text{min}}}) = 0$ where $\\Theta (\\cos ^2\\theta )$ is defined in ().",
"The resulting minimal angle is displayed in Fig.",
"REF for a large range of spins including nearly extremal.",
"This completes a similar plot drawn in Ref.",
"[60] for spins far from extremality.",
"Figure: cosθ bad hbox \\cos \\theta _\\text{\\text{min}} as a function of ISSO radius for several black hole spins aa.Figure: Euclidean embedding of the ISSO using the Boyer-Lindquist radius r ^\\hat{r}, azimuthal angle φ\\phi and polar angle θ\\theta for a=0.9999a=0.9999.",
"A cone is drawn at the critical polar angle beyond which the ISSO lies in the NHEK region.",
"In the extremal limit, this polar angle is θ≈65 ∘ \\theta \\approx 65^\\circ .We note that for high-spins, the radius asymptotes to $\\hat{r} = M$ and the minimal angle reaches a critical value around $0.42$ radians or $65^\\circ $ .",
"When the motion reaches regions sufficiently far from the equatorial plane, the ISSO radius increases steeply and leaves the near-horizon region ${\\hat{r}}\\simeq M$ .",
"Another graphical representation of this behavior is shown in Fig.",
"REF .",
"We will explain these features in the next section."
],
[
"The NHEK spherical orbit and the high-spin ISSOs",
"In the high-spin limit $a \\rightarrow M$ , the prograde ISSOs are characterized by the following Boyer-Lindquist energies and angular momentum: $\\hat{E}_{\\text{ISSO}}= \\frac{1}{\\sqrt{3}M}\\sqrt{M^2 \\mu ^2 +Q},\\qquad \\ell _{\\text{ISSO}} = + 2M \\hat{E}_{\\text{ISSO}}$ and the following Boyer-Lindquist radius: $\\hat{r}_{\\text{ISSO}} = M + M\\left( \\frac{Q+M^2 \\mu ^2}{-Q+\\frac{M^2 \\mu ^2}{2}} \\right)^{1/3} \\lambda ^{2/3}+\\mathcal {O}(\\lambda ^{4/3}).",
"$ Given the scaling in $\\lambda $ , for the range $0 \\le Q \\le \\frac{M^2 \\mu ^2}{2},$ the ISSOs belong to the NHEK geometry and admit the NHEK radius $R = R_{\\text{ISSO}} \\triangleq \\left( \\frac{Q+M^2 \\mu ^2}{-Q+\\frac{M^2 \\mu ^2}{2}} \\right)^{1/3}.$ In particular, the ISCO has the minimal radius $R_{\\text{ISCO}}=2^{1/3}$ .",
"In terms of NHEK quantities, the orbits admit a critical angular momentum and a vanishing NHEK energy, $\\ell = \\ell _* \\triangleq \\frac{2}{\\sqrt{3}}\\sqrt{Q+M^2 \\mu ^2},\\qquad E = 0.$ In the high-spin limit, the prograde ISSOs in the range (REF ) are therefore exactly the $\\text{Spherical}_*(Q)$ orbits in the classification of Sec.",
"REF .",
"The prograde ISSOs outside the range (REF ) and the retrograde ISSOs do not belong to the near-horizon geometry and will not be described here.",
"In terms of polar behavior, $\\text{Spherical}_*(Q)$ orbits are instances of $\\text{Pendular}(Q,\\ell _*)$ motion (except for $Q=0$ , where they are just equatorial orbits).",
"In the range (REF ), they admit an $\\epsilon _0$ as defined in (REF ) given by $\\epsilon _0 = \\frac{Q-2 M^2 \\mu ^2}{3} < 0$ , and the angular momentum lies below the value $\\ell _\\circ $ : $\\ell _* \\le \\sqrt{2} M \\mu < \\ell _\\circ .$ The main property of $\\text{Pendular}(Q,\\ell _*)$ motion is that the polar angle $\\theta $ is bounded in an interval around the equator (see (REF ) and (REF )) : $\\theta \\in [\\theta _{\\text{min}},\\pi -\\theta _{\\text{min}}]$ where $\\cos \\theta _{\\text{min}}=\\sqrt{z_+}=\\sqrt{\\frac{Q}{\\frac{3}{4}\\ell ^2_*+\\sqrt{\\frac{9}{16}\\ell _*^4-\\frac{\\ell _*^2 Q}{2}+Q^2}}}.$ At fixed $M\\mu $ , $\\theta _{\\text{min}}(Q)$ is a monotonic function interpolating between the equator $\\theta =90^\\circ $ at $Q=0$ and $\\theta _{\\text{VLS}} \\triangleq \\arccos {\\sqrt{2\\sqrt{3}-3}} \\approx 47^\\circ $ for $Q \\rightarrow \\infty $ .",
"The special angle $\\theta _{\\text{VLS}}$ is the velocity-of-light surface in the NHEK geometry (REF ) as described in Sec.",
"REF .",
"The ISSO therefore always lies in the region of NHEK spacetime around the equator, where there is no timelike Killing vector.",
"This is depicted in Fig.",
"REF .",
"Figure: For increasing Q≥0Q \\ge 0, Spherical * (Q)\\text{Spherical}_*(Q) orbits can explore a equator-centered band whose width becomes larger, finally reaching for Q→∞Q \\rightarrow \\infty the angular range θ∈[θ VLS ,π-θ VLS ]\\theta \\in [\\theta _{\\text{VLS}},\\pi -\\theta _{\\text{VLS}}] bounded by the velocity-of-light surface.",
"The prograde IBSOs lie in the near-NHEK region for Q≤2M 2 μ 2 Q \\le 2 M^2\\mu ^2, which further bounds the angular range.However, since the ISSO admits the range (REF ) due to its relationship to the asymptotically flat Boyer-Lindquist radius (REF ), the limiting angle is reached first for $Q=\\frac{M^2 \\mu ^2}{2}$ at $\\arccos {\\sqrt{3-2\\sqrt{2}}}\\approx 65^\\circ $ .",
"This explains the behavior depicted in Fig.",
"REF .",
"At the time of finalizing this draft, Ref.",
"[21] found an identical result (see their Eq.",
"(46)), the limiting angle of the ISSO is given by $\\arcsin {\\sqrt{2(\\sqrt{2}-1)}} = \\arccos {\\sqrt{3-2\\sqrt{2}}} \\approx 65^\\circ $ ."
],
[
"The near-NHEK spherical orbits and the high-spin IBSOs",
"The innermost bound spherical orbits (IBSOs) are determined by the equations $\\hat{R}(\\hat{r}) = \\partial _{\\hat{r}} \\hat{R}(\\hat{r}) = 0,\\qquad \\hat{E} = \\mu .$ In the high-spin limit $\\lambda \\rightarrow 0$ , the angular momentum and Boyer-Lindquist radius of the prograde IBSOs are given by $\\ell &=& \\ell _\\circ \\left(1+\\frac{\\lambda }{\\sqrt{2}}\\sqrt{1-\\frac{Q}{2M^2 \\mu ^2}}+\\mathcal {O}(\\lambda ^2)\\right), \\\\\\hat{r} &=& M \\left(1+ \\frac{\\sqrt{2}\\lambda }{\\sqrt{1-\\frac{Q}{2M^2 \\mu ^2}}}+\\mathcal {O}(\\lambda ^2)\\right) $ where $\\ell _\\circ \\equiv 2M \\mu $ .",
"In particular, for $Q=0$ we recover the scaling of the innermost bound circular orbit (IBCO) [59].",
"Given the scaling $\\sim \\lambda $ , the prograde IBCOs therefore lie in the near-NHEK region for all $Q < 2M^2 \\mu ^2$ .",
"Using (REF )–(REF ), the angular momentum, near-NHEK energy and near-NHEK radius are given in the high-spin limit by $\\ell &=& \\ell _\\circ ,\\\\\\frac{e}{\\kappa } &=& -\\sqrt{2M^2 \\mu ^2 -Q}, \\\\\\frac{r}{\\kappa } &=& \\frac{\\sqrt{2}\\lambda }{\\sqrt{1-\\frac{Q}{2M^2 \\mu ^2}}}.$ The prograde IBCOs in the range $0 \\le Q < 2M^2 \\mu ^2$ are described by instances of Spherical$(\\ell )$ orbits.",
"In terms of polar motion, $Q=0$ are equatorial and $Q>0$ are pendular of class Pendular$_\\circ (Q)$ ; see Table REF .",
"The polar range is determined as $\\theta _{\\text{\\text{min}}} \\le \\theta \\le \\pi - \\theta _{\\text{\\text{min}}}$ where $\\theta _{\\text{\\text{min}}} = \\arccos \\sqrt{\\frac{Q}{Q+\\ell _\\circ ^2}}.$ The maximal polar angle reachable within the near-NHEK region by IBSOs is obtained for the limiting value $Q = 2M^2 \\mu ^2$ at $\\theta _{\\text{\\text{min}}} = \\arccos \\sqrt{1/3} = \\arcsin {\\sqrt{2/3}} \\approx 55^\\circ .$ This critical angle was also previously obtained in Refs.",
"[42], [21].",
"Finally, note that spherical photon orbits in the high-spin limit were also discussed in Refs.",
"[61], [62]."
],
[
"Conformal mappings between radial classes",
"The near-horizon region of near-extremal Kerr black holes admits four Killing vectors forming the group $SL(2,\\mathbb {R}) \\times U(1)$ , hereafter denoted as the conformal group $G$ .",
"The geodesic equations are invariant under $G$ and the geodesics therefore transform under the action of $G$ .",
"Moreover, a group generated by four $\\mathbb {Z}_2$ symmetries exists that preserve the geodesic equations.",
"The subgroup preserving the domain $R > 0$ for NHEK (or $r > 0$ for near-NHEK) is generated by the $\\uparrow \\!\\downarrow $ -flip (REF ), which flips the geodesic orientation, and two additional $\\mathbb {Z}_2$ transformations that preserve the geodesic orientation: namely, the parity flip $\\theta \\rightarrow \\pi - \\theta ,\\qquad \\Phi \\rightarrow \\Phi + \\pi ,\\qquad s_\\theta ^i\\rightarrow -s_\\theta ^i, $ and the $\\rightleftarrows $ -flip $T \\rightarrow -T,\\qquad \\Phi \\rightarrow -\\Phi ,\\qquad \\lambda \\rightarrow -\\lambda ,\\qquad s_R^i \\rightarrow -s_R^i,\\qquad s_\\theta ^i \\rightarrow -s_\\theta ^i .$ The last discrete transformation that we use as a basis is the $\\rightleftarrows $-flip $R \\rightarrow -R, \\qquad \\Phi \\rightarrow -\\Phi , \\qquad \\ell \\rightarrow -\\ell , \\qquad s^i_R \\rightarrow -s^i_R.$ The parity transformation defined in (REF ) leaves each motion invariant and will not be considered further.",
"The $\\rightleftarrows $ -flip changes the boundary conditions of the geodesics, which may affect their denomination.",
"It maps bounded orbits to bounded orbits, and deflecting orbits to deflecting orbits, but plunging orbits to outward orbits, as illustrated in Fig.",
"REF .",
"For bounded orbits, the part before the turning point is mapped to the part after the turning point, and vice-versa.",
"The $\\rightleftarrows $-flip can be used as follows: one first continues a geodesic defined in $R > 0$ beyond the horizon $R = 0$ and the resulting geodesic with $R < 0$ is then mapped to a geodesic in the $R > 0$ region using the $\\rightleftarrows $-flip.",
"Together with the action of (REF ), it allows us to map plunging orbits with $\\ell > 0$ to bounded orbits with $\\ell < 0$ .",
"This process is illustrated in Fig.",
"REF .",
"Figure: Penrose diagram of NHEK spacetime depicting the action of the ⇄\\rightleftarrows -flip on (a) plunging and (b) bounded geodesics.",
"Under this transformation, a trajectory belonging to the patch I is mapped to an orbit of the patch I'.",
"While plunging geodesics become outward ones, bounded motion remains bounded.",
"The energy and angular momentum of the trajectory are unchanged.Figure: Penrose diagram representation of the construction of a critical NHEK bounded geodesic from a plunging one.",
"(a) Continuation of the trajectory beyond the horizon in patch I' until the radial potential root (depicted with dashes); (b) ⇄\\rightleftarrows -flip which brings the part of the bounded geodesic before the turning point in the NHEK Poincaré patch I; (c) ⇄\\rightleftarrows -flip, which maps the part of the bounded geodesic before the turning point to the part after the turning point.The equivalence classes of equatorial critical and supercritical prograde timelike geodesics under the action of $SL(2,\\mathbb {R}) \\times U(1) \\times \\uparrow \\!\\downarrow $ symmetry were derived in Ref.",
"[17] following earlier work [37], [38], [39], [41].",
"In this section, we will perform the decomposition of arbitrary geodesics into equivalence classes under the action of $SL(2,\\mathbb {R}) \\times U(1) \\times \\uparrow \\!\\downarrow \\times \\rightleftarrows \\times $$\\rightleftarrows $.",
"The Casimir $\\mathcal {C}$ of $SL(2,\\mathbb {R})$ cannot vary upon acting with $G \\equiv SL(2,\\mathbb {R}) \\times U(1)$ transformations.",
"Moreover, the action of the group $G$ acts trivially on the polar coordinate $\\theta $ .",
"These two properties imply that both $Q$ and $\\ell $ are invariant under the action of $G$ .",
"In particular, critical, supercritical or subcritical geodesics form distinct classes under $G$ .",
"On the contrary, the (near-)NHEK energy $E$ (or $e$ ) can vary under conformal transformations.",
"Conformal transformations can map NHEK to near-NHEK orbits, and vice-versa.",
"As a result of Propositions REF and REF , null geodesics can be treated on the same footing as timelike geodesics.",
"A conformal transformation belonging to $SL(2,\\mathbb {R}) \\times U(1)$ maps (near)-NHEK spacetime parametrized by $(T,R,\\theta ,\\Phi )$ to (near-)NHEK spacetime parametrized by $(\\bar{T},\\bar{R},\\theta ,\\bar{\\Phi })$We denote here without distinction NHEK and near-NHEK coordinates with capital letters.",
"where $\\overline{T}&=\\overline{T}(T,R),\\nonumber \\\\\\overline{R}&=\\overline{R}(T,R),\\\\\\overline{\\Phi }&=\\Phi +\\delta \\bar{\\Phi }(T,R)\\nonumber .$ The geodesic equations in (near)-NHEK imply $T=T(R)$ .",
"Therefore, the action of conformal symmetries reduces to an action on the radial motion, leaving the polar motion unchanged.",
"More precisely, in the decomposition of $\\Phi (\\lambda )$ ()–() in terms of a radial part and a polar part, the polar part will remain untouched by conformal transformations.",
"It was shown in Ref.",
"[17] that each equivalence class of equatorial prograde critical (respectively, supercritical) geodesics with incoming boundary conditions under $G \\times \\uparrow \\!\\downarrow $ admits a distinguished simple representative, namely the NHEK (respectively, near-NHEK) circular orbits.",
"After analysis, we obtain that each geodesic equivalence class under $G \\times \\uparrow \\!\\downarrow \\times \\rightleftarrows \\times $$\\rightleftarrows $ admits a spherical orbit as the simplest representative as illustrated in Fig.",
"REF .",
"Past directed geodesics must be considered as intermediate steps in order to relate each future directed geodesic to spherical geodesics.",
"Supercritical orbits ($\\ell ^2 > \\ell ^2_*$ ) admit the near-NHEK Spherical$(\\ell )$ orbit as a representative and critical orbits ($\\ell = \\pm \\ell _*$ ) admit the NHEK Spherical$_*$ orbit as a representative.",
"No subcritical spherical geodesic exists.",
"However, we introduce an analytically continued complex subcritical geodesic by continuing the radius $R_0 \\mapsto i R_0$ and show that it generates the subcritical class.",
"The explicit formulas for the three categories of equivalence classes of orbits under $G \\times \\uparrow \\!\\downarrow \\times \\rightleftarrows \\times $$\\rightleftarrows $ are given in the following sections.",
"We will denote the final coordinates and orbital parameters reached by the conformal mappings with bars."
],
[
"Spherical$_*$ {{formula:37563e23-cbdf-4a65-af27-9a6eec194f05}} Plunging{{formula:a2459a41-3c18-478d-ac8a-2762250daa7e}} (NHEK/NHEK).",
"The conformal mapping is given by $\\bar{T}&= - \\frac{R^2T}{R^2T^2-1},\\nonumber \\\\\\bar{R}&=\\frac{R^2T^2-1}{R},\\\\\\bar{\\Phi }&=\\Phi +\\log \\frac{RT+1}{RT-1}-i\\pi .\\nonumber $ It maps a (future-directed) NHEK spherical trajectory of radius $R_0$ to a (future-directed) critical plunge of energy $\\bar{E}=\\frac{2\\ell _*}{R_0} > 0$ ."
],
[
"Spherical$_*$ {{formula:0283f27c-6303-46ee-8b14-62ba1cd03fce}} Plunging{{formula:4c1a03ed-1564-4e79-811e-93a3edf08729}} (NHEK/near-NHEK).",
"One performs the NHEK/near-NHEK diffeomorphism $(T,R,\\theta ,\\Phi )\\rightarrow (\\bar{t},\\bar{R},\\theta ,\\bar{\\phi })$ , whose explicit form is $T&=-\\exp (-\\kappa \\bar{t})\\frac{\\bar{R}}{\\sqrt{\\bar{R}^2-\\kappa ^2}}\\nonumber ,\\\\R&=\\frac{1}{\\kappa }\\exp (\\kappa \\bar{t})\\sqrt{\\bar{R}^2-\\kappa ^2},\\\\\\Phi &=\\phi -\\frac{1}{2}\\log \\frac{\\bar{R}-\\kappa }{\\bar{R}+\\kappa }.\\nonumber $ Its inverse is $\\bar{t} &= \\frac{1}{\\kappa }\\log \\frac{R}{\\sqrt{R^2T^2-1}},\\nonumber \\\\\\bar{R} &=-\\kappa RT,\\\\\\bar{\\phi }&= \\Phi +\\frac{1}{2}\\log \\frac{RT+1}{RT-1}\\nonumber $ for $R>0$ and $RT<-1$ .",
"The orbital parameters are related as $R_0=\\frac{1}{\\kappa }\\exp (\\kappa t_0),\\qquad \\Phi _0=\\phi _0-\\frac{3}{4}.$"
],
[
"Plunging$_*$ {{formula:bb0a8a94-e9aa-4574-a7d8-8e962d12826e}} Outward{{formula:7a968b8f-b475-44bf-8ed2-a1a952cb209c}} (near-NHEK/near-NHEK).",
"The orbits are related by the $\\rightleftarrows $ -flip (REF )."
],
[
"Plunging$_*$ {{formula:be530885-b75b-4349-881e-8d4f8b02917e}} Plunging{{formula:854ef1e7-e729-4593-bd69-9945104d9ed7}} (near-NHEK/near-NHEK).",
"The two (future-directed) orbits are related via the diffeomorphism $\\bar{t} &= \\frac{1}{2\\kappa }\\log \\frac{\\sqrt{R^2-\\kappa ^2}\\cosh {\\kappa t}-R}{\\sqrt{R^2-\\kappa ^2}\\cosh {\\kappa t}+R}-\\frac{i\\pi }{\\kappa },\\nonumber \\\\\\bar{R} &= \\sqrt{R^2-\\kappa ^2}\\sinh {\\kappa t},\\\\\\bar{\\phi }&= \\phi + \\frac{1}{2}\\log \\frac{R\\sinh \\kappa t+\\kappa \\cosh \\kappa t}{R\\sinh \\kappa t-\\kappa \\cosh \\kappa t}\\nonumber .$ The energy of the new trajectory is a function of the initial time $t_0$ of the former one: $\\bar{e}=\\kappa ^2\\ell _*\\exp (-\\kappa t_0) > 0.$"
],
[
"Plunging$_*(e)$ {{formula:0e1cb68f-1bad-4d21-8865-3cbda5fe5edd}} Outward{{formula:8f2fcb09-7fcd-49d8-9bd0-59f64367070d}} (near-NHEK/near-NHEK).",
"The orbits are related by the $\\rightleftarrows $ -flip."
],
[
"Plunging$_*(E)$ {{formula:1182e34f-5621-4219-9465-9951e2092e3a}} Bounded{{formula:15adeaa2-e505-4abb-8457-97180032404b}} (NHEK/NHEK).",
"The critical bounded orbit is obtained from the plunging orbit by a continuation of the trajectory beyond the horizon ($R<0$ ) combined with $\\mathbb {Z}_2$ flips.",
"One must proceed in three steps: Continue the plunge defined from the physical domain $0\\le R\\le \\infty $ to its whole domain of definition $R_0\\le R\\le \\infty $ (i.e., up to the root of the radial potential $R_0=-\\frac{e}{2\\ell _*}$ ) and consider now only the part of the trajectory located beyond the horizon $R_0 \\le R\\le 0$ .",
"Apply the $\\rightleftarrows $-flip to the latter part of the solution.",
"This transformation restores the positivity of the radial coordinate.",
"It preserves the time orientation of the geodesic but flips the sign of its angular momentum $\\ell _*\\rightarrow -\\ell _*$ .",
"The new domain of definition of the trajectory is consequently $0\\le R \\le \\frac{E}{2\\ell _*}$ .",
"The procedure outlined above only leads to the part of the geodesic with $R^{\\prime }(\\lambda )>0$ , which is located before the turning point.",
"As outlined in Appendix , the part of a bounded trajectory located after the turning point can be obtained from the one located before it by a $\\rightleftarrows $ -flip.",
"This whole procedure is represented in Fig.",
"REF ."
],
[
"Plunging$_*(e)$ {{formula:41412868-9c76-4b4e-aedf-0ad98b72e82c}} Bounded{{formula:9a9b825e-17f2-4027-962e-2a539b119f3c}} (near-NHEK/near-NHEK).",
"The mapping is similar to the one outlined above using the $\\rightleftarrows $-flip.",
"One subtlety is that one should start with the Plunging$_*(e)$ orbit with $e > \\kappa \\ell _*$ in order to obtain the future-directed Bounded$_*^-(e)$ orbit."
],
[
"Plunging$_*(e)$ {{formula:8d3d7aa3-ec02-4ed3-afae-4e282a506e0f}} Bounded{{formula:1628901c-4d61-404f-bf7f-f307f7d28ebd}} (near-NHEK/near-NHEK).",
"We apply the $\\rightleftarrows $-flip as outlined in the previous paragraph, but now choosing $0< e < \\kappa \\ell _*$ .",
"This leads to a retrograde past-directed bounded orbit.",
"The future-directed prograde geodesic is then reached using the $\\uparrow \\!\\downarrow $ -flip."
],
[
"Spherical$(\\ell )$ {{formula:2f48a2ca-e91f-4906-b780-94a67e9ca983}} Marginal{{formula:e18a6881-f7d7-4ebb-b949-60882f9b85a4}} (near-NHEK/NHEK).",
"One applies the NHEK/near-NHEK diffeomorphism $T&=-\\exp (-\\kappa \\bar{t})\\frac{\\bar{R}}{\\sqrt{\\bar{R}^2-\\kappa ^2}}\\nonumber ,\\\\R&=\\frac{1}{\\kappa }\\exp (\\kappa \\bar{t})\\sqrt{\\bar{R}^2-\\kappa ^2},\\\\\\Phi &=\\phi -\\frac{1}{2}\\log \\frac{\\bar{R}-\\kappa }{\\bar{R}+\\kappa }\\nonumber $ which maps the orbit Spherical$(\\ell )$ on the past-directed Marginal$(-\\ell )$ orbit.",
"The future-directed Marginal$(\\ell )$ orbit is recovered by composing this transformation with a $\\uparrow \\!\\downarrow $ -flip."
],
[
"Marginal$(\\ell )$ {{formula:25412015-70c1-4039-98fd-7b1779cd8a9a}} Plunging{{formula:ffb3c427-41e6-4d84-8a5d-2955caf03286}} or Deflecting{{formula:2e7d7524-aa02-4b10-94cd-0d98ae8721eb}} (NHEK/NHEK).",
"One performs the transformation ($\\zeta \\ne 0$ ) $\\bar{T} &=\\frac{1}{\\bar{R}}\\frac{2R^2T\\cos \\zeta -(1+R^2(1-T^2))\\sin \\zeta }{2 R},\\nonumber \\\\\\bar{R} &=\\frac{R^2(1+T^2)-1+(1+R^2(1-T^2))\\cos \\zeta +2R^2T\\sin \\zeta }{2R},\\\\\\bar{\\Phi }&= \\Phi +\\log \\frac{\\cos \\frac{\\zeta }{2}R+\\sin \\frac{\\zeta }{2}(RT+1)}{\\cos \\frac{\\zeta }{2}R+\\sin \\frac{\\zeta }{2}(RT-1)}.\\nonumber $ As outlined in Ref.",
"[17], this mapping can be viewed as the action on Poincaré NHEK coordinates of a shift of the global NHEK time $\\tau \\rightarrow \\tau -\\zeta $ .",
"The energy of the final orbit is $\\bar{E}=\\sqrt{-\\mathcal {C}}(\\sin \\zeta +T_0(\\cos \\zeta -1)).$ We directly see that any energy $E\\ne 0$ can be reached by conveniently choosing the values of $T_0$ and $\\zeta $ ."
],
[
"Plunging$(E,\\ell )$ {{formula:f2aee122-47e1-452f-a342-4dec30d399be}} Outward{{formula:117b4416-1d5e-418d-b907-9d904fabd11e}} (NHEK/NHEK).",
"The orbits are related by the $\\rightleftarrows $ -flip."
],
[
"Plunging$(E,\\ell )$ {{formula:5e10e275-2514-4d5c-ba8f-5c05ad65dd18}} Bounded{{formula:b901aae9-e421-44e2-a2ac-5cd3b756d32c}} (NHEK/NHEK).",
"The mapping consists in extending the radial range of the plunging orbit beyond the horizon, $R<0$ , then using the $\\rightleftarrows $-flip, which leads to the Bounded$_>(E,-\\ell )$ orbit."
],
[
"Spherical$(\\ell )$ {{formula:afd6db8c-e810-4095-9820-79b20616d61e}} Plunging{{formula:c162d093-76cf-47e1-9ca4-fda233d442b4}} or Deflecting{{formula:cbc350a9-7b15-40bc-b4a3-fb780132ca48}} (near-NHEK/ near-NHEK).",
"One uses the diffeomorphism ($\\chi \\ne \\pm 1$ ) $t &= \\frac{1}{\\kappa }\\log \\frac{\\sqrt{\\bar{R}^2-\\kappa ^2}\\cosh \\kappa \\bar{t}-\\bar{R}}{\\sqrt{ R^2-\\kappa ^2}}\\nonumber ,\\\\R &=\\sqrt{\\bar{R}^2-\\kappa ^2}(\\sinh \\kappa \\bar{t}+\\chi \\cosh \\kappa \\bar{t})-\\chi \\bar{R},\\\\\\phi &= \\bar{\\phi }-\\frac{1}{2}\\log [\\frac{\\sqrt{\\bar{R}^2-\\kappa ^2}-\\bar{R}\\cosh \\kappa \\bar{t}+\\kappa \\sinh \\kappa \\bar{t}}{\\sqrt{\\bar{R}^2-\\kappa ^2}-\\bar{R}\\cosh \\kappa \\bar{t}-\\kappa \\sinh \\kappa \\bar{t}}\\frac{R+\\kappa }{R-\\kappa }]\\nonumber .$ This mapping can be seen as a NHEK global time shift written in near-NHEK coordinates; see Refs.",
"[17], [41].",
"The explicit inversion formula can be found in Ref.",
"[41].",
"The energy of the new trajectory reads as $\\bar{e}=\\kappa \\sqrt{-\\mathcal {C}}\\,\\chi .$ For $-\\frac{\\ell }{\\sqrt{-\\mathcal {C}}}<\\chi <-1$ , the orbit reached is future-directed and deflecting.",
"The trajectory becomes plunging for $\\chi >-1$ .",
"Note that for ${\\chi }>1$ , $\\bar{t}_0=-\\frac{1}{2\\kappa }\\log \\frac{1+\\chi }{1-\\chi }$ is complex and one has to perform an additional shift on $\\bar{t}$ to make it real."
],
[
"Plunging$(e,\\ell )$ {{formula:f6676101-8128-41b0-92ef-8d4f248eb333}} Outward{{formula:a793fb12-1c64-4194-8339-e22ebf312837}} (near-NHEK/near-NHEK).",
"The orbits are related by the $\\rightleftarrows $ -flip."
],
[
"Plunging$(e,\\ell )$ {{formula:5a0ffab7-7794-40f3-a5c6-39dfbf1e98a7}} Bounded{{formula:f70988ec-537b-4eca-bbbe-bf9ee834b741}} (near-NHEK/near-NHEK).",
"The mapping consists in extending the radial range of the plunging orbit with $e > \\kappa \\ell $ beyond the horizon, $r<0$ , then using the $\\rightleftarrows $-flip, which leads to the Bounded$_>(e,-\\ell )$ orbit."
],
[
"Complex class of spherical geodesics.",
"There is no near-NHEK spherical geodesic for $\\mathcal {C}>0$ .",
"We can nevertheless introduce the formal class of complex spherical trajectories $t(\\lambda ) &= -i\\frac{\\ell }{R_0}\\lambda ,\\\\R(\\lambda ) &= i R_0,\\qquad R_0\\triangleq \\frac{\\kappa \\ell }{\\sqrt{\\mathcal {C}}},\\\\\\phi (\\lambda ) &= \\phi _0-\\frac{3}{4}\\ell \\lambda +\\ell \\Phi _\\theta (\\lambda )$ which is a formal (but nonphysical) solution of the near-NHEK geodesic equations, of complex near-NHEK “energy” $e=-i\\kappa \\sqrt{\\mathcal {C}}$ .",
"We will denote this class of solutions as Spherical$_\\mathbb {C}(\\ell )$ and show that it can be used to generate all subcritical bounded trajectories by acting on it with properly chosen conformal transformations.",
"The parametrized form of the orbit reads as $R &= i R_0,\\\\\\phi (t) &= \\phi _0-\\frac{3}{4}iR_0t+\\ell \\Phi _\\theta (\\lambda (t)).$"
],
[
"Spherical$_\\mathbb {C}(\\ell )$ {{formula:47c73548-ed30-40ce-bd74-2f0d191f0a14}} Bounded{{formula:09ddfa8f-8726-4967-9e3e-674c891d9521}} .",
"One has to proceed in two steps, mimicking the procedure used to obtain the NHEK Plunging$(E,\\ell )$ class: [label=$\\diamond $ ] We apply the near-NHEK/NHEK diffeomorphism (REF ) to a Spherical$_\\mathbb {C}(\\ell )$ orbit, leading to another complex NHEK geodesic of null energy parametrized by $T(R) &= -\\frac{i\\ell }{\\sqrt{C}R},\\\\\\Phi (R) &= \\Phi _0-\\frac{3i\\ell }{8\\sqrt{C}}\\log \\frac{\\mathcal {C}R^2}{\\mathcal {C}+\\ell ^2}$ with the initial azimuthal angle $\\Phi _0\\triangleq \\phi _0-\\frac{3\\pi \\ell }{8\\sqrt{C}}-\\frac{1}{2}\\log (1-\\frac{2\\sqrt{C}}{\\sqrt{C}+i\\ell })$ .",
"We denote this class as Marginal$_\\mathbb {C}(\\ell )$ .",
"Second, we apply to the trajectory found above the global time shift (REF ), but upgraded with an imaginary parameter $\\zeta \\rightarrow i\\zeta $ .",
"This leads to the Bounded$_<(E,\\ell )$ class with orbital parameters $\\bar{E} &= \\sqrt{\\mathcal {C}}\\sinh \\zeta ,\\\\\\bar{\\Phi }_0 &= \\phi _0-\\frac{3\\pi \\ell }{8 \\sqrt{\\mathcal {C}}}-\\log (\\sqrt{\\mathcal {C}}-i\\ell )+\\frac{3i\\ell }{8\\sqrt{\\mathcal {C}}}\\log [\\mathcal {C}(\\mathcal {C}+\\ell ^2)(1+\\sqrt{\\frac{\\mathcal {C}+E^2}{\\mathcal {C}}})^2]\\nonumber \\\\&~-\\frac{3\\ell }{8\\sqrt{C}}\\log [E^2(\\mathcal {C}+\\ell ^2)]+\\arctan \\frac{\\sqrt{\\mathcal {C}}}{\\ell }.$ Note that choosing $\\zeta >0$ is sufficient to reach the full range of energies allowed for such a geodesic ($E>0$ ).",
"Any geodesic of orbital parameters ($T_0,\\tilde{\\Phi }_0$ ) can finally be obtained by performing the transformation $T\\rightarrow T+T_0$ , $\\Phi \\rightarrow \\Phi -\\bar{\\Phi }_0+\\tilde{\\Phi }_0$ , which also removes the unphysical imaginary part of the azimuthal coordinate."
],
[
"Spherical$_\\mathbb {C}(\\ell )$ {{formula:9266a540-2e46-4d00-be99-d4ffb95143d1}} Bounded{{formula:1e1d762a-eb93-47ca-9b0d-a09c589277fb}} .",
"We apply to the Spherical$_\\mathbb {C}(\\ell )$ class the near-NHEK global time shift (REF ) upgraded with an imaginary parameter $\\chi \\rightarrow i\\chi $ ($\\chi \\ne \\pm 1$ ), leading to a Bounded$_<(e,\\ell )$ orbit of parameters $\\bar{e} &= \\kappa \\sqrt{\\mathcal {C}}\\,\\chi ,\\\\\\bar{t}_0 &= t_0+\\frac{i}{\\kappa }\\arctan \\frac{\\kappa \\sqrt{\\mathcal {C}}}{e},\\\\\\bar{\\phi }_0 &= \\bar{\\phi }_0(\\phi _0,e,\\ell ,\\mathcal {C},\\kappa ).$ The explicit value of $\\bar{\\phi }_0$ is easily calculable, but too long to be reproduced here.",
"To reach a manifestly real orbit of orbital parameters $(\\tilde{t}_0,\\tilde{\\phi }_0)$ , one has to perform the final shift $t\\rightarrow t-\\bar{t}_0+\\tilde{t}_0,\\qquad \\phi \\rightarrow \\phi -\\bar{\\phi }_0+\\tilde{\\phi }_0.$ Figure: Schematical overview of the action of the group SL(2,ℝ)×U(1)×↑↓×⇄×SL(2,\\mathbb {R})\\times U(1) \\times \\uparrow \\!\\downarrow \\times \\rightleftarrows \\times \\!⇄\\rightleftarrows on near-horizon geodesics.",
"The equation numbers refer to the conformal maps described in the text."
],
[
"Acknowledgements",
"We thank Alex Lupsasca and Edward Teo for pointing out mistakes in previous versions of this paper.",
"A. D. acknowledges Kwinten Fransen for the help he provided computing complicated expressions using Mathematica.",
"A. D. is a Research Fellow and G. C. is a Research Associate of the F.R.S.-FNRS.",
"G. C. acknowledges support from the FNRS research credit No.",
"J003620F, the IISN convention No.",
"4.4503.15, and the COST Action GWverse No.",
"CA16104.",
"toc"
],
[
"Elliptic integrals and Jacobi functions",
"In this appendix, we set our conventions for the elliptic integrals and Jacobi functions used in the main text.",
"The incomplete elliptic integrals of the first, second, and third kind are defined as $\\!\\!F(\\varphi , m) &\\!\\!\\!\\triangleq \\!\\!\\!",
"& \\int _0^\\varphi \\frac{\\text{d} \\theta }{\\sqrt{1- m \\sin ^2 \\theta }} = \\int _0^{\\sin \\varphi } \\frac{\\text{d} t}{\\sqrt{(1-t^2)(1-m t^2)}}, \\\\\\!\\!E(\\varphi , m) &\\!\\!\\!\\triangleq \\!\\!\\!& \\int _0^\\varphi d\\theta \\sqrt{1- m \\sin ^2\\theta }= \\int _0^{\\sin \\varphi } \\text{d} t \\sqrt{\\frac{1-m t^2}{1-t^2}}, \\\\\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\Pi (n, \\varphi ,m)& \\!\\!\\!\\triangleq \\!\\!\\!",
"& \\int _0^\\varphi \\frac{1}{1- n \\sin ^2 \\theta }\\frac{\\text{d} \\theta }{\\sqrt{1- m \\sin ^2 \\theta }}= \\int _0^{\\sin \\varphi } \\frac{1}{1- n t^2} \\frac{\\text{d} t}{\\sqrt{(1-m t^2)(1-t^2)}},$ respectively.",
"We also define $E^{\\prime }(\\varphi , m ) = \\partial _m E(\\varphi ,m) = \\frac{1}{2m} [E(\\varphi ,m) - F(\\varphi , m)] $ .",
"The complete elliptic integrals of the first, second, and third kind are defined as $K(m) &\\triangleq & F(\\frac{\\pi }{2},m), \\\\E(m) &\\triangleq & E(\\frac{\\pi }{2},m), \\\\\\Pi (n,m) &\\triangleq & \\Pi (n,\\frac{\\pi }{2},m),$ respectively, and $E^{\\prime }(m)=\\partial _m E(m)$ .",
"Jacobi functions are defined as the inverse of the incomplete elliptic integrals of the first kind.",
"More precisely, one can invert $u = F(\\varphi ,m)$ into $\\varphi =\\text{am}(u,m)$ in the interval $-K(m) \\le u \\le K(m)$ .",
"The elliptic sinus, elliptic cosinus, and delta amplitude are defined as $\\text{sn}(u,m) &\\triangleq &\\sin \\text{am}(u,m), \\qquad \\text{cn}(u,m) \\triangleq \\cos \\text{am}(u,m),\\\\\\text{dn}(u,m) &\\triangleq & \\sqrt{1-m (\\sin \\text{am}(u,m))^2},$ respectively.",
"They have the periodicity ($k,l \\in \\mathbb {Z}$ ) $\\text{sn}(u + 2 k K(m)+2ilK(1-m) ,m) &= (-1)^k \\text{sn}(u,m) ,\\\\\\text{dn}(u + 2 k K(m)+2ilK(1-m) ,m) &= (-1)^l \\text{sn}(u,m)$ and obey the properties $\\text{cn}^2(u,m)+\\text{sn}^2(u,m) &=&1 ,\\qquad \\text{dn}^2(u,m)+m\\,\\text{sn}^2(u,m) =1, \\\\\\frac{\\partial }{\\partial u}\\text{am}(u,m) &=& \\text{dn}(u,m), \\\\\\frac{\\partial }{\\partial u}\\text{sn}(u,m) &=& \\text{cn}(u,m)\\text{dn}(u,m),\\\\\\frac{\\partial }{\\partial u}\\text{cn}(u,m) &=& - \\text{sn}(u,m)\\text{dn}(u,m),\\\\\\frac{\\partial }{\\partial u}\\text{dn}(u,m) &=& - m\\, \\text{sn}(u,m)\\text{cn}(u,m).$"
],
[
"Elementary Polar Integrals",
"For the pendular and equator-attractive cases, one has to compute the following integrals: $&\\hat{I}^{(0)}(x)\\triangleq \\int _{x_0}^x\\frac{t}{\\sqrt{\\Theta (t^2)}},\\quad \\hat{I}^{(1)}(x)\\triangleq \\int _{x_0}^x\\frac{t^2t}{\\sqrt{\\Theta (t^2)}}, \\quad \\hat{I}^{(2)}(x)\\triangleq \\int _{x_0}^x\\frac{t}{\\sqrt{\\Theta (t^2)}} \\frac{1}{1-t^2} $ where $-1 \\le x \\le 1$ .",
"In the main text, $x$ will be substituted by $\\cos \\theta $ , where $\\theta $ is a polar angle."
],
[
"$\\epsilon _0=0$",
"In this case, $\\Theta (t^2)=\\sqrt{\\frac{z_0}{Q}}(z_0-t^2)$ .",
"Choosing $x_0=0$ , one finds directly $\\hat{I}^{(0)}(x)&=\\sqrt{\\frac{z_0}{Q}}\\arcsin {\\frac{x}{\\sqrt{z_0}}},\\\\\\hat{I}^{(1)}(x)&=\\frac{1}{2}\\sqrt{\\frac{z_0}{Q}}[z_0\\arcsin {\\frac{x}{\\sqrt{z_0}}}-x\\sqrt{z_0-x^2}],\\\\\\hat{I}^{(2)}(x)&=\\sqrt{\\frac{z_0}{Q(1-z_0)}}\\arcsin \\sqrt{\\frac{x}{z_0}\\frac{1-z_0}{1-x}}$ and the particular values $\\hat{I}^{(0)}(\\sqrt{z_0})&=\\frac{\\pi }{2}\\sqrt{\\frac{z_0}{Q}},\\\\\\hat{I}^{(1)}(\\sqrt{z_0})&=\\frac{\\pi }{4}\\sqrt{\\frac{z_0}{Q}},\\\\\\hat{I}^{(2)}(\\sqrt{z_0})&=\\frac{\\pi }{2}\\sqrt{\\frac{z_0}{Q(1-z_0)}}.$ One inverts (REF ) as $x=\\sqrt{z_0}\\sin (\\frac{Q}{z_0}\\hat{I}^{(0)}).$"
],
[
"$\\epsilon _0\\ne 0$ , {{formula:c3c7a196-97d4-460a-a823-087365ecdd26}}",
"In this case, $\\Theta (t^2)=\\epsilon _0(t^2-z_-)(z_+-t^2)$ , where $t=\\cos \\theta $ .",
"Instead of solving separately the pendular and vortical cases as in Ref.",
"[18], we will introduce a formal notation enabling us to treat both cases simultaneously.",
"Let us define $q\\triangleq \\,\\text{sign}(Q),\\qquad z_{\\pm 1}\\triangleq z_\\pm ,\\qquad m\\triangleq \\left\\lbrace \\begin{array}{cc}\\frac{z_+}{z_-}, & q=+1 \\\\1-\\frac{z_-}{z_+}, & q=-1\\end{array}\\right.$ and $y(t)\\triangleq \\left\\lbrace \\begin{array}{cc}\\frac{t}{\\sqrt{z_+}}, & q=+1 \\\\\\,\\text{sign}(t) \\sqrt{\\frac{z_+-t^2}{z_+-z_-}}, & q=-1\\end{array}\\right.,\\qquad \\Psi ^q(x)\\triangleq \\arcsin {y(x)}.$ Pendular motion corresponds to $q = 1$ , which has $0 \\le t^2 \\le z_+ <1$ and $\\epsilon _0 z_- <0$ , while vortical motion corresponds to $q = -1$ , which has $0< z_- \\le t^2 \\le z_+ <1$ and $\\epsilon _0 >0$ .",
"One can then rewrite $\\frac{t}{\\sqrt{\\epsilon _0(t^2-z_-)(z_+-t^2)}}=\\frac{q}{\\sqrt{-q\\epsilon _0z_{-q}}}\\frac{y}{\\sqrt{(1-y^2)(1-my^2)}}.$ Both factors of the right side of this equation are real, either for pendular or for vortical motions.",
"The lower bound of the integral will be chosen as $x_0=0$ for $Q\\ge 0$ and $x_0=\\,\\text{sign}(x)\\sqrt{z_+}$ for $Q<0$ .",
"Then $y(x_0)=0$ for both values of $q$ .",
"This allows us to solve directly the integrals in terms of elliptic integrals (see Appendix for definitions and conventions used): $\\hat{I}^{(0)}(x)&=\\frac{q}{\\sqrt{-q\\epsilon _0z_{-q}}}F(\\Psi ^q(x),m),\\\\\\hat{I}^{(1)}(x)&=\\left\\lbrace \\begin{array}{ll}\\frac{-2z_+}{\\sqrt{-\\epsilon _0 z_-}}E^{\\prime }(\\Psi ^+(x),m), & q=+1\\\\- \\sqrt{\\frac{z_+}{\\epsilon _0}}E(\\Psi ^-(x),m),& q=-1\\end{array}\\right., \\\\\\hat{I}^{(2)}(x)&=\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\sqrt{-\\epsilon _0 z_-}}\\Pi (z_+,\\Psi ^+(x),m), & q=+1,\\\\\\frac{-1}{(1-z_+)\\sqrt{\\epsilon _0 z_+}}\\Pi (\\frac{z_--z_+}{1-z_+},\\Psi ^-(x),m), & q=-1\\end{array}\\right.",
".$ For $x=\\sqrt{z_q}$ , $\\Psi ^q(\\sqrt{z_q})=\\frac{\\pi }{2}$ for both $q= \\pm 1$ and the incomplete elliptic integrals are replaced by complete ones: $\\hat{I}^{(0)}(\\sqrt{z_q})&=\\frac{q}{\\sqrt{-q\\epsilon _0z_{-q}}}K(m),\\\\\\hat{I}^{(1)}(\\sqrt{z_q})&=\\left\\lbrace \\begin{array}{ll}\\frac{-2z_+}{\\sqrt{-\\epsilon _0 z_-}}E^{\\prime }(m), & q=+1\\\\- \\sqrt{\\frac{z_+}{\\epsilon _0}}E(m),& q=-1\\end{array}\\right.,\\\\\\hat{I}^{(2)}(\\sqrt{z_q})&=\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\sqrt{-\\epsilon _0 z_-}}\\Pi (z_+,m), & q=+1,\\\\\\frac{-1}{(1-z_+)\\sqrt{\\epsilon _0 z_+}}\\Pi (\\frac{z_--z_+}{1-z_+},m), & q=-1\\end{array}\\right.",
".$ For $q=-1$ , all integrals (REF ), (), and () vanish when evaluated at $x=\\pm \\sqrt{z_+}$ because $\\Psi ^-(\\sqrt{z_+})=0$ .",
"Finally, (REF ) can be inverted as $x=y^{-1}(\\,\\text{sn}\\,(\\sqrt{-q\\epsilon _0z_{-q}}\\hat{I}^{(0)},m))$ with $y^{-1}$ the inverse function of $y$ : $y^{-1}(t)=\\left\\lbrace \\begin{array}{ll}\\sqrt{z_+}t, & q=+1 \\\\\\,\\text{sign}(t)\\sqrt{z_+(1-mt^2)},& q=-1\\end{array}\\right.$ leading to the explicit formula $x=\\left\\lbrace \\begin{array}{ll}\\sqrt{z_+}\\,\\text{sn}\\,(\\sqrt{-\\epsilon _0z_{-}}\\hat{I}^{(0)},m), & q=+1 \\\\\\,\\text{sign}(x) \\sqrt{z_+}\\,\\text{dn}\\,(\\sqrt{\\epsilon _0z_{+}}\\hat{I}^{(0)},m),& q=-1.\\end{array}\\right.$ In the context of this paper, we will always have $x=\\cos \\theta $ ."
],
[
"$\\epsilon _0>0$ , {{formula:7bc13a51-ca11-4dd4-9576-567335c4dfff}}",
"This case is relevant for equator-attractive orbits.",
"In this case, the potential reduces to $\\Theta (t^2)=\\epsilon _0\\,t^2(z_+-t^2)$ .",
"One needs the following integrals (see also Ref.",
"[18]): $\\hat{\\mathcal {I}}^{(0)}&\\triangleq \\int _x^{\\sqrt{z_+}}\\frac{t}{\\sqrt{\\Theta (t^2)}}=\\frac{1}{\\sqrt{\\epsilon _0z_+}}\\text{arctanh}\\sqrt{1-\\frac{x^2}{z_+}},\\\\\\hat{\\mathcal {I}}^{(1)}&\\triangleq \\int _x^{\\sqrt{z_+}}\\frac{t^2t}{\\sqrt{\\Theta (t^2)}}=\\sqrt{\\frac{z_+-x^2}{\\epsilon _0}},\\\\\\hat{\\mathcal {I}}^{(2)}&\\triangleq \\int _x^{\\sqrt{z_+}}\\frac{t}{\\sqrt{\\Theta (t^2)}}(\\frac{1}{1-t^2}-1)=\\frac{1}{\\sqrt{\\epsilon _0(1-z_+)}}\\arctan \\sqrt{\\frac{z_+-x^2}{1-z_+}}$ for any $x\\le \\sqrt{z_+}$ .",
"All such integrals are obviously vanishing when evaluated at $x=\\sqrt{z_+}$ .",
"Note that $ \\hat{\\mathcal {I}}^{(2)}$ contains a $-1$ integrand for simplicity of the final answer."
],
[
"Explicit form of prograde NHEK geodesics",
"We provide here the explicit motion in Mino time and the parametrized form of all classes of future-oriented geodesics in NHEK in the region outside the horizon $R > 0$ .",
"Without loss of generality, we can use the existence of the $\\rightleftarrows $ and $\\rightleftarrows $-flips to restrict ourselves to the subclasses of NHEK future-oriented geodesics, determined as [label=$\\diamond $ ] Prograde ($\\ell \\ge 0$ ) geodesics; Partially ingoing geodesics, i.e., trajectories whose radial coordinate is decreasing on at least part of the total motion.",
"Because the near-horizon motion admits at most one turning point, we will only encounter spherical, plunging, marginal, bounded and deflecting motions."
],
[
"General integrals.",
"One must provide explicit solutions to equations (REF ) to ().",
"The main point one has to deal with consists of solving the radial integrals (REF ).",
"Notice that the primitives $\\mathbb {T}^{(i)}(R)=\\int \\frac{R}{R^i\\sqrt{E^2+2E\\ell R-\\mathcal {C}R^2}},\\qquad i=0,1,2$ can be directly integrated as $\\mathbb {T}^{(0)}(R)&=\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\sqrt{-\\mathcal {C}}}\\log [E\\ell -\\mathcal {C} R+\\sqrt{-\\mathcal {C}}\\sqrt{v_R(R)}], & \\qquad \\mathcal {C}\\ne 0 \\\\\\frac{\\sqrt{E^2+2E\\ell _*R}}{E\\ell _*}, & \\qquad \\mathcal {C}=0,~E\\ne 0,\\end{array}\\right.\\\\\\mathbb {T}^{(1)}(R)&=\\frac{\\log R-\\log [E+\\ell R+\\sqrt{v_R(R)}]}{E}\\qquad E\\ne 0,\\\\\\mathbb {T}^{(2)}(R)&=-\\frac{1}{E}(\\frac{\\sqrt{v_R(R)}}{ER}+\\ell \\,\\mathbb {T}^{(1)}(R)),\\qquad E\\ne 0.$ We treat each type of motion cited above separately.",
"We consider a geodesic path linking two events as described in the main text.",
"We consider here a future-oriented path, $\\Delta T>0$ and $\\Delta \\lambda >0$ .",
"Let us proceed systematically: [label=$\\diamond $ ] Spherical motion has $R$ constant, and the radial integrals are ill defined and irrelevant.",
"One can directly integrate the basic geodesic equations in this case.",
"For plunging motion, we will fix the final conditions at the final event $(T_f,R_f,\\theta _f,\\Phi _f)$ such that $R_f<0$ (We will choose $R_f$ to be a root of the radial potential in all cases where they are real.",
"Otherwise, we will chose it for convenience.)",
"We will drop the subscript $i$ of the initial event.",
"We denote here $\\Delta \\lambda =\\lambda _f-\\lambda $ , $\\Delta T=T_f-T(\\lambda )$ ,...and $T^{(j)}_R(R)=\\mathbb {T}^{(j)}(R)-\\mathbb {T}^{(j)}(R_f),\\qquad j=0,1,2.$ For bounded motion, we identify the physically relevant root of the radial potential $R_{\\text{turn}}>0$ at $T_{\\text{turn}}$ (which is an integration constant) that represents the turning point of the motion.",
"We consider the motion either before or after the turning point: [label=$\\rightarrow $ ] $\\underline{T(\\lambda )>T_{\\text{turn}}}:$ We choose the initial event as the turning point and the final one as $(T_>(\\lambda ),R_>(\\lambda ),$ $\\theta _>(\\lambda ),\\Phi _>(\\lambda )).$ This leads to $T^{(j)}_R(R_>)=\\mathbb {T}^{(j)}(R_{\\text{turn}})-\\mathbb {T}^{(j)}(R_>),\\qquad j=0,1,2.$ $\\underline{T(\\lambda )<T_{\\text{turn}}}:$ We choose the initial event as $(T_<(\\lambda ),R_<(\\lambda ),\\theta _<(\\lambda ),\\Phi _<(\\lambda ))$ and the final event as the turning point.",
"This leads to $T^{(j)}_R(R_<)=\\mathbb {T}^{(j)}(R_{\\text{turn}})-\\mathbb {T}^{(j)}(R_<),\\qquad j=0,1,2.$ The motion being symmetric in $R$ with respect to $R_{\\text{turn}}$ , one must only determine $T_>(R)$ and $\\Phi _>(R)$ , because $T_<(R)=-T_>(R),\\qquad \\Phi _<(R)=-\\Phi _>(R).$ For deflecting motion, there is also a turning point and (REF ) remains true.",
"This case is treated similarly to the bounded case but with a minus-sign change.",
"We get here $T_{R}^{(j)}(R_>)=\\mathbb {T}^{(j)}(R_>)-\\mathbb {T}^{(j)}(R_{\\text{turn}}), \\qquad T^{(j)}_R(R_<)=\\mathbb {T}^{(j)}(R_<)-\\mathbb {T}^{(j)}(R_{\\text{turn}}).$ Finally, for marginal motion, there is no turning point and the integral is immediate.",
"In what follows, for the motions with one turning point, we will only make explicit the motion after the turning point, $T>T_{\\text{turn}}$ , and we will denote $T=T_>$ and $\\Phi =\\Phi _>$ in order to simplify the notations."
],
[
"The explicit form is $T(\\lambda ) &=& T_0 + \\frac{\\ell _*}{R_0}(\\lambda -\\lambda _0) , \\\\R(\\lambda ) &=& R_0 , \\\\\\Phi (\\lambda ) &=&\\Phi _0 - \\frac{3}{4}\\ell _* (\\lambda -\\lambda _0) + \\ell _* \\Phi _\\theta (\\lambda -\\lambda _0).$ The parametrized form is [box=]align R = R0 , = 0 -34R0 (T-T0) + * ( R0* (T-T0))."
],
[
"The Casimir obeys here $\\mathcal {C}<0$ ; we denote $q\\triangleq \\sqrt{-\\mathcal {C}}$ and consider the initial condition $R(\\lambda _m)=R_m$ .",
"The explicit form of the solution is given (as a function of the Mino time) by $T(\\lambda ) &=& T_0+\\frac{\\ell }{R_m q}\\exp (q(\\lambda -\\lambda _m)), \\\\R(\\lambda ) &=& R_m \\exp (-q(\\lambda -\\lambda _m)) , \\\\\\Phi (\\lambda ) &=& \\Phi _0 -\\frac{3}{4}\\ell (\\lambda -\\lambda _m)+\\ell \\Phi _\\theta (\\lambda -\\lambda _m).$ The parametrized form is [box=]align T(R) = T0+q R, (R) = 0+34qRRm+((RRm)).",
"Here, the constants $T_0$ and $\\Phi _0$ remain arbitrary."
],
[
"The energy satisfies $E>0$ and the initial condition is $R(\\lambda _0)=R_0=-\\frac{E}{2\\ell _*}$ , leading to $\\lambda (R) -\\lambda _0= - \\frac{1}{\\ell _*} \\sqrt{1+\\frac{2 \\ell _* R}{E}}.$ The explicit form is $T(\\lambda ) &= \\frac{2 \\ell _*^2 (\\lambda - \\lambda _0)}{E(1-\\ell _*^2 (\\lambda -\\lambda _0)^2)}, \\\\R(\\lambda ) &= \\frac{E}{2\\ell _*} \\left( \\ell _*^2 (\\lambda -\\lambda _0)^2 - 1 \\right) , \\\\\\Phi (\\lambda ) &= -\\frac{3}{4}\\ell _* (\\lambda -\\lambda _0) +2 \\, \\text{arctanh}(\\ell _* (\\lambda - \\lambda _0)) \\nonumber \\\\&+ \\ell _* \\Phi _\\theta (\\lambda -\\lambda _0) .$ The parametrized form is [box=]align T(R) = 1R 1+2 * RE, (R) = 34 1+2 * RE-2 arctanh1+2 * RE+* ((R)-0)."
],
[
"The parameters obey $\\mathcal {C} < 0$ and $E>0$ .",
"We denote $q = \\sqrt{-\\mathcal {C}}$ and $\\lambda _-$ the value of the Mino time such that $R(\\lambda _-)= R_-$ .",
"The orbit is given by $\\lambda (R) - \\lambda _- = \\frac{1}{q} \\log \\left( \\frac{E \\sqrt{\\mathcal {C}+\\ell ^2}}{E \\ell - \\mathcal {C} R +q \\sqrt{v_R}} \\right)=\\frac{i}{q}\\arccos (\\frac{E\\ell -\\mathcal {C}R}{E\\sqrt{\\mathcal {C}+\\ell ^2}}).$ The last form is not explicitly real, but it allows us to find easily $T(\\lambda ) &= \\frac{q}{E} \\frac{{\\sinh (q(\\lambda -\\lambda _-))}}{\\frac{\\ell }{\\sqrt{\\mathcal {C} + \\ell ^2}}-\\cosh (q(\\lambda -\\lambda _-))}, \\\\R(\\lambda ) &= \\frac{E}{\\mathcal {C}} \\left( \\ell - \\sqrt{\\mathcal {C} + \\ell ^2} \\cosh (q (\\lambda - \\lambda _-))\\right), \\\\\\Phi (\\lambda ) &= - \\frac{3 \\ell }{4}(\\lambda -\\lambda _-) +2\\, \\text{arctanh}\\left( \\frac{\\ell + \\sqrt{\\mathcal {C} + \\ell ^2}}{q} \\tanh (\\frac{q}{2}(\\lambda -\\lambda _-))\\right)\\nonumber \\\\&+\\ell \\, \\Phi _\\theta (\\lambda -\\lambda _-) .$ The parametrized form can be simplified as [box=]align T(R) = vRE R, (R) = 0-E+R+vR(R) R +34q(E-CR+qvR(R))+((R)--) where $\\Phi _0\\triangleq \\log \\frac{E+\\ell R_-}{R_-}-\\frac{3\\ell }{4q}\\log (E\\ell -\\mathcal {C} R_-)$ .",
"The orbit start at $R=+\\infty $ at $\\lambda =-\\infty $ and reaches the black hole horizon $R=0$ at $\\lambda = \\lambda _- - \\lambda _H$ , where $\\lambda _H \\equiv \\frac{1}{q}\\text{arccosh}(\\frac{\\ell }{\\sqrt{\\mathcal {C} + \\ell ^2}})$ .",
"It never reaches $R_- < 0$ ."
],
[
"We have $\\mathcal {C} > 0$ and $E>0$ , leading to $\\lambda (R) -\\lambda _+ = \\frac{1}{q}\\text{arccos}\\left( \\frac{\\mathcal {C} R - E \\ell }{E \\sqrt{\\mathcal {C}+\\ell ^2}} \\right).$ We normalized Mino time such that $R(\\lambda _+) = R_+$ and denoted $q \\triangleq \\sqrt{\\mathcal {C}}$ .",
"The orbit starts at $\\lambda = \\lambda _+$ at the turning point $R_+$ and plunges inside the black hole $R=0$ at $\\lambda = \\lambda _++\\frac{1}{q}\\text{arccos}(\\frac{-\\ell }{\\sqrt{\\mathcal {C}+\\ell ^2}})$ .",
"We have $T (\\lambda ) &= \\frac{q}{E} \\frac{\\sin (q(\\lambda -\\lambda _+))}{\\frac{\\ell }{\\sqrt{\\mathcal {C} + \\ell ^2}}+\\cos (q(\\lambda -\\lambda _+))} , \\\\R(\\lambda ) &= \\frac{E}{\\mathcal {C}} \\left( \\ell + \\sqrt{\\mathcal {C} + \\ell ^2} \\cos (q (\\lambda -\\lambda _+) )\\right) , \\\\\\Phi (\\lambda ) &= - \\frac{3 \\ell }{4}(\\lambda -\\lambda _+) +2\\,\\text{arctanh}\\left( \\frac{\\ell - \\sqrt{\\mathcal {C} + \\ell ^2}}{q} \\tan (\\frac{q}{2}(\\lambda -\\lambda _+))\\right)\\nonumber \\\\&+\\ell \\Phi _\\theta (\\lambda -\\lambda _+).$ The parametrized form is given by (REF ) and (REF ), but with $q\\rightarrow iq$ .",
"One can write a manifestly real form of the azimuthal coordinate by shifting the initial value $\\Phi _0$ , leading to [box=]align (R)=0'-E+R+vR(R) R+34qqvRE-C R+((R)--) with $\\Phi ^{\\prime }_0\\triangleq \\log \\frac{E+\\ell R_-}{R_-}.$"
],
[
"Deflecting$(E,\\ell )$ .",
"We have $\\mathcal {C} < 0$ and $E<0$ , leading to ($q\\triangleq \\sqrt{-\\mathcal {C}}$ ) $\\lambda (R)-\\lambda _+=-\\frac{i}{q}\\arccos {\\frac{E\\ell -\\mathcal {C}R}{E\\sqrt{\\mathcal {C}+\\ell ^2}}}.$ The initial condition $R(\\lambda _+)=R_+$ corresponds to the minimal radius reached by the trajectory.",
"The orbit starts and ends at $R=+\\infty $ at Mino time $\\lambda =\\pm \\infty $ .",
"We have $T(\\lambda )&=\\frac{q}{E}\\frac{\\sinh {q(\\lambda -\\lambda _+)}}{\\frac{\\ell }{\\sqrt{\\mathcal {C}+\\ell ^2}}-\\cosh {q(\\lambda -\\lambda _+)}}\\\\R(\\lambda )&=\\frac{E}{\\mathcal {C}}(\\ell -\\sqrt{\\mathcal {C}+\\ell ^2}\\cosh {q(\\lambda -\\lambda _+)})\\\\\\Phi (\\lambda )&=-\\frac{3\\ell }{4}(\\lambda -\\lambda _+)+2\\,\\text{arctanh}\\left( \\frac{\\ell - \\sqrt{\\mathcal {C} + \\ell ^2}}{q} \\tanh (\\frac{q}{2}(\\lambda -\\lambda _+))\\right)\\nonumber \\\\&+\\ell \\Phi _\\theta (\\lambda -\\lambda _+).$ We finally have the parametrized form [box=]align T(R) = - vR(R)E R,(R) = 0+E+R+vR(R) R -34q(E-CR+qvR(R))+((R)-+) with $\\Phi _0\\triangleq -\\log \\frac{E+\\ell R_+}{R_+}+\\frac{3\\ell }{4q}\\log (E\\ell -\\mathcal {C}R_+)$ ."
],
[
"Explicit form of prograde near-NHEK geodesics",
"We follow the same procedure as the one in Appendix .",
"As before, we only focus, without loss of generality, on future-oriented partially ingoing prograde orbits.",
"For convenience, we also include one class of retrograde bounded geodesics."
],
[
"Spherical$(\\ell )$ .",
"The explicit form reads as $R&=R_0=\\frac{\\kappa \\ell }{\\sqrt{-\\mathcal {C}}},\\\\t(\\lambda )&=t_0+\\frac{\\ell }{R_0}(\\lambda -\\lambda _0),\\\\\\phi (\\lambda )&=\\phi _0-\\frac{3}{4}\\ell (\\lambda -\\lambda _0)+\\ell \\Phi _\\theta (\\lambda -\\lambda _0).$ The parametrized form is [box=]align R=R0=-C,(t)=0-34R0(t-t0)+(-0).",
"Note that $R_0\\ge (\\frac{2}{\\sqrt{3}}-1)\\kappa $ ."
],
[
"Plunging$_*$ .",
"One has $\\lambda -\\lambda _i=-\\frac{R-R_i}{\\kappa \\ell _*}.$ The explicit form is $R(\\lambda )&=R_i-\\kappa \\ell _*(\\lambda -\\lambda _i),\\\\t(\\lambda )&=-\\frac{1}{2\\kappa }\\log [1+\\kappa \\ell _*(\\lambda -\\lambda _i)\\frac{\\kappa \\ell _*(\\lambda -\\lambda _i)-2R_i}{R_i^2-\\kappa ^2}],\\\\\\phi (\\lambda )&=-\\frac{3}{4}\\ell _*(\\lambda -\\lambda _i)+\\frac{1}{2}\\log \\frac{1-\\frac{\\kappa \\ell _*(\\lambda -\\lambda _i)}{R_i-\\kappa }}{1-\\frac{\\kappa \\ell _*(\\lambda -\\lambda _i)}{R_i+\\kappa }}+\\ell _*\\Phi _\\theta (\\lambda -\\lambda _i).$ The parametrized form is [box=]align t(R)=-12R2-2Ri2-2,(R)= i+34R+12R-R++*((R)-i) with $\\phi _i\\triangleq -\\frac{3R_i}{4\\kappa }-\\frac{1}{2}\\log \\frac{R_i-\\kappa }{R_i+\\kappa }$ .",
"The geodesic starts from $R=+\\infty $ at $\\lambda =-\\infty $ and reaches the horizon at Mino time $\\lambda _H=\\lambda _i+\\frac{1}{\\ell _*}(\\frac{R_i}{\\kappa }-1)$ ."
],
[
"Bounded$_*(e)$ and Plunging{{formula:30dbfa97-2283-49bb-b8ef-2fcc48d5e19b}} .",
"The orbital parameters satisfy $\\mathcal {C}=0$ and $e<0$ (bounded) or $e>0$ (plunging).",
"The potential is simply $v_{R;\\kappa }(R)=e^2+\\kappa ^2\\ell _*^2+2e\\ell _* R$ and the initial conditions are imposed at $R(\\lambda _0)=R_0$ , where $R_0$ is the (unique) root of the radial potential.",
"One directly obtains $\\lambda (R)-\\lambda _0=-\\frac{\\sqrt{v_{R;\\kappa (R)}}}{e\\ell _*},$ leading to $R(\\lambda )&=R_0+\\frac{e\\ell _*}{2}(\\lambda -\\lambda _0)^2,\\\\t(\\lambda )&= -\\frac{1}{\\kappa }\\,\\text{arctanh}\\frac{2\\kappa e \\ell _*^2(\\lambda -\\lambda _0)}{\\kappa ^2\\ell _*^2+e^2[\\ell ^2_*(\\lambda -\\lambda _0)^2-1]},\\\\\\phi (\\lambda )&=-\\frac{3}{4}\\ell _*(\\lambda -\\lambda _0)+\\,\\text{sign}(e)\\, \\text{arctanh}\\frac{2e^2\\ell _*(\\lambda -\\lambda _0)}{-\\kappa ^2\\ell _*^2+e^2[\\ell _*^2(\\lambda -\\lambda _0)^2+1]}\\nonumber \\\\&~+\\ell _*\\Phi _\\theta (\\lambda -\\lambda _0).$ The parametric form is [box=]align t(R)=1 arccosh R+2*eR2-2,(R)=-34evR;(R)+arctanhvR;(R)e+* R+*((R)-0).",
"Notice that the requirements $v_{R;\\kappa }\\ge 0$ and $R\\ge \\kappa $ are sufficient to guarantee the reality of the inverse hyperbolic functions involved.",
"The trajectory reaches the horizon at Mino time $\\lambda _H=\\lambda _0-\\,\\text{sign}(e) \\sqrt{\\frac{2(\\kappa -R_0)}{e\\ell _*}}$ , which is smaller than $\\lambda _0$ for plunging motion and greater for bounded motion, as expected."
],
[
"(Retrograde) Bounded$_>(e,\\ell )$ .",
"The geodesic parameters satisfy $\\mathcal {C}<0$ , $\\ell < 0$ , and $e>0$ .",
"Therefore, $R_-$ is positive, and we choose the initial condition as $R(\\lambda _-)=R_-$ .",
"Defining $q\\triangleq \\sqrt{-\\mathcal {C}}$ , the explicit form reads as $R(\\lambda )&= \\frac{1}{\\mathcal {C}}[e\\ell -\\sqrt{(\\mathcal {C}+\\ell ^2)(e^2+\\kappa ^2\\mathcal {C})}\\cosh {q(\\lambda -\\lambda _-)}].$ The parametrized form is [box=]align t(R)=14F+(R)F-(R),(R)= 34q[e-CR+qvR;(R)]-14G+(R)G-(R)+((R)--) where we define $F_\\pm (R)&\\triangleq [eR+\\kappa (\\kappa \\ell \\pm \\sqrt{v_{R;\\kappa }(R)})]^2,\\\\G_\\pm (R)&\\triangleq (e+\\ell R\\pm \\sqrt{v_{R;\\kappa }(R)})^2.$ Note that using the identities $F_+(R)F_-(R)&=(e^2+\\kappa ^2\\mathcal {C})^2(R^2-\\kappa ^2)^2,\\\\G_+(R)G_-(R)&=(\\mathcal {C}+\\ell ^2)^2(R^2-\\kappa ^2)^2,$ one can rewrite (REF ) and (REF ) as $t(R)&=-\\frac{1}{2\\kappa }\\log \\frac{F_+(R)}{(e^2+\\kappa ^2\\mathcal {C})(R^2-\\kappa ^2)},\\\\\\phi (R)&= \\frac{3\\ell }{4q}\\log [e\\ell -\\mathcal {C}R+q\\sqrt{v_{R;\\kappa }(R)}]-\\frac{1}{2}\\log \\frac{G_+(R)}{(\\mathcal {C}+\\ell ^2)(R^2-\\kappa ^2)}+\\ell \\Phi _\\theta (\\lambda (R)-\\lambda _-).$ The geodesic motion starts from the past horizon, reaches $R_-$ at Mino time $\\lambda _-$ and crosses the future horizon at $\\lambda _H-\\lambda _-=\\frac{1}{q}\\text{arccosh}\\frac{e\\ell -\\kappa \\mathcal {C}}{\\sqrt{(\\mathcal {C}+\\ell ^2)(e^2+\\kappa ^2\\mathcal {C})}}$ ."
],
[
"Bounded$_<(e,\\ell )$ .",
"The parameters obey $\\mathcal {C}>0$ and $e\\ne 0$ ; therefore, $R_+$ is positive and the initial condition is chosen as $R(\\lambda _+)=R_+$ .",
"One defines $q\\triangleq \\sqrt{\\mathcal {C}}$ and gets the explicit form: $R(\\lambda )&= \\frac{1}{\\mathcal {C}}[e\\ell +\\sqrt{(\\mathcal {C}+\\ell ^2)(e^2+\\kappa ^2\\mathcal {C})}\\cos (q(\\lambda -\\lambda _+))].$ The parametrized form is given by (REF ) and (REF ), with the replacement rule $q\\rightarrow iq$ .",
"A manifestly real form of $\\phi $ is [box=]align (R)=34qqvR;e-C R-12G+(R)(C+2)(R2-2)+((R)--).",
"The geodesic motion starts from the white hole past horizon, reaches $R_+$ at Mino time $\\lambda _+$ and crosses the future horizon at $\\lambda _H-\\lambda _+=\\frac{1}{q}\\,\\text{arccos}\\frac{\\kappa \\mathcal {C}-e\\ell }{\\sqrt{(\\mathcal {C}+\\ell ^2)(e^2+\\kappa ^2\\mathcal {C})}}$ ."
],
[
"Plunging$(e,\\ell )$ .",
"The parameters satisfy $\\mathcal {C}<0$ and $e>-\\kappa q$ , where $q\\triangleq \\sqrt{-\\mathcal {C}}$ .",
"The roots of the radial potential are consequently either complex or negative.",
"In the complex case ($e^2+\\kappa ^2\\mathcal {C} < 0$ ), we define the real quantity $R_f \\triangleq \\frac{1}{\\mathcal {C}}[e\\ell -\\sqrt{-(\\mathcal {C}+\\ell ^2)(e^2+\\kappa ^2\\mathcal {C})}]$ and impose the final condition $R({\\lambda }_f)={R}_f$ , leading to $R(\\lambda )&= \\frac{1}{\\mathcal {C}}[e\\ell -\\sqrt{-( \\mathcal {C}+\\ell ^2)(e^2+\\kappa ^2\\mathcal {C})}(\\cosh {q(\\lambda -{\\lambda }_f)}-\\sqrt{2}\\sinh {q(\\lambda -{\\lambda }_f)})].$ Note that $\\lim _{x\\rightarrow \\pm \\infty }\\cosh x -\\sqrt{2}\\sinh x=\\mp \\infty $ leads to the expected behavior.",
"For negative roots, $R=R_-$ can be used as the final condition.",
"In both cases, the parametrized form is again as given in (REF ) and (REF ).",
"The orbit starts from $R=+\\infty $ at $\\lambda =-\\infty $ and reaches the horizon at Mino time $\\lambda _H-{\\lambda }_f=\\frac{1}{q}\\,\\text{arcsinh}\\, 1+\\log \\frac{e\\ell -\\kappa \\mathcal {C}+q((e+\\kappa \\ell )}{\\sqrt{-(\\mathcal {C}+\\ell ^2)(e^2+\\kappa ^2\\mathcal {C})}}$ ."
],
[
"Deflecting$(e,\\ell )$ .",
"One has $\\mathcal {C}<0$ and $e<0$ .",
"Choosing the initial condition as $R(\\lambda _-)=R_-$ and defining again $q\\triangleq \\sqrt{-\\mathcal {C}}$ , we get $R(\\lambda )&=\\frac{1}{\\mathcal {C}}[e\\ell -\\sqrt{(\\mathcal {C}+\\ell ^2)(e^2+\\kappa ^2\\mathcal {C})}\\cosh (q(\\lambda -\\lambda _-))].$ The parametrized form is [box=]align t(R)=-14F+(R)F-(R),(R)=- 34q[e-CR+qvR;(R)]+14G+(R)G-(R)+((R)--).",
"The orbit starts from $R=+\\infty $ at $\\lambda =-\\infty $ , reaches its minimal radial value $R_-$ at Mino time $\\lambda _-$ , and goes back to the asymptotic region at $\\lambda \\rightarrow +\\infty $ ."
],
[
"Comparison with Kapec-Lupsasca",
"An analysis of the (near-)NHEK geodesic motion with $\\ell \\ne 0$ was also performed by D. Kapec and A. Lupsasca in Ref.",
"[18].",
"This appendix aims to compare our radial taxonomy to the classification proposed in their paper, where the following classes of trajectories are defined: [label=$\\diamond $ ] Type I trajectories, for which $\\mathcal {C}>0$ .",
"Type II, for which $-\\ell ^2<\\mathcal {C}<0$ .",
"Type III, for which $\\mathcal {C}=0$ ."
],
[
"(Poincaré) NHEK.",
"Two subclasses of Type II are defined: Type IIA, with $E\\ell >0$ and Type IIB, with $E\\ell <0$ .",
"The comparison is displayed in Table REF .",
"In Ref.",
"[18], the cases $E=0$ were not considered.",
"Table: Comparison between our classification and Ref.",
"for NHEK geodesics."
],
[
"Near-NHEK.",
"For Type II, in addition to Types IIA and IIB, Ref.",
"[18] defines two additional subcases: [label=$\\diamond $ ] Case 1: $-(\\frac{e}{\\kappa })^2<\\mathcal {C}$ .",
"Case 2: $-(\\frac{e}{\\kappa })^2>\\mathcal {C}$ .",
"This is relevant for distinguishing geodesic classes with $e<0$ , $\\ell > \\ell _*$ .",
"The comparison is provided in Table REF .",
"The zero-measure cases $e=0$ and $e=-\\kappa \\sqrt{-\\mathcal {C}}$ were not considered in Ref.",
"[18].",
"Table: Comparison between our classification and for near-NHEK geodesics."
]
] | 2001.03478 |
[
[
"Real-Time RFI Mitigation for the Apertif Radio Transient System"
],
[
"Abstract Current and upcoming radio telescopes are being designed with increasing sensitivity to detect new and mysterious radio sources of astrophysical origin.",
"While this increased sensitivity improves the likelihood of discoveries, it also makes these instruments more susceptible to the deleterious effects of Radio Frequency Interference (RFI).",
"The challenge posed by RFI is exacerbated by the high data-rates achieved by modern radio telescopes, which require real-time processing to keep up with the data.",
"Furthermore, the high data-rates do not allow for permanent storage of observations at high resolution.",
"Offline RFI mitigation is therefore not possible anymore.",
"The real-time requirement makes RFI mitigation even more challenging because, on one side, the techniques used for mitigation need to be fast and simple, and on the other side they also need to be robust enough to cope with just a partial view of the data.",
"The Apertif Radio Transient System (ARTS) is the real-time, time-domain, transient detection instrument of the Westerbork Synthesis Radio Telescope (WSRT), processing 73 Gb of data per second.",
"Even with a deep learning classifier, the ARTS pipeline requires state-of-the-art real-time RFI mitigation to reduce the number of false-positive detections.",
"Our solution to this challenge is RFIm, a high-performance, open-source, tuned, and extensible RFI mitigation library.",
"The goal of this library is to provide users with RFI mitigation routines that are designed to run in real-time on many-core accelerators, such as Graphics Processing Units, and that can be highly-tuned to achieve code and performance portability to different hardware platforms and scientific use-cases.",
"Results on the ARTS show that we can achieve real-time RFI mitigation, with a minimal impact on the total execution time of the search pipeline, and considerably reduce the number of false-positives."
],
[
"Introduction",
"For a little over a decade [1], Fast Radio Bursts (FRBs) have represented the source of highly discussed open questions among astronomy academics.",
"FRBs are millisecond-duration, highly energetic, dispersed radio pulses emitted from distant regions of space-time, far beyond the vicinity of the Milky Way (MW).",
"Without considering propagation effects like scintillation and scattering, FRBs follow a relatively simple time-frequency structure described as being broadband and of narrow width [2].",
"Similarly to what is observed for single pulses from pulsarsPulsars are neutron stars of a few kilometers in radius, with a few times the mass of our Sun, and stable rotation periods duration varying between seconds [3] to milliseconds [4]., as an FRB travels through regions of ionized medium, a frequency-dependent time delay is imposed on its signal, where longer wavelengths of the propagating signal are slowed down more intensely than shorter ones.",
"This delay is quantified by a dispersion measure (DM), in unit of parsec per cubic centimeter (pc/cm$^3$ ), proportional to the number of free electrons along the line of sight: $\\textrm {DM} = \\int ^{d}_{0} n_e(l) \\mathit {dl},$ where $n_e$ is the electron number density, $l$ is a path length, and $d$ is the distance to the FRB [5].",
"While DM values for most known pulsars place them within the MW when compared to electron-density models [6], [7], the large DM values measured for FRBs suggest an extragalactic origin.",
"Among other international efforts, the observation program ALERTALERT stands for Apertif Legacy Exploration of the Radio Transient Sky.",
"searches for FRBs using the Westerbork Synthesis Radio Telescope (WSRT) located in the Netherlands.",
"The receivers of the WSRT were recently upgraded with phased-array feeds called Apertif, which along with the Apertif Radio Transient System (ARTS [8])—a hybrid machine of combined Field-Programmable Gate Arrays (FPGAs) and Graphics Processing Units (GPUs)—allow to form $\\sim 10^3$ beams on the sky with a high duty cycle.",
"The resulting data stream is searched in real-time using the AMBER pipeline [9] on the ARTS cluster, generating a list of single-pulse candidates for any detection having a Signal-to-Noise Ratio (SNR) greater than a specified threshold.",
"The common FRB searching strategy (such as [10], [9]) consists in incoherently de-dispersing the signal to correct for the time-delay among frequencies caused by dispersion (Figure REF ).",
"ARTS searches for FRBs by forming many different time series on the GPU corresponding to a wide range of dispersion measures.",
"The amount of dispersion in seconds is quantified by the time delay of the pulse between the lowest and highest radio frequencies of the observation in MHz, $v_{low}$ and $v_{high}$ respectively, as $\\mathit {\\Delta t} = k_{\\textrm {DM}} \\, (v_{low}^{-2}-v_{high}^{-2}) \\, \\textrm {DM} \\;\\textrm {s},$ where $k_{\\textrm {DM}}=\\frac{e^2}{2\\pi m_e c}\\approx 4.15\\times 10^3\\, \\textrm {MHz}^2\\,pc^{-1}\\,\\textrm {cm}^3\\,\\textrm {s}$ is the dispersion constant, with $m_e$ being the mass of the electron, and $c$ the speed of light.",
"ARTS processes 73 Gb/s of Apertif data, in real-time, searching for faint pulsars and FRBs, while at the same time aiming to mitigate the effects of Radio Frequency Interference (RFI) in a densely populated environment like the Netherlands.",
"Hence, our real-time requirement translates to the need for RFI mitigation methods that are fast and simple, while also being robust enough to cope with a partial view of the data.",
"Figure: A signal dispersed in time as a function of frequency in zero-DM plane (blue), and the same pulse after incoherent de-dispersion (orange).",
"The amounts of de-dispersion (Eq. )",
"applied to different frequency channels are shown by the black arrows.Even with a radio quiet zone around WSRT, the generation of RFI is becoming increasingly part of anthropic activities.",
"Furthermore, our sky is populated by a growing number of satellites for world-wide telecommunications.",
"In fact, RFI can be stronger than astrophysical signals due to the inverse-square law of propagation which stipulates that electromagnetic radiation dissipates at a quadratic rate with respect to distance [11].",
"As opposed to the prototypical FRB signal, RFI is generated on or near Earth and displays a wide range of time and frequency structures not generally found in nature.",
"Without RFI mitigation, spurious bright individual RFI signals at different frequencies can artificially align at higher DMs, in which case the sum of these signals can become greater than the detection threshold, therefore causing artificial detections.",
"These false positives can rapidly increase the size of the single-pulse candidates list, which needs to be verified by astronomers.",
"Not only can RFI cause false-positive detections (i.e.",
"non-astrophysical pulses erroneously classified as FRBs), but can also mask real, weak astrophysical signals and reduce the rate of true positives.",
"While a deep learning classifier [12] is being used a posteriori to reduce the number of false-positive detections that need to be verified by eye, removing RFI early on in the pipeline helps to reduce the overall compute time."
],
[
"Related work",
"RFI mitigation has previously been tackled via different methods, including linear methods (e.g.",
"Singular Value Decomposition [13] and Principal Component Analysis [14]), test of non-Gaussianity (e.g.",
"Spectral Kurtosis [15], [16]), threshold-based methods (e.g.",
"SumThreshold algorithm [17], where RFI is defined as values above a defined threshold in the time-frequency plane), and more recently via machine learning methods like K-nearest neighbour, Gaussian mixture model, and convolutional neural network [18], [19], [20]."
],
[
"RFIm: a RFI Mitigation Library",
"Our proposed solution to the challenge of real-time RFI mitigation is RFIm, a high-performance, open-source, tuned and extensible RFI mitigation library.",
"The goal of this library is to provide highly-tuned RFI mitigation routines that are designed to run in real-time on many-core accelerators, such as the many GPUs installed on ARTS, while at the same time be accurate enough to reduce false positives and increase the measured SNR of real astrophysical events.",
"Although RFIm is designed to be the main ARTS RFI mitigation library, it was also designed to be used on its own outside of the ARTS pipeline.",
"All mitigation methods are implemented in C++ and OpenCL, the Open Computing Language [21].",
"The C++ implementations are reference implementations, optimized for testing and readability, and are not used in production, while the OpenCL implementations are meant to be used in production, and are therefore highly optimized to run in real-time on GPUs.",
"The library is open source, released under version 2.0 of the Apache licence, and can be downloaded from GitHubAvailable at https://github.com/AA-ALERT/RFIm.",
"One of the main characteristics of RFIm is that it is designed with both code and performance portability in mind, and this is achieved using code generation and auto-tuning.",
"This means not only that the same code can be executed, without changes, on different hardware platforms, but also that the expected efficiency should be similar on these platforms.",
"In fact, the OpenCL code of the computational kernels is generated at run-time from a set of templates, so that it can be automatically tailored to the specific use-cases of the user, and it can take advantage of run-time knowledge of system and input parameters.",
"Moreover, a variety of parameters, such as the number of threads, the amount of work per thread, or synchronization schemes, can be tuned during code generation, so that the resulting code can achieve high performance running on different hardware.",
"This also allows processing efficiently data generated by different telescopes, or simply when observational parameters are changed because of different observations.",
"To help the user define all these tunable parameters, RFIm includes a tuning routine for each of the mitigation algorithms included.",
"By using these auto-tuning routines, a user can identify the best code generation parameters to use for a specific observational setup and computing platform, and save the resulting configurations for use in production.",
"RFIm currently contains two mitigation methods, both of them real-time thresholding methods: time-domain sigma cut (TDSC), and frequency-domain sigma cut (FDSC).",
"Further methods are being developed and will be integrated into RFIm in the future, including the Sum-threshold [17] and Edge-threshold [11] algorithms."
],
[
"Time and Frequency Domain Sigma Cut",
"During observations, AMBER splits the input time-series in order to process data in batches, where the number of time samples inside a batch varies depending on the number of samples per second and the DM range being searched.",
"Therefore, RFI mitigation within RFIm is performed on a per-batch basis.",
"Below, we briefly describe the two algorithms currently implemented within RFIm.",
"The TDSC method (Figure REF ) targets bright broadband events of low-DM.",
"For a given frequency channel (f_i), each time sample (t_i) of a batch is compared to the global statistics of the batch: the mean ($\\mu $ ), and the standard deviation ($\\sigma $ ).",
"If t_i is an outlier within the batch (e.g.",
"$|\\textrm {t\\_i}-\\mu | > n\\sigma $ , where $n$ is the user-defined threshold) it is replaced by the mean of the batch.",
"Figure: Schematic diagram of the TDSC algorithm computed within an OpenCL kernel.",
"For a given frequency channel (f_i), each time sample (t_i) of a batch is compared to the statistics of the batch: mean (μ\\mu ), and nn times the standard deviation (σ\\sigma ).",
"If a sample is an outlier within the batch, it is replaced by the mean of the batch.",
"We note that in practice, the applied comparison is symmetric.The FDSC method (Figure REF ) targets bright narrow-band pulses of short duration.",
"FDSC comprises two steps.",
"Firstly, for cases where no bandpass correction was applied to the data, frequency channels at a given time (t_i) are grouped into $n$ bins, and the mean of the bin is subtracted from t_i, resulting in a mean-corrected sample.",
"Secondly, each mean-corrected frequency channel is compared to the mean-corrected statistics of all frequency channels following a similar process as described for TDSC.",
"If a mean-corrected sample is an outlier, the original channel value is replaced by the mean of its respective bin.",
"Figure: Schematic diagram of the FDSC algorithm computed within an OpenCL kernel.",
"The algorithm proceeds in two steps: (1) for a given time sample (t_i), each frequency channel (f_i) is first attributed a bin, and a difference between each frequency channel and the mean of the bin is computed; (2) each mean-corrected frequency channel is compared to the mean-corrected statistics of all frequency channels following a similar process as in TDSC.",
"If a mean-corrected sample is an outlier, the original channel value is replaced by the mean of its respective bin.",
"We note that in practice, the applied comparison is symmetric.Although the currently implemented replacement strategy for both TDSC and FDSC is to use the mean, this is done purely to reduce the computational cost of these algorithms.",
"In fact, both TDSC and FDSC support a parameter representing the replacement strategy for the flagged samples, and RFIm users can implement their own strategies, such as using the median, random unflagged samples, or any other user-defined strategy."
],
[
"Accuracy",
"To evaluate how the methods proposed in the previous section perform we test the algorithms on both artificial and real data.",
"We begin with simulated data based on surveying parameters of ARTS (time sampling, total bandwidth, channelization, etc.).",
"Using this canvas, we generate Gaussian noise data with $\\mu =0$ and $\\sigma =1$ .",
"We then add to the noise a faint, broadband highly dispersed pulse representing an FRB.",
"Finally, we add strong RFI signals (two bright low-DM broadband pulses, and two narrow-band periodic pulses).",
"Figure REF shows the test data generation procedure.",
"To highlight the behaviour of each mitigation technique, we proceed in applying our mitigation algorithms on the dispersed data (DM=0, Figure REF ): TDSC only (top panel), FDSC only (central panel), and both methods combined (bottom panel).",
"As expected, bright broadband RFI is cleaned by the TDSC, and bright narrow-band RFI is cleaned by FDSC.",
"When applying both techniques on the test data, the faint FRB becomes apparent.",
"Figure: Time-frequency plots of simulated data in the Apertif frequency band.",
"Top: gaussian noise; middle: one faint high DM pulse, with DM = 500 pc/cm 3 ^3 and SNR = 15 is added to the noise; bottom: two low DM pulses with DM ∈{1,10}\\textrm {DM}\\in \\lbrace 1, 10\\rbrace and S/N∈{100,150}\\textrm {S/N}\\in \\lbrace 100, 150\\rbrace respectively, and two periodic narrow band pulses with SNR = 125 covering 10, 15 frequency channels respectively.",
"Data intensity (as highlighted in the colorbars) is in an arbitrary unit.Figure: RFI mitigation applied to data from the bottom panel of Fig.",
".",
"From top to bottom: mitigation in time with threshold=3.25σ\\sigma ; mitigation in frequency with a bin size of 32 channels, and threshold=2.75σ\\sigma ; both mitigation methods applied (zero-DM plane); and dedispersed window (DM = 500 pc/cm 3 ^3) with both mitigation methods applied.",
"The faint dispersed pulse can be seen starting from t=0.04s at the highest frequency.We then de-disperse the data to the FRB's DM value, and compare the result with and without RFI mitigation (Figure REF ).",
"In this specific case, the faint FRB would not have been detected by the pipeline as the SNR of the dedispersed burst is below the detection threshold (top two panels).",
"On the other hand, the time-series cleaned with RFIm shows the dedispersed burst having a SNR greater than the detection threshold.",
"Figure: Time series from Fig.",
"with and without RFI mitigation dedispersed at FRB's DM.",
"The dashed line shows the threshold separating a trigger (signal going above the threshold) from a non-trigger.",
"After RFI mitigation, this specific FRB is detected.Finally, to assure we do not radically remove high-DM pulses from observations, we evaluate our method on real data.",
"We apply TDSC and FDSC to a 299 seconds Apertif observation of PSR B0531+21PSR B0531+21 is commonly known as the Crab pulsar.",
"The data is sampled at 81.92 $\\mu $ s, with 300 MHz of bandwidth at L-Band, and a channel width of 0.195 MHz.",
"It is common to see variations in SNR between individual pulses.",
"using the OpenCL implementation of AMBER.",
"To simulate the iterative mitigation process used in the production pipeline, we perform three iterations of the algorithms.",
"Figure: Effect of applying TDSC and FDSC to detection's signal-to-noise ratio (SNR) with AMBER to an Apertif observation of PSR B0531+21.",
"The plot compares the SNR ratio as a function of the SNR computed using the original data (without RFI mitigation).",
"The ratio is computed between SNR detected with and without RFI mitigation (X/Original), where X corresponds to one of three experimental cases, where we apply RFI mitigation using threshold values of: TDSC=3.25σ\\sigma , combined TDSC=3.25σ\\sigma and FDSC=2.75σ\\sigma , and TDSC=4.00σ\\sigma respectively.Figure REF highlights the effect of mitigation on SNR for three experimental cases, where we apply RFI mitigation using threshold values of: TDSC=3.25$\\sigma $ , combined TDSC=3.25$\\sigma $ and FDSC=2.75$\\sigma $ , and TDSC=4.00$\\sigma $ respectively.",
"The scatter plot shows the SNR ratio—the SNR computed after applying RFI mitigation divided by the SNR computed on the original data—for multiple single pulse detections, as a function of the original SNR (no mitigation applied).",
"A ratio of 1 means there was no difference after applying mitigation for a given set of parameters; a ratio greater than 1 means an improved SNR after applying mitigation; and a ratio smaller than 1 means a reduced SNR after applying mitigation.",
"It is interesting to note that applying TDSC with 3.25$\\sigma $ (blue circles) improves the SNR in most cases (74%), and that the largest differences are found close to our detection threshold of 8.",
"Similarly, applying both TDSC with 3.25$\\sigma $ and FDSC with 2.75$\\sigma $ (orange x symbols) gives the same results as when using only TDSC with 3.25$\\sigma $ (the blue circles and orange x symbols overlap)—which is expected as FDSC should not affect broadband signal.",
"Unsurprisingly, applying TDSC with a poorly constraining 4.00$\\sigma $ threshold (green + symbols) gives identical results as in the case where no RFI mitigation is being applied.",
"We keep further analysis of trigger reduction factors, causes of specific SNR improvements or reductions, and the effect on true- and false-positive detections (e.g.",
"precision and recall) as future work.",
"However, we note that the total trigger counts prior to the machine learning classification step vary between the various cases: 200 triggers without RFI mitigation, 195 when applying TDSC=3.25$\\sigma $ , and 93 with the combination of TDSC=3.25$\\sigma $ and FDSC=2.75$\\sigma $ —hence reducing the amount of data to be post-processed.",
"Finally, we note some events detected in the original data are not detected after RFI mitigation with 3.25$\\sigma $ (10 missed detections), and vice-versa (14 new detections previously undetected in the original data)—which is in agreement with our simulation results."
],
[
"Performance",
"One of the design goals of RFIm is to provide accurate results in real-time, and while accuracy has been already discussed in Section REF , this section focuses on performance.",
"Figure REF presents the scalability of the TDSC algorithm, while Figure REF shows the same experiment for the FDSC algorithm.",
"Figure: Scalability experiment for the TDSC algorithm.Figure: Scalability experiment for the FDSC algorithm.In these experiments, tuned OpenCL implementations of the two algorithms are run on one of the NVIDIA GTX 1080Ti cards of the ARTS cluster, and execution time is measured; presented results are the average of multiple executions, to remove statistical fluctuations from the data.",
"The parameters that are varied in the experiment are: (1) the number of frequency channels, (2) the number of time samples, and (3) the number of input beams; to improve the readability of the figures, a representative selection of the results is plotted.",
"In both Figure REF and REF , each color coded line represents the execution time for a given combination of number of frequency channels and time samples, in correspondence with the number of beams on the $x$ axis; both the $x$ and the $y$ axes are in logarithmic scale.",
"The first noticeable result is that both algorithms run in real-time, with the execution time being, in the most complex cases, at most a tenth of a second for a whole batch of input data.",
"This is important because, in the context of an FRB searching pipeline, most of the processing time can still be dedicated to the search itself.",
"Another result is that both algorithms scale linearly not just in terms of number of beams, but also in terms of number of frequency channels and time samples.",
"Therefore, an increase in sensitivity or resolution in WSRT, or any other telescope for which RFIm is used, can more easily be accommodated, without requiring major changes in the software.",
"Figure: Median execution time of three different instances of AMBER.",
"RFI mitigation (in red) takes between 0.3% and 1% of the total execution time.Another way to look at the performance of RFIm is to measure its contribution to the total execution time of AMBER, the ARTS FRB pipeline.",
"The timing data was recorded during a survey observation in July 2019.",
"Figure REF presents an overview of the time, in seconds, spent inside each component of the AMBER pipeline, for this three hours observation; the three columns in the figure correspond to three different instances of AMBER, each running on a subset of the DM space.",
"During this observation, the TDSC algorithm of RFIm was enabled, and three iterations with 3.25$\\sigma $ were executed for each batch; the number of frequency channels is 1536, the number of beams 12, and the number of samples per second 12,208.",
"Results show that the contribution of TDSC to AMBER's execution time varies between the 0.33% for the low DMs search, to the 1.02% of the high DMs search.",
"Therefore, using RFIm we can mitigate the effects of RFI for ARTS, without significantly affecting the execution time of the whole pipeline."
],
[
"Conclusions",
"RFI will be an important problem for all upcoming FRB surveys, including the Square Kilometre Array.",
"As such, there is a need for real-time RFI mitigation.",
"In this paper we introduced RFIm, a high-performance, open-source, tuned and extensible RFI mitigation library, and showed how it can improve the quality of results with real and simulated data.",
"Moreover, we showed that RFIm achieves real-time performance in different scenarios (varying number of frequency channels, number of time samples per batch, number of input beams).",
"RFIm is already used in production for ARTS, mitigating RFI while barely affecting the execution time of AMBER.",
"While we aim to mitigate RFI, there exists a trade-off between pre-dedispersion RFI cleaning and false-positive rejection post-triggering by the deep-learning classifier.",
"To all current and future actions to remove RFI during real-time pulsar and FRB searches, the risk of over-cleaning the data and removing astronomical events needs to be considered [12].",
"With this in mind, we note that RFIm mitigation actions are currently performed in GPU memory only, leaving the original data unmodified, while the FRB search is performed on the mitigated data stream in memory.",
"Hence, provided the raw data is stored to disk during observation, one can also process the data a posteriori using different cleaning methods for further analysis."
],
[
"Acknowledgments",
"This work is supported by the Netherlands eScience Center under the project “AA-ALERT: Access and Acceleration of the Apertif Legacy Exploration of the Radio Transient Sky”, filenumber 027.015.G09.",
"We would like to thank Liam Connor, Joeri van Leeuwen, and Yogesh Maan for valuable discussions during the development of RFIm."
]
] | 2001.03389 |
[
[
"Ruin probability for the bi-seasonal discrete time risk model with\n dependent claims"
],
[
"Abstract The discrete time risk model with two seasons and dependent claims is considered.",
"An algorithm is created for computing the values of the ultimate ruin probability.",
"Theoretical results are illustrated with numerical examples."
],
[
"input@pathstyle/graphics/"
]
] | 2001.03431 |
[
[
"Quantum Anomalies in String-Inspired Running Vacuum Universe: Inflation\n and Axion Dark Matter"
],
[
"Abstract In this letter, we elaborate further on a Cosmological \"Running-Vacuum\" type model for the Universe, suggested previously by the authors within the context of a string-inspired effective theory in the presence of a Kalb-Ramond (KR) gravitational axion field which descends from the antisymmetric tensor of the massless gravitational string multiplet.",
"In the presence of this field, which has anomalous CP violating interactions with the gravitons, primordial gravitational waves induce gravitational anomalies, which in turn are responsible for the appearance of $H^2$ and $H^4$ contributions to the vacuum energy density, these terms being characteristic of generic \"running-vacuum-model (RVM) type\", where $H$ is the Hubble parameter.",
"In this work we prove in detail the appearance of the $H^4$ terms due to gravitational-anomaly-induced condensates in the energy density of the primordial Universe, which can self-consistently induce inflation, and subsequent exit from it, according to the generic features of RVM.",
"We also argue in favour of the robustness of our results, which were derived within an effective low-energy field theory approach, against Ultra Violet completion of the theory.",
"During the radiation and matter-dominated eras, gravitational anomalies cancel, as required for the consistency of the quantum matter/radiation field theory.",
"However, chiral and QCD-axion-type anomalies survive and have important consequences for both cosmic magnetogenesis and axionic dark matter in the Universe.",
"Finally, the stringy RVM scenario presented here predicts quintessence-like dynamical dark energy for the current Universe, which is compatible with the existing fitting analyses of such model against observations"
],
[
"Acknowledgements",
"We thank the referee for instructive comments and remarks on string theory considerations, which helped improving the presentation.",
"SB acknowledges support from the Research Center for Astronomy of the Academy of Athens in the context of the program “Tracing the Cosmic Acceleration”.",
"The work of NEM is supported in part by the UK Science and Technology Facilities research Council (STFC) under the research grants ST/P000258/1 and ST/T000759/1.",
"The work of JS has been partially supported by projects FPA2016-76005-C2-1-P (MINECO), 2017-SGR-929 (Generalitat de Catalunya) and MDM-2014-0369 (ICCUB).",
"This work is also partially supported by the COST Association Action CA18108 “Quantum Gravity Phenomenology in the Multimessenger Approach (QG-MM)”.",
"NEM acknowledges a scientific associateship (“Doctor Vinculado”) at IFIC-CSIC-Valencia University, Valencia, Spain.",
"In this Appendix we would like to justify the use of the effective action (REF ), based only on antisymmetric tensor and graviton degrees of freedom.",
"What we shall argue below is that a constant (or slowly moving) dilaton configuration, as assumed above and in [2], can be consistently implemented, and all our conclusions are not affected by the inclusion of the dilaton dynamics, upon certain reasonable assumptions that we shall outline explicitly.",
"Our starting point is the string effective action of the massless string gravitational multiplet (graviton, dilaton and KR antisymmetric tensor fields) in the Einstein frame [16], [17], [18] Our conventions and definitions used throughout this work are: signature of metric $(+, -,-,- )$ , Riemann Curvature tensor $R^\\lambda _{\\,\\,\\,\\,\\mu \\nu \\sigma } = \\partial _\\nu \\, \\Gamma ^\\lambda _{\\,\\,\\mu \\sigma } + \\Gamma ^\\rho _{\\,\\, \\mu \\sigma } \\, \\Gamma ^\\lambda _{\\,\\, \\rho \\nu } - (\\nu \\leftrightarrow \\sigma )$ , Ricci tensor $R_{\\mu \\nu } = R^\\lambda _{\\,\\,\\,\\,\\mu \\lambda \\nu }$ , and Ricci scalar $R = R_{\\mu \\nu }g^{\\mu \\nu }$ .",
"$S_B =\\; \\int d^{4}x\\sqrt{-g}\\Big ( \\dfrac{1}{2\\kappa ^{2}} [-R + 2\\, \\partial _{\\mu }\\Phi \\, \\partial ^{\\mu }\\Phi ] - \\frac{1}{6}\\, e^{-4\\Phi }\\, {\\mathcal {H}}_{\\lambda \\mu \\nu }{\\mathcal {H}}^{\\lambda \\mu \\nu } - V(\\Phi ) \\Big ) + S_B^{(\\alpha ^\\prime )} ,$ where $V(\\Phi )$ is a (non-perturbartively) generated potential for the the dilaton field $\\Phi $ , which we leave unspecified for the purposes of this work.",
"The $S_B^{(\\alpha ^\\prime )} $ represent two classes of string low-energy effective action terms, of higher order in $\\alpha ^\\prime $ [17], with $\\alpha ^\\prime = M_s^{-2}$ the Regge slope of the string and $M_s$ the string mass scale, which is not necessarily the same as the four dimensional gravitational constant $\\kappa ^2 = 8\\pi \\, {\\rm G} = M_{\\rm Pl}^{-2}$ : (i) higher (than two) derivative terms, including couplings of dilatons to the Gauss-Bonnet quadratic curvature invariant and (ii) terms involving higher than quadratic powers of ${\\mathcal {H}}_{\\mu \\nu \\rho }$ .",
"Their explicit form to $\\mathcal {O}(\\alpha ^\\prime )$ reads [17]: $S_B^{(\\alpha ^\\prime )} &= \\; \\frac{ \\alpha ^\\prime }{g_s^{(0)2}\\, \\kappa ^2} \\, \\int d^{4}x\\sqrt{-g}\\Big [- c_1 \\frac{1}{8} e^{-2\\Phi }\\Big (R_{\\mu \\nu \\rho \\sigma } \\, R^{\\mu \\nu \\rho \\sigma } - 4 R_{\\mu \\nu }\\, R^{\\mu \\nu } + R^2\\Big )+ c_2 e^{-2\\Phi } (\\partial _\\rho \\Phi \\,\\partial ^\\rho \\Phi )^2 + \\nonumber \\\\&+ c_3 \\, e^{-2\\Phi }\\,(R^{\\mu \\nu } -\\frac{1}{2} \\, g^{\\mu \\nu } \\, R)\\, \\partial _\\mu \\Phi \\, \\partial _\\nu \\Phi + c_4 \\, \\, e^{-2\\Phi }\\,\\nabla ^\\mu \\partial ^\\nu \\Phi \\, \\partial _\\mu \\Phi \\, \\partial _\\nu \\Phi \\nonumber \\\\&+\\; c_5\\, e^{-6\\Phi }\\,(-\\nabla ^\\mu \\mathcal {H}^{\\nu \\rho \\sigma } \\nabla _\\nu \\mathcal {H}_{\\sigma \\mu \\rho } +\\nabla _\\mu \\mathcal {H}^{\\mu \\rho \\sigma } \\, \\nabla ^\\nu \\mathcal {H}_{\\nu \\rho \\sigma } ) \\nonumber \\\\& + c_6 \\, e^{-6\\Phi }\\,( 5R^{\\mu \\nu \\rho \\sigma }\\, \\mathcal {H}_{\\mu \\nu \\lambda }\\, \\mathcal {H}^\\lambda _{\\,\\,\\rho \\sigma }-8R^{\\mu \\nu }\\, \\mathcal {H}_{\\mu \\rho \\sigma } \\,\\mathcal {H}_\\nu ^{\\,\\,\\rho \\sigma } + R\\,\\mathcal {H}_{\\mu \\nu \\rho }\\, \\mathcal {H}^{\\mu \\nu \\rho } ) \\nonumber \\\\&+ c_7 \\, e^{-6\\Phi }\\,\\mathcal {H}^{\\mu \\nu \\rho } \\, \\partial _\\rho \\Phi \\, \\partial ^\\lambda \\Phi \\, \\mathcal {H}_{\\lambda \\mu \\nu }+c_8 \\, e^{-6\\Phi }\\,\\mathcal {H}_{\\mu \\nu \\rho } \\, \\mathcal {H}^{\\mu \\nu \\rho } \\, \\partial _\\lambda \\Phi \\, \\partial ^\\lambda \\Phi \\nonumber \\\\& + c_9 \\, e^{-10\\Phi }\\,\\Big (-3\\mathcal {H}^{\\mu \\nu \\rho } \\, \\mathcal {H}_{\\lambda \\nu \\rho } \\, \\mathcal {H}^{\\lambda \\sigma \\tau } \\mathcal {H}_{\\mu \\sigma \\tau } + \\mathcal {H}^{\\mu \\nu \\rho } \\, \\mathcal {H}_{\\mu }^{\\,\\,\\lambda \\sigma } \\, \\mathcal {H}_{\\nu \\lambda }^{\\,\\,\\,\\,\\,\\,\\kappa }\\mathcal {H}_{\\kappa \\rho \\sigma } + \\frac{2}{3} (\\mathcal {H}_{\\mu \\alpha \\beta }\\, \\mathcal {H}_{\\nu \\alpha \\beta } )^2 \\Big )+ \\dots \\Big ]$ where $g_s^{(0)}$ is the string coupling when the dilaton $\\Phi =0$ , $\\nabla _\\mu $ denotes the gravitational covariant derivative with respect to the torsion-free connection, the $\\dots $ denote higher order terms and $c_i$ , $i=1,\\dots 9$ are numerical coefficients that can be determined by matching with string scattering amplitudes or $\\sigma $ -model conformal invariant conditions [17], [16].",
"For constant (or sufficiently slowly moving) dilatons, or weak KR field strengths $\\mathcal {H}_{\\mu \\nu \\rho }$ , slowly evolving with the cosmic time, all of the terms in (REF ) (and those of higher order) are subleading or not contributing to our discussion Thus we can neglect them and the relevant field equations can be highly simplified in our case.",
"Apart from the graviton equation, which we do not write explicitly here, variation of the effective action with respect to both the antisymmetric tensor and dilaton, yields the following field equations: ${\\rm antisymmetric~tensor}: \\qquad \\nabla ^\\mu \\Big (e^{-4\\Phi } {\\mathcal {H}}_{\\mu \\nu \\rho } \\Big )=0,$ ${\\rm dilaton}: \\frac{2}{\\kappa ^2}\\, \\nabla ^\\mu \\partial _\\mu \\Phi - \\frac{2}{3}\\, e^{-4\\Phi }\\, {\\mathcal {H}}_{\\lambda \\mu \\nu }{\\mathcal {H}}^{\\lambda \\mu \\nu } + \\dfrac{\\partial V(\\Phi )}{\\partial \\Phi } = 0~.$ In four space-time dimensions, a general solution of (REF ) is $e^{-4\\Phi } H_{\\mu \\nu \\rho } \\propto \\varepsilon _{\\mu \\nu \\rho \\sigma } \\, \\partial ^\\sigma \\overline{b}(x)\\,,$ where $\\overline{b}(x)$ is the background of the KR axion field $b(x)$ , which was introduced in the main text, as a Lagrange multiplier for the constraint (REF ).",
"Indeed, if one ignores higher than quadratic $\\mathcal {H}_{\\mu \\nu \\rho }$ terms in the path integral, after the introduction of the Lagrange multiplier $b$ field [2], and considers a saddle point of the action in the (exact) path integration over $\\mathcal {H}_{\\mu \\nu \\rho }$ fields, the mere use of the equations of motion for the KR field strength yields (REF ).",
"Let us now more specifically address the constant dilaton situation on which we based our discussion in this work.",
"Upon considering cosmic $b(t)$ fields, with canonically normalised kinetic terms, using (REF ), and noting that in our conventions [2] $\\frac{2}{3}\\, e^{-4\\Phi }\\,\\mathcal {H}_{\\mu \\nu \\rho } \\, \\mathcal {H}^{\\mu \\nu \\rho } = -2(\\dot{\\overline{b}})^2 < 0$ , we find from (REF ) that $\\partial _\\mu \\Phi \\simeq 0$ can be sustained provided the following relation is reached asymptotically: $(\\dot{\\overline{b}})^2 = - \\frac{1}{2}\\,\\dfrac{\\partial V(\\Phi )}{\\partial \\Phi } \\Big |_{\\Phi \\rightarrow \\Phi _0 \\simeq {\\rm const}}\\gtrsim 0\\,.$ A typical scenario which could satisfy (REF ), with $\\dot{b} \\simeq {\\rm constant}$ , hence fulfilling Eq.",
"(REF ) of interest to us here, would be, for instance, that of a `run away' pre big bang type [37] dilaton potential, in the range where the dilaton slowly approaches a constant value asymptotically and with a decaying trend $\\partial V(\\Phi )/\\partial \\Phi <0$ .",
"Interestingly, this situation characterises N=1 globally supersymmetric theories that can be embedded in a supergravity/superstring framework [38].It must be noted that the simplest supercritical strings exponential dilaton potential [24], does not satisfy (REF ), but more complicated brane models could [39], [40].",
"It is then straightforward to see that, under the above conditions, the higher order terms in (REF ) have either vanishing contributions to the equations of motion, or are subleading, for sufficiently small $\\dot{b} \\ll M^2_{\\rm Pl}$ , as required in the approach of [2], for our cosmic background solutions in a FLRW space time.",
"The upshot is that $\\Phi \\simeq $ const.",
"appears to be a viable assumption within our framework both as a self-consistent field theory of graviton and KR degrees of freedom or within a generic string inspired approach."
]
] | 2001.03465 |
[
[
"Perfect Quantum State Transfer in Glauber-Fock Cavity Array"
],
[
"Abstract We study transfer of single photon in an one-dimensional finite Glauber-Fock cavity array whose coupling strengths satisfy a square root law.",
"The evolved state in the array can be mapped to an upper truncated coherent state if the cavities are resonant.",
"Appropriate choices of resonance frequencies provide perfect transfer of a photon between any two cavities in the array.",
"Perfect transfer of the photon allows perfect quantum state transfer between the cavities.",
"Our findings may help in realizing quantum communication and information processing in photonic lattices."
],
[
"Introduction",
"Quantum state transfer is an essential task for realizing quantum communication [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].",
"This protocol requires transfer of an unknown qubit state between two nodes [11], [12], [13].",
"Many physical systems such as spin chain [1], [7], [6], Josephson-junction array [14], quantum dots [15], [16], coupled cavities [3], [8], [17], [18], photonic lattices [19], [20], etc have been investigated to realize quantum state transfer.",
"Possibility in precise control of resonance frequencies and coupling strengths of the array, coupled cavities provides a suitable physical system to realize controlled quantum state transfer [3], [8], [17], [21], [22], [23].",
"Cavities have been used to realize many interesting phenomena such as entanglement generation [24], [25], [26], [27], quantum state preparation [28], [29], [30], [31], localization-delocalization [32], [33], photon blockade [34], [35], [36], [37], [38] etc by properly tuning the resonance frequencies or including Kerr nonlinearity.",
"In addition, coupling the cavities and suitably modifying the coupling strengths provide perfect transfer of photons between the cavities [3], [8], [39], [40].",
"Such an array of cavities that has attracted a lot of attention is Glauber-Fock cavity array, whose coupling strengths satisfy $J_{k}=J\\sqrt{k}$ .",
"This array is used for realizing classical analogue of coherent state and displaced Fock state [41], dynamic localization and self imaging [42], transport and revival of squeezed light [43], Bloch-like revival [44], etc.",
"In this article, we investigate perfect quantum state transfer between any two cavities in a Glauber-Fock cavity array.",
"A single photon in the first cavity evolves to a state which can be mapped to a truncated coherent state.",
"Suitable choices of resonance frequencies of the cavities allow perfect transfer of single photon between the cavities.",
"This shows the possibility of transferring a qubit state of the form $\\alpha \\mathinner {|{0}\\rangle }+\\beta \\mathinner {|{1}\\rangle }$ , where $\\mathinner {|{0}\\rangle }$ is the vacuum state and $\\mathinner {|{1}\\rangle }$ is the single photon state.",
"This article is organized as follows: In Sec.",
", we briefly overview Glauber-Fock cavity array.",
"We show the possibility of perfect transfer of a photon between the cavities in the array in Sec.",
".",
"In Sec.",
", we establish perfect transfer of a qubit state.",
"Effect of dissipation on quantum state transfer is presented in Sec.",
".",
"Finally we summarized our results in Sec.",
"."
],
[
"Glauber-fock cavity array",
"Consider a system of $N$ cavities described by the Hamiltonian $H=\\sum _{k=1}^{N}\\omega _k a_k^\\dagger a_k +\\sum _{k=1}^{N-1} J_{k}(a_k^\\dagger a_{k+1}+a_k a_{k+1}^\\dagger ),$ where $\\omega _k$ is the resonance frequency of the $k$ th cavity.",
"The operator $a_k (a_k^\\dagger )$ is the annihilation (creation) operator for the $k$ th cavity.",
"The strength of coupling between $k$ th cavity and $(k+1)$ th cavity is $J_{k}$ , which satisfies the square root law, i.e, $J_{k}=J\\sqrt{k}$ .",
"These coupling strengths can be achieved by adjusting the separation between the cavities with $J_{k}\\sim e^{-\\eta d_k}$ [41].",
"Here $d_k$ is the distance between $k$ th cavity and $(k+1)$ th cavity, and $\\eta $ is a constant.",
"A key feature of $H$ given in Eqn.",
"REF is that it conserves total number of quanta ,i.e., $[H,\\sum _k a_k^\\dagger a_k]=0$ .",
"This signifies that there are invariant subspaces for the unitary dynamics generated by the Hamiltonian.",
"Consider the cavities are resonant, then the Hamiltonian given in Eqn.",
"REF in the interaction picture is $H_{int}=\\sum _{k=1}^{N-1} J_{k}(a_k^\\dagger a_{k+1}+a_k a_{k+1}^\\dagger ).$ We restrict the total number of quanta to be one.",
"Then, the set $\\mathinner {|{1_1,0_2,0_3,0_4...0_N}\\rangle },\\mathinner {|{0_1,1_2,0_3,0_4,...0_N}\\rangle }$ , ............, $\\mathinner {|{0_1,0_2,0_3,...1_N}\\rangle }$ forms a basis, where $n_m$ inside the ket represents $n$ photons in $m$ th cavity.",
"For simplicity, the state $\\mathinner {|{0_1,0_2,0_3,..,1_k,...0_N}\\rangle }$ which represents single photon in $k$ th cavity and other cavities are in vacuum, is denoted as $\\mathinner {|{k}\\rangle }\\rangle $ .",
"Now, the interaction Hamiltonian given in Eqn.",
"REF can be written in a matrix form using the aforementioned basis set.",
"The matrix form of $H_{int}$ is $H_{int}=\\left[\\begin{array}{cccccc}0 & J & 0 & 0 & \\dots \\\\J & 0 & \\sqrt{2}J & 0 & \\dots \\\\0 & \\sqrt{2}J & 0 & \\sqrt{3}J & \\dots \\\\\\vdots & \\vdots &\\vdots &\\ddots & \\sqrt{N-1}J\\\\0 & 0 & \\dots & \\sqrt{N-1}J & 0\\\\\\end{array}\\right],$ which can be written in the form $H_{int}=J(A+A^\\dagger ),$ where $A=\\left[\\begin{array}{cccccc}0 & 1 & 0 & 0 & \\dots \\\\0 & 0 & \\sqrt{2} & 0 & \\dots \\\\0 & 0 & 0 & \\sqrt{3} & \\dots \\\\\\vdots & \\vdots &\\vdots &\\ddots & \\sqrt{N-1}\\\\0 & 0 & \\dots & 0 &0\\\\\\end{array}\\right].$ It is to be noted that $A\\mathinner {|{k}\\rangle }\\rangle =\\sqrt{k}\\mathinner {|{k-1}\\rangle }\\rangle $ , which is analogous to annihilation operator.",
"Hence, the interaction Hamiltonian given in Eqn.",
"REF can be mapped to a driven oscillator with the driving strength $J$ .",
"Now, a single photon in the first cavity evolves to $\\mathinner {|{\\psi (t)}\\rangle }&=e^{-iH_{int}t}\\mathinner {|{1_1,0_2,0_3,0_4...}\\rangle },\\nonumber \\\\&=e^{-iH_{int}t}\\mathinner {|{1}\\rangle }\\rangle ,\\nonumber \\\\&=e^{-\\frac{1}{2}J^2t^2[A,A^\\dagger ]}e^{-iJt A^\\dagger }e^{-iJtA}\\mathinner {|{1}\\rangle }\\rangle ,$ where we have used the Baker-Hausdorf-Campbell theorem [45].",
"Using $[A,A^\\dagger ]=1$ and $A\\mathinner {|{1}\\rangle }\\rangle =0$ , the above equation can be written as $\\mathinner {|{\\psi (t)}\\rangle }&=e^{-\\frac{1}{2}J^2t^2}\\sum _{j} \\frac{(-iJtA^\\dagger )^j}{j!",
"}\\mathinner {|{1}\\rangle }\\rangle .$ Now, using the action of $A^\\dagger $ , i.e., $A^\\dagger \\mathinner {|{k}\\rangle }\\rangle =\\sqrt{k+1}\\mathinner {|{k+1}\\rangle }\\rangle $ , the evolved state becomes $\\mathinner {|{\\psi (t)}\\rangle }&=N_c\\sum _{k=1}^N \\frac{(-iJt)^{(k-1)}}{\\sqrt{(k-1)!",
"}}\\mathinner {|{k}\\rangle }\\rangle ,$ where $N_c=\\frac{e^{-(Jt)^2/2}}{\\sqrt{\\left[e^{-(Jt)^2}\\sum _{j=0}^{N-1}\\frac{(Jt)^{2j}}{j!",
"}\\right]}},$ is the normalization constant.",
"Using the definition of incomplete gamma function [46] $\\gamma (N,x)=(N-1)!\\left[1-e^{-x}\\sum _{j=0}^{N-1}\\frac{x^j}{j!",
"}\\right],$ normalization constant becomes $N_c=\\frac{e^{-(Jt)^2/2}}{\\sqrt{1-\\frac{\\gamma (N,(Jt)^2)}{(N-1)!",
"}}}.$ The evolved state given in Eqn.",
"REF is equivalent to an upper truncated coherent state [47], [48], [49].",
"In the limit of $N\\rightarrow \\infty $ , the state is analogous to a coherent state $\\mathinner {|{\\alpha }\\rangle }$ .",
"Probability of detecting the single photon in $m$ th cavity is $P_m=|\\langle \\langle m\\mathinner {|{\\psi (t)}\\rangle }|^2=\\left|N_c\\frac{(-iJt)^{(m-1)}}{\\sqrt{(m-1)!",
"}}\\right|^2.$ Fig.",
"REF shows the probability distribution $P_m$ at various values of $\\omega t$ .",
"This probability distribution is analogous to the photon number distributions in truncated coherent states with an appropriate amplitude.",
"In the context of perfect transfer, it can be infer from the figure that complete transfer of the photon to a specific cavity with unit probability is not possible.",
"Hence controlling the photon transfer is not achievable in resonant cavities for this choice of coupling strength.",
"Figure: Probability distribution P m P_m at (a)ωt=0,(b)ωt=30,(c)ωt=40(a)\\omega t=0,(b)\\omega t=30,(c)\\omega t=40 and (d)ωt=84(d) \\omega t=84.",
"We set J/ω=0.05,N=10J/\\omega =0.05, N=10."
],
[
"Perfect transfer of a photon",
"As seen in previous section, perfect transfer of a photon between any two cavities in the array is not possible if the cavities are resonant.",
"A feature that can be used for controlling the transfer is to suitably choosing the resonance frequencies.",
"Perfect transfer between any two cavities is already been achieved in an array by suitably modifying the resonance frequencies where the coupling strengths are symmetric about the centre [3].",
"A parabolic form of resonance frequencies are used for controlling the transfer [3].",
"In this article, we choose the resonance frequencies as $\\omega _k=C+\\left((k-1)-\\frac{(k-1)^2}{(m+n-2)}\\right),$ which are also parabolic (inverted).",
"This choice of resonance frequencies guarantees $\\omega _m=\\omega _n$ , i.e., $\\langle \\mathinner {\\langle {m}|}H\\mathinner {|{m}\\rangle }\\rangle =\\langle \\mathinner {\\langle {n}|}H\\mathinner {|{n}\\rangle }\\rangle ,$ for any $m$ and $n$ .",
"This is the basic requirement for transferring a photon from $m$ th cavity to $n$ th cavity.",
"These resonance frequencies are shown in Fig.",
"REF , where we choose $m=3$ and $n=7$ .",
"The frequencies that is given in Eqn.",
"REF can be realized in photonic crystal cavities where precisely controlling the resonance frequencies is possible by various techniques such as nanofluidic tuning [50], nanomechanical tuning [51], photochromic tuning [52], to mention a few.",
"Figure: Resonance frequencies ω k /C\\omega _k/C as a function of kk.",
"We choose m=3m=3 and n=7n=7 with N=10N=10.",
"Note that ω 3 =ω 7 \\omega _3=\\omega _7.Using degenerate perturbation theory, it can be seen that the states $\\mathinner {|{\\pm }\\rangle }\\rangle &=\\frac{1}{\\sqrt{2}}(\\mathinner {|{m}\\rangle }\\rangle \\pm \\mathinner {|{n}\\rangle }\\rangle )+\\mathcal {O}(J/\\Delta _{m,m+1})\\nonumber \\\\&+\\mathcal {O}(J/\\Delta _{m,m-1})+\\mathcal {O}(J/\\Delta _{n,n+1})+\\mathcal {O}(J/\\Delta _{n,n-1}),$ become the approximate eigenstates of $H$ given in Eqn.",
"REF .",
"Here, $\\Delta _{x,y}=\\omega _y-\\omega _x$ .",
"The corrections that is given in the above equation are small for smaller $J$ .",
"If the initial state is $\\mathinner {|{m}\\rangle }\\rangle $ , i.e., a single photon in the $m$ th cavity, then the state at later time is $\\mathinner {|{\\psi (t)}\\rangle }&=e^{-iHt}\\mathinner {|{m}\\rangle }\\rangle ,\\nonumber \\\\&\\approx \\frac{1}{\\sqrt{2}} e^{-iHt}[\\mathinner {|{+}\\rangle }\\rangle +\\mathinner {|{-}\\rangle }\\rangle ]+\\mathcal {O}(J/\\Delta _{m,m+1})\\nonumber \\\\&+\\mathcal {O}(J/\\Delta _{m,m-1})+\\mathcal {O}(J/\\Delta _{n,n+1})+\\mathcal {O}(J/\\Delta _{n,n-1}),\\nonumber \\\\&\\approx e^{-i\\lambda t}\\left[ \\cos \\theta t\\mathinner {|{m}\\rangle }\\rangle -i\\sin \\theta t\\mathinner {|{n}\\rangle }\\rangle \\right]+\\mathcal {O}(J/\\Delta _{m,m+1})\\nonumber \\\\&+\\mathcal {O}(J/\\Delta _{m,m-1})+\\mathcal {O}(J/\\Delta _{n,n+1})+\\mathcal {O}(J/\\Delta _{n,n-1}),$ with $\\theta =(\\lambda _+-\\lambda _-)/2$ and $\\lambda =(\\lambda _++\\lambda _-)/2$ .",
"Here $\\lambda _{\\pm }$ are the eigenvalues of $H$ correspond to the approximate eigenstates $\\mathinner {|{\\pm }\\rangle }\\rangle $ .",
"If $J$ is very small, the above equation can be written as $\\mathinner {|{\\psi (t)}\\rangle }\\approx e^{-i\\lambda t}\\left[ \\cos \\theta t\\mathinner {|{m}\\rangle }\\rangle -i\\sin \\theta t\\mathinner {|{n}\\rangle }\\rangle \\right].$ Note that the photon is exchanged periodically between $m$ th cavity and $n$ th cavity.",
"The probability of transferring a photon from $m$ th cavity to $n$ th cavity is $P_{mn}\\approx \\sin ^2\\theta t.$ At time $t=\\pi /(\\lambda _+-\\lambda _-)$ , photon is completely transferred from $m$ th cavity to $n$ th cavity.",
"This is the minimum time to transfer a photon from $m$ th cavity to $n$ th cavity.",
"Thus the condition given in Eqn.",
"REF ensures perfect transfer of a photon between any two cavities in the array.",
"Fig.",
"REF $(a)$ shows the probability of transferring a single photon from first cavity $(m=1)$ to fifth cavity $(n=5)$ and Fig.",
"REF $(b)$ shows the transfer probability from second cavity $(m=2)$ to fourth cavity $(n=4)$ .",
"Total number of cavities in the array is six.",
"It can be seen in figure that there is perfect transfer of a photon from $m$ th cavity to $n$ th cavity.",
"Figure: Probability of single photon transfer from mmth cavity to nnth cavity (P mn )(P_{mn}) as a function of ω 1 t\\omega _1t.",
"Here we set J/ω 1 =0.0013J/\\omega _1=0.0013.",
"All the system parameters are in the unit of ω 1 \\omega _1.",
"Resonance frequencies are satisfying the relation given in Eqn.",
".An interesting aspect of this transfer is that the other cavities in the array are not populated significantly during the evolution.",
"This comes from the fact that the states other than $\\mathinner {|{m}\\rangle }\\rangle $ and $\\mathinner {|{n}\\rangle }\\rangle $ do not contribute appreciably to $\\mathinner {|{\\psi (t)}\\rangle }$ given in Eqn.",
"REF ."
],
[
"Quantum state transfer",
"Perfect transfer of a photon between any two cavities in the array is possible if the resonance frequencies of the array satisfy the relation given in Eqn.",
"REF .",
"In this section, we establish perfect transfer of a quantum state of the form $\\alpha \\mathinner {|{0}\\rangle }+\\beta \\mathinner {|{1}\\rangle }$ , which is the superposition of vacuum state and a single photon state.",
"Consider the interaction term in $H$ to be of the form $H_{int}=\\sum _{k=1}^{N-1} J_{k+1}(e^{i\\eta } a_k^\\dagger a_{k+1}+ e^{-i\\eta } a_k a_{k+1}^\\dagger ),$ where the coupling strengths are assume to be complex.",
"Complex coupling strengths can be realized experimentally using wave-guide delay line [53].",
"Now, the approximate eigenvectors of $H$ are $\\mathinner {|{\\pm }\\rangle }\\rangle \\approx \\frac{1}{\\sqrt{2}}\\left[\\mathinner {|{m}\\rangle }\\rangle \\pm e^{i(n-m)\\eta }\\mathinner {|{n}\\rangle }\\rangle \\right].$ Consider the initial state of the cavity array to be $\\alpha \\mathinner {|{\\text{vac}}\\rangle }\\rangle +\\beta \\mathinner {|{m}\\rangle }\\rangle $ , which corresponds to the $m$ th cavity in the superposition $\\alpha \\mathinner {|{0}\\rangle }+\\beta \\mathinner {|{1}\\rangle }$ and the other cavities are in their respective vacuua.",
"The target state is $\\mathinner {|{\\Psi }\\rangle }=\\alpha \\mathinner {|{\\text{vac}}\\rangle }\\rangle +\\beta \\mathinner {|{n}\\rangle }\\rangle $ , which corresponds to the $n$ th cavity in the superposition $\\alpha \\mathinner {|{0}\\rangle }+\\beta \\mathinner {|{1}\\rangle }$ and the other cavities are in vacuum.",
"Now the initial state evolves as $\\mathinner {|{\\psi (t)}\\rangle }=\\alpha \\mathinner {|{\\text{vac}}\\rangle }\\rangle &+\\beta e^{-i\\lambda t}[\\cos \\theta t\\mathinner {|{m}\\rangle }\\rangle \\nonumber \\\\&-ie^{-i(n-m)\\eta }\\sin \\theta t \\mathinner {|{n}\\rangle }\\rangle ].$ Fidelity for the quantum state transfer from $m$ th cavity to $n$ th cavity is $F_{mn}=&\\left|\\langle \\Psi |\\psi (t)\\rangle \\right|^2,\\nonumber \\\\\\approx &\\left| |\\alpha |^2-i|\\beta |^2 e^{-i\\lambda t}e^{-i(n-m)\\eta }\\sin \\theta t \\right|^2,$ where $\\lambda =(\\lambda _++\\lambda _-)/2$ .",
"If $\\eta =\\frac{1}{(m-n)}(\\frac{\\pi }{2}-\\frac{\\lambda \\pi }{2\\theta })$ , the fidelity becomes nearly unity at $\\theta t=\\pi /2$ .",
"In other words, the state of the $n$ th cavity is in the superposition $\\alpha \\mathinner {|{0}\\rangle }+\\beta \\mathinner {|{1}\\rangle }$ and rest of the cavities are in vacuum.",
"Figure: Fidelity of quantum state transfer from 1st cavity to 4th cavity (F 14 )(F_{14}) where (a)α=1/2,β=3/2(a)\\alpha =1/2, \\beta =\\sqrt{3}/2, (b)α=1/3,β=i2/3(b)\\alpha =1/\\sqrt{3}, \\beta =i\\sqrt{2/3}, (c)α=1/2,β=1/2(c)\\alpha =1/\\sqrt{2}, \\beta =1/\\sqrt{2} and (d)α=3/2,β=1/2(d)\\alpha =\\sqrt{3}/2, \\beta =1/2.",
"Here we set J/ω 1 =0.0013J/\\omega _1=0.0013.",
"Resonance frequencies are satisfying the relation given in Eqn.",
".",
"All the system parameters are in the unit of ω 1 \\omega _1.Fig.",
"REF shows the fidelity $F_{14}$ , i.e., fidelity of quantum state transfer from first cavity to fourth cavity as a function of $\\omega _1 t$ for various choices of $\\alpha $ and $\\beta $ .",
"It is to be noted that the fidelity is unity for all choices of $\\alpha $ and $\\beta $ .",
"Hence, the condition given in Eqn.",
"REF ensures perfect transfer of a quantum state between any two cavities in the array."
],
[
"Effect of dissipation",
"An ideal system is characterized by complete isolation of the system from environment.",
"However, dissipation is unavoidable.",
"Effect of dissipation on quantum state transfer fidelity is studied by analyzing the master equation.",
"Master equation that includes dissipation in the cavity array is [54] $\\frac{\\partial \\rho }{\\partial t}=-i[H,\\rho ]+\\frac{\\gamma }{2}\\sum _{k=1}^N \\mathcal {L}(a_i)\\rho ,$ where $\\mathcal {L}(o)\\rho =(2o\\rho o^\\dagger -o^\\dagger o \\rho -\\rho o^\\dagger o),\\\\$ is the Lindblad superoperator [55].",
"For a single cavity, master equation can be solved analytically by super-operator method [56], [57].",
"However, for coupled cavities, the number of equations of motion depends on the dimension of density matrix and hence, an analytical solution for master equation is difficult to derive.",
"Here, we solve the master equation numerically.",
"The equations of motion for the density matrix elements $\\rho _{m,n}$ are $\\frac{\\partial \\rho _{m,n}}{\\partial t}&=\\frac{\\partial }{\\partial t}\\langle \\mathinner {\\langle {m}|}\\rho \\mathinner {|{n}\\rangle }\\rangle , \\nonumber \\\\&=-i[(\\omega _m-\\omega _n)\\rho _{m,n}+J_m(\\rho _{m+1,n}+\\rho _{m-1,n})\\nonumber \\\\&+J_n(\\rho _{m,n+1}+\\rho _{m,n-1})]+\\gamma (2\\rho _{m+1,n+1}-\\rho _{m,n}).$ By considering the initial state $\\mathinner {|{\\psi _{in}}\\rangle }=\\alpha \\mathinner {|{\\text{vac}}\\rangle }\\rangle +\\beta \\mathinner {|{m}\\rangle }\\rangle $ and numerically solving the above equations of motion one can obtain the density matrix at various instances of time.",
"Figure: Average fidelities of quantum state transfer F 13 F_{13} (continuous) and F 25 F_{25} (dashed) as a function of γ/J\\gamma /J.",
"All the resonance frequencies satisfy the relation given in Eqn.",
".Fidelity for the quantum state transfer in the presence of dissipation can be calculated as $F_{mn}=\\left[\\text{Tr}\\left(\\sqrt{\\sqrt{\\sigma }\\rho \\sqrt{\\sigma }}\\right)\\right]^2.$ Here $\\sigma =\\mathinner {|{\\Psi }\\rangle }\\mathinner {\\langle {\\Psi }|}$ , where $\\mathinner {|{\\Psi }\\rangle }=\\alpha \\mathinner {|{\\text{vac}}\\rangle }\\rangle +\\beta \\mathinner {|{n}\\rangle }\\rangle $ .",
"Fig.",
"REF shows average of fidelities of quantum state transfer between the cavities as a function of $\\gamma /J$ .",
"Here the average is taken over ensemble of initial states, i.e., $\\alpha $ and $\\beta $ are chosen randomly.",
"In the absence of dissipation, the fidelities are nearly unity.",
"Average fidelities decrease due to photon loss from the array."
],
[
"Summary",
"Quantum state transfer is essential for future quantum communication.",
"We studied single photon transfer and quantum state transfer in Glauber-Fock cavity array whose coupling strengths satisfy square root law.",
"If the cavities are resonant, the evolved state in the array can be mapped to an upper truncated coherent state.",
"By suitably modifying resonance frequencies of the cavities, we achieve perfect transfer of single photon between any two cavities in the array.",
"Transfer of a photon allows us to transfer of a qubit state between the cavities.",
"High fidelity of quantum state transfer is achieved even in the presence of dissipation.",
"These results are useful in the context of quantum information processing and communication.",
"The ideas presented here are also applicable for spin chains if the coupling strengths are suitably modified.",
"NM acknowledges Indian Institute of Technology Kanpur for postdoctoral fellowship."
],
[
"Conflict of interest",
"The authors declare that they have no conflict of interest."
]
] | 2001.03467 |
[
[
"A short proof of the blow-up lemma for approximate decompositions"
],
[
"Abstract Kim, K\\\"uhn, Osthus and Tyomkyn (Trans.",
"Amer.",
"Math.",
"Soc.",
"371 (2019), 4655--4742) greatly extended the well-known blow-up lemma of Koml\\'os, S\\'ark\\\"ozy and Szemer\\'edi by proving a `blow-up lemma for approximate decompositions' which states that multipartite quasirandom graphs can be almost decomposed into any collection of bounded degree graphs with the same multipartite structure and slightly fewer edges.",
"This result has already been used by Joos, Kim, K\\\"uhn and Osthus to prove the tree packing conjecture due to Gy\\'arf\\'as and Lehel from 1976 and Ringel's conjecture from 1963 for bounded degree trees as well as implicitly in the recent resolution of the Oberwolfach problem (asked by Ringel in 1967) by Glock, Joos, Kim, K\\\"uhn and Osthus.",
"Here we present a new and significantly shorter proof of the blow-up lemma for approximate decompositions.",
"In fact, we prove a more general theorem that yields packings with stronger quasirandom properties so that it can be combined with Keevash's results on designs to obtain results of the following form.",
"For all $\\varepsilon>0$, $r\\in \\mathbb{N}$ and all large $n$ (such that $r$ divides $n-1$), there is a decomposition of $K_n$ into any collection of $r$-regular graphs $H_1,\\ldots,H_{(n-1)/r}$ on $n$ vertices provided that $H_1,\\ldots,H_{\\varepsilon n}$ contain each at least $\\varepsilon n$ vertices in components of size at most $\\varepsilon^{-1}$."
],
[
"Introduction",
"The theme of decomposing `large' objects into `smaller' objects or finding a maximal number of specified `small' objects in a `larger' object is among the key topics in mathematics.",
"In discrete mathematics, it appears in Euler's question from 1782 for which $n$ there exist pairs of orthogonal Latin squaresThis is equivalent to a $K_4$ -decomposition of the 4-partite graph $K_{n,n,n,n}$ .",
"of order $n$ , in Steiner's questions for Steiner systems from the 1850s which cumulated in the `existence of designs' question, in Walecki's theorem on decompositions of complete graphs into edge-disjoint Hamilton cycles from the 1890s, and in Kirkman's famous `school girl problem'.",
"These questions and results set off an entire branch of combinatorics and design theory.",
"Several decades later in the 1970s, Wilson [27], [28], [29] famously proved that (the edge set of) the complete graph on $n$ vertices can be decomposed into any fixed graph provided necessary divisibility conditions are satisfied and $n$ is large, thereby solving the `existence of designs' question for graphs.",
"In 2014, Keevash verified the `existence of designs' for hypergraphs [14].",
"This has been reproved and generalised by Glock, Kühn, Lo and Osthus in [11], [12].",
"Keevash extended his results to a more general framework in [15].",
"In contrast to these questions and results where we aim to decompose a large graph into graphs of fixed size, one can also ask for decompositions into larger pieces, for example into graphs with the same number of vertices as the host graph.",
"A prime example is the Oberwolfach problem where Ringel asked in 1967 whether one can decompose (the edge set of) $K_{2n+1}$ into $n$ copies of any 2-regular graph on $2n+1$ vertices.",
"This problem received considerable attention and Glock, Joos, Kim, Kühn and Osthus [10] solved it for large $n$ .",
"Possibly equally well-known is Ringel's conjecture from 1963 stating that $K_{2n+1}$ can be decomposed into any tree with $n$ edges, as well as the tree packing conjecture due to Gyárfás and Lehel from 1976 stating that $K_n$ can be decomposed into any collection of trees $T_1,\\ldots ,T_n$ , where $T_i$ has $i$ vertices.",
"Ringel's conjecture has been solved by Montgomery, Pokrovskiy and Sudakov [24], [25] and both conjectures have been solved for bounded degree trees by Joos, Kim, Kühn and Osthus [13]; Allen, Böttcher, Clemens and Taraz [1] have solved these conjectures for trees with many leaves and maximum degree $o(n/\\log n)$ (in fact, they proved a more general result on degenerate graphs) – all mentioned results apply only when $n$ is sufficiently large.",
"We refer the reader to [3], [8], [18], [23] for earlier results regarding these conjectures and to [2], [4], [9], [21] for further developments in the field.",
"On a very high level, numerous decomposition results combine approximate decomposition results with certain absorbing techniques.",
"This includes [1], [11], [12], [13], [14], [15].",
"For many questions in extremal combinatorics, the blow-up lemma due to Komlós, Sárközy and Szemerédi [19] in combination with Szemerédi's regularity lemma has demonstrated its power and usefulness.",
"Having this in mind and in need of a powerful approximate decomposition result, Kim, Kühn, Osthus and Tyomkyn [18] proved a far-reaching generalisation of the blow-up lemma, a `blow-up lemma for approximate decompositions', which can also be combined with the regularity lemma to obtain almost decompositions of graphs into bounded degree graphs.",
"The blow-up lemma for approximate decompositions has already exhibited its versatility.",
"It has been applied in [13], [16], [20] and in [4] for a `bandwidth theorem for approximate decompositions', which in turn is one of the key ingredients for the resolution of the Oberwolfach problem in [10].",
"However, its very complex and long proof is an obstacle for further generalisations.",
"One main aim of this paper is to overcome this by presenting a new and significantly shorter proof.",
"Our approach makes it possible to include some more features including an easier handling of exceptional vertices, which results in a substantially easier applicability of the theorem, and the approximate decompositions share stronger quasirandom properties.",
"To be more precise, the first yields shorter proofs of the main results in [4] and [13] as certain technically involved preprocessing steps are no longer needed; the latter permits to combine our main result with Keevash's recent results on designs [15].",
"We dedicate a section at the end of the paper to demonstrate this and obtain new results on decomposing quasirandom graphs into regular spanning graphs."
],
[
"The blow-up lemma for approximate decompositions",
"In this section, we first introduce some terminology and then state the blow-up lemma for approximate decompositions.",
"We say that a collection/multiset of graphs $\\mathcal {H}=\\lbrace H_1,\\ldots , H_s\\rbrace $ packs into a graph $G$ if there is a function $\\phi : \\bigcup _{H\\in \\mathcal {H}}V(H)\\rightarrow V(G)$ such that $\\phi |_{V(H)}$ is injective and $\\phi $ injectively maps edges onto edges.",
"In such a case, we call $\\phi $ a packing of $\\mathcal {H}$ into $G$.",
"Our general aim is to pack a collection $\\mathcal {H}$ of multipartite graphs in a host graph $G$ having the same multipartite structure which is captured by a so-called `reduced graph' $R$ .",
"To this end, let $(H,G,R,\\mathcal {X},\\mathcal {V})$ be a blow-up instance if $H,G,R$ are graphs where $V(R)=[r]$ for some $r\\ge 2$ ; $\\mathcal {X}=(X_i)_{i\\in [r]}$ is a vertex partition of $H$ into independent sets, $\\mathcal {V}=(V_i)_{i\\in [r]}$ is a vertex partition of $G$ such that $|V_i|=|X_i|$ for all $i\\in [r]$ ; $H[X_{i},X_j]$ is empty whenever $ij\\in \\binom{[r]}{2}\\setminus E(R)$ .",
"We also refer to ${B}=(\\mathcal {H},G,R,\\mathcal {X},\\mathcal {V})$ as a blow-up instance if $\\mathcal {H}$ is a collection of graphs and $\\mathcal {X}$ is a collection of vertex partitions $(X_i^H)_{i\\in [r],H\\in \\mathcal {H}}$ so that $(H,G,R,(X_i^H)_{i\\in [r]},\\mathcal {V})$ is a blow-up instance for every $H\\in \\mathcal {H}$ .",
"The quasirandom notion mostly used in this paper coincides with the notion used in Szemeredi's regularity lemma.",
"For a bipartite graph $G$ with vertex partition $(V_1,V_2)$ , we define the density of $W_1,W_2$ with $W_i\\subseteq V_i$ by $d_G(W_1,W_2):=e_G(W_1,W_2)/|V_1||V_2|$ .",
"We say $G$ is $({\\varepsilon },d)$ -regular if $d_G(W_1,W_2)=d\\pm {\\varepsilon }$ for all $W_i\\subseteq V_i$ with $|W_i|\\ge {\\varepsilon }|V_i|$ , and $G$ is $({\\varepsilon },d)$ -super-regular if in addition $|N_G(v)\\cap V_{3-i}|= (d\\pm {\\varepsilon })|V_{3-i}|$ for each $i\\in [2]$ and $v\\in V_i$ .",
"The blow-up instance ${B}$ is $({\\varepsilon },d)$ -super-regular if $G[V_i,V_j]$ is $({\\varepsilon },d)$ -super-regular for all $ij\\in E(R)$ , and ${B}$ is $\\Delta $ -bounded if $\\Delta (R),\\Delta (H) \\le \\Delta $ for each $H\\in \\mathcal {H}$ .",
"Now we are ready to state the blow-up lemma for approximate decompositions.",
"Theorem (Kim, Kühn, Osthus, Tyomkyn [18]) For all $\\alpha \\in (0,1]$ and $r\\ge 2$ , there exist ${\\varepsilon }={\\varepsilon }(\\alpha ) > 0$ and $n_0 = n_0(\\alpha , r)$ such that the following holds for all $n \\ge n_0$ and $d\\ge \\alpha $ .",
"Suppose $(\\mathcal {H},G,R,\\mathcal {X},\\mathcal {V})$ is an $({\\varepsilon },d)$ -super-regular and $\\alpha ^{-1}$ -bounded blow-up instance such that $|V_i|=n$ for all $i\\in [r]$ , $|\\mathcal {H}|\\le \\alpha ^{-1}n$ , and $\\sum _{H\\in \\mathcal {H}} e_H(X_{i}^H,X_j^H) \\le (1-\\alpha )dn^2$ for all $ij \\in E(R)$ .",
"Then there is a packing $\\phi $ of $\\mathcal {H}$ into $G$ such that $\\phi (X_i^H)= V_i$ for all $i\\in [r]$ and $H\\in \\mathcal {H}$ .",
"We remark that there are more general versions of Theorem REF in [18], but omit the more technical statements here.",
"Instead we state our main result and the interested reader can easily check that it generalisesObserve that we do not allow different densities between the cluster pairs in $G$ .",
"However, this technical complication could very easily be implemented by adding at numerous places extra indices.",
"As this feature has never been used so far in applications, we omitted it for the sake of a clearer presentation.",
"the more technical versions in [18]."
],
[
"Main result",
"Most blow-up lemmas exhibit their power if they are applied in conjunction with Szemerédi's regularity lemma.",
"This, however, comes with the expense of a small set of vertices over which we have no control.",
"Consequently, in such a setting, when embedding a graph $H$ into $G$ , it is often the case that some vertices of $H$ are already embedded and the blow-up lemma is applied only to some nice part of $G$ .",
"To deal with such scenarios we consider extended blow-up instances.",
"We say $(H,G,R,\\mathcal {X},\\mathcal {V},\\phi _0)$ is an extended blow-up instance if $H,G,R$ are graphs where $V(R)=[r]$ for some $r\\ge 2$ ; $\\mathcal {X}=(X_i)_{i\\in [r]_0}$ is a vertex partition of $H$ into independent sets, $\\mathcal {V}=(V_i)_{i\\in [r]_0}$ is a vertex partition of $G$ such that $|V_i|=|X_i|$ for all $i\\in [r]_0$ ; $H[X_{i},X_j]$ is empty whenever $ij\\in \\binom{[r]}{2}\\setminus E(R)$ ; $\\phi _0$ is an injective embedding of $X_0$ into $V_0$ .",
"This definition also extends as above to the case when $H$ is replaced by a collection of graphs $\\mathcal {H}$ in the obvious way as before.",
"An extended blow-up instance is $({\\varepsilon },d)$ -super-regular if $G[V_i,V_j]$ is $({\\varepsilon },d)$ -super-regular for all $ij\\in E(R)$ .",
"Let ${B}=(\\mathcal {H},G,R,\\mathcal {X},\\mathcal {V},\\phi _0)$ be an extended blow-up instance.",
"We say ${B}$ is $({\\varepsilon },\\alpha )$ -linked if at most ${\\varepsilon }|X^H_i|$ vertices in $X^H_i$ have a neighbour in $X_0^H$ for all $i\\in [r]$ , $H\\in \\mathcal {H}$ ; $|V_i\\cap \\bigcap _{x_0\\in X_0^H\\cap N_H(x)} N_G(\\phi _0(x_0))| \\ge \\alpha |V_i|$ for all $x\\in X_i^H, i\\in [r], H\\in \\mathcal {H}$ ; $|\\phi _0^{-1}(v)|\\le {\\varepsilon }|\\mathcal {H}|$ for all $v\\in V_0$ ; $\\sum _{H\\in \\mathcal {H}}|N_H(x_0^H)\\cap N_H(x_0^{\\prime H}) \\cap X_i^H|\\le {\\varepsilon }|V_i|^{1/2}$ for all $i\\in [r]$ and distinct $v_0, v_0^{\\prime } \\in V_0$ where $x_0^H=\\phi _0^{-1}(v_0)\\cap X_0^H$ and $x_0^{\\prime H}=\\phi _0^{-1}(v_0^{\\prime })\\cap X_0^H$ for $H\\in \\mathcal {H}$ .",
"One feature of our result is that one can replace `blow-up instance' in Theorem REF by `extended blow-up instance that is $({\\varepsilon },\\alpha )$ -linked'.",
"We remark that the above conditions are easily met in applications known to us and are similar to conditions found elsewhere for this purpose.",
"Next, we define two types of structures for ${B}$ and our main result yields a packing that behaves as we would expect it from an idealised typical random packing with respect to these structures.",
"We say $(W,Y_1,\\ldots ,Y_k)$ is an $\\ell $ -set tester for ${B}$ if $k\\le \\ell $ and there exist $i\\in [r]$ and distinct $H_1,\\ldots ,H_k\\in \\mathcal {H}$ such that $W\\subseteq V_i$ and $Y_j\\subseteq X_i^{H_j}$ for all $j\\in [k]$ .",
"We say $(v,\\omega )$ is an $\\ell $ -vertex tester for ${B}$ if $v\\in V_i$ and $\\omega :\\bigcup _{H\\in \\mathcal {H}}X_i^H \\rightarrow [0,\\ell ]$ for some $i\\in [r]$ .",
"For a weight function $\\omega $ on a finite set $X$ , we define $\\omega (X^{\\prime }):=\\sum _{x\\in X^{\\prime }}\\omega (x)$ for any $X^{\\prime }\\subseteq X$ .",
"The following theorem is our main result.",
"Theorem For all $\\alpha \\in (0,1]$ and $r\\ge 2$ , there exist ${\\varepsilon }={\\varepsilon }(\\alpha ) > 0$ and $n_0 = n_0(\\alpha , r)$ such that the following holds for all $n \\ge n_0$ and $d\\ge \\alpha $ .",
"Suppose $(\\mathcal {H},G,R,\\mathcal {X},\\mathcal {V},\\phi _0)$ is an $({\\varepsilon },d)$ -super-regular, $\\alpha ^{-1}$ -bounded and $({\\varepsilon },\\alpha )$ -linked extended blow-up instance, $|V_i|=(1\\pm {\\varepsilon })n$ for all $i\\in [r]$ , $|\\mathcal {H}|\\le \\alpha ^{-1}n$ , and $\\sum _{H\\in \\mathcal {H}} e_H(X_{i}^H,X_j^H) \\le (1-\\alpha )dn^2$ for all $ij \\in E(R)$ .",
"Suppose $\\mathcal {W}_{set},\\mathcal {W}_{ver}$ are sets of $\\alpha ^{-1}$ -set testers and $\\alpha ^{-1}$ -vertex testers of size at most $n^{\\log n}$ , respectively.",
"Then there is a packing $\\phi $ of $\\mathcal {H}$ into $G$ which extends $\\phi _0$ such that $\\phi (X_i^H)= V_i$ for all $i\\in [r]_0$ and $H\\in \\mathcal {H}$ ; $|W\\cap \\bigcap _{j\\in [\\ell ]}\\phi (Y_j)|= |W||Y_1|\\cdots |Y_\\ell |/n^\\ell \\pm \\alpha n$ for all $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}$ ; $\\omega (\\bigcup _{H\\in \\mathcal {H}}X_i^H\\cap \\phi ^{-1}(v))=\\omega (\\bigcup _{H\\in \\mathcal {H}}X_i^H)/n\\pm \\alpha n$ for all $(v,\\omega )\\in \\mathcal {W}_{ver}$ ."
],
[
"Applications",
"The multipartite framework can be used to obtain results for the non-partite setting.",
"The next theorem applies to graphs $G$ that are $({\\varepsilon },d)$ -quasirandom; that is, if $n$ is the order of $G$ , then $|N_G(u)|=(d\\pm {\\varepsilon })n$ and $|N_G(u)\\cap N_G(v)|=(d^2\\pm {\\varepsilon })n$ for all distinct $u,v\\in V(G)$ .",
"Given $G$ and a collection of graphs $\\mathcal {H}$ on at most $n$ vertices, we say $(W,Y_1,\\ldots ,Y_k)$ is an $\\ell $ -set tester if $k\\le \\ell $ and there exist distinct $H_1,\\ldots ,H_k\\in \\mathcal {H}$ such that $W\\subseteq V(G)$ and $Y_i\\subseteq V(H_i)$ for all $i\\in [k]$ .",
"We say $(v,\\omega )$ is an $\\ell $ -vertex tester if $v\\in V(G)$ and $\\omega :\\bigcup _{H\\in \\mathcal {H}} V(H) \\rightarrow [0,\\ell ]$ .",
"Theorem For all $\\alpha >0$ , there exist ${\\varepsilon }>0$ and $n_0\\in \\mathbb {N}$ such that the following holds for all $n\\ge n_0$ and $d\\ge \\alpha $ .",
"Suppose $G$ is an $({\\varepsilon },d)$ -quasirandom graph on $n$ vertices and $\\mathcal {H}$ is a collection of graphs on at most $n$ vertices with $|\\mathcal {H}|\\le \\alpha ^{-1}n$ and $\\sum _{H\\in \\mathcal {H}}e(H)\\le (1-\\alpha )e(G)$ as well as $\\Delta (H)\\le \\alpha ^{-1}$ for all $H\\in \\mathcal {H}$ .",
"Suppose $\\mathcal {W}_{set},\\mathcal {W}_{ver}$ are sets of $\\alpha ^{-1}$ -set testers and $\\alpha ^{-1}$ -vertex testers of size at most $n^{\\log n}$ , respectively.",
"Then there is a packing $\\phi $ of $\\mathcal {H}$ into $G$ such that $|W\\cap \\bigcap _{i\\in [\\ell ]}\\phi (Y_i)|= |W||Y_1|\\cdots |Y_\\ell |/n^\\ell \\pm \\alpha n$ for all $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}$ ; $\\omega (\\bigcup _{H\\in \\mathcal {H}} V(H))\\cap \\phi ^{-1}(v))=\\omega (\\bigcup _{H\\in \\mathcal {H}} V(H))/n\\pm \\alpha n$ for all $(v,\\omega )\\in \\mathcal {W}_{ver}$ .",
"In many scenarios when one applies approximate decomposition results, as for example Theorem REF , it is important that the graph $G-\\phi (\\mathcal {H})$ has `small' maximum degree.",
"Here, this can be easily achieved by utilising vertex testers $(v,\\omega )$ where $\\omega $ assigns to all $x\\in \\bigcup _{H\\in \\mathcal {H}}V(H)$ the degree of $x$ .",
"We remark that set and vertex testers in our main result are very flexible and capture many desirable properties.",
"For example, Theorem REF implies the approximate decomposition result in [1] when restricted to graphs of bounded maximum degree.",
"In this paper, we give one example how to apply Theorem REF .",
"By exploiting set testers, we can combine it with Keevash's results on hypergraph decompositions to decompose pseudorandom graphs into regular spanning graphs as long as only a few graphs contain a few vertices in components of bounded size.",
"This is stronger as some results in [10] where a few graphs with almost all vertices in components of bounded size are required.The results in [10] consider only 2-regular graphs.",
"However, their proof for the part where they consider collections of graphs $\\mathcal {H}$ that contain a few graphs with almost all vertices in components of bounded size carries over verbatim to $r$ -regular graphs for any $r$ if $n$ is large in terms of $r$ .",
"For this result, we need a stronger notation of pseudorandomness as used by Keevash in [15].",
"We say a graph $G$ on $n$ vertices is $({\\varepsilon },s,d)$ -typical if $|\\bigcap _{u\\in U}N_G(u)|=(1\\pm {\\varepsilon })d^{|U|}n$ for all sets $U\\subseteq V(G)$ with $|U|\\le s$ .",
"Theorem For all $\\alpha >0$ , there exist ${\\varepsilon }>0$ and $s,n_0\\in \\mathbb {N}$ such that the following holds for all $n\\ge n_0$ and $d\\ge \\alpha $ .",
"Suppose $G$ is a regular $({\\varepsilon },s,d)$ -typical graph on $n$ vertices and $\\mathcal {H}$ is a collection of regular graphs on $n$ vertices with $\\sum _{H\\in \\mathcal {H}}e(H)=e(G)$ as well as $\\Delta (H)\\le \\alpha ^{-1}$ for all $H\\in \\mathcal {H}$ .",
"Suppose there are at least $\\alpha n$ graphs $H\\in \\mathcal {H}$ such that at least $\\alpha n$ vertices in $H$ belong to components of order at most $\\alpha ^{-1}$ .",
"Then there is a decomposition of the edge set of $G$ into $\\mathcal {H}$ .",
"Theorem REF makes progress on a conjecture by Glock, Joos, Kim, Kühn and Osthus who conjecture in [10] that $K_n$ can be decomposed into any collection $\\mathcal {H}$ of regular bounded degree graphs with $\\sum _{H\\in \\mathcal {H}}e(H)=\\binom{n}{2}$ ."
],
[
"Proof sketch",
"Before we explain our approach, we briefly sketch the approach of Kim, Kühn, Osthus and Tyomkyn in [18].",
"Their first step is to stack several graphs $H\\in \\mathcal {H}$ together to a new graph $\\widetilde{H}$ such that $\\widetilde{H}[X_i^{\\widetilde{H}},X_j^{\\widetilde{H}}]$ is essentially regular for all $ij\\in E(R)$ .In fact, their main theorem only applies to collections of graphs that are essentially regular and this stacking had to be performed again in [4] and [13] which made the application in both cases technically involved.",
"Let $\\widetilde{\\mathcal {H}}$ be the collection of these graphs $\\widetilde{H}$ .",
"They prove that such graphs $\\widetilde{H}$ can be embedded into $G$ by a probabilistic algorithm in a very uniform way.",
"For some $\\gamma \\ll \\alpha $ , they apply this algorithm to $\\gamma n$ graphs in $\\widetilde{\\mathcal {H}}$ in turn.",
"Observe that this may cause edge overlaps in $G$ .",
"Nevertheless, after embedding $\\gamma n$ graphs, they remove all `used' edges from $G$ and repeat.",
"At the end, they eliminate all edge overlaps by unembedding several vertices and complete the packing by utilising a thin edge slice put aside at the beginning.",
"Our approach is somewhat perpendicular to their approach.",
"We proceed cluster by cluster and find a function $\\phi _i$ which maps almost all vertices in $\\bigcup _{H\\in \\mathcal {H}}X_i^H$ into $V_i$ and which is consistent with our partial packing so far.",
"Our `Approximate Packing Lemma', stated in Section , performs one such step using an auxiliary hypergraph where we aim to find a large matching which is pseudorandom with respect to certain weight functions.",
"At the end, we complete the packing by also using a thin edge slice similar to [18].",
"At the beginning, we partition the clusters of our blow-up instance into many smaller clusters with the only purpose to ensure that $H[X_i^H,X_j^H]$ is a matching (see Section REF ).",
"This preprocessing is comparably simple and first used in [26].",
"Both the approach in [18] and ours draw on ideas from an alternative proof of the blow-up lemma by Rödl and Ruciński [26].",
"In spirit, our approach is again closer to the procedure in [26] as they also embed the clusters of $H$ in turn.",
"Many generalisations of the original blow-up lemma build on this alternative proof.",
"We hope that our alternative proof of the blow-up lemma for approximate decompositions paths the way for further developments in the field.",
"Some ideas in this paper are taken from our recent article with Glock [7] on rainbow embeddings in graphs.",
"As a crucial part of our proof, we apply the main result from another paper with Glock [6], where we prove the existence of quasirandom hypergraph matchings in hypergraphs with small codegree.",
"The idea that hypergraph matchings can be used to embed (almost) spanning graphs has been brought to our attention by [17]."
],
[
"Notation",
"For $k\\in \\mathbb {N}$ , we write $[k]_0:=[k]\\cup \\lbrace 0\\rbrace =\\lbrace 0,1,\\ldots ,k\\rbrace $ , where $[0]=\\emptyset $ .",
"For a finite set $S$ and $k\\in \\mathbb {N}$ , we write $\\binom{S}{k}$ for the set of all subsets of $S$ of size $k$ and $2^S$ for the powerset of $S$ .",
"For a set $\\lbrace i,j\\rbrace $ , we sometimes simply write $ij$ .",
"For $a,b,c\\in \\mathbb {R}$ , we write $a=b\\pm c$ whenever $a\\in [b-c,b+c]$ .",
"For $a,b,c\\in (0,1]$ , we sometimes write $a\\ll b \\ll c$ in our statements meaning that there are increasing functions $f,g:(0,1]\\rightarrow (0,1]$ such that whenever $a\\le f(b)$ and $b \\le g(c)$ , then the subsequent result holds.",
"For $a\\in (0,1]$ and $\\mathbf {b}\\in (0,1]^k$ , we write $a\\ll \\mathbf {b}$ whenever $a\\ll b_i$ for all $b_i\\in \\mathbf {b}$ , $i\\in [k]$ .",
"For the sake of a clearer presentation, we avoid roundings whenever it does not affect the argument.",
"For a graph $G$ , let $V(G)$ and $E(G)$ denote the vertex set and edge set, respectively.",
"We say $u\\in V(G)$ is a $G$ -neighbour of $v\\in V(G)$ if $uv\\in E(G)$ and let $N_G(u)$ be the set of all $G$ -neighbours of $u$ as well as $\\operatorname{deg}_G(u):=|N_G(u)|$ .",
"As usual, $\\Delta (G)$ denotes the maximum degree of $G$ .",
"For $u,v\\in V(G)$ , let $N_G(u,v):=N_G(u)\\cap N_G(v)$ denote the common neighbourhood of $u$ and $v$ , and let $N_G[u]:=N_G(u)\\cup \\lbrace u\\rbrace $ .",
"For a set $S$ , let $N_G(S):=\\bigcup _{v\\in S\\cap V(G)}N_G(v)$ .",
"We frequently treat collections of graphs as the graph obtained by taking the disjoint union of all members; for example, for a collection $\\mathcal {H}$ of graphs, we define $N_\\mathcal {H}(v):=\\bigcup _{H\\in \\mathcal {H}}N_H(v)$ for all $v\\in V(H)$ with $H\\in \\mathcal {H}$ .",
"For disjoint subsets $A,B\\subseteq V(G)$ , let $G[A,B]$ denote the bipartite subgraph of $G$ between $A$ and $B$ and $G[A]$ the subgraph of $G$ induced by $A$ .",
"For convenience, let $G[A,A]:=G[A]$ .",
"Let $e(G)$ denote the number of edges of $G$ and let $e_G(A,B)$ denote the number of edges of $G[A,B]$ .",
"For $k\\in \\mathbb {N}$ , let $G^k$ denote the $k$ -th power of $G$ , that is, the graph obtained from $G$ by adding all edges between vertices whose distance in $G$ is at most $k$ .",
"For graphs $G,H$ , we write $G-H$ to denote the graph with vertex set $V(G)$ and edge set $E(G)\\setminus E(H)$ ."
],
[
"Probabilistic tools and graph regularity",
"To verify the existence of subgraphs with certain properties we frequently consider random subgraphs and use McDiarmid's inequality to verify that specific random variables are highly concentrated around their mean.",
"Theorem (McDiarmid's inequality, see [22]) Suppose $X_1,\\dots ,X_m$ are independent Bernoulli random variables and suppose $b_1,\\dots ,b_m\\in [0,B]$ .",
"Suppose $X$ is a real-valued random variable determined by $X_1,\\dots ,X_m$ such that changing the outcome of $X_i$ changes $X$ by at most $b_i$ for all $i\\in [m]$ .",
"Then, for all $t>0$ , we have $\\mathrm {\\mathbb {P}}\\left[|X-\\mathrm {\\mathbb {E}}\\left[X\\right]|\\ge t\\right] \\le 2 \\exp \\left({-\\frac{2t^2}{B\\sum _{i=1}^m b_i}}\\right).$ We will also need the following two standard results concerning the robustness of ${\\varepsilon }$ -regular graphs.",
"Fact Suppose $G$ is an $({\\varepsilon },d)$ -regular bipartite graph with vertex partition $(A,B)$ and $Y\\subseteq B$ with $|Y|\\ge {\\varepsilon }|B|$ .",
"Then all but at most $2{\\varepsilon }|A|$ vertices of $A$ have $(d\\pm {\\varepsilon })|Y|$ neighbours in $Y$ .",
"Fact Suppose $1/n \\ll {\\varepsilon }\\ll d$ .",
"Suppose $G$ is an $({\\varepsilon },d)$ -super-regular bipartite graph with vertex partition $(A,B)$ , where ${\\varepsilon }^{1/6} n\\le |A|,|B|\\le n$ .",
"If $\\Delta (H)\\le {\\varepsilon }n$ and $X\\subseteq A\\cup B$ with $|X|\\le {\\varepsilon }n$ , then $G[A\\setminus X,B\\setminus X]-H$ is $({\\varepsilon }^{1/3},d)$ -super-regular.",
"We will also use the next result from [5].",
"(In [5] it is proved in the case when $|A|=|B|$ with $16{\\varepsilon }^{1/5}$ instead of ${\\varepsilon }^{1/6}$ .",
"The version stated below can be easily derived from this.)",
"Theorem Suppose $1/n\\ll {\\varepsilon }\\ll \\gamma ,d$ .",
"Suppose $G$ is a bipartite graph with vertex partition $(A,B)$ such that $|A|=n$ , $\\gamma n \\le |B|\\le \\gamma ^{-1}n$ and at least $(1-5{\\varepsilon })n^2/2$ pairs $u,v\\in A$ satisfy $\\operatorname{deg}_G(u),\\operatorname{deg}_G(v)\\ge (d- {\\varepsilon })|B|$ and $|N_G(u,v)|\\le (d+ {\\varepsilon })^2|B|$ .",
"Then $G$ is ${\\varepsilon }^{1/6}$ -regular.",
"At the end of our packing algorithm we apply the following version of the blow-up lemma due to Komlós, Sarközy, and Szemerédi.",
"Theorem (Komlós, Sarközy, and Szemerédi [19]) Suppose $1/n\\ll {\\varepsilon }\\ll 1/\\Delta ,d$ and $1/n\\ll 1/r$ .",
"Suppose $(H,G,R, (X_i)_{i\\in [r]},(V_i)_{i\\in [r]})$ is an $({\\varepsilon },d^{\\prime })$ -super-regular and $\\Delta $ -bounded blow-up instance, with $d^{\\prime }\\ge d$ as well as $|V_i|=(1\\pm {\\varepsilon })n$ for all $i \\in [r]$ and $(A_i)_{i\\in [r]}$ is a collection of graphs such that $A_i$ is bipartite with vertex partition $(X_i,V_i)$ and $({\\varepsilon },d_i)$ -super-regular for some $d_i\\ge d$ .",
"Then there is a packing $\\phi $ of $H$ into $G$ such that $\\phi (x)\\in N_{A_i}(x)$ for all $x\\in X_i$ and $i\\in [r]$ ."
],
[
"Pseudorandom hypergraph matchings",
"A key ingredient in the proof of our `Approximate Packing Lemma' in Section is the main result from [6] on pseudorandom hypergraph matchings.",
"For this we need some more notation.",
"Given a set $X$ and an integer $\\ell \\in \\mathbb {N}$ , an $\\ell $ -tuple weight function on $X$ is a function $\\omega \\colon \\binom{X}{\\ell }\\rightarrow \\mathbb {R}_{\\ge 0}$ .",
"For a subset $X^{\\prime }\\subseteq X$ , we then define $\\omega (X^{\\prime }):=\\sum _{S\\in \\binom{X^{\\prime }}{\\ell }}\\omega (S)$ .",
"For $k\\in [\\ell ]_0$ and a tuple $T\\in \\binom{X}{k}$ , define $\\omega (T):=\\sum _{S\\supseteq T}\\omega (S)$ and let $\\Vert \\omega \\Vert _{k}:=\\max _{T\\in \\binom{X}{k}}\\omega (T)$ .",
"Suppose $\\mathcal {H}$ is an $r$ -uniform hypergraph and $\\omega $ is an $\\ell $ -tuple weight function on $E(\\mathcal {H})$ .",
"Clearly, if $\\mathcal {M}$ is a matching, then a tuple of edges which do not form a matching will never contribute to $\\omega (\\mathcal {M})$ .",
"We thus say that $\\omega $ is clean if $\\omega (\\mathcal {E})=0$ whenever $\\mathcal {E}\\in \\binom{E(\\mathcal {H})}{\\ell }$ is not a matching.",
"Theorem ([6]) Suppose $1/\\Delta \\ll \\delta ,1/r,1/L$ , $r\\ge 2$ , and let ${\\varepsilon }:=\\delta /50L^2r^2$ .",
"Let $\\mathcal {H}$ be an $r$ -uniform hypergraph with $\\Delta (\\mathcal {H})\\le \\Delta $ and $\\Delta ^c(\\mathcal {H})\\le \\Delta ^{1-\\delta }$ as well as $e(\\mathcal {H})\\le \\exp (\\Delta ^{{\\varepsilon }^2})$ .",
"Suppose that for each $\\ell \\in [L]$ , we are given a set $\\mathcal {W}_\\ell $ of clean $\\ell $ -tuple weight functions on $E(\\mathcal {H})$ of size at most $\\exp (\\Delta ^{{\\varepsilon }^2})$ , such that $\\omega (E(\\mathcal {H}))\\ge \\Vert \\omega \\Vert _{k}\\Delta ^{k+\\delta }$ for all $\\omega \\in \\mathcal {W}_\\ell $ and $k\\in [\\ell ]$ .",
"Then there exists a matching $\\mathcal {M}$ in $\\mathcal {H}$ such that $\\omega (\\mathcal {M})=(1\\pm \\Delta ^{-{\\varepsilon }}) \\omega (E(\\mathcal {H}))/\\Delta ^\\ell $ for all $\\ell \\in [L]$ and $\\omega \\in \\mathcal {W}_\\ell $ ."
],
[
"Refining partitions",
"Here, we provide a useful result to refine the vertex partition of a blow-up instance such that every $H\\in \\mathcal {H}$ only induces a matching between its refined partition classes.",
"While in [26] this procedure was easily obtained by applying the classical Hajnal–Szemerédi Theorem, we perform a random procedure to obtain more control on the mass distribution of a weight function with respect to the refined partition.",
"Lemma Suppose $1/n\\ll \\beta \\ll \\alpha $ and $1/n\\ll 1/r$ .",
"Suppose $\\mathcal {H}$ is a collection of at most $\\alpha ^{-1}n$ graphs, $(X_i^H)_{i\\in [r]}$ is a vertex partition of $H$ , and $\\Delta (H)\\le \\alpha ^{-1}$ for every $H \\in \\mathcal {H}$ .",
"Suppose $n/2\\le |X_i^H|=|X_i^{H^{\\prime }}| \\le 2n$ for all $H,H^{\\prime }\\in \\mathcal {H}$ and $i\\in [r]$ .",
"Suppose $\\mathcal {W}$ is a set of weight functions $\\omega \\colon \\bigcup _{H\\in \\mathcal {H}, i\\in [r]} X_i^H\\rightarrow [0,\\alpha ^{-1}]$ with $|\\mathcal {W}|\\le {\\rm e}^{\\sqrt{n}}$ .",
"Then for all $H\\in \\mathcal {H}$ and $i\\in [r]$ , there exists a partition $(X_{i,j}^H)_{j\\in [\\beta ^{-1}]}$ of $X^H_i$ such that for all $H\\in \\mathcal {H},\\omega \\in \\mathcal {W}, i,i^{\\prime }\\in [r],j,j^{\\prime }\\in [\\beta ^{-1}]$ where $i\\ne i^{\\prime }$ or $j\\ne j^{\\prime }$ , we have that $X^H_{i,j}$ is independent in $H^2$ ; $|X_{i,1}^H|\\le \\ldots \\le |X_{i,\\beta ^{-1}}^H|\\le |X_{i,1}^H|+1$ ; $\\omega (X_{i,j}^H)=\\beta \\omega (X_i^H) \\pm \\beta ^{3/2}n$ .",
"$\\sum _{H\\in \\mathcal {H}}e_H(X_{i,j}^H,X_{i^{\\prime },j^{\\prime }}^H)= \\beta ^2\\sum _{H\\in \\mathcal {H}}e_H(X_{i}^H,X_{i^{\\prime }}^H) \\pm n^{5/3}$ ; Proof.",
"Our strategy is as follows.",
"We first consider every $H\\in \\mathcal {H}$ in turn and construct a partition $(Y_{i,j})_{j\\in [\\beta ^{-1}]}$ that essentially satisfies REF –REF with $Y_{i,j}$ playing the role of $X_{i,j}^H$ .",
"Then we perform a vertex swapping procedure to resolve some conflicts in $Y_{i,j}$ and obtain $Z_{i,j}^H$ .",
"In the end, we permute the ordering of $(Z_{i,j}^H)_{j\\in [\\beta ^{-1}]}$ for each $H\\in \\mathcal {H},i\\in [r]$ to also ensure REF .",
"To simplify notation, we assume from now on that $|X_i^H|$ is divisible by $\\beta ^{-1}$ for all $i\\in [r]$ and at the end we explain how very minor modifications yield the general case.",
"Fix some $H\\in \\mathcal {H}$ and write $X_i$ for $X_i^H$ .",
"We claim that there exist partitions $(Y_{i,j})_{j\\in [\\beta ^{-1}]}$ of $X_i$ for each $i\\in [r]$ such that for all $i\\in [r],j\\in [\\beta ^{-1}]$ $|Y_{i,j}|=\\beta |X_i|\\pm \\beta ^2n$ ; at most $2\\alpha ^{-2}\\beta ^2 n$ pairs of vertices in $Y_{i,j}$ are adjacent in $H^2$ ; $\\omega (Y_{i,j})=\\beta \\omega (X_i) \\pm \\beta ^2 n$ for all $\\omega \\in \\mathcal {W}$ .",
"Indeed, the existence of such partitions can be seen by assigning every vertex in $X_i$ uniformly at random to some $Y_{i,j}$ for $j\\in [\\beta ^{-1}]$ .",
"Together with a union bound and Theorem REF we conclude that REF –REF hold simultaneously with positive probability.",
"Next, we slightly modify these partitions to obtain a new collection of partitions.",
"These modifications can be performed for each $i\\in [r]$ independently.",
"Hence, we fix some $i\\in [r]$ .",
"For all $j\\in [\\beta ^{-1}]$ , let $W_{j}\\subseteq Y_{i,j}$ be such that $|Y_{i,j} \\setminus W_j|=\\beta |X_i|- \\beta ^{5/3}n$ and $W_j$ contains all vertices in $Y_{i,j}$ that contain an $H^2$ -neighbour in $Y_{i,j}$ (the sets $W_j$ clearly exist by REF ,REF ).",
"Let $\\lbrace w_1,\\ldots ,w_s\\rbrace =W:=\\bigcup _{j\\in [\\beta ^{-1}]}W_j$ and observe that $s=|X_i|-\\sum _{[\\beta ^{-1}]}|Y_{i,j}\\setminus W_j|=\\beta ^{2/3}n$ .",
"Now arbitrarily assign labels in $[\\beta ^{-1}]$ to the vertices in $W$ such that each label is used exactly $\\beta ^{5/3}n$ times.",
"Let $Z_j(0):=Y_{i,j}\\setminus W_j$ for all $j\\in [\\beta ^{-1}]$ .",
"To obtain the desired partitions we perform the following swap procedure for every $t\\in [s]$ in turn.",
"Say $w_t\\in W_j$ and $w_t$ receives label $j^{\\prime }$ .",
"We select $j^{\\prime \\prime }\\in [\\beta ^{-1}]\\setminus \\lbrace j,j^{\\prime }\\rbrace $ such that $w_t$ has no $H^2$ -neighbour in $Z_{j^{\\prime \\prime }}(t-1)$ and such that $Z_{j^{\\prime \\prime }}(t-1)$ contains a vertex $w$ that has no $H^2$ -neighbour in $Z_{j^{\\prime }}(t-1)$ .",
"In such a case we say that $j^{\\prime \\prime }$ is selected in step $t$ .",
"Then we define $Z_{j^{\\prime \\prime }}(t):= (Z_{j^{\\prime \\prime }}(t-1)\\cup \\lbrace w_t\\rbrace )\\setminus \\lbrace w\\rbrace $ , $Z_{j^{\\prime }}(t):=Z_{j^{\\prime }}(t)\\cup \\lbrace w\\rbrace $ and $Z_k(t):=Z_k(t-1)$ for all $k\\in [\\beta ^{-1}]\\setminus \\lbrace j^{\\prime },j^{\\prime \\prime }\\rbrace $ .",
"Note that $\\beta |X_i| - \\beta ^{5/3}n \\le |Z_k(t)|\\le \\beta |X_i|$ for all $t\\in [s],k\\in [\\beta ^{-1}]$ .",
"Observe also that we have always at least $\\beta ^{-1}/2$ choices to select $j^{\\prime \\prime }$ in step $t$ .",
"As $s= \\beta ^{2/3}n$ , we can ensure that each $j^{\\prime \\prime }\\in [\\beta ^{-1}]$ is selected, say, at most $10\\beta ^{5/3}n$ times.",
"We write $Z_{i,j}^H:=Z_j(s)$ and then it is straightforward to see that for all $j\\in [\\beta ^{-1}]$ we have that $|Z_{i,j}^H|= \\beta |X_i^H|$ ; $Z_{i,j}^H$ is independent in $H^2$ ; $\\omega (Z_{i,j}^H)=\\beta \\omega (X_i^H) \\pm \\beta ^{3/2} n/2$ for all $\\omega \\in \\mathcal {W}$ .",
"As $i\\in [r]$ and $H\\in \\mathcal {H}$ are chosen arbitrarily, the statements REF –REF hold for all $i\\in [r],H\\in \\mathcal {H}$ .",
"In the remainder of the proof we show how to find permutations $\\lbrace \\pi _{i}^H\\rbrace _{H\\in \\mathcal {H},i\\in [r]}$ such that REF holds for $X_{i,j}^H:=Z_{i,\\pi _i^H(j)}^H$ .",
"The answer is simple, random permutations yield REF with probability, say, at least $1/2$ .",
"To see this, fix $i,i^{\\prime }\\in [r],j,j^{\\prime }\\in [\\beta ^{-1}]$ where $i\\ne i^{\\prime }$ or $j\\ne j^{\\prime }$ , and then Theorem REF implies that $\\sum _{H\\in \\mathcal {H}}e_H(X_{i,j}^H,X_{i^{\\prime },j^{\\prime }}^H)= \\beta ^2\\sum _{H\\in \\mathcal {H}}e_H(X_{i}^H,X_{i^{\\prime }}^H) \\pm n^{5/3}/2$ with probability at least $1-1/n$ .",
"A union bound over all $i,i^{\\prime },j,j^{\\prime }$ completes the proof.",
"In the beginning we made the assumption that $\\beta ^{-1}$ divides $X_i^H$ .",
"To avoid this assumption, we simply remove a set $\\widetilde{X}_i^H$ of size at most $\\beta ^{-1}-1$ from $X_i^H$ such that $\\beta ^{-1}$ divides $X_i^H\\setminus \\widetilde{X}_i^H$ and perform the entire procedure with $X_i^H\\setminus \\widetilde{X}_i^H$ instead of $X_i^H$ .",
"That is, for all $H\\in \\mathcal {H}$ and $i\\in [r]$ , we obtain a partition $(\\widetilde{X}_{i,j}^H)_{j\\in [\\beta ^{-1}]}$ of $X^H_i\\setminus \\widetilde{X}^H_i$ satisfying REF –REF , where $|\\widetilde{X}_{i,j}^H|=|\\widetilde{X}_{i,j^{\\prime }}^H|$ for all $i\\in [r],j,j^{\\prime }\\in [\\beta ^{-1}]$ , and REF and REF hold with error terms `$\\pm n^{5/3}/2$ ' and `$\\pm \\beta ^{3/2}n/2$ ', respectively.",
"At the very end we add the vertices in $\\widetilde{X}_i^H$ to the partition $(\\widetilde{X}_{i,j}^H)_{j\\in [\\beta ^{-1}]}$ while preserving REF and REF .",
"We may do so by performing a swap argument as before.",
"Observe that our error bounds give us enough room to spare.",
"$\\square $"
],
[
"Approximate packings",
"The goal of this section is to provide an `Approximate Packing Lemma' (Lemma REF ).",
"Given a blow-up instance $(\\mathcal {H},G,R,\\mathcal {X},\\mathcal {V})$ , it allows us to embed almost all vertices of $\\bigcup _{H\\in \\mathcal {H}}X_i^H$ into $V_i$ , while maintaining crucial properties for future embedding rounds of other clusters.",
"To describe this setup we define a packing instance and collect some more notation."
],
[
"Packing instances",
"Given a graph $G$ and a set $\\mathcal {E}$ , we call $\\psi \\colon E(G)\\rightarrow 2^\\mathcal {E}$ an edge set labelling of $G$ .",
"A label $\\alpha \\in \\mathcal {E}$ appears on an edge $e$ if $\\alpha \\in \\psi (e)$ .",
"We define the maximum degree $\\Delta _\\psi (G)$ of $\\psi $ as the maximum number of edges of $G$ on which any fixed label appears.",
"We define the maximum codegree $\\Delta ^c_\\psi (G)$ of $\\psi $ as the maximum number of edges of $G$ on which any two fixed labels appear together.",
"Let $r\\in \\mathbb {N}_0$ .",
"We say $(\\mathcal {H},G,R,\\mathcal {A},\\psi )$ is a packing-instance of size $r$ if $\\mathcal {H}$ is a collection of graphs, and $G$ and $R$ are graphs, where $V(R)=[r]_0$ ; $\\mathcal {A}=\\bigcup _{H\\in \\mathcal {H},i\\in [r]_0}A^H_i$ is a union of balanced bipartite graphs $A^H_i$ with vertex partition $(X^H_i,V_i)$ ; $(X^H_i)_{i\\in [r]_0}$ is a partition of $H$ into independent sets for every $H\\in \\mathcal {H}$ , and $(V_i)_{i\\in [r]_0}$ is a partition of $V(G)$ ; $R=R_A\\cup R_B$ is the union of two edge-disjoint graphs with $N_R(0)=[r]$ ; for all $H\\in \\mathcal {H}$ , the graph $H[X^H_i,X^H_j]$ is a matching if $ij\\in E(R)$ and empty otherwise; $\\psi \\colon E(\\mathcal {A})\\rightarrow 2^\\mathcal {E}$ is an edge set labelling such that $\\Delta _\\psi (A^H_i)\\le 1$ for all $H\\in \\mathcal {H},i\\in [r]_0$ , and for every label $\\alpha \\in \\mathcal {E}$ with $\\alpha \\in \\psi (xv)\\cap \\psi (x^{\\prime }v^{\\prime })$ and $v,v^{\\prime }\\in V(G)$ , we have $v=v^{\\prime }$ .",
"In such a case, we write for simplicity ${X}_i:=\\bigcup _{H\\in \\mathcal {H}}X^H_i$ and ${A}_i:=\\bigcup _{H\\in \\mathcal {H}}A^H_i$ for each $i\\in [r]_0$ , and whenever we write $xv\\in E({A}_i)$ , we tacitly assume that $x\\in {X}_i, v\\in V_i$ .",
"The only reason why $R$ is the disjoint union of two graphs lies in the nature of our approach; while $R_A$ represents parts of $R$ as in the statement of our main result (Lemma REF , which is very similar to Theorem REF ), the edges of $R_B$ represent copies of edge slices of $G$ that in the end will be used to complete the approximate packing.",
"We use copies here to obtain a unified setup for the Approximate Packing Lemma, alternatively, we could have used parallel edges in the reduced graph.",
"The aim of this section is to map almost all vertices of ${X}_0$ into $V_0$ by defining a function $\\sigma \\colon {X}_0^\\sigma \\rightarrow V_0$ in ${A}_0$ (that is, $x\\sigma (x)\\in E({A}_0)$ ) where ${X}_0^\\sigma \\subseteq {X}_0$ .",
"(Hence, we refer to subgraphs of ${A}_i$ as candidacy graphs.)",
"For convenience, we identify such a function $\\sigma $ with its corresponding edge set $M$ defined as $M=M(\\sigma ):=\\lbrace xv\\colon x\\in {X}_0^\\sigma , v\\in V_0, \\sigma (x)=v \\rbrace $ .",
"We say $\\begin{minipage}[c]{0.8}\\em \\sigma \\colon {X}_0^\\sigma \\rightarrow V_0 is a \\emph {conflict-free packing} if \\sigma |_{{X}_0^\\sigma \\cap X_0^H} is injective for all H\\in \\mathcal {H} and \\psi (e)\\cap \\psi (f)=\\emptyset for all distinct e,f\\in M(\\sigma ).\\end{minipage}$ The set $\\psi (xv)$ will encode the set of edges of $G$ that are used for the embedding when mapping $x$ to $v$ .",
"The property that $\\psi (e)\\cap \\psi (f)=\\emptyset $ for all distinct $e,f\\in M(\\sigma )$ will guarantee that in the proof of our main result (Lemma REF ) every edge in $G$ is used at most once.",
"Given a conflict-free packing $\\sigma \\colon {X}_0^\\sigma \\rightarrow V_0$ in ${A}_0$ , we update the remaining candidacy graphs and their edge set labelling according to the following two definitions.",
"Definition (Updated candidacy graphs) For a conflict-free packing $\\sigma \\colon {X}_0^\\sigma \\rightarrow V_0$ in ${A}_0$ and all $H\\in \\mathcal {H},i\\in [r]$ , let $A^H_i[\\sigma ]$ be the updated candidacy graph (with respect to $\\sigma $ ) which is defined by the spanning subgraph of $A^H_i$ that contains precisely those edges $xv\\in E(A^H_i)$ for which the following holds: if $x$ has an $H$ -neighbour $x_0\\in {X}_0^\\sigma $ (which would be unique), then $\\sigma (x_0)v\\in E(G[V_0,V_i])$ .",
"Definition (Updated labelling) For a conflict-free packing $\\sigma \\colon {X}_0^\\sigma \\rightarrow V_0$ in ${A}_0$ , let $\\psi [\\sigma ]$ be the updated edge set labelling (with respect to $\\sigma $ ) defined as follows: for all $H\\in \\mathcal {H}, i\\in [r]$ and $xv\\in E(A^H_i[\\sigma ])$ , if $x$ has an $H$ -neighbour $x_0\\in {X}_0^\\sigma $ , then set $\\psi [\\sigma ](xv):=\\psi (xv)\\cup \\lbrace \\sigma (x_0),v\\rbrace $ , and otherwise set $\\psi [\\sigma ](xv):=\\psi (xv)$ .",
"In order to be able to analyse our packing process in Section , we carefully maintain quasirandom properties of the candidacy graphs throughout the procedure.",
"To this end, we refer to a packing instance $(\\mathcal {H},G, R,\\mathcal {A},\\psi )$ of size $r$ as an $({\\varepsilon },\\mathbf {d})$ -packing-instance, where $\\mathbf {d}=(d_A,d_B,d_0,\\ldots ,d_r)$ , if $G[V_i,V_j]$ is $({\\varepsilon },d_Z)$ -super-regular for all $ij\\in E(R_Z), Z\\in \\lbrace A,B\\rbrace $ ; $A^H_i$ is $({\\varepsilon },d_i)$ -super-regular for all $H\\in \\mathcal {H},i\\in [r]_0$ ; $e_{H}(N_{A^H_i}(v_i),N_{A^H_j}(v_j))=(d_id_j\\pm {\\varepsilon })e_{H}(X^H_i,X^H_j)$ for all $H\\in \\mathcal {H}, ij\\in E(R_A), v_iv_j\\in E(G[V_i,V_j])$ ; $\\Delta _\\psi ({A}_i)\\le (1+{\\varepsilon })d_i|V_i|$ for all $i\\in [r]_0$ .",
"Property REF ensures that no edge is a potential candidate for too many graphs in $\\mathcal {H}$ and REF enables us to maintain this property for future embedding rounds (see Lemma REFREF below).",
"Let ${P}=(\\mathcal {H},G, R,\\mathcal {A},\\psi )$ be an $({\\varepsilon },\\mathbf {d})$ -packing-instance of size $r$ .",
"Similarly as for a blow-up instance, we say $(W,Y_1,\\ldots ,Y_k)$ is an $\\ell $ -set tester for ${P}$ if $k\\le \\ell $ and there exist distinct $H_1,\\ldots ,H_k\\in \\mathcal {H}$ such that $W\\subseteq V_0$ and $Y_j\\subseteq X_0^{H_j}$ for all $j\\in [k]$ .",
"For $i\\in N_{R_A}[0]$ and $v\\in V_i$ , we say $\\omega \\colon E({A}_i)\\rightarrow [0,\\ell ]$ is an $\\ell $ -edge tester with centre $v$ for ${P}$ if $\\omega (x^{\\prime }v^{\\prime })=0$ for all $x^{\\prime }v^{\\prime }\\in E({A}_i)$ with $v^{\\prime }\\in V_i, v^{\\prime }\\ne v$ .",
"We say $\\omega \\colon E({A}_0)\\rightarrow [0,\\ell ]$ is an $\\ell $ -edge tester with centres in ${X}_0$ if there exist vertices $\\lbrace x_H\\rbrace _{H\\in \\mathcal {H}}$ with $x_H\\in X_0^H$ for each $H\\in \\mathcal {H}$ such that $\\omega (x^{\\prime }v^{\\prime })=0$ for all $x^{\\prime }v^{\\prime }\\in E({A}_0)$ with $x^{\\prime }\\notin \\lbrace x_H\\rbrace _{H\\in \\mathcal {H}}$ .",
"Further, let $\\dim (\\omega )$ be the dimension of $\\omega $ defined as $\\dim (\\omega )={\\left\\lbrace \\begin{array}{ll}1 &\\text{ if $\\omega (E({A}_i))=\\omega (E(A_i^H))$ for some $H\\in \\mathcal {H}$,}\\\\ 2 & \\text{ otherwise.}\\end{array}\\right.",
"}$ Moreover, for every $H\\in \\mathcal {H}$ , let $H_+$ be an auxiliary supergraph of $H$ that is obtained by adding a maximal number of edges between $X^H_0$ and $X^H_i$ for every $i\\in [r]$ subject to $H_+[X^H_0,X^H_i]$ being a matching.",
"We call $\\mathcal {H}_+:=\\bigcup _{H\\in \\mathcal {H}}H_+$ an enlarged graph of $\\mathcal {H}$.",
"We say that ${P}$ is nice (with respect to $\\mathcal {H}_+$ ) if $|N_{A^{H}_i}(x_i)\\cap N_{G}(v_j)|=(d_id_Z\\pm {\\varepsilon })|V_i|$ for all $x_jv_j\\in E(A^H_j)$ whenever $\\lbrace x_i\\rbrace =N_{H_+}(x_j)\\cap X^H_i$ , $H\\in \\mathcal {H},ij\\in E(R_Z), Z\\in \\lbrace A,B\\rbrace $ ; $|N_G(v_i,v_j)\\cap V_0|=(d_A^2\\pm {\\varepsilon })|V_0|$ for all $ij\\in E(R_A-\\lbrace 0\\rbrace )$ and $v_iv_j\\in E(G[V_i,V_j])$ .",
"Using standard regularity methods (see Facts REF and REF ), it is straightforward to verify the following: $\\begin{minipage}[c]{0.85}\\em For every ({\\varepsilon },\\mathbf {d})-packing-instance (\\mathcal {H},G, R,\\mathcal {A},\\psi ) of size r and every enlarged graph \\mathcal {H}_+ of \\mathcal {H}, there exist spanning subgraphs G^{\\prime }\\subseteq G and \\mathcal {A}^{\\prime }\\subseteq \\mathcal {A}such that (\\mathcal {H},G^{\\prime }, R,\\mathcal {A}^{\\prime },\\psi ) is a nice ({\\varepsilon }^{\\prime },\\mathbf {d})-packing-instance of size r with respect to \\mathcal {H}_+ for some {\\varepsilon }^{\\prime } with {\\varepsilon }\\ll {\\varepsilon }^{\\prime }\\ll 1/r.\\end{minipage}$"
],
[
"Approximate Packing Lemma",
"We now state our Approximate Packing Lemma.",
"Roughly speaking it states that given a packing instance, we can find a conflict-free packing such that the updated candidacy graphs are still super-regular, albeit with a smaller density.",
"Moreover, with respect to certain weight functions on the candidacy graphs, the updated candidacy graphs behave as we would expect this by a random and independent deletion of the edges.",
"Lemma (Approximate Packing Lemma) Let $1/n\\ll {\\varepsilon }\\ll {\\varepsilon }^{\\prime }\\ll \\mathbf {d},1/r,1/s$ .",
"Suppose $(\\mathcal {H},G,R,\\mathcal {A},\\psi )$ is an $({\\varepsilon },\\mathbf {d})$ -packing-instance of size $r$ , $\\Vert \\psi \\Vert \\le s$ , $|\\mathcal {H}|\\le sn$ , $|V_i|=(1\\pm {\\varepsilon })n$ for all $i\\in [r]_0$ , $\\sum _{H\\in \\mathcal {H}}e_{H}(X_0^H,X_i^H)\\le d_An^2$ for all $i\\in N_{R_A}(0)$ , and $e_{H}(X^H_i,X^H_j)\\ge {{\\varepsilon }^{\\prime 2}n}$ for all $H\\in \\mathcal {H}, ij\\in E(R)$ .",
"Suppose $\\Delta _\\psi ^c({A}_i)\\le \\sqrt{n}$ for all $i\\in N_{R_A}[0]$ , and suppose $\\mathcal {W}_{set},\\mathcal {W}_{edge}$ are sets of $s$ -set testers and $s$ -edge testers of size at most $n^{3\\log n}$ , respectively.",
"Then there is a conflict-free packing $\\sigma \\colon {X}_0^\\sigma \\rightarrow V_0$ in ${A}_0$ such that for all $H\\in \\mathcal {H}$ , we have $|{X}_0^\\sigma \\cap X_0^H|\\ge (1-{\\varepsilon }^{\\prime })n$ and for all $i\\in [r]$ there exists a spanning subgraph $A^{H,new}_i$ of the updated candidacy graph $A^H_i[\\sigma ]$ (where ${A}_i^{new}:=\\bigcup _{H\\in \\mathcal {H}}A^{H,new}_i$ ) with $A^{H,new}_i$ is $({\\varepsilon }^{\\prime },d_id_Z)$ -super-regular for all $i\\in N_{R_Z}(0), Z\\in \\lbrace A,B\\rbrace $ ; $e_{H}(N_{A_i^{H,new}}(v_i),N_{A_j^{H,new}}(v_j))=(d_id_j d_A^2\\pm {\\varepsilon }^{\\prime })e_{H}(X^H_i,X^H_j)$ for all $ij\\in E(R_A-\\lbrace 0\\rbrace )$ and $v_iv_j\\in E(G[V_i,V_j])$ ; $\\omega (E({A}_i^{new}))=(1\\pm {\\varepsilon }^{\\prime 2})d_A\\omega (E({A}_i))\\pm {\\varepsilon }^{\\prime 2}n^{\\dim (\\omega )}$ for all $\\omega \\in \\mathcal {W}_{edge}$ and $i\\in N_{R_A}(0)$ ; $\\Delta _{\\psi [\\sigma ]}({A}_i^{new})\\le (1+{\\varepsilon }^{\\prime })d_id_A|V_i|$ for all $i\\in N_{R_A}(0)$ ; $\\Delta _{\\psi [\\sigma ]}^c({A}_i^{new})\\le \\sqrt{n}$ for all $i\\in N_{R_A}(0)$ ; $|W\\cap \\bigcap _{j\\in [\\ell ]}\\sigma (Y_j\\cap {X}_0^\\sigma )|= |W||Y_1|\\cdots |Y_\\ell |/n^\\ell \\pm {\\varepsilon }^{\\prime } n$ for all $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}$ ; $\\omega (M(\\sigma ))=(1\\pm {\\varepsilon }^{\\prime }){\\omega (E({A}_0))}/{d_0n}\\pm {\\varepsilon }^{\\prime }n$ for all $\\omega \\in \\mathcal {W}_{edge}$ with centre in $V_0$ or centres in ${X}_0$ .",
"Properties REF , REF and REF ensure that REF –REF are also satisfied for the updated candidacy graphs ${A}_i^{new}$ , respectively, and REF ensures that the codegree of the updated labelling $\\psi [\\sigma ]$ is still small on ${A}_i^{new}$ .",
"Property REF states that the weight of the edge testers on the updated candidacy graphs ${A}_i^{new}$ is what we would expect by a random sparsification of the edges in ${A}_i$ , and REF and REF guarantee that $\\sigma $ behaves like a random packing with respect to the set and edge testers.",
"We split the proof into two steps.",
"In Step REF , we construct an auxiliary hypergraph and apply Theorem REF to obtain the required conflict-free packing $\\sigma $ .",
"By defining suitable weight functions in Step REF , we employ the conclusions of Theorem REF to establish REF –REF .",
"Proof.",
"Let $\\mathcal {H}_+$ be an enlarged graph of $\\mathcal {H}$ , and for $i\\in [r]$ , let ${A}_i^{bad}$ and ${A}_i^{good}$ be spanning subgraphs of ${A}_i$ such that ${A}_i^{bad}$ contains precisely those edges $xv\\in E({A}_i)$ where $N_{\\mathcal {H}_+}(x)\\cap {X}_0=\\emptyset $ , and $E({A}_i^{good}):=E({A}_i)\\setminus E({A}_i^{bad})$ .",
"We may assume that $|\\mathcal {H}|=sn$ and $\\sum _{H\\in \\mathcal {H}}e_{H}(X_0^H,X_i^H)\\le (d_A+{\\varepsilon }^{\\prime 3/2})n^2$ for all $i\\in N_{R_A}(0)$ , where the last inequality will be only used in (REF ).",
"(Otherwise we artificially add some graphs to $\\mathcal {H}$ subject to the condition that still $e_{H}(X^H_i,X^H_j)\\ge {{\\varepsilon }^{\\prime 2}n}$ for all $H\\in \\mathcal {H}, ij\\in E(R)$ , and accordingly we add some graphs to $\\mathcal {A}$ satisfying REF –REF .)",
"We may also assume that $\\psi \\colon E(\\mathcal {A})\\rightarrow 2^\\mathcal {E}$ is such that $|\\psi (e)|=s$ for all $e\\in E({A}_0)$ (otherwise we add artificial labels that we delete at the end again), and $(\\mathcal {H},G,R,\\mathcal {A},\\psi )$ is a nice $({\\varepsilon },\\mathbf {d})$ -packing-instance with respect to $\\mathcal {H}_+$ (otherwise we may employ (REF ) and replace ${\\varepsilon }$ by some $\\tilde{{\\varepsilon }}$ , where ${\\varepsilon }\\ll \\tilde{{\\varepsilon }}\\ll {{\\varepsilon }^{\\prime }}$ ; observe also that this does not cause problems with the weight of the edge testers in REF and REF , as the operation in (REF ) only deletes few edges of $\\mathcal {A}$ incident to every vertex).",
"Step 1.",
"Constructing an auxiliary hypergraph We want to use Theorem REF to find the required conflict-free packing $\\sigma $ in ${A}_0$ .",
"To this end, let $(V_0^H)_{H\\in \\mathcal {H}}$ be disjoint copies of $V_0$ , and for $H\\in \\mathcal {H}$ and $e=x_0v_0\\in E(A_0^{H})$ , let $e^H:=x_0v_0^H$ where $v_0^H$ is the copy of $v_0$ in $V_0^H$ .",
"Let $f_e:=e^H\\cup \\psi (e)$ for each $e\\in E(A_0^H), H\\in \\mathcal {H}$ and let $\\mathcal {H}^{aux}$ be the $(s+2)$ -uniform hypergraph with vertex set $\\bigcup _{H\\in \\mathcal {H}}(X_0^H\\cup V_0^H)\\cup \\mathcal {E}$ and edge set $\\lbrace f_e\\colon e\\in E({A}_0) \\rbrace $ .",
"A key property of the construction of $\\mathcal {H}^{aux}$ is a bijection between conflict-free packings $\\sigma $ in ${A}_0$ and matchings $\\mathcal {M}$ in $\\mathcal {H}^{aux}$ by assigning $\\sigma $ to $\\mathcal {M}=\\lbrace f_e\\colon e\\in M(\\sigma ) \\rbrace $ .",
"(Recall that $M=M(\\sigma )$ is the edge set corresponding to $\\sigma $ .)",
"It is easy to estimate $\\Delta (\\mathcal {H}^{aux})$ and $\\Delta ^c(\\mathcal {H}^{aux})$ in order to apply Theorem REF .",
"Since $A_0^{H}$ is $({\\varepsilon },d_0)$ -super-regular for each $H\\in \\mathcal {H}$ , $|X_0^H|=|V_0|=(1\\pm {\\varepsilon })n$ , and $\\Delta _\\psi ({A}_0)\\le (1+{\\varepsilon })d_0|V_0|$ , we conclude that $\\Delta (\\mathcal {H}^{aux})\\le (d_0+3{\\varepsilon })n=:\\Delta .$ Note that the codegree in $\\mathcal {H}^{aux}$ of two vertices in $\\bigcup _{H\\in \\mathcal {H}}(X_0^H\\cup V_0^H)$ is at most 1, and similarly, the codegree in $\\mathcal {H}^{aux}$ of a vertex in $\\bigcup _{H\\in \\mathcal {H}}(X_0^H\\cup V_0^H)$ and a label in $\\mathcal {E}$ is at most 1 because $\\Delta _\\psi (A_0^H)\\le 1$ for all $H\\in \\mathcal {H}$ .",
"By assumption, $\\Delta _\\psi ^c({A}_0)\\le \\sqrt{n}$ .",
"Altogether, this implies that $\\Delta ^c(\\mathcal {H}^{aux})\\le \\sqrt{n}\\le \\Delta ^{1-{\\varepsilon }^2}.$ Suppose $\\mathcal {W}=\\bigcup _{\\ell \\in [s]}\\mathcal {W}_\\ell $ is a set of size at most $n^{4\\log n}$ of given weight functions $\\omega \\in \\mathcal {W}_\\ell $ for $\\ell \\in [s]$ with $\\omega \\colon \\binom{E({A}_0)}{\\ell }\\rightarrow [0,s]$ .",
"Note that every weight function $\\omega \\colon \\binom{E({A}_0)}{\\ell }\\rightarrow [0,s]$ naturally corresponds to a weight function $\\omega _{\\mathcal {H}^{aux}}\\colon \\binom{E(\\mathcal {H}^{aux})}{\\ell }\\rightarrow [0,s]$ by defining $\\omega _{\\mathcal {H}^{aux}}(\\lbrace f_{e_1},\\ldots ,f_{e_\\ell }\\rbrace ):=\\omega (\\lbrace e_1,\\ldots ,e_\\ell \\rbrace )$ .",
"We will explicitly specify $\\mathcal {W}$ in Step REF and it is simple to check that for each $\\omega \\in \\mathcal {W}$ the corresponding weight function $\\omega _{\\mathcal {H}^{aux}}$ will be clean.",
"Our main idea is to find a hypergraph matching in $\\mathcal {H}^{aux}$ that behaves like a typical random matching with respect to $\\lbrace \\omega _{\\mathcal {H}^{aux}}\\colon \\omega \\in \\mathcal {W}\\rbrace $ in order to establish REF –REF .",
"Suppose $\\ell \\in [s]$ and $\\omega \\in \\mathcal {W}_\\ell $ .",
"If $\\omega (E({A}_0))\\ge n^{1+{\\varepsilon }/2}$ or $\\ell \\ge 2$ , define $\\tilde{\\omega }:=\\omega $ .",
"Otherwise, choose $\\tilde{\\omega }\\colon E({A}_0)\\rightarrow [0,s]$ such that $\\omega \\le \\tilde{\\omega }$ and $\\tilde{\\omega }(E({A}_0))= n^{1+{\\varepsilon }/2}$ .",
"By (REF ) and (REF ), we can apply Theorem REF (with $(d_0+3{\\varepsilon })n,{\\varepsilon }^2,s+2,s,\\lbrace \\tilde{\\omega }_{\\mathcal {H}^{aux}}\\colon {\\omega }\\in \\mathcal {W}_\\ell \\rbrace $ playing the roles of $\\Delta ,\\delta ,r,L,\\mathcal {W}_\\ell $ ) to obtain a matching $\\mathcal {M}$ in $\\mathcal {H}^{aux}$ that corresponds to a conflict-free packing $\\sigma \\colon {X}_0^\\sigma \\rightarrow V_0$ in ${A}_0$ with its corresponding edge set $M=M(\\sigma )$ that satisfies the following properties (where ${\\hat{\\varepsilon }}:={\\varepsilon }^{1/2}$ ): $\\omega (M)&=(1\\pm {\\hat{\\varepsilon }})\\frac{\\omega (E({A}_0))}{(d_0n)^\\ell } \\text{ for } \\omega \\in \\mathcal {W}_\\ell , \\ell \\in [s]\\text{ where } \\omega (E({A}_0))\\ge \\Vert \\omega \\Vert _{k}\\Delta ^{k+{\\varepsilon }^2} \\text{ for all } k\\in [\\ell ];\\\\\\omega (M)&\\le \\max \\Big \\lbrace (1+{\\hat{\\varepsilon }})\\frac{\\omega (E({A}_0))}{d_0n}, n^{\\varepsilon }\\Big \\rbrace \\text{ for all }\\omega \\in \\mathcal {W}_1.$ One way to exploit (REF ) is to control the number of edges in $M$ between sufficiently large sets of vertices.",
"To this end, for subsets $S\\subseteq X_0^H$ and $T\\subseteq V_0$ for some $H\\in \\mathcal {H}$ with $|S|,|T|\\ge 2{\\varepsilon }n$ , we define a weight function $\\omega _{S,T}\\colon E(A_0^{H})\\rightarrow \\lbrace 0,1\\rbrace $ with $\\omega _{S,T}(e):=\\mathbb {1}_{\\lbrace e\\in E(A_0^{H}[S,T])\\rbrace }.$ That is, $\\omega _{S,T}(M)$ counts the number of edges in $A_0^{H}$ between $S$ and $T$ that lie in $M$ .",
"Since $A_0^H$ is $({\\varepsilon },d_0)$ -super-regular we have that $e_{A_0^{H}}(S,T)=(d_0\\pm {\\varepsilon })|S||T|\\ge {\\varepsilon }^3n^2$ which implies together with (REF ) that whenever $\\omega _{S,T}\\in \\mathcal {W}$ , then $\\sigma $ is chosen such that $|\\sigma (S\\cap {X}_0^\\sigma )\\cap T|=\\omega _{S,T}(M)=(1\\pm 2{\\hat{\\varepsilon }})\\frac{|S||T|}{n}.$ Step 2.",
"Employing weight functions to conclude REF – REF By Step REF , we may assume that (REF ) and () hold for a set of weight functions $\\mathcal {W}$ that we will define during this step.",
"We will show that for this choice of $\\mathcal {W}$ the conflict-free packing $\\sigma \\colon {X}_0^\\sigma \\rightarrow V_0$ as obtained in Step REF satisfies REF –REF .",
"Similarly as in Definition REF (here, $\\mathcal {H}$ is replaced by $\\mathcal {H}_+$ ), we define subgraphs $A^{H,\\ast }_i$ of $A^{H}_i$ as follows.",
"$\\begin{minipage}[c]{0.85}\\em For all H\\in \\mathcal {H}, i\\in [r], let A^{H,\\ast }_i be the spanning subgraph of A^{H}_i containing precisely those edges xv\\in E(A^{H}_i) for which the following holds: if \\lbrace x_0\\rbrace =N_{\\mathcal {H}_+}(x)\\cap {X}_0^\\sigma , then \\sigma (x_0)v\\in E(G[V_0,V_i]).\\end{minipage}$ Observe that $A^{H,\\ast }_i$ is a spanning subgraph of the updated candidacy graph $A^H_i[\\sigma ]$ as in Definition REF .",
"By taking a suitable subgraph of $A^{H,\\ast }_i$ , we will in the end obtain the required candidacy graph $A^{H,new}_i$ .",
"First, we show that $|{X}_0^\\sigma \\cap X_0^H|\\ge (1-3{\\hat{\\varepsilon }})n$ for each $H\\in \\mathcal {H}$ .",
"Adding $\\omega _{X_0^H,V_0}$ as defined in (REF ) for every $H\\in \\mathcal {H}$ to $\\mathcal {W}$ and using (REF ) yields $|{X}_0^\\sigma \\cap X_0^H|=\\omega _{X_0^H,V_0}(M)\\ge (1-3{\\hat{\\varepsilon }})n.$ Step 2.1.",
"Checking REF For all $H\\in \\mathcal {H}$ and $i\\in N_{R_Z}(0), Z\\in \\lbrace A,B\\rbrace $ we proceed as follows.",
"Let $Y^{H}_i:=N_{H_+}({X}_0^\\sigma )\\cap X^H_i$ .",
"We first show that $A^{H,\\ast }_i[Y^{H}_i,V_i]$ is $({\\hat{\\varepsilon }}^{1/18},d_id_Z)$ -super-regular (see (REF )).",
"We do so by showing that every vertex in $Y^{H}_i\\cup V_i$ has the appropriate degree, and that the common neighbourhood of most pairs of vertices in $V_i$ have the correct size such that we can employ Theorem REF to guarantee the super-regularity of $A^{H,\\ast }_i[Y^{H}_i,V_i]$ .",
"Note that $|Y^{H}_i|\\ge |{X}_0^\\sigma \\cap X_0^H|-2{\\varepsilon }n\\ge (1-4{\\hat{\\varepsilon }})n$ by (REF ).",
"For every vertex $x\\in Y^{H}_i$ with $\\lbrace x_0\\rbrace =N_{H_+}(x)\\cap {X}_0^\\sigma $ , we have $\\operatorname{deg}_{A^{H,\\ast }_i}(x)=|N_{A^{H}_i}(x)\\cap N_{G}(\\sigma (x_0))|$ .",
"Since our packing-instance is nice, REF implies that $\\operatorname{deg}_{A^{H,\\ast }_i}(x)=(d_id_Z\\pm {\\varepsilon })|V_i|$ .",
"For $v\\in V_i$ , let $N_v:=N_{A^{H}_i}(v)$ .",
"Observe that $\\operatorname{deg}_{A^{H,\\ast }_i[Y^{H}_i,V_i]}(v)=|\\sigma (N_{H_+}(N_v)\\cap {X}_0^\\sigma )\\cap N_G(v)|,$ and $|N_{H_+}(N_v)\\cap X_0^H|=|N_v|\\pm 2{\\varepsilon }n=(d_i\\pm 5{\\varepsilon })n$ , and $|N_G(v)\\cap V_0|=(d_Z\\pm 3{\\varepsilon })n$ .",
"Adding for every vertex $v\\in V_i$ , the weight function $\\omega _{S,T}$ as defined in (REF ) for $S:=N_{H_+}(N_v)\\cap X_0^H$ and $T:=N_G(v)\\cap V_0$ to $\\mathcal {W}$ , we obtain by (REF ) and (REF ) that $\\operatorname{deg}_{A^{H,\\ast }_i[Y^{H}_i,V_i]}(v)=(1\\pm 2{\\hat{\\varepsilon }})|N_{H_+}(N_v)\\cap X_0^H||N_G(v)\\cap V_0|n^{-1}=(d_id_Z\\pm {{\\hat{\\varepsilon }}}^{1/2})|Y^{H}_i|.$ Next, we use Theorem REF to show that $A^{H,\\ast }_i[Y^{H}_i,V_i]$ is $({\\hat{\\varepsilon }}^{1/18},d_id_Z)$ -super-regular.",
"We call a pair of vertices $u,v\\in V_i$ good if $|N_{A^{H}_i}(u,v)|=(d_i\\pm {\\varepsilon })^2|X^H_i|$ , and $|N_G(u,v)\\cap V_0|=(d_Z\\pm {\\varepsilon })^2|V_0|$ .",
"By the ${\\varepsilon }$ -regularity of $A^{H}_i$ and $G[V_0,V_i]$ , there are at most $2{\\varepsilon }|V_i|^2$ pairs $u,v\\in V_i$ which are not good.",
"For all good pairs $u,v\\in V_i$ , let $S_{u,v}:=N_{H_+}(N_{A^{H}_i}(u,v))\\cap X_0^H$ and $T_{u,v}:=N_G(u,v)\\cap V_0$ .",
"We add the weight function $\\omega _{S_{u,v},T_{u,v}}$ as defined in (REF ) to $\\mathcal {W}$ .",
"Observe that $|S_{u,v}|=|N_{A^{H}_i}(u,v)|\\pm 2{\\varepsilon }n=(d_i\\pm {\\varepsilon }^{1/2})^2n$ and $|T_{u,v}|=(d_Z\\pm {\\varepsilon }^{1/2})^2n.$ By (REF ), we obtain for all good pairs $u,v\\in V_j$ that $|N_{A^{H,\\ast }_i[Y^{H}_i,V_i]}(u,v)|&=|\\sigma (S_{u,v}\\cap {X}_0^\\sigma )\\cap T_{u,v}|=(1\\pm 2{\\hat{\\varepsilon }})|S_{u,v}||T_{u,v}|n^{-1}\\nonumber \\\\&\\le (d_id_Z+{\\hat{\\varepsilon }}^{1/3})^2|Y^{H}_i|.$ Now, by (REF ) and (REF ), we can apply Theorem REF , and obtain that $\\text{$A^{H,\\ast }_i[Y^{H}_i,V_i]$ is $({\\hat{\\varepsilon }}^{1/18},d_id_Z)$-super-regular.",
"}$ In order to complete the proof of REF , we show that we can find a spanning subgraph $A^{H,new}_i$ of $A^{H,\\ast }_i$ that is $({\\varepsilon }^{\\prime },d_id_Z)$ -super-regular.",
"Let $E(A^{H,new}_i[Y^{H}_i,V_i]):=E(A^{H,\\ast }_i[Y^{H}_i,V_i]).$ For every vertex $x\\in X^H_i\\setminus Y^{H}_i$ , we have that $\\operatorname{deg}_{A^{H,*}_i}(x)=(d_i\\pm {\\varepsilon })|V_i|$ because $A^{H}_i$ is $({\\varepsilon },d_i)$ -super-regular.",
"Suppose $\\mathcal {W}^{bad}$ is a collection of at most $n^{4\\log n}$ weight functions $\\omega ^{bad}\\colon E({A}_i^{bad})\\rightarrow [0,s]$ ; we will specify $\\mathcal {W}^{bad}$ explicitly when we establish REF .",
"We claim that we can delete for every vertex $x\\in X^H_i\\setminus Y^{H}_i$ some incident edges in $A^{H,*}_i$ and obtain a subgraph $A_i^{H,new}$ such that $\\operatorname{deg}_{A^{H,new}_i}(x)&=(d_id_Z\\pm 2{\\varepsilon })|V_i| \\text{ for every $x\\in X^H_i\\setminus Y^{H}_i$;}\\\\ \\omega ^{bad}(E({A}_i^{new}))&=(1\\pm {\\varepsilon })d_Z\\omega ^{bad}(E({A}_i^{bad}))\\pm {\\varepsilon }n\\text{ for every $\\omega ^{bad}\\in \\mathcal {W}^{bad}$.",
"}$ This can be easily seen by a probabilistic argument: For all $H\\in \\mathcal {H}$ and $x\\in X^H_i\\setminus Y^{H}_i$ , we keep each edge incident to $x$ in $A^{H}_i$ independently at random with probability $d_Z$ .",
"Then, McDiarmid's inequality (Theorem REF ) together with a union bound yields that (REF ) and () hold simultaneously with probability at least, say, $1/2$ .",
"Since $|X^H_i|=(1\\pm {\\varepsilon })n$ , we have that $|X^H_i\\setminus Y^{H}_i|\\le 4{\\hat{\\varepsilon }}n$ by (REF ).",
"Hence, (REF ) implies together with (REF ) that $A^{H,new}_i$ is $({\\varepsilon }^{\\prime },d_id_Z)$ -super-regular, which establishes REF .",
"Step 2.2.",
"Checking REF For all $H\\in \\mathcal {H},ij\\in E(R_A-\\lbrace 0\\rbrace )$ , and $v_iv_j\\in E(G[V_i,V_j])$ we proceed as follows.",
"Let $\\widetilde{E}:=E(H[N_{A^H_i}(v_i),N_{A^H_j}(v_j)])$ and $S&:=\\lbrace \\lbrace {x_i^{\\prime }},{x_j^{\\prime }} \\rbrace \\subseteq X_0^H \\colon x_ix_i^{\\prime },x_jx_j^{\\prime }\\in E(H_+), x_ix_j\\in \\widetilde{E}\\rbrace ,\\\\S_1&:=\\lbrace S^{\\prime }\\in S\\colon |S^{\\prime }|=1 \\rbrace , \\quad \\text{and}\\quad S_2:=\\lbrace S^{\\prime }\\in S\\colon |S^{\\prime }|=2 \\rbrace ,\\\\E_1&:=\\lbrace xv\\in E(A_0^H)\\colon x\\in S_1,v\\in N_G(v_i,v_j) \\rbrace ,\\\\E_2&:=\\lbrace \\lbrace xv,x^{\\prime }v^{\\prime } \\rbrace \\in \\textstyle \\binom{E(A_0^H)}{2}\\colon \\lbrace x,x^{\\prime } \\rbrace \\in S_2, v\\in N_G(v_i), v^{\\prime }\\in N_G(v_j), v\\ne v^{\\prime }\\rbrace .$ By assumption (see REF ), we have that $|\\widetilde{E}|=(d_id_j\\pm {\\varepsilon })e_{H}(X^H_i,X^H_j)$ .",
"Since $e_{H}(X^H_i,X^H_j)\\ge {\\varepsilon }^{\\prime 2}n$ , we conclude that $|S|=|S_1|+|S_2|=(d_id_j\\pm {\\varepsilon })e_{H}(X^H_i,X^H_j)\\pm 4{\\varepsilon }n=(d_id_j\\pm {\\hat{\\varepsilon }})e_{H}(X^H_i,X^H_j).$ Note that the term of `$\\pm 4{\\varepsilon }n$ ' in (REF ) accounts for possible vertices $x_i\\in N_{A_i^H}(v_i)$ and $x_j\\in N_{A^H_j}(v_j)$ that do not have an $H_+$ -neighbour in $X_0^H$ .",
"We define the following weight functions $\\omega _1\\colon E(A_0^H)\\rightarrow \\lbrace 0,1\\rbrace $ and $\\omega _2\\colon \\binom{E(A_0^H)}{2}\\rightarrow \\lbrace 0,1\\rbrace $ by setting $\\omega _1(e):=\\mathbb {1}_{\\lbrace e\\in E_1 \\rbrace }$ and $\\omega _2(\\lbrace e_1,e_2\\rbrace ):=\\mathbb {1}_{\\lbrace \\lbrace e_1,e_2\\rbrace \\in E_2 \\rbrace }$ and add them to $\\mathcal {W}$ .",
"By the definition of $A_i^{H,new}$ (recall (REF ) and (REF )), we crucially observe that $e_{H}(N_{A_i^{H,new}}(v_i),N_{A_j^{H,new}}(v_j))=\\omega _1(M)+\\omega _2(M)\\pm 5{\\hat{\\varepsilon }}n.$ Note that the term of `$\\pm 5{\\hat{\\varepsilon }}n$ ' in (REF ) accounts for possible vertices $x_i\\in N_{A_i^H}(v_i)$ and $x_j\\in N_{A^H_j}(v_j)$ that do not have an $H_+$ -neighbour in $X_0^H$ (at most $4{\\varepsilon }n$ ), and possible vertices in $S$ that are left unembedded (at most $4{\\hat{\\varepsilon }}n$ by (REF )).",
"Let us for the moment assume that $|S_1|,|S_2|\\ge { {\\varepsilon }^{\\prime 5} n}$ (otherwise the claimed estimations in (REF ) and () below are trivially true).",
"Since $A_0^H$ is $({\\varepsilon },d_0)$ -super-regular and $|N_G(v_i,v_j)\\cap V_0|=(d_A^2\\pm 3{\\varepsilon })n$ by REF , we obtain that $\\omega _1(E(A_0^H))=|E_1|=(d_0\\pm {\\varepsilon })|S_1||N_G(v_i,v_j)\\cap V_0|=(d_0d_A^2\\pm {\\hat{\\varepsilon }})|S_1|n.$ By Fact REF , all but at most $6{\\varepsilon }n$ elements $\\lbrace x_i^{\\prime },x_j^{\\prime }\\rbrace \\in S_2$ are such that $x_k^{\\prime }$ has $(d_0\\pm {\\varepsilon })|N_G(v_k)\\cap V_0|$ neighbours in $N_G(v_k)\\cap V_0$ for both $k\\in \\lbrace i,j\\rbrace $ because $A_0^H$ is $({\\varepsilon },d_0)$ -super-regular and $G[V_0,V_k]$ is $({\\varepsilon },d_A)$ -super-regular.",
"Each of these $6{\\varepsilon }n$ exceptional elements contributes at most $|N_G(v_i)\\cap V_0||N_G(v_j)\\cap V_0|\\le 3n^2$ to $\\omega _2(E(A_0^H))$ .",
"This implies that $\\omega _2(E(A_0^H))=|E_2|&=(d_0\\pm 2{\\varepsilon })^2(|S_2|\\pm 6{\\varepsilon }n)|N_G(v_i)\\cap V_0||N_G(v_j)\\cap V_0|\\pm 18{\\varepsilon }n^3\\nonumber \\\\&=(d_0^2d_A^2\\pm {\\hat{\\varepsilon }})|S_2|n^2.$ For all $e_1\\in E(A_0^H)$ , the number of edges $e_2$ for which $\\lbrace e_1,e_2\\rbrace \\in E_2$ is at most $2n$ , implying $\\Vert \\omega _2\\Vert _{1}\\Delta ^{1+{\\varepsilon }^2}\\le 2n\\Delta ^{1+{\\varepsilon }^2}\\le \\omega _2(E(A_0^H))$ , and clearly, $\\Vert \\omega _2\\Vert _{2}\\Delta ^{2+{\\varepsilon }^2}\\le \\Delta ^{2+{\\varepsilon }^2}\\le \\omega _2(E(A_0^H))$ and $\\Vert \\omega _1\\Vert _{1}\\Delta ^{1+{\\varepsilon }^2}\\le \\Delta ^{1+{\\varepsilon }^2}\\le \\omega _1(E(A_0^H))$ .",
"(Recall that $\\Delta =(d_0+3{\\varepsilon })n$ .)",
"Hence, by (REF ), we conclude that $\\omega _1(M)&=(1\\pm {\\hat{\\varepsilon }})\\frac{\\omega _1(E(A_0^H))}{d_0n}\\stackrel{(\\ref {eq:weight E_1})}{=}d_A^2|S_1|\\pm {{\\varepsilon }^{\\prime 2}} e_{H}(X^H_i,X^H_j),\\\\\\omega _2(M)&=(1\\pm {\\hat{\\varepsilon }})\\frac{\\omega _2(E(A_0^H))}{(d_0n)^2}\\stackrel{(\\ref {eq:weight E^neq})}{=}d_A^2|S_2|\\pm {{\\varepsilon }^{\\prime 2}} e_{H}(X^H_i,X^H_j).$ Clearly, the final equalities in (REF ) and () are also true if $|S_1|,|S_2|<{\\varepsilon }^{\\prime 5} n$ because $e_{H}(X^H_i,X^H_j)\\ge {\\varepsilon }^{\\prime 2}n$ .",
"Now, together with (REF ) and (REF ) this implies that $e_{H}(N_{A_i^{H,new}}(v_i),N_{A_j^{H,new}}(v_j))=d_A^2|S|\\pm 3{{\\varepsilon }^{\\prime 2}} e_{H}(X^H_i,X^H_j)=(d_id_jd_A^2\\pm {\\varepsilon }^{\\prime })e_{H}(X^H_i,X^H_j),$ which establishes REF .",
"Step 2.3.",
"Checking REF We will even show that REF holds for all $\\omega \\in \\mathcal {W}_{edge}\\cup \\mathcal {W}_{edge}^{\\prime }$ with $\\omega \\colon E({A}_i)\\rightarrow [0,s]$ and centre $v\\in V_i, i\\in N_{R_A}(0)$ , where $\\mathcal {W}_{edge}^{\\prime }$ is a set of edge testers that we will explicitly specify in Step REF when establishing REF .",
"For all $\\omega \\in \\mathcal {W}_{edge}\\cup \\mathcal {W}_{edge}^{\\prime }$ with centre $v\\in V_i$ , $i\\in N_{R_A}(0)$ we define a weight function $\\omega _0\\colon E({A}_0)\\rightarrow [0,s]$ by $\\omega _0(x_0v_0):={\\left\\lbrace \\begin{array}{ll}\\omega (x_iv) &\\text{if $ \\lbrace x_i\\rbrace = N_{\\mathcal {H}_+}(x_0)\\cap {X}_i$, $x_iv\\in E({A}_i^{good})$ and $v_0v\\in E(G)$,}\\\\ 0 &\\text{otherwise,}\\end{array}\\right.",
"}$ and we add $\\omega _0$ to $\\mathcal {W}$ .",
"(Recall that ${A}_i^{good}$ is the spanning subgraph of ${A}_i$ containing precisely those edges $x_iv_i\\in E({A}_i)$ , where $N_{\\mathcal {H}_+}(x_i)\\cap {X}_0\\ne \\emptyset $ .)",
"For every edge $x_iv\\in E({A}_i^{good})$ with $\\lbrace x_0\\rbrace = N_{\\mathcal {H}_+}(x_i)\\cap {X}_0$ , property REF yields that $|N_{{A}_0}(x_0)\\cap N_G(v)|=(d_0d_A\\pm 3{\\varepsilon })n.$ Hence, every edge $x_iv\\in E({A}_i^{good})$ contributes weight $\\omega (x_iv)\\cdot (d_0d_A\\pm 3{\\varepsilon }) n$ to $\\omega _0(E({A}_0))$ , and we obtain $\\omega _0(E({A}_0))&= \\omega (E({A}_i^{good})) (d_0d_A\\pm 3{\\varepsilon })n.$ By the definition of ${A}_i^{new}$ (recall (REF ) and (REF )), if $\\sigma (x_0)\\in N_G(v)$ for $\\lbrace x_0\\rbrace = N_{\\mathcal {H}_+}(x_i)\\cap {X}_0$ , then the edge $x_iv\\in E({A}_i^{good})$ is in $E({A}_i^{new})$ .",
"Hence, if $x_0v_0\\in M(\\sigma )=M$ , then this contributes weight $\\omega _0(x_0v_0)$ to $\\omega (E({A}_i^{new}))$ .",
"If $\\omega (E({A}_i^{good}))\\ge {\\varepsilon }n$ , then $\\omega _0(E({A}_0))\\ge n^{1+{\\varepsilon }}\\ge s\\Delta ^{1+{\\varepsilon }^2}\\ge \\Vert \\omega _0\\Vert _{1}\\Delta ^{1+{\\varepsilon }^2}$ , and thus (REF ) implies that $\\omega _0(M)=(1\\pm {\\hat{\\varepsilon }})\\frac{\\omega _0(E({A}_0))}{d_0n}=(1\\pm 2{\\hat{\\varepsilon }})d_A\\omega (E({A}_i^{good}))\\pm {\\hat{\\varepsilon }}n.$ If $\\omega (E({A}_i^{good}))<{\\varepsilon }n$ , then () implies that $\\omega _0(M)\\le \\max \\Big \\lbrace (1+{\\hat{\\varepsilon }})\\frac{\\omega _0(E({A}_0))}{d_0n},n^{\\varepsilon }\\Big \\rbrace \\le {\\hat{\\varepsilon }}n,$ and hence, (REF ) also holds in this case.",
"We now make a key observation: $\\omega (E({A}_i^{new}))=\\omega _0(M)+\\omega (\\Lambda )\\pm \\omega (\\Gamma ),$ for $\\Gamma :=\\lbrace x_iv\\in E({A}_i^{good})\\colon N_{\\mathcal {H}_+}(x_i)\\cap {X}_0^\\sigma =\\emptyset \\rbrace $ and $\\Lambda :=E({A}_i^{bad})\\cap E({A}_i^{new})$ .",
"Next, we want to control $\\omega (\\Gamma )$ and $\\omega (\\Lambda )$ .",
"In order to bound $\\omega (\\Gamma )$ , we define a weight function $\\omega _\\Gamma \\colon E({A}_0)\\rightarrow [0,s]$ by $\\omega _\\Gamma (x_0v_0):={\\left\\lbrace \\begin{array}{ll}\\omega (x_iv) &\\text{if $\\lbrace x_i\\rbrace =N_{\\mathcal {H}_+}(x_0)\\cap {X}_i$, $x_iv\\in E({A}_i^{good})$,}\\\\0 &\\text{otherwise,}\\end{array}\\right.",
"}$ and we add $\\omega _\\Gamma $ to $\\mathcal {W}$ .",
"Observe that $\\omega _\\Gamma (M)$ accounts for the $\\omega $ -weight of edges $x_iv\\in E({A}_i^{good})$ such that $N_{\\mathcal {H}_+}(x_i)\\cap {X}_0\\in {X}_0^\\sigma $ and thus $x_iv\\notin \\Gamma $ .",
"Hence $\\omega (\\Gamma )=\\omega (E({A}_i^{good}))-\\omega _\\Gamma (M)$ .",
"For every vertex $x_0\\in {X}_0$ , we have $\\operatorname{deg}_{{A}_0}(x_0)=(d_0\\pm 3{\\varepsilon })n$ .",
"Hence, every edge $x_iv\\in E({A}_i^{good})$ contributes weight $\\omega (x_iv)\\cdot (d_0\\pm 3{\\varepsilon })n$ to $\\omega _\\Gamma (E({A}_0))$ , and we obtain $\\omega _\\Gamma (E({A}_0))=\\omega (E({A}_i^{good}))(d_0\\pm 3{\\varepsilon })n.$ If $\\omega (E({A}_i^{good}))\\ge {\\varepsilon }n$ , then $\\omega _\\Gamma (E({A}_0))\\ge n^{1+{\\varepsilon }}\\ge s\\Delta ^{1+{\\varepsilon }^2}\\ge \\Vert \\omega _\\Gamma \\Vert _{1}\\Delta ^{1+{\\varepsilon }^2}$ , and thus (REF ) implies that $\\omega _\\Gamma (M)=(1\\pm {\\hat{\\varepsilon }})\\frac{\\omega _\\Gamma (E({A}_0))}{d_0n}=(1\\pm 2{\\hat{\\varepsilon }})\\omega (E({A}_i^{good}))\\pm {\\hat{\\varepsilon }}n.$ Again, if $\\omega (E({A}_i^{good}))<{\\varepsilon }n$ , then () implies that (REF ) also holds in this case.",
"Hence, we conclude that $\\omega (\\Gamma )=\\omega (E({A}_i^{good}))-\\omega _\\Gamma (M)\\stackrel{(\\ref {eq:weight omega_Gamma})}{\\le } 2{\\hat{\\varepsilon }}\\omega (E({A}_i^{good}))+{\\hat{\\varepsilon }}n.$ In order to bound $\\omega (\\Lambda )$ , we use () and add $\\omega |_{E({A}_i^{bad})}$ to $\\mathcal {W}^{bad}$ .",
"Then () implies that $\\omega (\\Lambda )=(1\\pm {\\varepsilon })d_A\\omega (E({A}_i^{bad}))\\pm {\\varepsilon }n.$ Finally, equations (REF ), (REF ), (REF ) and (REF ) yield that $\\omega (E({A}_i^{new}))\\nonumber &= (1\\pm 2{\\hat{\\varepsilon }})d_A\\omega (E({A}_i^{good}))+(1\\pm {\\varepsilon })d_A\\omega (E({A}_i^{bad}))\\pm 2{\\hat{\\varepsilon }}\\omega (E({A}_i^{good}))\\pm 3{\\hat{\\varepsilon }}n\\\\&=(1\\pm {\\varepsilon }^{\\prime 2})d_A\\omega (E({A}_i))\\pm {\\varepsilon }^{\\prime 2}n.$ This establishes REF for all $\\omega \\in \\mathcal {W}_{edge}\\cup \\mathcal {W}_{edge}^{\\prime }$ .",
"Step 2.4.",
"Checking REF We show that for the updated edge set labelling $\\psi [\\sigma ]$ , we have $\\Delta _{\\psi [\\sigma ]}({A}_i^{new})\\le (1+{\\varepsilon }^{\\prime })d_id_A|V_i|$ for every $i\\in N_{R_A}(0)$ .",
"Recall that we defined $\\psi [\\sigma ]$ in Definition REF such that for $xv\\in E(A^{H,new}_i)$ , we have $\\psi [\\sigma ](xv)=\\psi (xv)\\cup \\lbrace \\sigma (x_0)v \\rbrace $ , if $x$ has an $H$ -neighbour $x_0\\in {X}_0^\\sigma $ , and otherwise $\\psi [\\sigma ](xv)=\\psi (xv)$ .",
"We split the proof of REF into two claims, where Claim REF bounds the number of edges on which an `old' label of $\\psi $ appears on the updated candidacy graph, and Claim REF bounds the number of edges on which a `new' label that we additionally added to $\\psi [\\sigma ]$ appears in the updated candidacy graph.",
"Let $\\psi _i\\colon E({A}_i)\\rightarrow 2^{\\mathcal {E}_i}$ be the (old) edge set labelling $\\psi $ restricted to ${A}_i$ and we may assume that $|\\mathcal {E}_i|\\le n^4$ .",
"Claim 1.",
"We can add at most $n^5$ weight functions to $\\mathcal {W}_{edge}^{\\prime }$ to ensure that $\\Delta _{\\psi _i}({A}_i^{new})\\le (1+{\\varepsilon }^{\\prime })d_id_A|V_i|$ for every $i\\in N_{R_A}(0)$ .",
"Proof of claim: For all $i\\in N_{R_A}(0)$ and $e\\in \\mathcal {E}_i$ , let $\\omega _e\\colon E({A}_i)\\rightarrow \\lbrace 0,1\\rbrace $ be such that $\\omega _e(x_iv_i):=\\mathbb {1}_{\\lbrace e\\in \\psi _i(x_iv_i) \\rbrace }$ and we add $\\omega _e$ to $\\mathcal {W}_{edge}^{\\prime }$ .",
"By assumption (see REF ), we have $\\Delta _\\psi ({A}_i)\\le (1+{\\varepsilon })d_i|V_i|$ , which implies that $\\omega _e(E({A}_i))\\le (1+{\\varepsilon })d_i|V_i|$ .",
"Since (REF ) in Step REF is also valid for $\\omega _e\\in \\mathcal {W}_{edge}^{\\prime }$ , we conclude that $e$ appears on at most $(1+{\\varepsilon }^{\\prime 2})d_A(1+{\\varepsilon })d_i|V_i|+{\\varepsilon }^{\\prime 2}n\\le (1+{\\varepsilon }^{\\prime })d_id_A|V_i|$ edges of ${A}_i^{new}$ , which completes the proof of Claim REF .",
"$-$ Claim 2.",
"We can add at most $n^3$ weight functions to $\\mathcal {W}$ to ensure that each $e\\in E(G[V_0,V_i])$ appears on at most $(1+{\\varepsilon }^{\\prime })d_id_A|V_i|$ edges of ${A}_i^{new}$ for every $i\\in N_{R_A}(0)$ .",
"Proof of claim: For all $i\\in N_{R_A}(0)$ and $e=v_0v_i\\in E(G[V_0,V_i])$ , we proceed as follows.",
"Let $N:= N_{{A}_0}(v_0)\\cap N_\\mathcal {H}(N_{{A}_i}(v_i))$ .",
"We define a weight function $\\omega _e\\colon E({A}_0)\\rightarrow \\lbrace 0,1\\rbrace $ by $\\omega _e(xv):=\\mathbb {1}_{\\lbrace v=v_0 \\text{ and }x\\in N\\rbrace }$ for every $xv\\in E({A}_0)$ , and we add $\\omega _e$ to $\\mathcal {W}$ .",
"Then, $e$ appears on $\\omega _e(M)$ edges of ${A}_i^{new}$ .",
"Observe that $\\omega _e(E({A}_0))=|N|&\\stackrel{\\hphantom{\\ref {item:P triple intersection}}}{=}\\sum _{H\\in \\mathcal {H}}e_{H}(N_{A_0^H}(v_0),N_{A^H_i}(v_i))\\nonumber \\\\&\\stackrel{\\ref {item:P triple intersection}}{=}\\sum _{H\\in \\mathcal {H}}(d_0d_i\\pm {\\varepsilon })e_{H}(X_0^H,X^H_i)\\le (d_0d_id_A+{2{\\varepsilon }^{\\prime 3/2}})n^2,$ where the last inequality holds because $\\sum _{H\\in \\mathcal {H}}e_{H}(X_0^H,X^H_i)\\le {(d_A+{\\varepsilon }^{\\prime 3/2})n^2}$ , by assumption.",
"With (), we obtain that $\\omega _e(M)\\le \\max \\Big \\lbrace (1+{\\hat{\\varepsilon }})\\frac{\\omega _e(E({A}_0))}{d_0n},n^{\\varepsilon }\\Big \\rbrace \\stackrel{(\\ref {eq:bound N})}{\\le }(1+{\\varepsilon }^{\\prime })d_id_A|V_i|,$ which completes the proof of Claim REF .",
"$-$ Step 2.5.",
"Checking REF Recall that $\\psi _i\\colon E({A}_i)\\rightarrow 2^{\\mathcal {E}_i}$ denotes the edge set labelling $\\psi $ restricted to ${A}_i$ .",
"For each $i\\in [r]$ , $e=v_0v_i\\in E(G[V_0,V_i])$ , and $f\\in \\mathcal {E}_i$ , we show that $\\lbrace e,f\\rbrace $ appears on at most $\\sqrt{n}$ edges of ${A}^{new}_i$ .",
"This will imply REF because any set $\\lbrace e^{\\prime },f^{\\prime }\\rbrace \\in \\binom{\\mathcal {E}_i}{2}$ appears also on at most $\\Delta ^c_{\\psi }({A}_i)\\le \\sqrt{n}$ edges of ${A}^{new}_i$ , and no two edges of $E(G[V_0,V_i])$ appear together as a label on an edge of ${A}^{new}_i$ .",
"Let ${X}_0^f:=N_{\\mathcal {H}_+}(\\lbrace x_i\\in {X}_i\\colon x_iv_i\\in E({A}_i),f\\in \\psi _i(x_iv_i)\\rbrace )\\cap {X}_0.$ We define a weight function $\\omega _{e,f}\\colon E({A}_0)\\rightarrow \\lbrace 0,1\\rbrace $ by $\\omega _{e,f}(xv):=\\mathbb {1}_{\\lbrace v=v_0 \\text{ and } x\\in {X}_0^f\\rbrace }$ for every $xv\\in E({A}_0)$ and add $\\omega _{e,f}$ to $\\mathcal {W}$ .",
"Since $\\Delta _{\\psi }({A}_i)\\le (1+{\\varepsilon })d_i|V_i|$ by REF , we obtain that $\\omega _{e,f}(E({A}_0))\\le 2n.$ Note that $\\lbrace e,f\\rbrace $ appears on at most $\\omega _{e,f}(M)$ edges of ${A}^{new}_i$ .",
"Now, () implies that $\\omega _{e,f}(M)\\le n^{\\varepsilon }$ , which establishes REF .",
"Step 2.6.",
"Checking REF For each $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}$ with $H_1,\\ldots ,H_\\ell \\in \\mathcal {H}$ such that $Y_j\\subseteq X_0^{H_j}$ , we define $E_{(W,Y_1,\\ldots ,Y_\\ell )}:=\\left\\lbrace \\textstyle \\bigcup _{j\\in [\\ell ]}\\lbrace xy_j \\rbrace \\colon xy_j\\in E(A_0^{H_j}[W,Y_j])\\text{ for all $j\\in [\\ell ]$} \\right\\rbrace \\subseteq \\binom{E({A}_0)}{\\ell },$ and a weight function $\\omega _{(W,Y_1,\\ldots ,Y_\\ell )}\\colon \\binom{E({A}_0)}{\\ell }\\rightarrow \\lbrace 0,1\\rbrace $ by $\\omega _{(W,Y_1,\\ldots ,Y_\\ell )}(\\lbrace e_1,\\ldots ,e_\\ell \\rbrace ):=\\mathbb {1}_{\\lbrace \\lbrace e_1,\\ldots ,e_\\ell \\rbrace \\in E_{(W,Y_1,\\ldots ,Y_\\ell )} \\rbrace }$ and we add $\\omega _{(W,Y_1,\\ldots ,Y_\\ell )}$ to $\\mathcal {W}$ .",
"Observe that $\\omega _{(W,Y_1,\\ldots ,Y_\\ell )}(M)=\\left|W\\cap \\textstyle \\bigcap _{j\\in [\\ell ]}\\sigma (Y_j\\cap {X}_0^\\sigma )\\right|.$ In view of the statement, we may assume that $|W|,|Y_j|\\ge {{\\varepsilon }^{\\prime 2} n}$ for all $j\\in [\\ell ]$ .",
"Since $\\ell \\le s$ and $A_0^H$ is $({\\varepsilon },d_0)$ -super-regular for every $H\\in \\mathcal {H}$ , we obtain with Fact REF that are at most ${\\varepsilon }^{1/2}n$ vertices in $W$ that do not have $(d_0\\pm {\\varepsilon })|Y_j|$ many neighbours in $Y_j$ for every $j\\in [\\ell ]$ .",
"Hence we obtain that $\\omega _{(W,Y_1,\\ldots ,Y_\\ell )}(E({A}_0))=|E_{(W,Y_1,\\ldots ,Y_\\ell )}|=(d_0^\\ell \\pm {{\\varepsilon }^{\\prime 2}})|W||Y_1|\\cdots |Y_\\ell |.$ For $k\\in [\\ell ]$ , any set of $k$ edges $\\lbrace e_1,\\ldots ,e_{k}\\rbrace $ is contained in at most $(2n)^{\\ell -k}$ $\\ell $ -tuples in $E_{(W,Y_1,\\ldots ,Y_\\ell )}$ , which implies that $\\Vert \\omega _{(W,Y_1,\\ldots ,Y_\\ell )}\\Vert _{k}\\Delta ^{k+{\\varepsilon }^2}\\le (2n)^{\\ell -k}\\Delta ^{k+{\\varepsilon }^2}\\stackrel{(\\ref {eq:weight W_set})}{\\le } \\omega _{(W,Y_1,\\ldots ,Y_\\ell )}(E({A}_0)).$ Hence, by (REF ), we conclude that $\\omega _{(W,Y_1,\\ldots ,Y_\\ell )}(M)=(1\\pm {\\hat{\\varepsilon }})\\frac{\\omega _{(W,Y_1,\\ldots ,Y_\\ell )}(E({A}_0))}{(d_0n)^\\ell }\\stackrel{(\\ref {eq:weight W_set})}{=}\\frac{|W||Y_1|\\cdots |Y_\\ell |}{n^\\ell }\\pm {\\varepsilon }^{\\prime }n,$ which establishes REF by (REF ).",
"Step 2.7.",
"Checking REF We add $\\mathcal {W}_{edge}$ to $\\mathcal {W}$ and fix some $\\omega \\in \\mathcal {W}_{edge}$ .",
"If $\\omega (E({A}_0))\\le n^{1+{\\varepsilon }/2}$ , then we obtain by () that $\\omega (M)\\le n^{{\\varepsilon }}$ and thus, $\\omega (M)=(1\\pm {\\varepsilon }^{\\prime }){\\omega (E({A}_0))}/{d_0n}\\pm {\\varepsilon }^{\\prime }n$ .",
"If $\\omega (E({A}_0))\\ge n^{1+{\\varepsilon }/2}$ , then we obtain by (REF ) that $\\omega (M)=(1\\pm {\\hat{\\varepsilon }})\\omega (E({A}_0))/d_0n$ .",
"This establishes REF and completes the proof of Lemma REF .",
"$\\square $"
],
[
"Proof of the main result",
"The following lemma is very similar to Theorem REF .",
"We only require additionally that all graphs in $\\mathcal {H}$ only span a matching between two clusters that is either empty or not too small.",
"This reduction has already been used in [26] (and in several other extensions of the blow-up lemma) and it is also not complicated in our framework.",
"Lemma Let $1/n\\ll {\\varepsilon }\\ll \\alpha ,d$ and $1/n\\ll 1/r$ .",
"Suppose $(\\mathcal {H},G,R,\\mathcal {X},\\mathcal {V},\\phi ^\\ast )$ is an $({\\varepsilon },d)$ -super-regular, $\\alpha ^{-1}$ -bounded and $({\\varepsilon },\\alpha )$ -linked extended blow-up instance, $|V_i|=(1\\pm {\\varepsilon })n$ for all $i\\in [r]$ , and $|\\mathcal {H}|\\le \\alpha ^{-1}n$ .",
"Suppose that $\\sum _{H\\in \\mathcal {H}} e_H(X_{i}^H,X_j^H) \\le (1-\\alpha )dn^2$ for all $ij \\in E(R)$ and $H[X_i^H,X_j^H]$ is a matching of size at least $\\alpha ^{2}n$ if $ij\\in E(R)$ and empty if $ij\\in \\binom{[r]}{2}\\setminus E(R)$ for each $H\\in \\mathcal {H}$ .",
"Suppose $\\mathcal {W}_{set},\\mathcal {W}_{ver}$ are sets of $\\alpha ^{-1}$ -set testers and $\\alpha ^{-1}$ -vertex testers of size at most $n^{2\\log n}$ , respectively.",
"Then there is a packing $\\phi $ of $\\mathcal {H}$ in $G$ which extends $\\phi ^\\ast $ such that $\\phi (X_i^H)=V_i$ for all $i\\in [r]_0$ and $H\\in \\mathcal {H}$ ; $|W\\cap \\bigcap _{j\\in [\\ell ]}\\phi (Y_j)|= |W||Y_1|\\cdots |Y_\\ell |/n^\\ell \\pm \\alpha n$ for all $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}$ ; $\\omega (\\bigcup _{H\\in \\mathcal {H}}X_i^H\\cap \\phi ^{-1}(v))=\\omega (\\bigcup _{H\\in \\mathcal {H}}X_i^H)/n\\pm \\alpha n$ for all $(v,\\omega )\\in \\mathcal {W}_{ver}$ and $v\\in V_i$ .",
"We first prove our main result (Theorem REF ) assuming Lemma REF .",
"Proof of Theorem REF .",
"We choose a new constant $\\beta $ such that ${\\varepsilon }\\ll \\beta \\ll \\alpha ,d$ .",
"For each $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}$ with $W\\subseteq V_i$ , $i\\in [r]$ and $k\\in [\\ell ]$ , let $\\omega _{Y_{k}}\\colon \\bigcup _{H\\in \\mathcal {H}}X_i^H\\rightarrow \\lbrace 0,1\\rbrace $ be such that $\\omega _{Y_{k}}(x)=\\mathbb {1}_{\\lbrace x\\in Y_{k} \\rbrace }$ , and let $\\mathcal {W}_Y$ be the set containing all those weight functions.",
"We delete from every $H\\in \\mathcal {H}$ the set $X_0^H$ and apply Lemma REF to this collection of graphs and the set of weight functions $\\mathcal {W}^\\ast :=\\lbrace \\omega \\colon (v,\\omega )\\in \\mathcal {W}_{ver} \\rbrace \\cup \\mathcal {W}_Y$ , which yields a refined partition of $\\mathcal {H}$ ; to be more precise, for all $H\\in \\mathcal {H}$ and $i\\in [r]$ , we obtain a partition $(X^H_{i,j})_{j\\in [\\beta ^{-1}]}$ of $X^H_i$ satisfying REF –REF of Lemma REF .",
"Let $\\mathcal {X}^{\\prime }$ be the collection of vertex partitions of the graphs in $\\mathcal {H}$ given by $(X_0^H)_{H\\in \\mathcal {H}}$ and $(X^H_{i,j})_{H\\in \\mathcal {H},i\\in [r],j\\in [\\beta ^{-1}]}$ .",
"In particular, Lemma REFREF yields that $\\omega (X_{i,j}^H)=\\beta \\omega (X_i^H)\\pm \\beta ^{3/2}n, \\text{ for all $H\\in \\mathcal {H},\\omega \\in \\mathcal {W}^\\ast ,i\\in [r],j\\in [\\beta ^{-1}]$.",
"}$ Let $R^{\\prime }$ be the graph with vertex set $[r]\\times [\\beta ^{-1}]$ and two vertices $(i,j),(i^{\\prime },j^{\\prime })$ are joined by an edge if $ii^{\\prime }\\in E(R)$ .",
"Note that $\\Delta (R^{\\prime })\\le \\alpha ^{-1}\\beta ^{-1}$ because $\\Delta (R)\\le \\alpha ^{-1}$ .",
"According to the refinement $\\mathcal {X}^{\\prime }$ of $\\mathcal {X}$ , we claim that there exists a refined partition $\\mathcal {V}^{\\prime }$ of $\\mathcal {V}$ consisting of the collection of $V_0$ together with $(V_{i,j})_{i\\in [r],j\\in [\\beta ^{-1}]}$ , where $(V_{i,j})_{j\\in [\\beta ^{-1}]}$ is a partition of $V_i$ for every $i\\in [r]$ such that $|W\\cap V_{i,j}|=\\beta |W|\\pm \\beta ^{3/2}n$ for all $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}$ and $j\\in [\\beta ^{-1}]$ with $W\\subseteq V_i, \\ell \\in [\\alpha ^{-1}]$ ; $(\\mathcal {H},G,R^{\\prime },\\mathcal {X}^{\\prime },\\mathcal {V}^{\\prime },\\phi _0)$ is an $({\\varepsilon }^{1/2},d)$ -super-regular, $\\beta ^{-2}$ -bounded and $({\\varepsilon }^{1/2},\\alpha /2)$ -linked extended blow-up instance.",
"Indeed, the existence of $\\mathcal {V}^{\\prime }$ follows by a simple probabilistic argument.",
"For each $i\\in [r]$ , let $\\tau _i\\colon V_i\\rightarrow [\\beta ^{-1}]$ where $\\tau _i(v)$ is chosen uniformly at random for every $v\\in V_i$ , all independently, and let $V_{i,j}:=\\lbrace v\\in V_i\\colon \\tau _i(v)=j\\rbrace $ for every $j\\in [\\beta ^{-1}]$ .",
"Chernoff's inequality and a union bound imply that REF holds simultaneously together with the following properties with probability at least $1-{\\rm e}^{-\\sqrt{n}}$ : $G[V_{i,j},V_{i^{\\prime },j^{\\prime }}]$ is $({\\varepsilon }^{1/2},d)$ -super-regular for all $ii^{\\prime }\\in E(R), j,j^{\\prime }\\in [\\beta ^{-1}]$ ; $|\\bigcap _{x_0\\in X_0^H\\cap N_H(x)} N_G(\\phi _0(x_0))\\cap V_{i,j}| \\ge \\alpha /2 |V_{i,j}|$ for all $x\\in X_{i,j}^H, i\\in [r],j\\in [\\beta ^{-1}], H\\in \\mathcal {H}$ .",
"Standard properties of the multinomial distribution yield that $|V_{i,j}|=|X_{i,j}^H|$ for all $i\\in [r]$ , $j\\in [\\beta ^{-1}]$ , $H\\in \\mathcal {H}$ with probability at least $\\Omega (n^{-r\\beta ^{-1}})$ .",
"To see in REF that the instance is $({\\varepsilon }^{1/2},\\alpha /2)$ -linked, observe further that the number of vertices in $X_{i,j}^H$ that have a neighbour in $X_0^H$ is at most ${\\varepsilon }|X_i^H|\\le {\\varepsilon }^{1/2}|X_{i,j}^H|$ and $\\sum _{H\\in \\mathcal {H}}|N_H(\\phi _0^{-1}(v_0),\\phi _0^{-1}(v_0^{\\prime }))\\cap X_{i,j}^H|\\le {\\varepsilon }|V_i|^{1/2}\\le {\\varepsilon }^{1/2}|V_{i,j}|^{1/2}$ for all $i\\in [r]$ , $j\\in [\\beta ^{-1}]$ and distinct $v_0,v_0^{\\prime }\\in V_0$ .",
"Thus, for every $i\\in [r]$ , there exists a partition $(V_{i,j})_{j\\in [\\beta ^{-1}]}$ of $V_i$ satisfying REF and REF .",
"Let $n^{\\prime }:=\\beta n$ .",
"Next we show how to lift the vertex and set testers from the original blow-up instance to the just defined blow-up instance.",
"For each $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}$ and distinct $H_1,\\ldots ,H_\\ell \\in \\mathcal {H}$ such that $W\\subseteq V_i$ for some $i\\in [r]$ and $Y_{k}\\subseteq X_i^{H_{k}}$ for all $k\\in [\\ell ]$ , we define $(W_j,Y_{1,j},\\ldots ,Y_{\\ell ,j})$ by setting $W_j:=W\\cap V_{i,j}$ and $Y_{k,j}:=Y_{k}\\cap X_{i,j}^{H_{k}}$ for all $j\\in [\\beta ^{-1}], k\\in [\\ell ]$ .",
"By REF , we conclude that $|W_j|=\\beta |W|\\pm \\beta ^{3/2}n$ , and by (REF ), we have that $|Y_{k,j}|=\\omega _{Y_{k}}(X_{i,j}^{H_{k}})=\\beta \\omega _{Y_{k}}(X_i^{H_{k}}) \\pm \\beta ^{3/2}n =\\beta |Y_{k}|\\pm \\beta ^{3/2}n$ .",
"Let $\\mathcal {W}_{set}^{\\prime }:=\\lbrace (W_j,Y_{1,j},\\ldots ,Y_{\\ell ,j})\\colon j\\in [\\beta ^{-1}],(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set} \\rbrace $ .",
"For each $(v,\\omega )\\in \\mathcal {W}_{ver}$ with $v\\in V_{i,j}$ , let $\\omega ^{\\prime }:=\\omega |_{\\bigcup _{H\\in \\mathcal {H}}X_{i,j}^H}$ and $\\mathcal {W}_{ver}^{\\prime }:=\\lbrace (v,\\omega ^{\\prime })\\colon (v,\\omega )\\in \\mathcal {W}_{ver} \\rbrace $ .",
"Next, we add some edges to the graphs in $\\mathcal {H}$ ensuring that all matchings between two clusters are either empty or of small linear size.",
"To this end, we add a minimum number of edges to $H[X_{i,j}^H,X_{i^{\\prime },j^{\\prime }}^H]$ for all $\\lbrace (i,j),(i^{\\prime },j^{\\prime })\\rbrace \\in E(R^{\\prime })$ and $H\\in \\mathcal {H}$ such that the obtained supergraph $H^{\\prime }[X_{i,j}^{H^{\\prime }},X_{i^{\\prime },j^{\\prime }}^{H^{\\prime }}]$ is a matching of size at least $\\beta ^4n$ .",
"Note that $\\Delta (H^{\\prime })\\le \\Delta (R^{\\prime })+\\alpha ^{-1}\\le \\beta ^{-2}$ .",
"Let $\\mathcal {H}^{\\prime }$ be the collection of graphs $H^{\\prime }$ obtained in this manner.",
"Together with Lemma REFREF , we conclude for all $\\lbrace (i,j),(i^{\\prime },j^{\\prime })\\rbrace \\in E(R^{\\prime })$ that $\\sum _{H^{\\prime }\\in \\mathcal {H}^{\\prime }}e_{H^{\\prime }}(X_{i,j}^{H^{\\prime }},X_{i^{\\prime },j^{\\prime }}^{H^{\\prime }})&\\le {2\\beta ^4n\\cdot \\alpha ^{-1}n}+ \\sum _{H\\in \\mathcal {H}}e_H(X_{i,j}^{H},X_{i^{\\prime },j^{\\prime }}^{H})\\\\&\\le \\beta ^3n^2+ \\beta ^2\\sum _{H\\in \\mathcal {H}}e_H(X_{i}^H,X_{i^{\\prime }}^H) + n^{5/3}\\le (1-\\alpha /2)dn^{\\prime 2}.$ Obviously, it suffices to construct a packing of $\\mathcal {H}^{\\prime }$ into $G$ which extends $\\phi _0$ and satisfies Theorem REFREF –REF .",
"By REF and because $\\beta \\ll \\alpha $ , also $(\\mathcal {H}^{\\prime },G,R^{\\prime },\\mathcal {X}^{\\prime },\\mathcal {V}^{\\prime },\\phi _0)$ is an $({\\varepsilon }^{1/2},d)$ -super-regular, $\\beta ^{-2}$ -bounded and $({\\varepsilon }^{1/2},{\\beta ^2})$ -linked extended blow-up instance, and we can apply Lemma REF to $(\\mathcal {H}^{\\prime },G,R^{\\prime },\\mathcal {X}^{\\prime },\\mathcal {V}^{\\prime },\\phi _0)$ with set testers $\\mathcal {W}_{set}^{\\prime }$ and vertex testers $\\mathcal {W}_{ver}^{\\prime }$ as follows: Table: NO_CAPTIONHence, we obtain a packing $\\phi $ of $\\mathcal {H}^{\\prime }$ in $G$ which extends $\\phi _0$ such that for all $i\\in [r], j\\in [\\beta ^{-1}]$ $\\phi (X_{i,j}^H)\\subseteq V_{i,j}$ for all $H\\in \\mathcal {H}$ ; $|W_j\\cap \\bigcap _{k\\in [\\ell ]}\\phi (Y_{k,j})|= |W_j||Y_{1,j}|\\cdots |Y_{\\ell ,j}|/n^{\\prime \\ell }\\pm {\\beta ^2} n^{\\prime }$ for all $(W_j,Y_{1,j},\\ldots ,Y_{\\ell ,j})\\in \\mathcal {W}_{set}^{\\prime }$ ; $\\omega ^{\\prime }(\\bigcup _{H\\in \\mathcal {H}}X_{i,j}^H \\cap \\phi ^{-1}(v))=\\omega ^{\\prime }(\\bigcup _{H\\in \\mathcal {H}}X_{i,j}^H)/n^{\\prime }\\pm {\\beta ^2} n^{\\prime }$ for all $(v,\\omega ^{\\prime })\\in \\mathcal {W}_{ver}^{\\prime }$ with $v\\in V_{i,j}$ .",
"Observe that REF establishes Theorem REFREF .",
"For $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}$ , we conclude that $\\left|W\\cap \\bigcap _{k\\in [\\ell ]}\\phi (Y_{k})\\right|&\\stackrel{\\hphantom{\\rm \\ref {item:II},(\\ref {eq:thm split 4})}}{=} \\sum _{j\\in [\\beta ^{-1}]}\\left|W_j\\cap \\bigcap _{k\\in [\\ell ]}\\phi (Y_{k,j})\\right|\\\\&\\stackrel{\\rm \\ref {item:II},(\\ref {eq:thm split 4})}{=} \\sum _{j\\in [\\beta ^{-1}]}\\left(\\frac{\\beta ^{\\ell +1}\\left(|W||Y_1|\\cdots |Y_\\ell |\\pm \\beta ^{1/3}n^{\\ell +1}\\right)}{(\\beta n)^\\ell }\\pm {\\beta ^2} n^{\\prime }\\right)\\\\&\\stackrel{\\hphantom{\\rm \\ref {item:II},(\\ref {eq:thm split 4})}}{=}|W||Y_1|\\cdots |Y_\\ell |/n^\\ell \\pm \\alpha n.$ Hence, Theorem REFREF holds.",
"For $(v,\\omega )\\in \\mathcal {W}_{ver}$ with $v\\in V_{i,j}$ and its corresponding tuple $(v,\\omega ^{\\prime })\\in \\mathcal {W}_{ver}^{\\prime }$ , we conclude that $\\omega (\\textstyle \\bigcup _{H\\in \\mathcal {H}}X_{i}^H\\cap \\phi ^{-1}(v))&\\stackrel{~\\rm \\ref {item:I}~}{=}\\omega ^{\\prime }(\\textstyle \\bigcup _{H\\in \\mathcal {H}}X_{i,j}^H\\cap \\phi ^{-1}(v))\\stackrel{\\rm \\ref {item:III}}{=}\\omega ^{\\prime }(\\bigcup _{H\\in \\mathcal {H}}X_{i,j}^H)/n^{\\prime }\\pm {\\beta ^2} n^{\\prime }\\\\&\\stackrel{(\\ref {eq:thm split 4})}{=}\\frac{\\beta \\omega (\\textstyle \\bigcup _{H\\in \\mathcal {H}} X_i^H)\\pm \\beta ^{4/3}n^2}{\\beta n}\\pm {\\beta ^2} n^{\\prime }= \\omega (\\textstyle \\bigcup _{H\\in \\mathcal {H}} X_i^H)/n\\pm \\alpha n.$ This yields Theorem REFREF and completes the proof.",
"$\\square $ Theorem REF can be easily deduced from Theorem REF by randomly partitioning $G$ and applying Lemma REF to $\\mathcal {H}$ with $r=1$ .",
"In particular, the proof is very similar to the proof of Theorem REF and therefore omitted.",
"We proceed with the proof of Lemma REF .",
"Proof of Lemma REF .",
"We split the proof into four steps.",
"In Step , we partition $G$ into two edge-disjoint subgraphs $G_A$ and $G_B$ .",
"In Step , we define `candidacy graphs' that we track for the partial packing in Step , where we iteratively apply Lemma REF to consider the clusters in turn.",
"We only use the edges of $G_A$ for the partial packing in Step such that we can complete the packing in Step using the edges of $G_B$ and the ordinary blow-up lemma.",
"We will proceed cluster by cluster in Step to find a function that packs almost all vertices of $\\mathcal {H}$ into $G$ .",
"Since $r$ may be much larger than ${\\varepsilon }^{-1}$ , we need to carefully control the growth of the error term.",
"We do so, by considering a proper vertex colouring $c\\colon V(R)\\rightarrow [T]$ of $R^3$ where $T:=\\alpha ^{-3}$ , and choose new constants ${\\varepsilon }_0,{\\varepsilon }_1,\\ldots ,{\\varepsilon }_T,\\mu ,\\gamma $ such that ${\\varepsilon }\\ll {\\varepsilon }_0\\ll {\\varepsilon }_1\\ll \\cdots \\ll {\\varepsilon }_T\\ll \\mu \\ll \\gamma \\ll \\alpha ,d$ .",
"To obtain the order in which we consider the clusters in turn, we simply relabel the cluster indices such that the colour values are non-decreasing; that is, $c(1)\\le \\cdots \\le c(r)$ .",
"Note that the sets $(c^{-1}(k))_{k\\in [T]}$ are independent in $R^3$ .",
"For $i\\in [r], t\\in [r]_0$ , let $c_i(t):=\\max \\lbrace \\lbrace 0\\rbrace \\cup \\lbrace c(j)\\colon j\\in N_{R}[i]\\cap [t]\\rbrace \\rbrace ,\\quad \\text{and} \\quad m_i(t):=|N_R(i)\\cap [t]|.$ That is, if we think of $[t]$ as the indices of clusters that have already been embedded, then $c_i(t)$ denotes the largest colour of an already embedded cluster in the closed neighbourhood of $i$ in $R$ , and $m_i(t)$ denotes the number of neighbours of $i$ in $R$ that have already been embedded.",
"For $t\\in [r]_0$ , let ${X}_{t}&:=\\textstyle \\bigcup _{H\\in \\mathcal {H}}X^H_{t},& \\mathcal {X}_t&:=\\textstyle \\bigcup _{\\ell \\in [t]_0}{X}_{\\ell },& \\mathcal {V}_t&:=\\textstyle \\bigcup _{\\ell \\in [t]_0}V_{\\ell }.$ For every vertex tester $(v,\\omega )\\in \\mathcal {W}_{ver}$ with $v\\in V_i$ for some $i\\in [r]$ , we define its corresponding function $\\omega _v$ on $\\lbrace x_iv_i\\colon x_i\\in {X}_i,v_i\\in V_i \\rbrace $ by setting $\\omega _v(x_iv_i):=\\omega (x_i)\\mathbb {1}_{\\lbrace v_i=v\\rbrace }$ .",
"Let $\\mathcal {W}_{edge}^i:=\\lbrace \\omega _v \\colon (v,\\omega )\\in \\mathcal {W}_{ver}, v\\in V_i \\rbrace .$ Step 1.",
"Partitioning the edges of $G$ In order to reserve an exclusive set of edges for the completion in Step , we partition the edges of $G$ into two subgraphs $G_A$ and $G_B$ .",
"For each edge $e$ of $G$ independently, we add $e$ to $G_B$ with probability $\\gamma $ and otherwise to $G_A$ .",
"Let $d_A:=(1-\\gamma )d$ , $d_B:=\\gamma d$ , $\\alpha _A:=(1-\\gamma )^{\\alpha ^{-1}}\\alpha /2$ and $\\alpha _B:=\\gamma ^{\\alpha ^{-1}}\\alpha /2$ .",
"Using Chernoff's inequality, we can easily conclude that with probability at least $1-1/n$ we have for all $Z\\in \\lbrace A,B\\rbrace $ that $\\begin{minipage}[c]{0.8}\\em G_Z[V_i,V_j] is (2{\\varepsilon },d_Z)-super-regular for all ij\\in E(R),\\end{minipage}\\\\\\begin{minipage}[c]{0.8}\\em |V_i\\cap \\bigcap _{x_0\\in X_0^H\\cap N_H(x)} N_{G_Z}(\\phi ^\\ast (x_0))|\\ge \\alpha _Z|V_i| for all x\\in X_i^H, i\\in [r], H\\in \\mathcal {H}.\\end{minipage}$ Hence, we may assume that $G$ is partitioned into $G_A$ and $G_B$ such that (REF ) and () hold.",
"Step 2.",
"Candidacy graphs For $t\\in [r]_0$ , we call $\\phi \\colon \\bigcup _{H\\in \\mathcal {H},i\\in [t]_0}\\widehat{X}^H_i\\rightarrow \\mathcal {V}_t$ a $t$ -partial packing if $\\widehat{X}^H_i\\subseteq X^H_i$ , $\\phi |_{\\widehat{X}^H_0}=\\phi ^\\ast |_{X_0^H}$ , and $\\phi (\\widehat{X}^H_i)\\subseteq V_i$ for all $H\\in \\mathcal {H}, i\\in [t]_0$ such that $\\phi $ is a packing of $(H[\\widehat{X}^H_0\\cup \\ldots \\cup \\widehat{X}^H_t])_{H\\in \\mathcal {H}}$ into $G_A[\\mathcal {V}_t]$ .",
"Note that $\\widehat{X}^H_0=X^H_0$ , and $\\phi |_{\\widehat{X}^H_i}$ is injective for all $H\\in \\mathcal {H}$ and $i\\in [t]_0$ .",
"For convenience, we often write $\\mathcal {X}_t^{\\phi } :=\\textstyle \\bigcup _{H\\in \\mathcal {H},i\\in [t]_0}\\widehat{X}^H_i.$ Suppose $t\\in [r]_0$ and $\\phi _t\\colon \\mathcal {X}_t^{\\phi _t}\\rightarrow \\mathcal {V}_t$ is a $t$ -partial packing.",
"We introduce the notion of candidates (with respect to $\\phi _t$ ) for future packing rounds and track those relations in two kinds of bipartite auxiliary graphs that we call candidacy graphs: A graph $A^H_i(\\phi _t)$ with bipartition $(X^H_i,V_i)$ , $i\\in [r]$ that will be used to extend the $t$ -partial packing $\\phi _t$ to a $(t+1)$ -partial packing $\\phi _{t+1}$ via Lemma REF in Step , and a graph $B^H_i(\\phi _t)$ that will be used for the completion in Step .",
"For convenience, we define $B^H_i(\\phi _t)$ on a copy $(X^{H,B}_i,V_i^B)$ of $(X^H_i,V_i)$ .",
"That is, for all $H\\in \\mathcal {H}, i\\in [r]$ , let $X^{H,B}_i$ and $V_i^B$ be disjoint copies of $X^H_i$ and $V_i$ , respectively.",
"Let $\\pi $ be the bijection that maps a vertex in $\\bigcup _{H\\in \\mathcal {H},i\\in [r]}(X^H_i\\cup V_i)$ to its copy in $\\bigcup _{H\\in \\mathcal {H},i\\in [r]}(X^{H,B}_i\\cup V_i^B)$ .",
"Let $G_+$ and $H_+$ be supergraphs of $G_A$ and $H\\in \\mathcal {H}$ with vertex partitions $(V_0,\\ldots ,V_r,V_1^B,\\ldots ,V_r^B)$ and $(X^H_0,\\ldots , X^H_r,X^{H,B}_1,\\ldots , X^{H,B}_r)$ , respectively, and edge sets $E(G_+) &:= E(G_A)\\cup \\lbrace u\\pi (v)\\colon uv\\in E(G_B) \\rbrace , \\\\E(H_+) &:= E(H)\\cup \\lbrace u\\pi (v)\\colon uv\\in E(H) \\rbrace .$ Let $R_B$ be the graph on $[r]\\cup \\lbrace 1^B,,\\ldots ,r^B\\rbrace $ with edge set $E(R_B) :=\\lbrace ij^B\\colon ij\\in E(R) \\rbrace .$ By taking copies $(X^{H,B}_i,V_i^B)$ for all $(X^H_i,V_i)$ and defining the candidacy graphs $B^H_i(\\phi _t)$ on these copies, and by enlarging $G$ , $H$ and $R$ accordingly to $G_+$ , $H_+$ and $R\\cup R_B$ , we will be able to update the candidacy graphs $A^H_i(\\phi _t)$ and $B^H_i(\\phi _t)$ simultaneously in Step when we apply Lemma REF in order to extend $\\phi _t$ to a $(t+1)$ -partial packing $\\phi _{t+1}$ .",
"We now define $A^H_i(\\phi _t)$ and $B^H_i(\\phi _t)$ .",
"Let $X^{H,A}_i:=X^H_i$ and $V_i^A:=V_i$ for all $H\\in \\mathcal {H}$ , $i\\in [r]$ .",
"For $Z\\in \\lbrace A,B\\rbrace $ , $H\\in \\mathcal {H}$ and $i\\in [r]$ , we say that $v\\in V_i^Z$ is a candidate for $x\\in X_i^{H,Z}$ given $\\phi _t$ if $\\phi _t(N_{H_+}(x)\\cap \\mathcal {X}_t^{\\phi _t})\\subseteq N_{G_+}(v).$ For all $Z\\in \\lbrace A,B\\rbrace $ , let $Z^H_i(\\phi _t)$ be a bipartite graph with vertex partition $(X^{H,Z}_i,V_i^Z)$ and edge set $E(Z^H_i(\\phi _t)):=\\lbrace xv\\colon x\\in X^{H,Z}_i, v\\in V_i^Z,\\text{ and $v$ is a candidate for $x$ given $\\phi _t$} \\rbrace .$ We call every spanning subgraph of $Z^H_i(\\phi _t)$ a candidacy graph (with respect to $\\phi _t$ ).",
"Furthermore, for all $H\\in \\mathcal {H}$ and $i\\in [r]$ , we assign to every edge $xv\\in E(A^H_i(\\phi _t))$ an edge set labelling $\\psi _t(xv)$ of size at most $\\alpha ^{-1}$ .",
"This set encodes the edges between $v$ and $\\phi _t(N_{H}(x)\\cap \\mathcal {X}_t^{\\phi _t})$ in $G_A$ that are covered if we embed $x$ onto $v$ ; to be more precise, for all $H\\in \\mathcal {H}$ , $i\\in [r]$ , and every edge $xv\\in E(A^H_i(\\phi _t))$ , we set $\\psi _t(xv):=E\\big (G_A\\big [\\phi _t(N_{H}(x)\\cap \\mathcal {X}_t^{\\phi _t}),\\lbrace v\\rbrace \\big ]\\big ).$ Tracking this set enables us to extend a $t$ -partial packing $\\phi _t$ to a $(t+1)$ -partial packing $\\phi _{t+1}$ by finding a conflict-free embedding (see definition in (REF )) in $\\bigcup _{H\\in \\mathcal {H}}A^H_{t+1}(\\phi _t)$ via Lemma REF .",
"Since $|N_{H}(x)\\cap \\mathcal {X}_t^{\\phi _t}|\\le {\\alpha ^{-1}}$ , we have $|\\psi _t(xv)|\\le {\\alpha ^{-1}}$ .",
"Before we proceed to Step and extend $\\phi _t$ to $\\phi _{t+1}$ , we consider the candidacy graphs and their edge set labelling with respect to $\\phi ^\\ast $ .",
"Claim 1.",
"For all $H\\in \\mathcal {H}, i\\in [r], Z\\in \\lbrace A,B\\rbrace $ , there exists a candidacy graph $Z^H_i\\subseteq Z^H_i(\\phi ^\\ast )$ with respect to $\\phi ^\\ast $ (where ${A}_i:=\\bigcup _{H\\in \\mathcal {H}}A_i^H$ ) such that $Z^H_i$ is $({\\varepsilon }_{0},\\alpha _Z)$ -super-regular; $\\Delta _{\\psi _0}({A}_i)\\le {\\varepsilon }_0 n$ ; $\\Delta ^c_{\\psi _0}({A}_i)\\le \\sqrt{n}$ ; $e_{H}(N_{A^H_i}(v_i),N_{A^H_j}(v_j))=(\\alpha _A^2\\pm {\\varepsilon }_{0})e_{H}(X_i^H,X_j^H)$ for all $v_iv_j\\in E(G_A[V_i,V_j])$ , $ij\\in E(R)$ ; $\\omega _v(E({A}_i))=\\alpha _A\\omega ({X}_{i})\\pm {\\varepsilon }_0n^2$ for all $\\omega _v\\in \\mathcal {W}_{edge}^i$ .",
"Proof of claim: We fix $H\\in \\mathcal {H}, i\\in [r], ij\\in E(R), Z\\in \\lbrace A,B\\rbrace $ , $v_iv_j\\in E(G_A[V_i,V_j])$ and $\\omega _v\\in \\mathcal {W}_{edge}^i$ as defined in (REF ).",
"For each $k\\in \\lbrace i,j\\rbrace $ , let $\\widetilde{X}^H_k$ be the set of vertices in $X^H_k$ that have a neighbour in $X_0^H$ .",
"Observe that $Z^H_k(\\phi ^\\ast )[X^H_k\\setminus \\widetilde{X}^H_k,V_k]$ is a complete bipartite graph for $k\\in \\lbrace i,j\\rbrace $ , and $e_{H}(N_{Z^H_i(\\phi ^\\ast )[X^H_i\\setminus \\widetilde{X}^H_i,V_i]}(v_i),N_{Z^H_j(\\phi ^\\ast )[X^H_j\\setminus \\widetilde{X}^H_j,V_j]}(v_j))=e_{H}(X^H_i,X^H_j)\\pm 4{\\varepsilon }n$ because $|\\widetilde{X}^H_k|\\le {\\varepsilon }|X^H_k|\\le 2{\\varepsilon }n$ .",
"By () and the definition of candidates in (REF ), we obtain that $\\operatorname{deg}_{Z^H_i(\\phi ^\\ast )}(x_i)\\ge \\alpha _Z|V_i|$ for all $x_i\\in \\widetilde{X}^H_i$ .",
"Note further that $\\omega _v(E(\\bigcup _{H\\in \\mathcal {H}}A^H_i[X_i^H\\setminus \\widetilde{X}^H_i,V_i]))=\\omega ({X}_i)\\pm {\\varepsilon }^{1/2}n^2$ .",
"Hence, there exists a subgraph $Z^H_i\\subseteq Z^H_i(\\phi ^\\ast )$ that satisfies REF , REF and REF , which can be seen by keeping each edge in $Z^H_i(\\phi ^\\ast )[X^H_i\\setminus \\widetilde{X}^H_i,V_i]$ independently at random with probability $\\alpha _Z$ and by possibly removing some edges incident to $x_i\\in \\widetilde{X}_i^H$ in $Z^H_i(\\phi ^\\ast )$ deterministically.",
"In order to see REF , note that $|\\psi _0^{-1}(v_0v_i)|\\le |N_{\\mathcal {H}}(\\phi _0^{-1}(v_0))\\cap {X}_i|\\le {\\alpha ^{-1}}|\\phi _0^{-1}(v_0)|\\le {\\alpha ^{-1}}{\\varepsilon }|\\mathcal {H}|\\le {\\varepsilon }_0n.$ Since the blow-up instance is $({\\varepsilon },\\alpha )$ -linked, we have $\\sum _{H\\in \\mathcal {H}}|N_H(\\phi _0^{-1}(v_0),\\phi _0^{-1}(v_0^{\\prime }))\\cap X_i^H|\\le {\\varepsilon }|V_i|^{1/2}$ for all distinct $v_0v_i,v_0^{\\prime }v_i\\in E(G[V_0,V_i])$ , $i\\in [r]$ , which implies REF .",
"This completes the proof of the claim.",
"$-$ Step 3.",
"Induction We inductively prove that the following statement $\\mathbf {S}(t)$ holds for all $t\\in [r]_0$ , which will provide a partial packing of $\\mathcal {H}$ into $G_A$ .",
"$\\mathbf {S}(t)$ .",
"For all $H\\in \\mathcal {H}$ and $Z\\in \\lbrace A,B\\rbrace $ , there exists a $t$ -partial packing $\\phi _t\\colon \\mathcal {X}_t^{\\phi _t}\\rightarrow \\mathcal {V}_t$ with $|\\mathcal {X}_t^{\\phi _t}\\cap X^H_i|\\ge (1-{\\varepsilon }_{c_i(t)})n$ for all $i\\in [t]$ , and there exists a candidacy graph $Z^H_i\\subseteq Z^H_i(\\phi _t)$ (where ${A}_i:=\\bigcup _{H\\in \\mathcal {H}}A_i^H$ ) such that $Z^H_i$ is $({\\varepsilon }_{c_i(t)},\\alpha _Z d_Z^{m_i(t)})$ -super-regular for all $i\\in [r]\\setminus [t]$ if $Z=A$ and for all $i\\in [r]$ if $Z=B$ ; $\\Delta _{\\psi _t}({A}_i)\\le (1+{\\varepsilon }_{c_i(t)})\\alpha _Ad_A^{m_i(t)}|V_i|$ for all $i\\in [r]\\setminus [t]$ ; $\\Delta ^c_{\\psi _t}({A}_i)\\le \\sqrt{n}$ for all $i\\in [r]\\setminus [t]$ ; $e_{H}(N_{A^H_i}(v_i),N_{A^H_j}(v_j))=(\\alpha _A^2 d_A^{m_i(t)+m_j(t)}\\pm {\\varepsilon }_{\\max \\lbrace c_i(t),c_j(t)\\rbrace })e_{H}(X^H_i,X^H_j)$ for all $H\\in \\mathcal {H}$ , $ij\\in E(R-[t])$ and $v_iv_j\\in E(G_A[V_i,V_j])$ ; $|\\phi _t^{-1}(v)|\\ge (1-{\\varepsilon }_{c_i(t)}^{1/2})|\\mathcal {H}|-{\\varepsilon }_{c_i(t)}n$ and $|\\phi _t^{-1}(v)\\cap N_{\\mathcal {H}}(\\mathcal {X}_t\\setminus \\mathcal {X}_t^{\\phi _t})|\\le {\\varepsilon }_{c_i(t)}^{1/2}{n}$ for all $v\\in V_i$ , $i\\in [t]$ ; $\\omega _v(E({A}_i))=\\alpha _Ad_A^{m_i(t)}\\omega ({X}_{i})\\pm {\\varepsilon }_{c_i(t)}n^2$ for all $\\omega _v\\in \\mathcal {W}_{edge}^i$ and $i\\in [r]\\setminus [t]$ ; $|W\\cap \\bigcap _{j\\in [\\ell ]}\\phi _t(Y_j\\cap \\mathcal {X}_t^{\\phi _t})|= |W||Y_1|\\cdots |Y_\\ell |/n^\\ell \\pm \\alpha n/2$ for all $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}$ with $W\\subseteq V_i$ , $i\\in [t]$ ; $\\omega ({X}_i\\cap \\phi _t^{-1}(v))=\\omega ({X}_i)/n\\pm \\alpha n/2$ for all $(v,\\omega )\\in \\mathcal {W}_{ver}$ with $v\\in V_i$ , $i\\in [t]$ .",
"Properties $\\mathbf {S}(t)$REF –REF will be used particularly to establish $\\mathbf {S}(t+1)$ by applying Lemma REF .",
"Property REF enables us to establish REF , which together with REF basically implies Lemma REFREF and REF as we merely modify the $r$ -partial packing $\\phi _r$ for the completion in Step where we exploit REF (for $Z=B$ ) and REF .",
"The statement $\\mathbf {S}(0)$ holds for $\\phi _0=\\phi ^\\ast $ by Claim .",
"Hence, we assume the truth of $\\mathbf {S}(t)$ for some $t\\in [r-1]_0$ and let $\\phi _t\\colon \\mathcal {X}_t^{\\phi _t}\\rightarrow \\mathcal {V}_t$ and $A^H_i$ and $B^H_i$ be as in $\\mathbf {S}(t)$ ; we set ${A}_i:=\\bigcup _{H\\in \\mathcal {H}}A_i^H$ and ${B}_i:=\\bigcup _{H\\in \\mathcal {H}}B^H_i$ .",
"We will extend $\\phi _t$ such that $\\mathbf {S}(t+1)$ holds.",
"Any function $\\sigma \\colon {X}_{t+1}^\\sigma \\rightarrow V_{t+1}$ with ${X}_{t+1}^\\sigma \\subseteq {X}_{t+1}$ extends $\\phi _t$ to a function $\\phi _{t+1}\\colon \\mathcal {X}_t^{\\phi _t}\\cup {X}_{t+1}^\\sigma \\rightarrow \\mathcal {V}_{t+1}$ as follows: $\\phi _{t+1}(x):={\\left\\lbrace \\begin{array}{ll}\\phi _t(x) & \\text{if $x\\in \\mathcal {X}_t^{\\phi _t}$,}\\\\\\sigma (x) & \\text{if $x\\in {X}_{t+1}^\\sigma $.}\\end{array}\\right.",
"}$ We now make a key observation: By definition of the candidacy graphs ${A}_{t+1}$ and their edge set labellings as in (REF ), if $\\sigma $ is a conflict-free packing in ${A}_{t+1}$ as defined in (REF ), then $\\phi _{t+1}$ is a $(t+1)$ -partial packing.",
"We aim to apply Lemma REF in order to obtain a conflict-free packing $\\sigma $ in ${A}_{t+1}$ .",
"Let $\\mathcal {H}_{t+1} &:=\\bigcup _{H\\in \\mathcal {H}}H_+\\left[\\textstyle \\bigcup _{i\\in N_R[t+1]\\setminus [t]}X^H_i\\cup \\bigcup _{i\\in N_R(t+1)}X_i^{H,B}\\right],\\\\G_{t+1} &:=G_+\\left[\\textstyle \\bigcup _{i\\in N_R[t+1]\\setminus [t]}V_i\\cup \\bigcup _{i\\in N_R(t+1)}V_i^{B}\\right],\\\\ \\mathcal {A}_{t+1}&:=\\textstyle \\bigcup _{i\\in N_R[t+1]\\setminus [t]}{A}_i \\cup \\bigcup _{i\\in N_R(t+1)}{B}_i,\\\\ R_{t+1}&:=R[N_{R}[t+1]\\setminus [t]]\\cup R_B[N_{R_B}[t+1]].$ Note that ${P}:=(\\mathcal {H}_{t+1},G_{t+1},R_{t+1},\\mathcal {A}_{t+1},\\psi _t|_{E(\\mathcal {A}_{t+1})})$ is a packing instance of size $\\operatorname{deg}_{R_{t+1}}(t+1)$ with $t+1$ playing the role of 0, and we claim that ${P}$ is indeed an $({\\varepsilon }_{c(t+1)-1},\\mathbf {d})$ -packing instance, where $\\mathbf {d}=\\big (d_A,d_B,(\\alpha _A d_A^{m_i(t)})_{i\\in N_R[t+1]\\setminus [t]}, (\\alpha _B d_B^{m_i(t)})_{i\\in N_R(t+1)}\\big )$ .",
"Observe that by definition of $c_i(t)$ and $m_i(t)$ in (REF ), we have: $\\begin{minipage}[c]{0.9}\\em If i\\in N_R(t+1), then m_i(t+1)=m_i(t)+1, and c(t+1)=c_i(t+1)>\\max \\lbrace c_i(t),c_j(t)\\rbrace for all j\\in N_R(i).If i\\in [r]\\setminus N_R(t+1), then m_i(t+1)=m_i(t).\\end{minipage}$ Note that for the inequality in (REF ) we used that no pair of adjacent vertices in $R$ has two neighbours in $R$ that are coloured alike as we have chosen the vertex colouring as a colouring in $R^3$ .",
"In particular, we infer from (REF ) that ${\\varepsilon }_{c(t+1)-1}={\\varepsilon }_{c_i(t+1)-1}\\ge {\\varepsilon }_{c_i(t)}$ for all $i\\in N_R(t+1)$ .",
"Therefore, REF follows from (REF ), property REF follows from $\\mathbf {S}(t)$REF , property REF follows from $\\mathbf {S}(t)$REF with $R[N_R[t+1]\\setminus [t]]$ playing the role of $R_A$ , and REF follows from $\\mathbf {S}(t)$REF .",
"Observe further that $\\psi _t$ as defined in (REF ) satisfies $\\Vert \\psi _t\\Vert \\le \\alpha ^{-1}$ ; $\\sum _{H\\in \\mathcal {H}}e_{H}(X_{t+1}^H,X^H_i)\\le (1-\\alpha )dn^2\\le d_An^2$ for all $i\\in N_R(t+1)\\setminus [t]$ ; $\\Delta ^c_{\\psi _t}({A}_i)\\le \\sqrt{n}$ for all $i\\in N_R[t+1]\\setminus [t]$ by $\\mathbf {S}(t)$REF .",
"Hence, we can apply Lemma REF to ${P}$ with Table: NO_CAPTIONand with set testers $\\mathcal {W}_{set}^{t+1}$ where we denote by $\\mathcal {W}_{set}^{t+1}\\subseteq \\mathcal {W}_{set}$ the set of set testers $(W,Y_1,\\ldots ,Y_\\ell )$ with $W\\subseteq V_{t+1}$ , and with edge testers $\\mathcal {W}_{edge}^{t+1}\\cup \\mathcal {W}_{edge}^\\ast $ where we will define the set $\\mathcal {W}_{edge}^\\ast $ when proving $\\mathbf {S}(t+1)$REF and REF in Steps and .",
"Let $\\sigma \\colon {X}_{t+1}^\\sigma \\rightarrow V_{t+1}^\\sigma $ be the conflict-free packing in ${A}_{t+1}$ obtained from Lemma REF with $|{X}_{t+1}^\\sigma \\cap X_{t+1}^H|\\ge (1-{\\varepsilon }_{c_i(t+1)})n$ for all $H\\in \\mathcal {H}$ , which extends $\\phi _t$ to $\\phi _{t+1}$ as defined in (REF ).",
"Fix some $H\\in \\mathcal {H}$ .",
"By Definition REF , the updated candidacy graphs with respect to $\\sigma $ obtained from Lemma REF are also updated candidacy graphs with respect to $\\phi _{t+1}$ as defined in (REF ) in Step .",
"Hence, the graphs $A_i^{H,new}$ in Lemma REF correspond to subgraphs $\\widetilde{A}^H_i\\subseteq A^H_i(\\phi _{t+1})$ for all $i\\in N_R(t+1)\\setminus [t]$ , and $\\widetilde{B}^H_i\\subseteq B^H_i(\\phi _{t+1})$ for all $i\\in N_R(t+1)$ that satisfy REF –REF .",
"Step 3.1.",
"Checking $\\mathbf {S}(t+1)$ REF By REF and (REF ), we obtain that $\\widetilde{A}^H_i$ is $({\\varepsilon }_{c_i(t+1)},\\alpha _A d_A^{m_i(t+1)})$ -super-regular for all $i\\in N_R(t+1)\\setminus [t]$ , and $\\widetilde{B}^H_i$ is $({\\varepsilon }_{c_i(t+1)},\\alpha _B d_B^{m_i(t+1)})$ -super-regular for all $i\\in N_R(t+1)$ .",
"Note that for each $i\\in [r]\\setminus N_R(t+1)$ , we have $m_i(t)=m_i(t+1)$ and $A^H_i(\\phi _t)=A^H_i(\\phi _{t+1})$ and $B^H_i(\\phi _t)=B^H_i(\\phi _{t+1})$ .",
"For $i\\in [r]\\setminus [t+1], i^{\\prime }\\in [r]$ , let $\\widehat{A}^H_i:={\\left\\lbrace \\begin{array}{ll}\\widetilde{A}^H_i & \\text{if $i\\in N_R(t+1)\\setminus [t]$,}\\\\A^H_i & \\text{otherwise.}\\end{array}\\right.",
"}&&\\widehat{B}^H_{i^{\\prime }}:={\\left\\lbrace \\begin{array}{ll}\\widetilde{B}^H_{i^{\\prime }} & \\text{if $i^{\\prime }\\in N_R(t+1)$,}\\\\B^H_{i^{\\prime }} & \\text{otherwise.}\\end{array}\\right.",
"}$ Then the graphs $\\widehat{A}^H_i$ and $\\widehat{B}^H_{i^{\\prime }}$ are candidacy graphs satisfying $\\mathbf {S}(t+1)$REF .",
"Step 3.2.",
"Checking $\\mathbf {S}(t+1)$ REF and $\\mathbf {S}(t+1)$ REF The new edge set labelling $\\psi _{t+1}$ as defined in (REF ) corresponds to the updated edge set labelling as in Definition REF .",
"By REF , $\\mathbf {S}(t)$REF and (REF ), we obtain for every $i\\in [r]\\setminus [t+1]$ that $\\Delta _{\\psi _{t+1}}(\\bigcup _{H\\in \\mathcal {H}}\\widehat{A}^H_i)\\le (1+{\\varepsilon }_{c_i(t+1)})\\alpha _Ad_A^{m_i(t+1)}|V_i|$ .",
"This establishes $\\mathbf {S}(t+1)$REF .",
"Similarly, by REF with $R[N_R[t+1]\\setminus [t]]$ playing the role of $R_A$ and by $\\mathbf {S}(t)$REF , we obtain for every $i\\in [r]\\setminus [t+1]$ that $\\Delta ^c_{\\psi _{t+1}}(\\bigcup _{H\\in \\mathcal {H}}\\widehat{A}^H_i)\\le \\sqrt{n}$ , which establishes $\\mathbf {S}(t+1)$REF .",
"Step 3.3.",
"Checking $\\mathbf {S}(t+1)$ REF In order to show $\\mathbf {S}(t+1)$REF , fix $H\\in \\mathcal {H}$ , $ij\\in E(R-[t+1])$ , and $v_iv_j\\in E(G[V_i,V_j])$ .",
"Observe, that $|\\lbrace i,j\\rbrace \\cap N_R(t+1)|\\in \\lbrace 0,1,2\\rbrace .$ If $|\\lbrace i,j\\rbrace \\cap N_R(t+1)|=2$ , then this implies together with (REF ) that $c_i(t+1)=c_j(t+1)=c(t+1)>\\max \\lbrace c_i(t),c_j(t)\\rbrace $ , and $m_i(t)+1=m_i(t+1)$ as well as $m_j(t)+1=m_j(t+1)$ .",
"Hence we obtain by REF (with $R[N_R[t+1]\\setminus [t]]$ playing the role of $R_A$ ) and $\\mathbf {S}(t)$REF that $e_{H}(N_{\\widehat{A}^H_i}(v_i),N_{\\widehat{A}^H_j}(v_j))=\\big (\\alpha _A^2 d_A^{m_i(t+1)+m_j(t+1)}\\pm {\\varepsilon }_{\\max \\lbrace c_i(t+1),c_j(t+1)\\rbrace }\\big )e_{H}(X^H_i,X^H_j).$ If $|\\lbrace i,j\\rbrace \\cap N_R(t+1)|=1$ , say $i\\in N_R(t+1)$ , then this implies together with (REF ) that $\\begin{minipage}[c]{0.7}\\em c_i(t+1)=\\max \\lbrace c_i(t+1),c_j(t+1)\\rbrace =c(t+1)>\\max \\lbrace c_i(t),c_j(t)\\rbrace , and m_i(t)+1=m_i(t+1), m_j(t)=m_j(t+1).\\end{minipage}$ By (REF ), we have that $\\widehat{A}_j^H=A_j^H$ because $j\\notin N_R(t+1)$ .",
"Let $N:=N_{A^H_i}(v_i)\\cap N_{H}(N_{A^H_j}(v_j))$ and we define a weight function $\\omega _N\\colon E({A}_i)\\rightarrow \\lbrace 0,1\\rbrace $ by $\\omega _N(xv):=\\mathbb {1}_{\\lbrace v=v_i \\rbrace }\\mathbb {1}_{\\lbrace x\\in N \\rbrace }$ and add $\\omega _N$ to $\\mathcal {W}_{edge}^\\ast $ .",
"Note that $\\dim (\\omega _N)=1$ (with $\\dim (\\omega _N)$ defined as in (REF )) and that $\\omega _N(E({A}_i))=|N|=(\\alpha _A^2 d_A^{m_i(t)}d_A^{m_j(t)}\\pm {\\varepsilon }_{\\max \\lbrace c_i(t),c_j(t)\\rbrace })e_{H}(X^H_i,X^H_j)$ by $\\mathbf {S}(t)$REF .",
"This implies that $e_{H}(N_{\\widehat{A}^H_i}(v_i),N_{\\widehat{A}^H_j}(v_j))&\\stackrel{\\hphantom{\\ref {emb lem:edge weight}}}{=}|N_{\\widehat{A}^H_i}(v_i)\\cap N_{H}(N_{A^H_j}(v_j))|= |N_{\\widehat{A}^H_i}(v_i)\\cap N|= \\omega _N(E(\\widehat{A}^H_i))\\\\&\\stackrel{\\ref {emb lem:edge weight}}{=}(1\\pm {\\varepsilon }_{c(t+1)}^2)d_A \\omega _N(E({A}_i))\\pm {\\varepsilon }_{c(t+1)}^2n\\\\&\\stackrel{~(\\ref {eq:ij=1})~}{=}\\big (\\alpha _A^2 d_A^{m_i(t+1)+m_j(t+1)}\\pm {\\varepsilon }_{\\max \\lbrace c_i(t+1),c_j(t+1)\\rbrace }\\big )e_{H}(X^H_i,X^H_j).$ If $|\\lbrace i,j\\rbrace \\cap N_R(t+1)|=0$ , then this implies together with (REF ) and (REF ), that $m_i(t)=m_i(t+1)$ , $m_j(t)=m_j(t+1)$ , and $\\widehat{A}^H_i=A^H_i$ .",
"Consequently, (REF ) holds which establishes $\\mathbf {S}(t+1)$REF .",
"Step 3.4.",
"Checking $\\mathbf {S}(t+1)$ REF In order to establish $\\mathbf {S}(t+1)$REF , we first consider $v_i\\in V_i$ for $i\\in N_R(t+1)\\cap [t]$ .",
"We define a weight function $\\omega _i^\\ast \\colon E({A}_{t+1})\\rightarrow \\lbrace 0,1\\rbrace $ by $\\omega _{v_{i}}^\\ast (xv):=\\mathbb {1}_{\\lbrace x\\in {X}_{i}^\\ast \\rbrace }$ for ${X}_{i}^\\ast :=N_{\\mathcal {H}}(\\phi _t^{-1}(v_i))\\cap {X}_{t+1}$ and every $xv\\in E({A}_{t+1})$ , and we add $\\omega _{v_{i}}$ to $\\mathcal {W}_{edge}^\\ast $ .",
"By $\\mathbf {S}(t)$REF , we have $\\omega _{v_{i}}(E({A}_{t+1}))=(\\alpha _Ad_A^{m_{t+1}(t)}\\pm 3{\\varepsilon }_{c_{t+1}(t)})|{X}_i^\\ast |n,$ and by $\\mathbf {S}(t)$REF , we have $|\\phi _{t+1}^{-1}(v_i)\\cap N_{\\mathcal {H}}(\\mathcal {X}_{t+1}\\setminus \\mathcal {X}_{t+1}^{\\phi _{t+1}})|\\le {\\varepsilon }_{c_i(t)}^{1/2}{n} + |{X}_{i}^\\ast |-\\omega _{v_{i}}(M)$ with $M=M(\\sigma )$ being the corresponding edge set to $\\sigma $ .",
"By REF , we obtain that $\\omega _{v_{i}}(M)&\\stackrel{\\ref {emb lem:vertex testers}}{=}(1\\pm {\\varepsilon }_{c(t+1)})\\frac{\\omega _{v_{i}}(E({A}_{t+1}))}{\\alpha _Ad_A^{m_{t+1}(t)}n}\\pm {\\varepsilon }_{c(t+1)}n\\stackrel{(\\ref {eq:weight w_v_i})}{\\ge }(1-{\\varepsilon }_{c(t+1)}^{3/4})|{X}_i^\\ast |-{{\\varepsilon }_{c(t+1)}n}.$ Together with (REF ), this implies that $|\\phi _{t+1}^{-1}(v_i)\\cap N_{\\mathcal {H}}(\\mathcal {X}_{t+1}\\setminus \\mathcal {X}_{t+1}^{\\phi _{t+1}})|\\le {\\varepsilon }_{c(t+1)}^{1/2}{n}.$ Hence, it now suffices to establish $\\mathbf {S}(t+1)$REF for all $v_{t+1}\\in V_{t+1}$ by $\\mathbf {S}(t)$REF .",
"We define weight functions $\\omega _{v_{t+1}},\\omega _{v_{t+1}}^\\ast \\colon E({A}_{t+1})\\rightarrow \\lbrace 0,1\\rbrace $ by $\\omega _{v_{t+1}}(xv):=\\mathbb {1}_{\\lbrace v=v_{t+1} \\rbrace }$ and $\\omega _{v_{t+1}}^\\ast (xv):=\\mathbb {1}_{\\lbrace v=v_{t+1} \\text{ and } x\\in {X}_{t+1}^\\ast \\rbrace }$ for ${X}_{t+1}^\\ast :=N_{\\mathcal {H}}(\\mathcal {X}_t\\setminus \\mathcal {X}_t^{\\phi _t})\\cap {X}_{t+1} $ and every $xv\\in E({A}_{t+1})$ , and we add $\\omega _{v_{t+1}}$ and $\\omega _{v_{t+1}}^\\ast $ to $\\mathcal {W}_{edge}^\\ast $ .",
"Observe that $\\mathbf {S}(t)$ implies that $\\omega _{v_{t+1}}(E({A}_{t+1}))&=(\\alpha _Ad_A^{m_{t+1}(t)}\\pm 3{\\varepsilon }_{c_{t+1}(t)})|\\mathcal {H}|n,\\\\\\omega _{v_{t+1}}^\\ast (E({A}_{t+1}))&\\le |{X}_{t+1}^\\ast |\\le {\\varepsilon }_{c_{t+1}(t)}^{1/2} |\\mathcal {H}|n,$ and we have that $|\\phi _{t+1}^{-1}(v_{t+1})|=\\omega _{v_{t+1}}(M)$ and $|\\phi _{t+1}^{-1}(v_{t+1})\\cap {X}_{t+1}^\\ast |\\le \\omega _{v_{t+1}}^\\ast (M)$ .",
"By REF , we obtain that $\\omega _{v_{t+1}}(M)&\\stackrel{\\ref {emb lem:vertex testers}}{=}(1\\pm {\\varepsilon }_{c(t+1)})\\frac{\\omega _{v_{t+1}}(E({A}_{t+1}))}{\\alpha _Ad_A^{m_{t+1}(t)}n}\\pm {\\varepsilon }_{c(t+1)}n\\stackrel{(\\ref {eq:weight v_t+1})}{\\ge }(1-{\\varepsilon }_{c(t+1)}^{1/2})|\\mathcal {H}|-{{\\varepsilon }_{c(t+1)}n};\\\\\\omega _{v_{t+1}}^\\ast (M)&\\stackrel{\\ref {emb lem:vertex testers}}{=}(1\\pm {\\varepsilon }_{c(t+1)})\\frac{\\omega _{v_{t+1}}^\\ast (E({A}_{t+1}))}{\\alpha _Ad_A^{m_{t+1}(t)}n}\\pm {\\varepsilon }_{c(t+1)}n\\stackrel{(\\ref {eq:weight v_t+1^*})}{\\le }{\\varepsilon }_{c(t+1)}^{1/2}{n}.$ Note that ${\\varepsilon }_{c(t+1)}={\\varepsilon }_{c_{t+1}(t+1)}$ .",
"Altogether, this establishes $\\mathbf {S}(t+1)$REF .",
"Step 3.5.",
"Checking $\\mathbf {S}(t+1)$REF – REF In order to establish $\\mathbf {S}(t+1)$REF , consider $\\omega _v\\in \\mathcal {W}_{edge}^i$ for $i\\in N_R(t+1)\\setminus [t+1]$ .",
"By (REF ), it holds that $c(t+1)=c_i(t+1)$ .",
"With REF we obtain that $\\omega _v(E(\\textstyle \\bigcup _{H\\in \\mathcal {H}}\\widehat{A}^H_i))=(1\\pm {\\varepsilon }_{c(t+1)}^2)d_A\\omega _v(E({A}_i))\\pm {\\varepsilon }_{c(t+1)}^2n^2\\stackrel{\\text{$\\mathbf {S}(t)$\\ref {updated vertex testers}}}{=}\\alpha _A d_A^{m_i(t+1)}\\omega ({X}_i)\\pm {\\varepsilon }_{c_i(t+1)}n^2,$ which together with $\\mathbf {S}(t)$REF establishes $\\mathbf {S}(t+1)$REF .",
"Next we verify $\\mathbf {S}(t+1)$REF .",
"Note that REF implies that $|W\\cap \\bigcap _{j\\in [\\ell ]}\\sigma (Y_j\\cap {X}_{t+1}^\\sigma )|= |W||Y_1|\\cdots |Y_\\ell |/n^\\ell \\pm {\\varepsilon }_{c(t+1)}n$ for all $(W,Y_1,\\ldots ,Y_\\ell )\\in \\mathcal {W}_{set}^{t+1}$ , which together with $\\mathbf {S}(t)$REF yields $\\mathbf {S}(t+1)$REF .",
"In order to establish $\\mathbf {S}(t+1)$REF , let $\\mathcal {W}_{ver}^{t+1}\\subseteq \\mathcal {W}_{ver}$ be the set of vertex testers $(v,\\omega )$ with $v\\in V_{t+1}$ .",
"Hence, for all $(v,\\omega )\\in \\mathcal {W}_{ver}^{t+1}$ and and its corresponding edge tester $\\omega _v\\in \\mathcal {W}_{edge}^{t+1}$ as defined in (REF ), property REF implies that $\\omega ({X}_{t+1}\\cap \\sigma ^{-1}(v))= \\omega _v(M)&\\stackrel{\\ref {emb lem:vertex testers}}{=}(1\\pm {\\varepsilon }_{c(t+1)})\\frac{\\omega _v(E({A}_{t+1}))}{\\alpha _Ad_A^{m_{t+1}(t)}n} \\pm {\\varepsilon }_{c(t+1)}n\\\\&\\stackrel{\\text{~$\\mathbf {S}(t)$\\ref {updated vertex testers}~}}{=}(1\\pm {\\varepsilon }_{c(t+1)})\\frac{\\alpha _Ad_A^{m_{t+1}(t)}\\omega ({X}_{t+1})\\pm {\\varepsilon }_{c_{t+1}(t)}n^2}{\\alpha _Ad_A^{m_{t+1}(t)}n}\\pm {\\varepsilon }_{c(t+1)}n\\\\&\\stackrel{\\hphantom{\\text{~$\\mathbf {S}(t)$\\ref {updated vertex testers}~}}}{=}\\frac{\\omega ({X}_{t+1})}{n}\\pm \\alpha n/2.$ Together with $\\mathbf {S}(t)$REF , this yields $\\mathbf {S}(t+1)$REF .",
"Step 4.",
"Completion Let $\\phi _r\\colon \\textstyle {\\bigcup _{H\\in \\mathcal {H},i\\in [r]_0}\\widehat{X}^H_i \\rightarrow \\mathcal {V}_r}$ be an $r$ -partial packing satisfying $\\mathbf {S}(r)$ with $({\\varepsilon }_T,d_i)$ -super-regular candidacy graphs $B^H_i\\subseteq B^H_i(\\phi _r)$ where $d_i:=\\alpha _Bd_B^{\\operatorname{deg}_R(i)}$ for all $i\\in [r]$ .",
"We aim to apply iteratively the ordinary blow-up lemma in order to complete the partial packing $\\phi _r$ using the edges in $G_B$ .",
"Recall that ${\\varepsilon }_T\\ll \\mu \\ll \\gamma \\ll \\alpha ,d$ .",
"Our general strategy is as follows.",
"For every $H\\in \\mathcal {H}$ in turn, we choose a set $X_i\\subseteq X^H_i$ for all $i\\in [r]$ of size roughly $\\mu n$ by selecting every vertex uniformly at random with the appropriate probability and adding $X^H_i\\setminus \\widehat{X}_i^{H}$ deterministically.",
"Afterwards, we apply the blow-up lemma to embed $H[{X_1\\cup \\ldots \\cup X_r}]$ into $G_B$ , which together with $\\phi _r$ yields a complete embedding of $H$ into $G_A\\cup G_B$ .",
"Before we proceed with the details of our procedure (see Claim ), we verify in Claim that we can indeed apply the blow-up lemma to a subgraph of $H\\in \\mathcal {H}$ provided some easily verifiable conditions are satisfied.",
"Recall that we have defined the candidacy graph $B^H_i$ on a copy $(X^{H,B}_i,V_i^B)$ via the bijection $\\pi $ only to conveniently apply Lemma REF in Step .",
"That is, for all $H\\in \\mathcal {H}, i\\in [r]$ , we can identify $B^H_i$ with an isomorphic bipartite graph on $(X^H_i,V_i)$ and edge set $\\lbrace xv\\colon \\pi (x)\\pi (v)\\in E(B^H_i)\\rbrace $ .",
"Let $\\mathcal {B}$ be the union over all $H\\in \\mathcal {H}, i\\in [r]$ of these graphs.",
"For $H\\in \\mathcal {H}$ and a subgraph $G^\\circ $ of $G_B$ , we write $\\mathcal {B}_{G^\\circ }[{X_i^H,V_i}]$ for the graph that arises from $\\mathcal {B}[{X_i^H,V_i}]$ by deleting every edge $xv$ with $x\\in X_i^H, v\\in V_i$ for which there exists $x^{\\prime }x\\in E(H)$ such that $\\phi _r(x^{\\prime })v\\in E(G^\\circ )$ .",
"(We may think of $E(G^\\circ )$ as the edge set in $G_B$ that we have already used in our completion step for packing some other graphs of $\\mathcal {H}$ into $G_A\\cup G_B$ .)",
"For future reference, we observe that $\\begin{minipage}[c]{0.6}\\em \\Delta (\\mathcal {B}[X^H_i,V_i]-\\mathcal {B}_{G^\\circ }[X_i^H,V_i])\\le \\alpha ^{-1}\\Delta (G^\\circ ) {for all i\\in [r].",
"}\\end{minipage}$ Claim 2.",
"Suppose $H\\in \\mathcal {H}$ , $G^\\circ \\subseteq G_B$ and $W_i\\subseteq V_i$ for all $i\\in [r]$ such that the following hold: $V_i\\setminus \\phi _r(\\widehat{X}_i^H)\\subseteq W_i\\subseteq V_i$ and $|W_i|= (\\mu \\pm {\\varepsilon }_T^{1/2}) n$ for all $i\\in [r]$ ; $\\mathcal {B}_{G^\\circ }[X_i, W_i]$ is $(\\mu ^{1/31},d_i)$ -super-regular where $X_i:=(X_i^H\\cap \\phi ^{-1}_r(W_i\\cap \\phi _r(\\widehat{X}_i^{H})))\\cup (X_i^{H}\\setminus \\widehat{X}_i^{H})$ ; $G_B[W_i,W_j]$ is $({\\varepsilon }_T^{1/3},d_B)$ -super-regular for all $ij\\in E(R)$ ; $|N_{G^\\circ }(v)\\cap W_i|\\le \\mu ^{3/2}n$ for all $v\\in V_j$ and $ij\\in E(R)$ .",
"Then there exists an embedding $\\phi ^{\\widetilde{H}}$ of $\\widetilde{H}:=H[X_1,\\ldots ,X_r]$ into $\\widetilde{G}:=G_B[W_1,\\ldots ,W_r]-G^\\circ $ such that $\\phi ^{\\prime }:=\\phi ^{\\widetilde{H}} \\cup \\phi _r|_{V(H)\\setminus V(\\widetilde{H})}$ is an embedding of $H$ into $G$ where $\\phi ^{\\prime }(X_i^H)= V_i$ for all $i\\in [r]$ and all edges incident to a vertex in $\\bigcup _{i\\in [r]}X_i$ are embedded on an edge in $G_B-G^\\circ $ .",
"Proof of claim: Note that $|X_i|=|W_i|= (\\mu \\pm {\\varepsilon }_T^{1/2})n$ (by REF and REF ) and $\\widetilde{H}[X_i,X_j]$ is empty whenever $ij\\notin E(R)$ and a matching otherwise.",
"Moreover, by REF and REF , Fact REF yields that $\\widetilde{G}[W_i,W_j]$ is $(\\mu ^{1/31},d_B)$ -super-regular for all $ij\\in E(R)$ (with room to spare).",
"This shows that $(\\widetilde{H}, \\widetilde{G},R,(X_i)_{i\\in [r]},(W_i)_{i\\in [r]})$ is a $(\\mu ^{1/31},d_B)$ -super-regular blow-up instance.",
"We apply Theorem REF to this blow-up instance (with $(\\mathcal {B}_{G^\\circ }[X_i, W_i])_{i\\in [r]}$ playing the role of $(A_i)_{i\\in [r]}$ ) and obtain an embedding $\\phi ^{\\widetilde{H}}$ of $\\widetilde{H}$ into $\\widetilde{G}$ .",
"Recall that $\\phi _r|_{V(H)\\setminus V(\\widetilde{H})}$ is an embedding of $H-V(\\widetilde{H})$ into $G_A$ .",
"For $x\\in V(\\widetilde{H})$ , $x^{\\prime }\\in V(H)\\setminus V(\\widetilde{H})$ and $xx^{\\prime }\\in E(H)$ , we conclude that $\\phi ^{\\widetilde{H}}(x)\\phi _r(x^{\\prime })\\in E(G_B-G^\\circ )$ by the definition of the candidacy graphs in (REF ) and the definition of $\\mathcal {B}_{G^\\circ }$ .",
"$-$ Claim 3.",
"For all $H\\in \\mathcal {H}$ and $i\\in [r]$ , there exist sets $\\overline{X}_i^H\\subseteq \\widehat{X}_i^H$ with $|\\overline{X}_i^H|\\ge (1-2\\mu )n$ and a packing $\\phi $ of $\\mathcal {H}$ into $G$ that extends $\\phi ^\\ast $ such that $\\phi |_{\\overline{X}_i^H}=\\phi _r|_{\\overline{X}_i^H}$ as well as $\\phi (X_i^H)=V_i$ .",
"Proof of claim: We write $\\mathcal {H}=\\lbrace H_1,\\ldots ,H_{|\\mathcal {H}|}\\rbrace $ and let $\\mathcal {H}_h:=\\lbrace H_1,\\ldots ,H_h\\rbrace $ for all $h \\in [|\\mathcal {H}|]_0$ .",
"For all $v\\in V_i$ , $i\\in [r]$ , let $\\tau ^h(v):=|\\lbrace H\\in \\mathcal {H}_h\\colon v\\in V_i\\setminus \\phi _r(\\widehat{X}_i^{H})\\rbrace |$ and $\\sigma ^{h}(v):=|\\lbrace \\phi _r^{-1}(v)\\cap \\bigcup _{H\\in \\mathcal {H}_h, j\\in N_R(i)}N_{H}(X_j^H\\setminus \\widehat{X}_j^H)\\rbrace |$ .",
"We inductively prove that the following statement $\\mathbf {C}(h)$ holds for all $h\\in [|\\mathcal {H}|]_0$ .",
"$\\mathbf {C}(h)$ .",
"There exists a packing $\\phi ^{h}$ of $\\mathcal {H}_h$ into $G$ that extends $\\phi ^\\ast $ such that for $G^\\circ _{h}:=G_B \\cap \\phi ^h(\\mathcal {H}_h)$ we have $\\deg _{G^\\circ _{h}}(v)\\le \\alpha ^{-1}(\\tau ^{h}(v)+\\sigma ^{h}(v))+\\mu ^{2/3}n$ for all $v\\in V(G)$ ; for all $H\\in \\mathcal {H}_h$ and $i\\in [r]$ , there exist sets $\\overline{X}_i^H\\subseteq \\widehat{X}_i^H$ with $|\\overline{X}_i^H|\\ge (1-2\\mu )n$ such that $\\phi ^{h}|_{\\overline{X}_i^H}=\\phi _r|_{\\overline{X}_i^H}$ as well as $\\phi ^h(X_i^H)=V_i$ .",
"Let $G_0^\\circ $ be the edgeless graph on $V(G)$ and $\\phi ^0$ be the empty function; then $\\mathbf {C}(0)$ holds.",
"Hence, we may assume the truth of $\\mathbf {C}(h)$ for some $h\\in [|\\mathcal {H}|-1]_0$ and let $\\phi ^h$ and $G^\\circ _h$ be as in $\\mathbf {C}(h)$ .",
"By $\\mathbf {C}(h)$REF , there are at most $\\sum _{H\\in \\mathcal {H}_h,j\\in N_R[i]}\\alpha ^{-1}|X_j^H\\setminus \\overline{X}_j^H|\\le 5\\alpha ^{-3}\\mu n^2$ edges of $G_h^\\circ $ incident to a vertex in $V_i$ for each $i\\in [r]$ .",
"Hence, there are at most $10\\alpha ^{-3}\\mu ^{1/3}n$ vertices in $V_i$ of degree at least $\\mu ^{2/3} n/2$ in $G_h^\\circ $ .",
"Let $V_i^{high}\\subseteq V_i$ be a set of size $\\mu ^{1/4}n$ that contains all vertices of degree at least $\\mu ^{2/3} n/2$ in $G_h^\\circ $ .",
"For all $i\\in [r]$ , we select every vertex in $\\phi _r(\\widehat{X}^{H_{h+1}}_i)\\setminus V_i^{high}$ independently with probability $\\mu (1-\\mu ^{1/4})^{-1}$ and denote by $W_i$ their union together with $V_i\\setminus \\phi _r(\\widehat{X}^{H_{h+1}}_i)$ ; we define $X_i:=(X_i^{H_{h+1}}\\cap \\phi ^{-1}_r(W_i\\cap \\phi _r(\\widehat{X}_i^{H_{h+1}})))\\cup (X_i^{H_{h+1}}\\setminus \\widehat{X}_i^{H_{h+1}})$ .",
"Note that $\\mathbf {S}(r)$ implies that $\\begin{minipage}[c]{0.6}\\em {\\left|X_i^{H_{h+1}}\\setminus \\widehat{X}_i^{H_{h+1}}\\right|=\\left|V_i\\setminus \\phi _r(\\widehat{X}_i^{H_{h+1}})\\right|\\le 2{\\varepsilon }_Tn {for all i\\in [r].",
"}}\\end{minipage}$ In the following we will show that the assumptions of Claim are satisfied with probability at least 1/2, say, for $H_{h+1}$ and $G^\\circ _h$ playing the role of $H$ and $G^\\circ $ , respectively.",
"In particular, there is a choice for $W_i$ such that the assumptions of Claim hold.",
"To obtain REF , we apply Chernoff's inequality to the sum of indicator variables which indicate whether a vertex in $V_i$ is randomly selected.",
"Together with (REF ), this shows that REF holds with probability at least $1-1/n^3$ , say.",
"By $\\mathbf {C}(h)$REF and $\\mathbf {S}(r)$REF , we obtain that $\\Delta (G_h^\\circ )\\le 2\\mu ^{2/3}n$ .",
"We exploit (REF ) and conclude that $\\Delta (\\mathcal {B}[X^{H_{h+1}}_i, V_i]-\\mathcal {B}_{G_{h}^\\circ }[X^{H_{h+1}}_i, V_i])\\le 2\\alpha ^{-1}\\mu ^{2/3}n$ for all $i\\in [r]$ .",
"Thus Fact REF implies that $\\mathcal {B}_{G_{h}^\\circ }[X^{H_{h+1}}_i, V_i]$ is $(\\mu ^{1/5},d_i)$ -super-regular for all $i\\in [r]$ .",
"For all $i\\in [r]$ and $x\\in X^{H_{h+1}}_i$ , Chernoff's inequality implies that $|N_{\\mathcal {B}_{G_{h}^\\circ }}(x)\\cap W_i|= (d_i\\pm 2\\mu ^{1/5})|W_i|$ and similarly, for all $v\\in V_i$ , we have $|N_{\\mathcal {B}_{G_{h}^\\circ }}(v)\\cap X_i|= (d_i\\pm 2\\mu ^{1/5})|X_i|$ .",
"Moreover, for all distinct $v,v^{\\prime }$ with $|N_{\\mathcal {B}_{G_{h}^\\circ }}(v,v^{\\prime })|=(d_i\\pm 2\\mu ^{1/5})^2n$ (which we call good, and there are at least $(1-2\\mu ^{1/5})\\binom{n}{2}$ good pairs), we also obtain $|N_{\\mathcal {B}_{G_{h}^\\circ }}(v,v^{\\prime })\\cap X_i|=(d_i\\pm 3\\mu ^{1/5})^2|X_i|$ , all with probability at least $1-1/n^3$ .",
"Observe that Theorem REF implies that there at least $(1-3\\mu ^{1/5})\\binom{\\mu n}{2}$ good pairs in $W_i$ also with probability at least $1-1/n^3$ .",
"Therefore, we may apply Theorem REF and obtain that $\\mathcal {B}_{G_{h}^\\circ }[X_i, W_i]$ is $(\\mu ^{1/31},d_i)$ -super-regular for all $i\\in [r]$ which yields REF with probability at least $1-1/n^2$ , say.",
"To obtain REF , for each $ij\\in E(R)$ , we proceed as follows.",
"Observe first that $G_B[V_i,V_j]$ is $(2{\\varepsilon },d_B)$ -super-regular by (REF ).",
"Hence $G_B[W_i,W_j]$ is clearly ${\\varepsilon }_T^{1/3}$ -regular as $|W_i|,|W_j|\\ge \\mu n/2$ by REF .",
"Therefore, we only need to control the degrees of the vertices in $G_B[W_i,W_j]$ which follows directly by Chernoff's inequality with probability at least $1-1/n^3$ and because of (REF ).",
"Since $\\Delta (G_h^\\circ )\\le 2\\mu ^{2/3}n$ and because of (REF ), we conclude by Chernoff's inequality that REF holds with probability at least $1-1/n^3$ .",
"Therefore, the assumptions of Claim are achieved by our construction with probability at least $1/2$ .",
"Fix such a choice for $W_1,\\ldots ,W_r$ and apply Claim that returns an embedding $\\phi ^{\\widetilde{H}_{h+1}}$ of $\\widetilde{H}_{h+1}:=H_{h+1}[X_1,\\ldots ,X_r]$ into $G_B[W_1,\\ldots ,W_r]-G_h^\\circ $ such that $\\phi ^{\\prime }:=\\phi ^{\\widetilde{H}_{h+1}}\\cup \\phi _r|_{{V(H_{h+1})\\setminus V(\\widetilde{H}_{h+1})}}$ is an embedding of $H_{h+1}$ into $G$ where $\\phi ^{\\prime }(X_i^H)=V_i$ and all edges incident to a vertex in $\\bigcup _{i\\in [r]}X_i$ are embedded on an edge in $G_B-G^\\circ _h$ .",
"We define $\\phi ^{h+1}:=\\phi ^h\\cup \\phi ^{\\prime }$ and obtain $\\mathbf {C}(h+1)$REF with $\\overline{X}_i^{H_{h+1}}:=X_i^{H_{h+1}}\\setminus X_i$ .",
"It is straightforward to check that by our construction also $\\mathbf {C}(h+1)$REF holds.",
"$-$ Let $\\phi $ be as in Claim .",
"This directly implies conclusion REF of Lemma REF .",
"Conclusion REF of Lemma REF follows from Claim together with $\\mathbf {S}(r)$REF as we merely modified $\\phi _r$ to obtain $\\phi $ .",
"For a similar reason, Lemma REF REF follows from Claim and $\\mathbf {S}(r)$REF .",
"This completes the proof.",
"$\\square $"
],
[
"Applications",
"In what follows, we provide an illustration for an application of vertex and set testers so that the leftover is suitably well-behaved.",
"For a graph $H$ , let $H^+$ arise from $H$ by adding a labelled vertex $x$ and joining $x$ to all vertices of $H$ .",
"We call $x$ the apex vertex of $H^+$ .",
"Theorem (Keevash [15], cf.",
"[10]) Suppose $1/n\\ll {\\varepsilon }\\ll 1/s \\ll d_0, 1/m$ .",
"Suppose $H$ is an $r$ -regular graph on $m$ vertices.",
"Let $G$ be a graph with vertex partition $(V,W)$ such that $W$ is an independent set, $d_0n\\le |W|\\le |V|=n$ and $|\\bigcap _{x\\in V^{\\prime }\\cup W^{\\prime }}N_G(x)|=(1\\pm {\\varepsilon })d_V^{|V^{\\prime }|}d_W^{|W^{\\prime }|}n$ for all $V^{\\prime }\\subseteq V,W^{\\prime }\\subseteq W$ with $1\\le |V^{\\prime }|+|W^{\\prime }|\\le s$ where $d_V=rd|W|/n$ and $d_W=d$ for some $d\\ge d_0$ .",
"Suppose that $|N_G(v)\\cap V|=r|N_G(v)\\cap W|$ for all $v\\in V$ and $m$ divides $\\operatorname{deg}_G(w)$ for all $w\\in W$ .",
"Then there is a decomposition of the edge set of $G$ into copies of $H^+$ where the apex vertices are contained in $W$ .",
"Proof of Theorem REF .",
"Choose ${\\varepsilon }\\ll \\delta \\ll 1/s \\ll \\beta \\ll \\alpha $ .",
"Among the $\\alpha n$ graphs in $\\mathcal {H}$ that contain at least $\\alpha n$ vertices in components of size at most $\\alpha ^{-1}$ , there is a collection $\\mathcal {H}^{\\prime }$ of $\\beta n$ $r$ -regular graphs that contain each at least $\\beta n$ vertices in components all isomorphic to some graph $J$ where ${|V(J)|}\\le \\alpha ^{-1}$ and $r\\in [\\alpha ^{-1}]$ .",
"For all $H\\in \\mathcal {H}^{\\prime }$ , let $H^-$ arise from $H$ by deleting $\\beta n/|V(J)|$ components isomorphic to $J$ .",
"We denote by $I^H$ a set of $\\beta n$ isolated vertices disjoint from $V(H^-)$ .",
"Let $\\widetilde{\\mathcal {H}}:=(\\mathcal {H}\\setminus \\mathcal {H}^{\\prime })\\cup \\bigcup _{H\\in \\mathcal {H}^{\\prime }}(H^-\\cup I^H)$ .",
"Let $G_1$ be a $({2{\\varepsilon }}, s,d_1)$ -typical subgraph of $G$ where $d_1:=(1+\\delta )(d-\\beta ^2r)$ such that $G-G_1$ is $({2{\\varepsilon }}, s,d-d_1)$ -typical; that is, $e(\\widetilde{\\mathcal {H}}) \\le (1-\\delta /2)e(G_1)$ .",
"Clearly, $G_1$ exists by considering a random subgraph and then applying Chernoff's inequality.",
"Now we apply Theorem REF to obtain a packing $\\phi $ of $\\widetilde{\\mathcal {H}}$ in $G_1$ with $\\delta ^2$ playing the role of $\\alpha $ and sets of set and vertex testers $\\mathcal {W}_{set}$ , $\\mathcal {W}_{ver}$ defined as follows.",
"For all $H_1,\\ldots ,H_{\\ell _1}\\in \\mathcal {H}^{\\prime }$ and $v_1,\\ldots ,v_{\\ell _2}\\in V(G)$ with $1\\le \\ell _1$ and $\\ell _1+\\ell _2\\le s$ , we add the set tester $(V^{\\prime },I^{H_1},\\ldots ,I^{H_{\\ell _1}})$ to $\\mathcal {W}_{set}$ where $V^{\\prime }:=V(G)\\cap \\bigcap _{i\\in [\\ell _2]}N_{G-G_1}(v_i)$ .",
"Then Theorem REF implies that (where $d_2:=d-d_1$ ) $\\bigg |\\bigcap _{i\\in [\\ell _1]}\\phi (I^{H_i})\\cap \\bigcap _{i\\in [\\ell _2]}N_{G-G_1}(v_i)\\bigg |= (\\beta ^{\\ell _1}d_2^{\\ell _2}\\pm 2\\delta ^2)n.$ For each $v\\in V(G)$ , we define a vertex tester $(v,\\omega )$ where $\\omega $ assigns every vertex in $V(H)$ its degree for all $H\\in \\widetilde{H}$ and add $(v,\\omega )$ to $\\mathcal {W}_{ver}$ .",
"Then Theorem REF implies that $\\Delta (G_1-\\phi (\\widetilde{\\mathcal {H}}))\\le 2\\delta n$ .",
"Let $G_2:= G- \\phi (\\widetilde{\\mathcal {H}})$ .",
"Next, we add $\\beta n$ vertices $W:=\\lbrace v_H\\rbrace _{H\\in \\mathcal {H}^{\\prime }}$ to $G_2$ and join $v_H$ to all vertices in $\\phi (I^H)$ and denote this new graph by $G_3$ .",
"Let $d_V:=\\beta ^2r$ and $d_W:=\\beta $ .",
"Hence (REF ), the typicality of $G-G_1$ , and $\\Delta (G_1-\\phi (\\widetilde{\\mathcal {H}}))\\le 2\\delta n$ imply that for all $w_1,\\ldots ,w_{\\ell _1}\\in W$ and $v_1,\\ldots ,v_{\\ell _2}\\in V(G)$ $\\bigcap _{x\\in \\lbrace w_1,\\ldots ,w_{\\ell _1},v_1,\\ldots ,v_{\\ell _2}\\rbrace }N_{G_3}(x)=(1\\pm \\sqrt{\\delta })\\beta ^{\\ell _1}\\cdot (\\beta ^2 r)^{\\ell _2}n=(1\\pm \\sqrt{\\delta })d_W^{\\ell _1}d_V^{\\ell _2}n$ whenever $1\\le \\ell _1+\\ell _2\\le s$ .",
"We apply Theorem REF to $G_3$ to obtain a decomposition of $G_3$ into copies of $J^+$ where the apex vertices are contained in $W$ .",
"Observe that this yields the desired decomposition of $G$ into $\\mathcal {H}$ .",
"Indeed, for $H\\in \\mathcal {H}^{\\prime }$ , let $\\mathcal {J}_H$ be the set of all copies of $J^+$ in $G_3$ whose apex vertex is $v_H$ ; hence $|\\mathcal {J}_H|=|I^H|/|V(J)|=\\beta n/|V(J)|$ .",
"We define a packing $\\phi ^{\\prime }$ of $\\mathcal {H}$ in $G$ as follows.",
"For all $H\\in \\mathcal {H}\\setminus \\mathcal {H}^{\\prime }$ , let $\\phi ^{\\prime }\\mid _{V(H)}:=\\phi \\mid _{V(H)}$ .",
"For all $H\\in \\mathcal {H}^{\\prime }$ , let $\\phi ^{\\prime }\\mid _{V(H^-)}:=\\phi \\mid _{V(H^-)}$ and each component of $H-V(H^-)$ (which is isomorphic to $J$ ) is mapped to $C-v_H$ for some $C\\in \\mathcal {J}_H$ such that every $C-v_H$ is the image of exactly one component isomorphic to $J$ of $H-V(H^-)$ .",
"$\\square $"
],
[
"Acknowledgement",
"The second author thanks Jaehoon Kim for stimulating discussions at early stages of the project."
]
] | 2001.03506 |
[
[
"Static topological black hole with nonminimal derivative coupling and\n nonlinear electromagnetic field of Born-Infeld type"
],
[
"Abstract We consider scalar-tensor gravity with nonminimal derivative coupling and Born-Infeld electromagnetic field which is minimally coupled to gravity.",
"Since cosmological constant is taken into account it allowed us not only derive static black hole with spherical horizon but also to obtain topological solutions with non-spherical horizons.",
"The obtained metrics are thoroughly analyzed, namely for different distances and types of topology of horizon.",
"To investigate singularities of the metrics Kretschmann scalar is used and it is shown that the character of singularity depends on the type of topology of horizon and dimension of space.",
"We also investigate black hole's thermodynamics, namely we obtain and examine black hole's temperature.",
"To derive the first law of black hole's thermodynamics Wald's approach is applied.",
"Nonetheless this approach is well established, there is ambiguity in definition of black hole's entropy which can be resolved just by virtue of some independent approach."
],
[
"Introduction",
"General Relativity is extremely successful theory which explains vast range of gravitational phenomena starting from the planetary motion and up to the evolution of the Universe [1], [2].",
"Numerous observations in astrophysics go hand in hand with theoretically predicted values [3], but nonetheless there are still some open issues that give a chance for new theories which can be treated as generalization of Einsteinian General Relativity [4], [5].",
"The most puzzling questions which still remain unsolved are the problems related to the origin (or existence) of cosmological singularities, Dark Energy and Dark Matter issues, the problems of evolution of the early Universe, for instance the problem of Inflation which is explained by approaches that take into account terms of the higher order of curvature [6].",
"To find solutions of the mentioned above problems different sorts of modification of General Relativity were used.",
"Among them we distinguish for example $F(R)$ theory, Lovelock Gravity, nonlocal modifications of the gravitational action, approaches which incorporate torsion tensor, namely Teleparallel Gravity, Scalar-Tensor Gravity.",
"There are some interrelations between different of these approaches, but certainly any of them has its own peculiarities, more detailed description of the mentioned modifications of General Relativity is given in recent review [5].",
"Scalar-Tensor theories are probably the most conservative modification of General Relativity, because the departing point of all these theories is still General Relativity, but additional scalar fields are also included.",
"The scalar fields can be coupled with gravitational degrees of freedom in different ways and due to the character of coupling we can classify all these theories.",
"Important feature of the scalar-tensor theories is also coupling between the scalar and additional material fields which might be taken into consideration, but it should be pointed out here that usually it is supposed to be just minimal coupling between the material fields and gravity in order to obey the equivalence principle.",
"Among the various types of the scalar-tensor theories we would like to focus on the so-called Horndeski Gravity [7].",
"One of the most remarkable features of this theory is related to the fact that equations of motion in Horndeski Gravity are of the second order, so it is free from ghost instabilities.",
"Being beyond the scope in the gravity community for several decades it has been studied intensively since the time when relation between Horndeski Gravity and some scenarios in String Theory was established.",
"Namely, this revival of interest in Horndeski Gravity is caused by the investigation of the so called Galileon Theories [8], [9] which are the scalar field theories that posses shift or Galilean symmetry, they are ghost free and those studies also renewed interest to DGP-model [10], which originally suffered from the ghost instabilities.",
"The equivalence between Horndeski and Galileon Theories was established [11].",
"Another approach which also gives rise to Galileon Theory takes its origin in Kaluza-Klein dimensional reduction procedure [12], [13].",
"Some other approaches related to String Theory also give rise to Galileon-like models [14], [15].",
"We note that multiscalar versions of Horndeski Gravity were also studied [16], [17], [18], [19] and approaches that go beyond Horndeski Theory, but keeping their main attractive features were considered [20], [21], [22], [23], [24].",
"Since its second revival Hondeski Gravity has been applied to vast range of problems in Cosmology and Physics of Black Holes.",
"In particular, cosmological solutions and various cosmological scenarios were studied [25], [26], [27].",
"Dynamics of Dark Energy/ Dark Matter models was investigated [28], [29], [30].",
"Slow-roll inflation mechanism without violation of unitarity bounds can be achieved in case of nonminimally coupled theories [31].",
"Various aspects of inflation were studied in theories with nonminimal derivative coupling, in particular reheating process during rapid oscillations and curvaton scenario were examined [32], [33], [34], [35], [36].",
"Effective field theory of Dark Energy was considered and its relation to Horndeski Gravity was established [37].",
"Predictions of Horndeski Gravity and constraints on various modifications of General Relativity due to modern experimentally obtained results were discussed [5], [38], [39], [40].",
"A lot of attention has been paid to the investigation of compact objects such as neutron stars and black holes in Horndeski Gravity.",
"Horndeski Gravity in its most general setting is known to have very complicated structure, thus the studies of the compact objects in this case are very difficult to perform.",
"As a consequence, the investigation of the black holes and the neutron stars even in particular cases of Horndeski Gravity is of paramount importance.",
"Rinaldi was the first who derived a static black hole solution in four dimensional case [41].",
"Then static black holes' solutions were obtained and investigated for various dimensions and in more general setup of Horndeski Gravity [42], [43], [44], [45], [46], [47], [48], [49].",
"Some attention has been paid to the examination of slowly rotating neutron stars and black holes [50], [51], [52].",
"Some aspects of black hole thermodynamics were studied [53], [54], [55].",
"Stability problem for different types of black holes, for instance with respect to odd-parity perturbations, was investigated [56], [57], [58], [59], [60].",
"Important problem also related to stability of black holes, is the causal structure which was studied in [61].",
"Boson and neutron stars in Horndeski Gravity were examined [50], [56], [63], [64], [65].",
"Several other aspects of black hole's physics in Horndeski Theory such as the existence of hair [66], or influence of higher order terms over curvature [67] were investigated.",
"In this work we consider a particular case of Horndeski gravity, namely the theory with nonminimal derivative coupling and we also take into account nonlinear electromagnetic field of Born-Infeld type which is minimally coupled to gravity.",
"Within this framework we find static solutions of field equations which represent black holes.",
"We point out here that charged static black hole solution in case of standard linear Maxwell field was studied in [44], [54].",
"In our previous work [55] we studied charged black hole, but with nonlinearity of other type, namely the so-called power-law nonlinearity.",
"Some aspects of the present work were also examined in our recent work [68] where comparison between Born-Infeld and power-law cases was shown.",
"Here we give deeper analysis of Born-Infeld case, with thorough investigation of the obtained solutions.",
"We also study thermodynamics of the black holes, derive and examine their temperature and obtain the first law of black hole's thermodynamics.",
"We would like to point out here that Born-Infeld modification of gauge field Lagrangian [69] and even its gravitational generalization, namely the so-called Eddington-inspired Born-Infeld gravity [70] are the subjects of active investigation for recent years [71].",
"The revival of interest to the Born-Infeld theory was mainly inspired by studies in String Theory, namely it was shown [72] that Born-Infeld Lagrangian might be treated as a low energy effective theory which describes vector field coupled to the string ending on D-brane.",
"Born-Infeld theory also appears in a bosonic sector of an effective theory which describes spontaneous breaking of supersymmetry [73].",
"Apart of its importance for String Theory, the Born-Infeld theory possesses some attractive features which make this theory rather exceptional among the other nonlinear modifications of a linear theory.",
"Firstly, we would like to mention finiteness of a self-energy of a point particle, actually it was the aim which brought Born and Infeld to introduce such a modification of the linear Maxwell theory [69].",
"Similarly to the linear vacuum Maxwell equation, vacuum Born-Infeld equations possess zero birefringence and have exceptional causal behaviour [74].",
"Born-Infeld theory conserves helicity, that was shown in several independent ways [75], [76], [77], namely it was argued to be a consequence of self-duality of the Born-Infeld theory.",
"Born-Infeld theory was shown to be the only theory, except the standard Maxwell electrodynamics, with causal massless spin-1 propagation without supersymmetrization [78].",
"These peculiarities of the Born-Infeld theory were a good motivation to study Born-Infeld theory in different contexts [71] and in particular they give solid arguments for us to consider Born-Infeld theory in the framework of the theory with nonminimal derivative coupling.",
"We also note here that actually we consider not the general Born-Infeld theory, but rather its simplified form which nevertheless inherits the main features of the latter theory, this fact can be explained by complicated structures of both theories which we use here, namely the Scalar-Tensor theory of Gravity we use here and the Born-Infeld theory.",
"The organization of the present paper is the following: in the second section obtain and investigate the black holes solutions, it should be pointed out here that since we take into account cosmological constant apart of spherically symmetric black hole solutions with non-spherical topology of horizon are considered in the present work.",
"Small subsection of the second section is devoted to the investigation of the gauge potential for the corresponding black holes.",
"In the third section we investigate some aspects of thermodynamics of the black hole, namely the behaviour of its temperature is studied and it is compared with the results derived for the linear Maxwell field [54], [55], in the second part of this section we obtain the first law of black hole's thermodynamics using Wald's approach.",
"In the forth section we give some conclusions."
],
[
"Field equations and black hole's solution",
"The action for the system we consider comprises of several terms, namely the standard Einstein-Hilbert part with cosmological constant, the terms describing the scalar field minimally and nonminimally coupled to gravity and finally the electromagnetic part, minimally coupled to gravity and which is represented by the action of Born-Infeld type.",
"So this action can be written the form: $S=\\frac{1}{16\\pi }\\int d^{n+1}x\\sqrt{-g}\\left( R-2\\Lambda -\\frac{1}{2}\\left(\\alpha g^{\\mu \\nu }-\\eta G^{\\mu \\nu }\\right)\\partial _{\\mu }\\varphi \\partial _{\\nu }\\varphi +4\\beta ^2\\left(1-\\sqrt{1+\\frac{F_{\\mu \\nu }F^{\\mu \\nu }}{2\\beta ^2}}\\right)\\right)+S_{GHY}$ and here $g_{\\mu \\nu }$ is the metric tensor and $g=det{g_{\\mu \\nu }}$ denotes its determinant, $G_{\\mu \\nu }$ and $R$ are the Einstein tensor and Ricci scalar respectively, $\\Lambda $ is the cosmological constant and $\\varphi $ is the scalar field, nonminimally coupled to gravity, $F_{\\mu \\nu }=\\partial _{\\mu }A_{\\nu }-\\partial _{\\nu }A_{\\mu }$ is the Maxwell field tensor and $\\beta $ denotes the Born-Infeld coupling constant.",
"It should be noted that when $\\beta \\rightarrow \\infty $ the Born-Infeld term gets transformed into standard linear Maxwell term, we also remark that the parameter $\\beta $ is supposed to be positive to obtain reasonable physical results which are in agreement with linear field case.",
"Finally, $S_{GHY}$ denotes boundary Gibbons-Hawking-York term, which is taken into account to have the variational problem well defined, and in case of the theory with nonminimal derivative coupling it can be represented in the form: $S_{GHY}=\\frac{1}{8\\pi }\\int d^nx\\sqrt{|h|}\\left(K+\\frac{\\eta }{4}\\left[\\nabla ^{\\mu }\\varphi \\nabla ^{\\nu }\\varphi K_{\\mu \\nu }+(n^{\\mu }n^{\\nu }\\nabla _{\\mu }\\varphi \\nabla _{\\nu }\\varphi +(\\nabla \\varphi )^2)K\\right]\\right),$ and here $h$ denotes the determinant of the boundary metric $h_{\\mu \\nu }$ , $K_{\\mu \\nu }$ and $K$ are the extrinsic curvature tensor and its trace respectively, $n_{\\mu }$ is the normal to the boundary hypersurface.",
"As it is known Gibbons-Hawking-York term does not affect on the equations of motion in the bulk.",
"As we have noted in the introduction, the gauge field part of the Lagrangian in the action (REF ) should be treated as a particular case of a general Born-Infeld Lagrangian which can be cast in the form: ${\\cal L}_{BI}=\\sqrt{-det(\\beta g_{\\mu \\nu })}-\\sqrt{-det(\\beta g_{\\mu \\nu }+F_{\\mu \\nu })},$ where $g_{\\mu \\nu }$ is the metric tensor, $F_{\\mu \\nu }$ is the gauge field tensor introduced above and $\\beta $ again denotes the Born-Infeld coupling constant.",
"If one would like to take into account the lowest order contribution over $\\beta $ , but keeping the main features of the general Born-Infeld theory, the Lagrangian (REF ) can be rewritten in the form: ${\\cal L}_{BI}\\simeq \\beta ^{(n+1)/2}\\sqrt{-g}\\left(1-\\sqrt{1+\\frac{F_{\\mu \\nu }F^{\\mu \\nu }}{2\\beta ^2}}\\right).$ The latter relation coincides with the Lagrangian for the gauge field in the action (REF ) with respect to the constant factor $\\beta ^{(n-3)/2}$ .",
"We also point out here that under assumption about the form of the gauge potential $A_{\\mu }$ or consequently the field tensor $F_{\\mu \\nu }$ which we use below, the right hand side of the latter relation is exactly the same as in the case of general Born-Infeld theory (REF ), this fact can be verified easily.",
"The principle of the least action, applied to the action (REF ) gives rise to equations of motion for the system.",
"Namely, for the gravitational field the equations of motion can be written in the form: $G_{\\mu \\nu }+\\Lambda g_{\\mu \\nu }=\\frac{1}{2}(\\alpha T^{(1)}_{\\mu \\nu }+\\eta T^{(2)}_{\\mu \\nu })+T^{(3)}_{\\mu \\nu }$ where the following notations are used: $T^{(1)}_{\\mu \\nu }=\\nabla _{\\mu }\\varphi \\nabla _{\\nu }\\varphi -\\frac{1}{2}g_{\\mu \\nu }\\nabla ^{\\lambda }\\varphi \\nabla _{\\lambda }\\varphi ,$ $\\nonumber T^{(2)}_{\\mu \\nu }=\\frac{1}{2}\\nabla _{\\mu }\\varphi \\nabla _{\\nu }\\varphi R-2\\nabla ^{\\lambda }\\varphi \\nabla _{\\nu }\\varphi R_{\\lambda \\mu }+\\frac{1}{2}\\nabla ^{\\lambda }\\varphi \\nabla _{\\lambda }\\varphi G_{\\mu \\nu }-g_{\\mu \\nu }\\left(-\\frac{1}{2}\\nabla _{\\lambda }\\nabla _{\\kappa }\\varphi \\nabla ^{\\lambda }\\nabla ^{\\kappa }\\varphi \\right.\\\\\\left.+\\frac{1}{2}(\\nabla ^2\\varphi )^2-R_{\\lambda \\kappa }\\nabla ^{\\lambda }\\varphi \\nabla ^{\\kappa }\\varphi \\right)-\\nabla _{\\mu }\\nabla ^{\\lambda }\\varphi \\nabla _{\\nu }\\nabla _{\\lambda }\\varphi +\\nabla _{\\mu }\\nabla _{\\nu }\\varphi \\nabla ^2\\varphi -R_{\\lambda \\mu \\kappa \\nu }\\nabla ^{\\lambda }\\varphi \\nabla ^{\\kappa }\\varphi $ $T^{(3)}_{\\mu \\nu }=2\\beta ^2g_{\\mu \\nu }\\left(1-\\sqrt{1+\\frac{F_{\\kappa \\lambda }F^{\\kappa \\lambda }}{2\\beta ^2}}\\right)+\\frac{2F_{\\mu \\rho }{F_{\\nu }}^{\\rho }}{\\sqrt{1+\\frac{F_{\\kappa \\lambda }F^{\\kappa \\lambda }}{2\\beta ^2}}}$ It is worth being remarked that the term $T^{(1)}_{\\mu \\nu }$ is the standard form of stress-energy tensor for a minimally coupled scalar field and $T^{(2)}_{\\mu \\nu }$ corresponds to the stress-energy tensor of the nonminimally coupled part.",
"Finally, the term $T^{(3)}_{\\mu \\nu }$ denotes the stress-energy tensor for the electromagnetic field given by the Born-Infeld action.",
"Varying the action (REF ) with respect to the scalar field $\\varphi $ one arrives at the following equation: $(\\alpha g_{\\mu \\nu }-\\eta G_{\\mu \\nu })\\nabla ^{\\mu }\\nabla ^{\\nu }\\varphi =0.$ Taking the variation of the action (REF ) with respect to the gauge potential $A_{\\mu }$ we obtain the equations for the gauge field: $\\nabla _{\\mu }\\left(\\frac{F^{\\mu \\nu }}{\\sqrt{1+\\frac{F_{\\kappa \\lambda }F^{\\kappa \\lambda }}{2\\beta ^2}}}\\right)=0.$ It should be noted that in the limit $\\beta \\rightarrow \\infty $ standard Maxwell equations are recovered.",
"Since the action (REF ) includes the cosmological constant $\\Lambda $ and it means that topological solutions might exist.",
"Here we will consider static topological solutions and consequently suppose that the metric takes the form: $ds^2=-U(r)dt^2+W(r)dr^2+r^2d\\Omega ^{2(\\varepsilon )}_{(n-1)},$ where $d\\Omega ^{2(\\varepsilon )}_{n-1}$ is the line element of a $n-1$ –dimensional hypersurface of a constant curvature which can be represented in the form: $d\\Omega ^{2(\\varepsilon )}_{(n-1)}={\\left\\lbrace \\begin{array}{ll}d\\theta ^2+\\sin ^2{\\theta }d\\Omega ^2_{(n-2)}, \\quad \\varepsilon =1,\\\\d\\theta ^2+{\\theta }^2 d\\Omega ^2_{(n-2)},\\quad \\varepsilon =0,\\\\d\\theta ^2+\\sinh ^2{\\theta }d\\Omega ^2_{(n-2)},\\quad \\varepsilon =-1,\\end{array}\\right.",
"}$ and here $d\\Omega ^2_{(n-2)}$ is the line element of a $n-2$ –dimensional hypersphere.",
"Thus, the expression $d\\Omega ^{2(\\varepsilon )}_{(n-1)}$ represents the line element on a hypersurface of positive, null or negative curvature for given values of parameter $\\varepsilon $ .",
"We point out here that in the present work we consider dimensions $n\\geqslant 3$ .",
"As it has been mentioned above we are going to obtain the static solutions and it means that we can choose the gauge field form as follows: $A=A_{0}(r)dt$ (only a scalar component of electromagnetic potential is nonzero).",
"Taking the chosen form of gauge field and the metric (REF ) into account we solve the equations (REF ) and obtain the electromagnetic field tensor of the form: $F_{rt}=-\\frac{q\\beta }{\\sqrt{q^2+\\beta ^2r^{2(n-1)}}}\\sqrt{UW},$ where $q$ is an integration constant related to the black hole's charge.",
"We note here that the behaviour of the electromagnetic field depends on the product of metric functions $UW$ which is a function of the radial coordinate $r$ .",
"Taking into consideration the explicit form of the metric (REF ) and integrating the equation (REF ) for a once we arrive at the relation: $\\sqrt{\\frac{U}{W}}r^{n-1}\\left[\\alpha -\\eta \\frac{(n-1)}{2rW}\\left(\\frac{U^{\\prime }{}}{U}-\\frac{(n-2)}{r}(\\varepsilon W-1)\\right)\\right]\\varphi ^{\\prime }=C$ In the following we assume that obtained in the latter relation constant $C$ is equal to zero and it simplifies the procedure of solving of the field equations (REF ).",
"The imposed condition on the constant $C$ is equivalent to the assumption about the relation between a component of the metric tensor $g_{\\mu \\nu }$ and Einstein tensor $G_{\\mu \\nu }$ , namely: $\\alpha g_{rr}-\\eta G_{rr}=0$ We note that the relation (REF ) was used in most papers where black holes in the theory with nonminimal derivative coupling were examined, in particular it appears in [41], [42], [53], [54], [55].",
"Taking into account the evident form for the metric (REF ) and the obtained above expression for electromagnetic field (REF ) we can represent the equations (REF ) in the form: $\\nonumber \\frac{(n-1)}{2rW}\\left(\\frac{W^{\\prime }{}}{W}+\\frac{(n-2)}{r}(\\varepsilon W-1)\\right)\\left(1+\\frac{3}{4}\\eta \\frac{(\\varphi ^{\\prime }{})^2}{W}\\right)-\\Lambda =\\frac{\\alpha }{4W}(\\varphi ^{\\prime }{})^2+\\\\\\frac{\\eta }{2}\\left(\\frac{(n-1)}{rW^2}{\\varphi ^{\\prime \\prime }}\\varphi ^{\\prime }{}+\\frac{(n-1)(n-2)}{r^2W^2}(\\varphi ^{\\prime }{})^2\\left(\\varepsilon W-\\frac{1}{2}\\right)\\right)-2\\beta ^2+\\frac{2\\beta }{r^{n-1}}\\sqrt{q^2+\\beta ^2 r^{2(n-1)}};$ $\\nonumber \\frac{(n-1)}{2rW}\\left(\\frac{U^{\\prime }{}}{U}-\\frac{(n-2)}{r}(\\varepsilon W-1)\\right)\\left(1+\\frac{3}{4}\\eta \\frac{(\\varphi ^{\\prime }{})^2}{W}\\right)+\\Lambda =\\\\\\frac{\\alpha }{4W}(\\varphi ^{\\prime }{})^2-\\frac{\\eta }{2}\\varepsilon \\frac{(n-1)(n-2)}{2r^2W}(\\varphi ^{\\prime }{})^2+2\\beta ^2-\\frac{2\\beta }{r^{n-1}}\\sqrt{q^2+\\beta ^2 r^{2(n-1)}};$ $\\nonumber \\left[\\frac{1}{2UW}\\left(U^{\\prime }{}^{\\prime }{}-\\frac{(U^{\\prime }{})^2}{2U}-\\frac{U^{\\prime }{}W^{\\prime }{}}{2W}\\right)+\\frac{n-2}{2rW}\\left(\\frac{U^{\\prime }{}}{U}-\\frac{W^{\\prime }{}}{W}\\right)-\\frac{(n-2)(n-3)}{2r^2W}(\\varepsilon W-1)\\right]\\times \\\\\\nonumber \\left(1+\\frac{\\eta }{4}\\frac{(\\varphi ^{\\prime }{})^2}{W}\\right)+\\Lambda =-\\frac{\\alpha }{4W}(\\varphi ^{\\prime }{})^2-\\frac{\\eta }{2W^2}\\varphi ^{\\prime }{}^{\\prime }{}\\varphi ^{\\prime }{}\\left(\\frac{U^{\\prime }{}}{2U}+\\frac{n-1}{r}\\right)+\\\\\\frac{\\eta }{2}\\frac{(\\varphi ^{\\prime }{})^2}{W}\\left(\\frac{U^{\\prime }{}W^{\\prime }{}}{4UW^2}+\\frac{(n-2)W^{\\prime }{}}{2rW^2}-\\varepsilon \\frac{(n-2)(n-3)}{2r^2}\\right)+2\\beta ^2\\left(1-\\frac{\\beta r^{n-1}}{\\sqrt{q^2+\\beta ^2r^{2(n-1)}}}\\right).$ It is worth noting that we have four equations, namely (REF ), (REF ), (REF ) and (REF ) for three unknown functions $U(r)$ , $W(r)$ and $\\varphi ^{\\prime }$ , it means the we might choose three of them to find the unknown functions and the fourth equation should be satisfied as an identity for the obtained solution.",
"From the reason of simplicity we take the equations (REF ), (REF ) and (REF ).",
"Having used the equations (REF ) and (REF ) we can write: $(\\varphi ^{\\prime })^2=-\\frac{4r^2W}{\\eta (2\\alpha r^2+\\varepsilon \\eta (n-1)(n-2))}\\left(\\alpha +\\Lambda \\eta -2\\beta ^2\\eta +2\\beta \\eta r^{1-n}\\sqrt{q^2+\\beta ^2r^{2(n-1)}}\\right);$ $UW=\\frac{\\left((\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )r^2+\\varepsilon \\eta (n-1)(n-2)-2\\beta \\eta r^{3-n}\\sqrt{q^2+\\beta ^2r^{2(n-1)}}\\right)^2}{(2\\alpha r^2+\\varepsilon \\eta (n-1)(n-2))^2}.$ We point out here that the right hand side of the relation (REF ) has to take positive values outside black hole's horizon.",
"This condition leads to some restrictions on the parameters of coupling $\\alpha $ , $\\eta $ , cosmological constant $\\Lambda $ , charge parameter $q$ and Born-Infeld parameter $\\beta $ .",
"Having supposed that the coupling parameters $\\alpha $ and $\\eta $ are positive (and assuming that the expression $2\\alpha r^2+\\varepsilon \\eta (n-1)(n-2)$ is positive in the outer domain) one arrives at the conclusion that the expression $\\alpha +\\Lambda \\eta -2\\beta ^2\\eta +2\\beta \\eta r^{1-n}\\sqrt{q^2+\\beta ^2r^{2(n-1)}}$ should be negative outside the horizon, what can be achieved if the cosmological constant $\\Lambda $ is negative.",
"We point out here that similar condition on the cosmological constant was imposed in case of neutral [52] and charged [55] black holes in Horndeski gravity.",
"Since the metric function $W(r)$ diverges on the horizon changing its sign while crossing the horizon, the same is true for the function $(\\varphi ^{\\prime })^2$ , unless the expression $\\alpha +\\Lambda \\eta -2\\beta ^2\\eta +2\\beta \\eta r^{1-n}\\sqrt{q^2+\\beta ^2r^{2(n-1)}}$ also changes it sign on the horizon.",
"It leads to the consequence that in the inner domain $\\varphi ^{\\prime }$ becomes purely imaginary (phantom-like behaviour), similar situation takes place in case of neutral and charged black holes [52], [55].",
"We point out there that the kinetic energy of the scalar field $K=\\nabla _{\\mu }\\varphi \\nabla ^{\\mu }\\varphi $ is finite at the horizon and is positive in the inner domain up to the moment when the expression $\\alpha +\\Lambda \\eta -2\\beta ^2\\eta +2\\beta \\eta r^{1-n}\\sqrt{q^2+\\beta ^2r^{2(n-1)}}$ changes it sign.",
"Similar analysis performed under assumption that $\\eta <0$ while $\\alpha >0$ shows that in this case the cosmological constant $\\Lambda $ should be taken negative as well.",
"Here we also note that if $\\alpha +\\Lambda \\eta -2\\beta ^2\\eta =0$ , the right hand side of the relation (REF ) becomes negative in the outer domain and it is not acceptable from the physical point of view.",
"If one imposes the condition $\\alpha -\\Lambda \\eta +2\\beta ^2\\eta =0$ to provide positivity of the right hand side of the relation (REF ) the parameter $\\eta $ should take negative values, this condition allows to obtain the following relations for the metric function in a bit simpler form, but it does not change the character of asymptotic behaviour of the metric function $U(r)$ for small and large distances in comparison with the general case as it might be shown from the relations to be obtained.",
"The right hand side of the relation (REF ) is always nonnegative and it demonstrates that the metric functions $U(r)$ and $W(r)$ always have the same sign as it should be for a black hole's solution.",
"It is worth emphasizing that in the limit $\\beta \\rightarrow \\infty $ the relations (REF ), (REF ) as well as the relations for the metric function $U(r)$ that will be obtained below, are reduced to corresponding relations derived for linear field [44], [54], [55].",
"The metric function $U(r)$ can be written as follows: $\\nonumber U(r)=\\varepsilon -\\frac{\\mu }{r^{n-2}}-\\frac{2(\\Lambda -2\\beta ^2)}{n(n-1)}r^2-\\frac{2\\beta (\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{\\alpha (n-1)r^{n-2}}\\int \\sqrt{q^2+\\beta ^2r^{2(n-1)}}dr+\\frac{(\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )^2}{2(n-1)\\alpha \\eta r^{n-2}}\\times \\\\\\int \\frac{r^{n+1}}{r^2+d^2}dr-\\frac{2\\beta (\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )d^2}{\\alpha (n-1) r^{n-2}}\\int \\frac{\\sqrt{q^2+\\beta ^2r^{2(n-1)}}}{r^2+d^2}dr+\\frac{2\\beta ^2\\eta }{\\alpha (n-1)r^{n-2}}\\int \\frac{r^{3-n}(q^2+\\beta ^2r^{2(n-1)})}{r^2+d^2}dr,$ where $d^2=\\varepsilon \\eta (n-1)(n-2)/2\\alpha $ and we point out that in the following relations we assume that $d^2>0$ .",
"It should be noted that the second and the fourth integrals in the relation (REF ) have some differences for odd and even dimensions of space $n$ and the first and third ones cannot be expressed in terms of elementary functions.",
"Namely the first integral when $q^2r^{2(1-n)}/\\beta ^2<1$ can be represented as follows [79]: $\\int \\sqrt{q^2+\\beta ^2r^{2(n-1)}}dr=\\frac{\\beta }{n}r^n{_{2}F_{1}}\\left(-\\frac{1}{2},\\frac{n}{2(1-n)};\\frac{2-n}{2(1-n)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)$ and we additionally stress here that the right hand side of the written above relation is valid for large distances ($r^{2(n-1)}>q^2/\\beta ^2$ ), whereas for small distances $r^{2(n-1)}<q^2/\\beta ^2$ we can write: $\\int \\sqrt{q^2+\\beta ^2r^{2(n-1)}}dr=qr{_{2}F_{1}}\\left(-\\frac{1}{2},\\frac{1}{2(n-1)};\\frac{2n-1}{2(n-1)};-\\frac{\\beta ^2}{q^2}r^{2(n-1)}\\right).$ The third integral in (REF ) needs also special care and due to its importance we pay attention to it.",
"We point out here that the result of calculation of the mentioned integral depends on the parity of $n$ and similarly to the written above relations (REF ) and (REF ) its evident form is defined by the condition whether $r^{2(n-1)}>q^2/\\beta ^2$ or $r^{2(n-1)}<q^2/\\beta ^2$ , but it also depends on the relation between $r$ and $d$ (takes a bit different form for $r>d$ and $r<d$ ).",
"Namely, for odd $n$ when $r^{2(n-1)}>q^2/\\beta ^2$ and $r>d$ (large distances) this integral can be represented in the form: $\\int \\frac{\\sqrt{q^2+\\beta ^2r^{2(n-1)}}}{r^2+d^2}dr=\\beta r^{n-2}\\sum ^{+\\infty }_{j=0}\\frac{(-1)^j}{n-2(j+1)}\\left(\\frac{d}{r}\\right)^{2j}{_{2}}F_{1}\\left(-\\frac{1}{2},\\frac{n-2(j+1)}{2(1-n)};\\frac{n+2j}{2(n-1)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right),$ if $r<d$ , but still $r^{2(n-1)}>q^2/\\beta ^2$ we can represent the latter integral in the following form: $\\int \\frac{\\sqrt{q^2+\\beta ^2r^{2(n-1)}}}{r^2+d^2}dr=\\frac{\\beta r^{n}}{d^2}\\sum ^{+\\infty }_{j=0}\\frac{(-1)^j}{n+2j}\\left(\\frac{r}{d}\\right)^{2j}{_{2}}F_{1}\\left(-\\frac{1}{2},\\frac{n+2j}{2(1-n)};\\frac{n-2(j+1)}{2(n-1)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right).$ There is no difficulties with the latter integral when two other options for distance are given, namely when $r^{2(n-1)}<q^2/\\beta ^2$ and $r<d$ or $r>d$ .",
"In contrast to odd $n$ , the latter integral has some subtleties when $n$ is even.",
"If $r^{2(n-1)}>q^2/\\beta ^2$ and $r>d$ it can be written in the form: $\\nonumber \\int \\frac{\\sqrt{q^2+\\beta ^2r^{2(n-1)}}}{r^2+d^2}dr=\\beta r^{n-2}\\mathop {\\sum ^{+\\infty }_{j=0}}_{ j\\ne \\frac{n}{2}-1}\\frac{(-1)^j}{n-2(j+1)}\\left(\\frac{d}{r}\\right)^{2j}{_{2}}F_{1}\\left(-\\frac{1}{2},\\frac{n-2(j+1)}{2(1-n)};\\frac{n+2j}{2(n-1)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)\\\\+(-1)^{\\frac{n}{2}-1}\\beta d^{n-2}\\left(\\sum ^{+\\infty }_{l=1}\\frac{(-1)^l}{l!",
"}\\left(-\\frac{1}{2}\\right)_{l}\\left(\\frac{q}{\\beta }\\right)^{2l}\\frac{r^{2(1-n)l}}{2(1-n)l}+\\ln {\\left(\\frac{r}{d}\\right)}\\right),$ where $(a)_l$ denotes the Pochhammer symbol of $a$ .",
"When $r<d$ , but still $r^{2(n-1)}>q^2/\\beta ^2$ we can write: $\\nonumber \\int \\frac{\\sqrt{q^2+\\beta ^2r^{2(n-1)}}}{r^2+d^2}dr=\\beta \\frac{r^n}{d^2}\\mathop {\\sum ^{+\\infty }_{j=0,}\\sum ^{+\\infty }_{l=0}}_{j\\ne (n-1)l-\\frac{n}{2}}\\frac{(-1)^{j+l}}{l!",
"(n+2j+2(1-n)l)}\\left(-\\frac{1}{2}\\right)_l\\left(\\frac{r}{d}\\right)^{2l}\\left(\\frac{q}{\\beta }r^{1-n}\\right)^{2l}\\\\+\\beta d^{n-2}\\sum ^{+\\infty }_{l=1}\\frac{(-1)^{nl-\\frac{n}{2}}}{l!",
"}\\left(-\\frac{1}{2}\\right)_l\\left(\\frac{q}{\\beta }d^{1-n}\\right)^{2l}\\ln {\\left(\\frac{r}{d}\\right)}.$ If $r^{2(n-1)}<q^2/\\beta ^2$ and $r<d$ or $r>d$ one can also write the evident form for the above integral.",
"It should be pointed out that the main difference between odd and even $n$ cases for given above integral is the presence of logarithmic terms $\\sim \\ln (r/d)$ for even $n$ , whereas for odd $n$ such a term does not appear.",
"Considering large distance case when $r^{2(n-1)}>q^2/\\beta ^2$ and $r>d$ and taking into account the written above relations (REF ) and (REF ) or (REF ) we can write the explicit form for the metric function $U$ , namely for odd $n$ it takes the form: $\\nonumber U(r)=\\varepsilon -\\frac{\\mu }{r^{n-2}}-\\frac{2(\\Lambda -2\\beta ^2)}{n(n-1)}r^2-\\frac{2\\beta ^2(\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{\\alpha n(n-1)}r^2{_{2}F_{1}}\\left(-\\frac{1}{2},\\frac{n}{2(1-n)};\\frac{n-2}{2(n-1)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)+\\\\\\nonumber \\frac{(\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )^2}{2\\alpha \\eta (n-1)}\\left[\\sum ^{(n-1)/2}_{j=0}(-1)^j\\frac{d^{2j}r^{2(1-j)}}{n-2j}+(-1)^{\\frac{n+1}{2}}\\frac{d^n}{r^{n-2}}\\arctan {\\left(\\frac{r}{d}\\right)}\\right]-\\frac{2\\beta ^2(\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )d^2}{\\alpha (n-1)}\\times \\\\\\nonumber \\sum ^{+\\infty }_{j=0}\\frac{(-1)^j}{n-2(j+1)}\\left(\\frac{d}{r}\\right)^{2j}{_{2}F_{1}}\\left(-\\frac{1}{2},\\frac{n-2(j+1)}{2(1-n)};\\frac{n+2j}{2(n-1)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)+\\frac{2\\beta ^2\\eta }{\\alpha (n-1)}\\times \\\\\\left[q^2\\sum ^{(n-5)/2}_{j=0}\\frac{(-1)^jr^{6-2n+2j}}{(4-n+2j)d^{2(j+1)}}+\\frac{(-1)^{\\frac{n-3}{2}}}{r^{n-2}}\\left(\\frac{q^2}{d^{n-2}}+\\beta ^2d^n\\right)\\arctan \\left(\\frac{r}{d}\\right)+\\beta ^2\\sum ^{(n-1)/2}_{j=0}(-1)^j\\frac{d^{2j}r^{2(1-j)}}{n-2j}\\right];$ and for even $n$ we obtain: $\\nonumber U(r)=\\varepsilon -\\frac{\\mu }{r^{n-2}}-\\frac{2(\\Lambda -2\\beta ^2)}{n(n-1)}r^2-\\frac{2\\beta ^2(\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{\\alpha n(n-1)}r^2{_{2}F_{1}}\\left(-\\frac{1}{2},\\frac{n}{2(1-n)};\\frac{n-2}{2(n-1)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)+\\\\\\nonumber \\frac{(\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )^2}{2\\alpha \\eta (n-1)}\\left[\\sum ^{n/2-1}_{j=0}(-1)^j\\frac{d^{2j}r^{2(1-j)}}{n-2j}+(-1)^{\\frac{n}{2}}\\frac{d^n}{2r^{n-2}}\\ln {\\left(\\frac{r^2}{d^2}+1\\right)}\\right]-\\frac{2\\beta ^2(\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )d^2}{\\alpha (n-1)}\\times \\\\\\nonumber \\left[\\mathop {\\sum ^{+\\infty }_{j=0}}_{j\\ne \\frac{n}{2}-1}\\frac{(-1)^j}{n-2(j+1)}\\left(\\frac{d}{r}\\right)^{2j}{_{2}F_{1}}\\left(-\\frac{1}{2},\\frac{n-2(j+1)}{2(1-n)};-\\frac{n+2j}{2(1-n)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)+(-1)^{\\frac{n}{2}}\\frac{d^{n-2}}{r^{n-2}}\\times \\right.\\\\\\nonumber \\left.\\left(\\sum ^{+\\infty }_{j=1}\\frac{(-1)^j}{j!",
"}\\left(-\\frac{1}{2}\\right)_{j}\\left(\\frac{q}{\\beta }\\right)^{2j}\\frac{r^{2(1-n)j}}{2(n-1)j}-\\ln {\\left(\\frac{r}{d}\\right)}\\right)\\right]+\\frac{2\\beta ^2\\eta }{\\alpha (n-1)}\\left[q^2\\sum ^{(n-6)/2}_{j=0}\\frac{(-1)^jr^{6-2n+2j}}{(4-n+2j)d^{2(j+1)}}+\\right.\\\\\\left.\\frac{(-1)^{\\frac{n-2}{2}}}{2r^{n-2}}\\left(\\frac{q^2}{d^{n-2}}\\ln \\left(1+\\frac{d^2}{r^2}\\right)-\\beta ^2d^n\\ln \\left(1+\\frac{r^2}{d^2}\\right)\\right)+\\beta ^2\\sum ^{n/2-1}_{j=0}(-1)^j\\frac{d^{2j}r^{2(1-j)}}{n-2j}\\right].$ It should be stressed that the infinite sums in the written above relations (REF ) and (REF ) are convergent when $r>d$ and $r^{2(n-1)}>q^2/\\beta ^2$ (which also takes place for large distances), but there is no difficulty in writing the evident form for the metric function $U(r)$ when $r^{2(n-1)}>q^2/\\beta ^2$ and $d<r$ or for other two possible options for the distance (small distances).",
"As we have noted above, the condition $d^2>0$ is imposed after the integral form for the metric function $U(r)$ is written, we point out here that solutions with $d^2<0$ might be studied, but similarly as it was shown for neutral black hole [52] or power-law field [55] the corresponding solutions do not represent a black hole.",
"For the flat horizon case ($\\varepsilon =0$ ) the metric function $U(r)$ can be written in a simpler form: $\\nonumber U(r)=-\\frac{\\mu }{r^{n-2}}+\\frac{(\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )^2+4\\beta ^4\\eta ^2}{2\\alpha \\eta n(n-1)}r^2-\\frac{2\\beta ^2(\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{\\alpha n(n-1)}r^2\\times \\\\{_{2}F_{1}}\\left(-\\frac{1}{2},\\frac{n}{2(1-n)};\\frac{2-n}{2(1-n)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)-\\frac{2\\beta ^2\\eta q^2}{\\alpha (n-1)(n-2)}r^{2(2-n)}.$ We note that the written above relation takes place for large $r$ , for small $r$ instead of the relation (REF ) the relation (REF ) should be used, completely in the same way as it was done for the previously considered cases when $\\varepsilon \\ne 0$ .",
"Taking into account all the obtained expressions for the metric function $U$ and the product $UW$ we might investigate their behaviour for some specific values of $r$ .",
"Firstly, we examine the function $U(r)$ for small values of $r$ .",
"Having used the relations (REF ) and (REF ) rewritten for small distances and considering only the leading terms we write: $U(r)\\simeq \\frac{4\\beta ^2q^2}{\\varepsilon (n-1)^2(n-2)(4-n)}r^{2(3-n)}$ and here we omit all the subleading terms for small $r$ .",
"It should be pointed out that the relation (REF ) is valid when $n\\geqslant 5$ .",
"If $n=3$ the leading term is of the form: $U(r)\\simeq -\\frac{\\mu }{r},$ in this case the dominant term for small distances is the same as for the nonminimally coupled theory without any electromagnetic field [52].",
"This fact can be explained by nonsingular behaviour of the electromagnetic potential at the origin of coordinates for the case $n=3$ and what is not true for higher dimensions.",
"When $n=4$ the leading term can be written in the following form: $U(r)\\simeq -\\frac{\\beta ^2q^2}{9\\varepsilon r^2}\\ln {\\left(1+\\frac{d^2}{r^2}\\right)},$ as it is easy to conclude for $n=4$ the leading term (REF ) is caused by the term of the same origin as in the relation (REF ), but due to different powers of $r$ under integral from which they are derived they can have either power-law or logarithmic dependences.",
"The product of the metric functions $UW$ for small $r$ and $n\\geqslant 4$ takes the form: $UW\\simeq \\frac{4\\beta ^2q^2}{(n-1)^2(n-2)^2}r^{2(3-n)},$ whereas for $n=3$ the product $UW\\simeq \\left(1-\\frac{\\beta q}{\\varepsilon }\\right)^2$ and when $q=0$ it goes to the limit that is typical for static black holes in standard General Relativity.",
"We note that singular behaviour of the product $UW$ for small $r$ takes place for the black holes with linear and power-law nonlinear electromagnetic filed in the theory with nonminimal derivative coupling [54], [55].",
"In the above analysis we have not considered the solution with flat horizon surface (REF ).",
"As it is easy to see, the leading term for small distances takes the form as follows: $U(r)\\simeq -\\frac{2\\beta ^2\\eta q^2}{\\alpha (n-1)(n-2)}r^{2(2-n)},$ here we note that the latter equation takes place for all $n\\geqslant 3$ .",
"Having compared the relations (REF ) and (REF ) one can conclude that for the case of flat horizon ($\\varepsilon =0$ ) the singularity of the metric function $U(r)$ when $r\\rightarrow 0$ is stronger than for nonflat horizon surface ($\\varepsilon =\\pm 1$ ).",
"We also remark that for small distances for $\\varepsilon =0$ and $\\varepsilon =1$ the leading term is negative and for $\\varepsilon =-1$ it might be positive.",
"For large $r$ the metric functions (REF ), (REF ) and (REF ) have similar dependence given by the leading term of the form: $U\\simeq \\frac{(\\alpha -\\Lambda \\eta )^2}{2n(n-1)\\alpha \\eta }r^2,$ and here similarly to small distances we do not write subleading terms.",
"It is easy to see that for large $r$ the metric function is always of anti-de Sitter type.",
"The product of the metric functions for large $r$ is as follows $UW\\simeq (\\alpha -\\Lambda \\eta )^2/4\\alpha ^2$ .",
"We also consider the regime of large $\\eta $ , namely when the terms which correspond to the minimal coupling (terms related to the parameter $\\alpha $ ) are supposed to be considerably smaller than the terms which appear due to the presence of nonminimal coupling (terms related to $\\eta $ ).",
"The metric function $U(r)$ takes the form: $\\nonumber U(r)\\simeq \\varepsilon -\\frac{\\mu }{r^{n-2}}+\\frac{4\\beta ^2 q^2}{\\varepsilon (n-1)^2(n-2)(4-n)}r^{2(3-n)}-\\frac{2(\\Lambda -2\\beta ^2)}{n(n-1)}r^2+\\\\\\nonumber \\frac{(\\Lambda -2\\beta ^2)^2+4\\beta ^4}{\\varepsilon (n-1)^2(n^2-4)}r^4-\\frac{4\\beta ^2}{n(n-1)}r^2{_{2}F_{1}}\\left(-\\frac{1}{2},\\frac{n}{2(1-n)};\\frac{2-n}{2(1-n)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)+\\\\\\frac{4\\beta ^2(\\Lambda -2\\beta ^2)r^4}{\\varepsilon (n-1)^2(n^2-4)}{_{2}F_{1}}\\left(-\\frac{1}{2},\\frac{n+2}{2(1-n)};\\frac{4-n}{2(1-n)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)+{\\cal O}\\left(\\frac{1}{\\eta }\\right).$ Having compared the expression (REF ) with (REF ) one can conclude that for the cases of nonflat horizon ($\\varepsilon =\\pm 1$ ) the written above relation does not contain the terms proportional to $\\eta $ whereas the relation (REF ) does, so they have completely different behavior in this regime.",
"We also remark that similar situation took place for nonlinear electromagnetic field with power-law dependence [55].",
"The product of the metric functions in regime of large $\\eta $ for nonflat topology of horizon takes the form: $UW\\simeq \\left(1-\\frac{\\Lambda -2\\beta ^2}{\\varepsilon (n-1)(n-2)}r^2-\\frac{2\\beta r^{3-n}}{\\varepsilon (n-1)(n-2)}\\sqrt{q^2+\\beta ^2r^{2(n-1)}}\\right)^2+{\\cal O}\\left(\\frac{1}{\\eta }\\right).$ For flat horizon solution ($\\varepsilon =0$ ) we obtain: $UW\\simeq \\eta ^2\\left(\\frac{\\Lambda -2\\beta ^2}{2\\alpha }+\\frac{\\beta \\eta }{\\alpha }r^{1-n}\\sqrt{q^2+\\beta ^2r^{2(n-1)}}\\right)^2+{\\cal O}(\\eta ).$ We have analyzed the behaviour of the metric function $U(r)$ for different distances $r$ and various values of some parameters.",
"Here we also give graphical representation which might make the given above analysis more transparent.",
"The figure [REF ] shows that the change of the parameter $\\beta $ (parameter of nonlinearity of the gauge field) affects weakly on the behaviour of the metric function (at least this effect is small in the domain of change of the parameters we have used here).",
"The influence of the cosmological constant $\\Lambda $ as it follows even from the above analysis is more important for large distances, where the AdS-term becomes dominant, but for small distances the influence of the AdS-term is negligibly small.",
"Some particular interest might be also in considering of the regime of small $\\beta $ , namely for the metric function $U(r)$ we have: $U(r)\\simeq \\varepsilon -\\frac{\\mu }{r^{n-2}}-\\frac{2\\Lambda }{n(n-1)}r^2-\\frac{2\\beta (\\alpha -\\Lambda \\eta )}{\\alpha (n-1)}\\frac{q}{r^{n-3}}-\\frac{2\\beta (\\alpha +\\Lambda \\eta )}{\\alpha (n-1)}\\frac{qd}{r^{n-2}}\\arctan {\\left(\\frac{r}{d}\\right)}+{\\cal {O}}(\\beta ^2),$ and for the product of the metric functions we write: $UW\\simeq \\frac{\\left((\\alpha -\\Lambda \\eta )r^2+\\varepsilon \\eta (n-1)(n-2)\\right)}{(2\\alpha r^2+\\varepsilon \\eta (n-1)(n-2))^2}\\left((\\alpha -\\Lambda \\eta )r^2+\\varepsilon \\eta (n-1)(n-2)-4\\eta q r^{3-n}\\right)+{\\cal {O}}(\\beta ^2).$ We point out that for the flat horizon case ($\\varepsilon =0$ ) the last term in the asymptotic relation (REF ) does not appear.",
"It should be also noted that given above two relations are valid for small and intermediate distances, since the product $\\beta r^{2(n-1)}$ which is present in general relation for the metric functions might become large for corresponding large values of $r$ .",
"Figure: Metric functions U(r)U(r) for different values of parameter β\\beta (the left graph) and different values of the cosmological constant Λ\\Lambda (the right one).",
"For all the graphs we have n=3n=3, ε=1\\varepsilon =1, α=0.2\\alpha =0.2, η=0.4\\eta =0.4, μ=1\\mu =1, q=0.2q=0.2.",
"For the left graph the other parameters are equal to Λ=-2\\Lambda =-2, β=1\\beta =1 (solid curve) β=10\\beta =10 (dashed curve).",
"For the right graph β=1\\beta =1, and Λ=-1\\Lambda =-1 (dotted curve), Λ=-2\\Lambda =-2 (solid curve) and Λ=-3\\Lambda =-3 (dashed curve).To obtain information about coordinate and physical singularities of the metric Kretschmann scalar at different points should be examined.",
"In general it takes the form: $R_{\\mu \\nu \\kappa \\lambda }R^{\\mu \\nu \\kappa \\lambda }=\\frac{1}{UW}\\left(\\frac{d}{dr}\\left[\\frac{U^{\\prime }}{\\sqrt{UW}}\\right]\\right)^2+\\frac{(n-1)}{r^2W^2}\\left(\\frac{(U^{\\prime })^2}{U^2}+\\frac{(W^{\\prime })^2}{W^2}\\right)+\\frac{2(n-1)(n-2)}{r^4W^2}(\\varepsilon W-1)^2.$ One can verify easily that at the horizon point $r_+$ , namely when $U(r_+)=0$ the Kretschmann scalar (REF ) is nonsingular, it means that the horizon points are the points with a coordinate singularity as it should be for a black hole.",
"To investigate the behavior of the metric at the origin and at the infinity one should use corresponding asymptotic relations that has been obtained previously.",
"At large distances when $r\\rightarrow \\infty $ the Kretschmann scalar is as follows: $R_{\\mu \\nu \\kappa \\lambda }R^{\\mu \\nu \\kappa \\lambda }\\simeq \\frac{8(n+1)\\alpha ^2}{n(n-1)^2\\eta ^2}$ It should be noted that the relation for Kretschmann scalar at the infinity is completely the same as it is for a chargeless solution [52] or with nonlinear electromagnetic field [55].",
"This similarity is caused by the same asymptotic behaviour of all the metrics at the infinity.",
"Since the metric functions have different behaviour in the domain close to the origin of coordinates for diverse dimensions of space $n$ and values of $\\varepsilon $ it means that the Kretschmann scalar (REF ) should be examined separately for all these cases.",
"Firstly, we consider the situation when $n\\geqslant 5$ and $\\varepsilon =\\pm 1$ .",
"Having substituted the leading terms of the metric functions given by the relations (REF ) and (REF ) into the relation (REF ) and after little algebra we obtain: $R_{\\mu \\nu \\kappa \\lambda }R^{\\mu \\nu \\kappa \\lambda }\\simeq \\frac{4(n-1)^2(n-2)(n-3)^2}{(n-4)r^4}.$ It should be pointed out that the obtained relation does not contain the integration constant $q$ nor the parameter $\\beta $ which are present in the leading terms of the metric functions.",
"We also remark here that for power-law field as well as for the linear one [55] the Kretschmann scalar at the origin has similar dependence $\\sim 1/r^4$ and does not depend on the charge parameter $q$ .",
"Now we derive the Kretschmann scalar when $n=4$ and $\\varepsilon \\ne 0$ .",
"Here it is necessary to utilize the relations (REF ) and (REF ).",
"As a result we arrive at the expression: $R_{\\mu \\nu \\kappa \\lambda }R^{\\mu \\nu \\kappa \\lambda }\\simeq \\frac{40}{r^4}\\ln ^2\\left(1+\\frac{d^2}{r^2}\\right).$ We note here that the written above expression does not depend on the parameters $q$ and $\\beta $ analogously as it was in the previous case ($n\\geqslant 5$ ), but it has a bit stronger singular behaviour at the origin due to the presence of a divergent logarithmic factor.",
"In three dimensional space ($n=3$ ) we have to use the the relation (REF ) and the corresponding relation for the function $W$ and as a result the Kretschmann scalar in the vicinity of the origin takes the form: $R_{\\mu \\nu \\kappa \\lambda }R^{\\mu \\nu \\kappa \\lambda }\\simeq \\frac{\\mu ^2}{\\left(1-\\frac{\\beta q}{\\varepsilon }\\right)^4 r^6}$ We conclude that in three dimensional space the singularity of the Kretschmann scalar is the strongest and it is caused by the other term than in the previously analyzed cases when $n\\geqslant 4$ .",
"When $q=0$ the relation (REF ) is completely the same as it was for the neutral case [52].",
"Now we consider flat horizon geometry ($\\varepsilon =0$ ) and we should use the asymptotic relation (REF ) and corresponding relation for the metric function $W$ .",
"We remark that for the flat horizon geometry we do not consider the cases of various dimensions separately because the behaviour of the metric functions is defined by similar relation for all the dimensions.",
"As a result we obtain: $R_{\\mu \\nu \\kappa \\lambda }R^{\\mu \\nu \\kappa \\lambda }\\simeq \\frac{2(n-1)(n-2)}{r^4}$ and here similarly to the investigated above cases of nonflat geometry the behaviour of the Kretschmann scalar does not depend on the parameters $q$ and $\\beta $ , but in contrast with the nonflat cases the relation (REF ) is valid for all the dimensions $n\\geqslant 3$ .",
"We also note here that for the given solutions there is an additional point where the Kretschmann scalar is singular, namely this is the point where $W(r)=0$ .",
"It is easy to check that such a point exists, it can be verified by virtue of the relation (REF ), namely when the enumerator of that relation is equal to zero.",
"This singularity point takes place in the interior region, since the condition we impose when we set to zero the enumerator of the relation (REF ) is compatible with the condition we have imposed on the square of the derivative of the scalar field $(\\varphi ^{\\prime })^2$ (REF ).",
"It should be noted, that at this point the components of the metric tensor are not singular, but the singularity of the Kretschmann scalar appears due to the fact that the determinant of the metric tensor is equal to zero here."
],
[
"Electromagnetic field potential",
"The important feature of a charged black hole is its gauge potential, namely it has key role when one tries to derive the first law of black holes thermodynamics.",
"The evident form of the gauge potential can be found easily when one uses the relation for the gauge field (REF ).",
"So, we can write: $A_0(r)\\equiv \\psi =\\psi _0+\\int \\frac{q\\beta }{\\sqrt{q^2+\\beta ^2r^{2(n-1)}}}\\sqrt{UW}dr.$ Taking into account the relation for the product of the metric functions (REF ) one can perform the integration and write the gauge potential in the form: $\\nonumber A_0(r)\\equiv \\psi (r)=\\psi _0-\\frac{q}{2(n-2)\\alpha }\\frac{(\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{r^{n-2}}{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{n-2}{2(n-1)};\\frac{3n-4}{2(n-1)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)-\\\\\\nonumber \\frac{qd^2}{2\\alpha }\\frac{(\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )}{r^n}\\sum ^{+\\infty }_{j=0}\\frac{(-1)^j}{n+2j}\\left(\\frac{d}{r}\\right)^{2j}{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{n+2j}{2(n-1)};\\frac{3n+2(j-1)}{2(n-1)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)-\\\\\\frac{\\beta ^2\\eta q}{\\alpha }\\left(\\sum ^{(n-5)/2}_{j=0}\\frac{(-1)^jr^{4-n+2j}}{(4-n+2j)d^{2(j+1)}}+\\frac{(-1)^{\\frac{n-3}{2}}}{d^{n-2}}\\arctan \\left(\\frac{r}{d}\\right)\\right)$ for odd $n$ and large $r$ and $\\nonumber A_0(r)\\equiv \\psi (r)=\\psi _0-\\frac{q}{2(n-2)\\alpha }\\frac{(\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{r^{n-2}}{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{n-2}{2(n-1)};\\frac{3n-4}{2(n-1)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)-\\\\\\nonumber \\frac{qd^2}{2\\alpha }\\frac{(\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )}{r^n}\\sum ^{+\\infty }_{j=0}\\frac{(-1)^j}{n+2j}\\left(\\frac{d}{r}\\right)^{2j}{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{n+2j}{2(n-1)};\\frac{3n+2(j-1)}{2(n-1)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)-\\\\\\frac{\\beta ^2\\eta q}{\\alpha }\\left(\\sum ^{(n-6)/2}_{j=0}\\frac{(-1)^jr^{4-n+2j}}{(4-n+2j)d^{2(j+1)}}+\\frac{(-1)^{\\frac{n-2}{2}}}{2d^{n-2}}\\ln \\left(1+\\frac{d^2}{r^2}\\right)\\right)$ for even $n$ and large $r$ respectively.",
"We note that almost all the terms which depend on the radial coordinate $r$ go to zero at infinity and the only exclusion is the $\\arctan (r/d)$ term in the case of odd $n$ but this term is finite at the infinity, so the value of the gauge potential at the infinity is mainly defined by the constant of integration $\\psi _0$ that we have introduces in our relation.",
"This constant can be taken arbitrary, but one can fix it imposing a condition on the potential in arbitrary point.",
"We note that for small $r$ another representation for hypergeometric function should be used.",
"Namely, for odd $n$ we can write: $\\nonumber A_0(r)\\equiv \\psi (r)=\\psi _0+\\frac{\\beta (\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{2\\alpha }r{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{1}{2(n-1)};\\frac{2n-1}{2(n-1)};-\\frac{\\beta ^2}{q^2}r^{2(n-1)}\\right)+\\\\\\nonumber \\frac{\\beta (\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )}{2\\alpha }r\\sum ^{+\\infty }_{j=0}\\frac{(-1)^j}{2j+1}\\left(\\frac{r}{d}\\right)^{2j}{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{2j+1}{2(n-1)};\\frac{2(j+n)-1}{2(n-1)};-\\frac{\\beta ^2}{q^2}r^{2(n-1)}\\right)-\\\\\\frac{\\beta ^2\\eta q}{\\alpha }\\left(\\sum ^{(n-5)/2}_{j=0}\\frac{(-1)^jr^{4-n+2j}}{(4-n+2j)d^{2(j+1)}}+\\frac{(-1)^{\\frac{n-3}{2}}}{d^{n-2}}\\arctan \\left(\\frac{r}{d}\\right)\\right)$ and for even $n$ we obtain: $\\nonumber A_0(r)\\equiv \\psi (r)=\\psi _0+\\frac{\\beta (\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{2\\alpha }r{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{1}{2(n-1)};\\frac{2n-1}{2(n-1)};-\\frac{\\beta ^2}{q^2}r^{2(n-1)}\\right)+\\\\\\nonumber \\frac{\\beta (\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )}{2\\alpha }r\\sum ^{+\\infty }_{j=0}\\frac{(-1)^j}{2j+1}\\left(\\frac{r}{d}\\right)^{2j}{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{2j+1}{2(n-1)};\\frac{2(j+n)-1}{2(n-1)};-\\frac{\\beta ^2}{q^2}r^{2(n-1)}\\right)-\\\\\\frac{\\beta ^2\\eta q}{\\alpha }\\left(\\sum ^{(n-6)/2}_{j=0}\\frac{(-1)^jr^{4-n+2j}}{(4-n+2j)d^{2(j+1)}}+\\frac{(-1)^{\\frac{n-2}{2}}}{2d^{n-2}}\\ln \\left(1+\\frac{d^2}{r^2}\\right)\\right)$ The written above relations show that near the origin of coordinates the gauge potential $A_0(r)$ is singular when $n\\geqslant 4$ , namely when $n=4$ the potential has a logarithmic divergence at the origin whereas for $n>4$ we have power-law singularity.",
"It should be pointed out that even the gauge field (REF ) is singular at the origin if $n\\geqslant 4$ , the only case when the gauge field and the potential are not singular is $n=3$ .",
"We also emphasize that the behaviour of the gauge field and potential in our case is completely different from the situation which takes place in standard General Relativity with a gauge field described by a Born-Infeld type of action, where the gauge field and potential have nonsingular behaviour for all values of $n$ .",
"As we have noted above if we impose some condition on the potential it allows us to fix the integration constant $\\psi _0$ .",
"We impose that at the horizon $r_+$ the potential $A_0(r)$ is equal to zero ($A(r_+)=0$ ).",
"This requirement is not necessary, but it will be convenient for us when we use the Wald procedure in the following section.",
"Since we have imposed that $A(r_+)=0$ this condition allows us to represent the constant $\\psi _0$ as a function of the horizon radius $r_+$ and the other parameters such as $q$ , $\\Lambda $ , $\\alpha $ , $\\beta $ , $\\eta $ and $n$ , namely for the odd $n$ we can write: $\\nonumber \\psi _0=-\\frac{\\beta (\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{2\\alpha }r_+{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{1}{2(n-1)};\\frac{2n-1}{2(n-1)};-\\frac{\\beta ^2}{q^2}r^{2(n-1)}_+\\right)-\\\\\\nonumber \\frac{\\beta (\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )}{2\\alpha }r_+\\sum ^{+\\infty }_{j=0}\\frac{(-1)^j}{2j+1}\\left(\\frac{r_+}{d}\\right)^{2j}{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{2j+1}{2(n-1)};\\frac{2(j+n)-1}{2(n-1)};-\\frac{\\beta ^2}{q^2}r^{2(n-1)}_+\\right)+\\\\\\frac{\\beta ^2\\eta q}{\\alpha }\\left(\\sum ^{(n-5)/2}_{j=0}\\frac{(-1)^jr^{4-n+2j}_+}{(4-n+2j)d^{2(j+1)}}+\\frac{(-1)^{\\frac{n-3}{2}}}{d^{n-2}}\\arctan \\left(\\frac{r_+}{d}\\right)\\right).$ We point out that here we have used the relation (REF ), which gives the evident form of the gauge potential for relatively small distances, where the horizon $r_+$ is located.",
"For the case of even $n$ instead of (REF ) the relation (REF ) should be utilized and as a result we can write: $\\nonumber \\psi _0=-\\frac{\\beta (\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{2\\alpha }r_+{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{1}{2(n-1)};\\frac{2n-1}{2(n-1)};-\\frac{\\beta ^2}{q^2}r^{2(n-1)}_+\\right)-\\\\\\nonumber \\frac{\\beta (\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )}{2\\alpha }r_+\\sum ^{+\\infty }_{j=0}\\frac{(-1)^j}{2j+1}\\left(\\frac{r_+}{d}\\right)^{2j}{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{2j+1}{2(n-1)};\\frac{2(j+n)-1}{2(n-1)};-\\frac{\\beta ^2}{q^2}r^{2(n-1)}_+\\right)+\\\\\\frac{\\beta ^2\\eta q}{\\alpha }\\left(\\sum ^{(n-6)/2}_{j=0}\\frac{(-1)^jr^{4-n+2j}_+}{(4-n+2j)d^{2(j+1)}}+\\frac{(-1)^{\\frac{n-2}{2}}}{2d^{n-2}}\\ln \\left(1+\\frac{d^2}{r^2_+}\\right)\\right)$ If $\\varepsilon =0$ the gauge potential takes a simpler form, namely for large $n$ we can write: $A_0(r)\\equiv \\psi =\\psi _0+\\frac{\\beta ^2\\eta q}{(n-2)\\alpha }r^{2-n}-\\frac{(\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )q}{2(n-2)\\alpha r^{n-2}}{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{2-n}{2(1-n)};\\frac{4-3n}{2(1-n)};-\\frac{q^2}{\\beta ^2}r^{2(1-n)}\\right)$ and for small $r$ we obtain: $A_0(r)\\equiv \\psi =\\psi _0+\\frac{\\beta ^2\\eta q}{(n-2)\\alpha }r^{2-n}+\\frac{\\beta (\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{2\\alpha }r{_{2}F_{1}}\\left(\\frac{1}{2},\\frac{1}{2(n-1)};\\frac{2n-1}{2(n-1)};-\\frac{\\beta ^2}{q^2}r^{2(n-1)}\\right).$ It follows from the latter two relations that the gauge potential has power-law singularity at the origin for all $n\\geqslant 3$ and at the infinity it equals to the constant $\\psi _0$ .",
"Having the explicit form of the gauge field (REF ) and using the Gauss law we calculate total charge of the black hole which is of the crucial importance for black hole thermodynamics.",
"The Gauss law for the Born-Infeld electrodynamics takes the following form: $Q=\\frac{1}{4\\pi }\\int _{\\Sigma }\\left(1+\\frac{F_{\\kappa \\lambda }F^{\\kappa \\lambda }}{2\\beta }\\right)^{-\\frac{1}{2}}*F$ and here $*F$ denotes the Hodge dual of electromagnetic field form $F$ and the integral is taken over a closed $n-1$ –dimensional hypersurface $\\Sigma $ .",
"After calculation of the latter integral one arrives at: $Q=\\frac{\\omega _{n-1}}{4\\pi }q,$ where $\\omega _{n-1}$ is the hypersurface area of a “unit” hypersurface of constant curvature (it would be surface area of a unit hypersphere in case of spherical symmetry).",
"For nonspherical geometry of horizon it is convenient to define the total electric charge per unit area which can be written in the form: $\\bar{Q}=\\frac{1}{\\omega _{n-1}}Q.$ The electric potential measured at the infinity with respect to the horizon can be represented in the form: $\\Phi _q=A_{\\mu }\\chi ^{\\mu }\\Big |_{+\\infty }-A_{\\mu }\\chi ^{\\mu }\\Big |_{r_+},$ where $\\chi ^{\\mu }$ is a timelike Killing vector null on the event horizon and we have taken it to be the time translation vector $\\chi ^{\\mu }=\\partial /\\partial t$ .",
"After calculation we arrive at the relation: $\\Phi _q=\\psi _0,$ where now the parameter $\\psi _0$ is determined as a function of the horizon radius $r_+$ and the other parameters due to the relations (REF ) and (REF ) for odd and even $n$ respectively.",
"It should be pointed out that one can impose that $A_{\\mu }$ is zero at the infinity, but it would give rise to the same expression for the measured electric potential $\\Phi _q$ ."
],
[
"Black hole thermodynamics",
"In this section we derive and investigate main thermodynamic relations of the black hole solutions obtained in the previous section.",
"One of the most important thermodynamic quantities of the black hole is its temperature which can be obtained in the same way as it is done in General Relativity, namely its definition is based on the notion of surface gravity which can represented in the following form: $\\kappa ^2=-\\frac{1}{2}\\nabla _{a}{\\chi }_b\\nabla ^{a}{\\chi }^{b},$ and here $\\bar{\\chi }_a$ is a Killing vector, which should be null on the event horizon.",
"Similarly as in the previous section we take the vector of time translation ${\\chi }^a=\\partial /\\partial t$ .",
"One can calculate the surface gravity (REF ) and substituting it in the definition of the temperature one arrives at the relation: $T=\\frac{\\kappa }{2\\pi }=\\frac{1}{4\\pi }\\frac{U^{\\prime }{}(r_+)}{\\sqrt{U(r_+)W(r_+)}}$ where $r_+$ is the radius of the event horizon of the black hole.",
"It should be pointed out that nonetheless on the complicated structure of the metric function $U(r)$ the evident form of which contains hypergeometric functions, the temprature can be represented in a relatively compact form which comprises just of rational and irrational functions of the horizon radius $r_+$ .",
"After all the calculations the temperature can be represented in the following form: $\\nonumber T=\\frac{1}{4\\pi }\\frac{2\\alpha r^2_{+}+\\varepsilon \\eta (n-1)(n-2)}{(\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )r^2_{+}+\\varepsilon \\eta (n-1)(n-2)-2\\beta \\eta r^{3-n}_{+}\\sqrt{q^2+\\beta ^2r^{2(n-1)}_{+}}}\\times \\\\\\nonumber \\left[\\frac{(n-2)\\varepsilon }{r_+}-\\frac{2\\beta (\\alpha -\\Lambda \\eta +2\\beta ^2\\eta )}{\\alpha (n-1)r^{n-2}_+}\\sqrt{q^2+\\beta ^2r^{2(n-1)}_+}+\\frac{(\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )^2}{2(n-1)\\alpha \\eta }\\frac{r^3_+}{r^2_{+}+d^2}-\\right.\\\\\\left.\\frac{2(\\Lambda -2\\beta ^2)}{n-1}r_{+}-\\frac{2\\beta (\\alpha +\\Lambda \\eta -2\\beta ^2\\eta )d^2}{\\alpha (n-1)r^{n-2}_+}\\frac{\\sqrt{q^2+\\beta ^2r^{2(n-1)}_+}}{r^2_{+}+d^2}+\\frac{2\\beta ^2\\eta \\left(q^2+\\beta ^2r^{2(n-1)}_+\\right)}{\\alpha (n-1)r^{2n-5}_{+}(r^2_{+}+d^2)}\\right].$ In the limit when $\\beta \\rightarrow \\infty $ we recover the relation for the temperature for linear Maxwell field, namely we arrive at [55]: $\\nonumber T=\\frac{1}{4\\pi }\\frac{2\\alpha r^2_{+}+\\varepsilon \\eta (n-1)(n-2)}{(\\alpha -\\Lambda \\eta )r^2_{+}+\\varepsilon \\eta (n-1)(n-2)-\\eta q^2r^{2(2-n)}_{+}}\\left[\\frac{(n-2)\\varepsilon }{r_+}+\\frac{(\\alpha -\\Lambda \\eta )^2}{2\\alpha \\eta }r_{+}-\\frac{2q^2}{(n-1)r^{2n-3}_{+}}\\right.\\\\\\left.+\\frac{r_+}{\\alpha (n-1)(r^2_{+}+d^2)}\\left((\\alpha +\\Lambda \\eta )q^2r^{2(2-n)}_{+}-\\frac{(\\alpha +\\Lambda \\eta )^2}{2\\eta }d^2+\\frac{\\eta q^4}{2}r^{2(3-2n)}_{+}\\right)\\right].$ Here we remark, that the relation for the temperature for linear Maxwell field given in our previous work [55] was written in a bit different form, but some simple transformations allows us to obtain the relation (REF ).",
"The obtained relations for the temperature (REF ) for arbitrary $\\beta $ , as well as its particular case (REF ) for linear field are not so simple to comprehend their behaviour in full details, but nevertheless some general conclusions can be made just looking at the given above relations.",
"First of all for large horizon radii $r_+$ the dominant term in both cases is linear over $r_+$ , namely we arrive at the relation $T\\simeq (\\alpha -\\Lambda \\eta )r_+/4\\pi \\eta $ for arbitrary $\\beta $ .",
"The fact that the leading term of the asymptotic relation does not depend on the parameter $\\beta $ can be explained by the domination of the AdS-term in this case and also thanks to the circumstance that Born-Infeld modification of gauge action was introduced to eliminate divergent behaviour of the gauge field for small distances, whereas for the large ones the electromagnetic field behaves almost in the same way as for the linear field case.",
"For small horizon radius the behaviour of the temperature is different for arbitrary finite $\\beta $ and for the limit case of linear field ($\\beta \\rightarrow \\infty $ ), namely for the first case we have $T\\sim \\beta r^{2-n}_+$ whereas for the latter one it behaves as $T\\sim r^{3-2n}_+$ so we conclude that for linear field the dependence of the temperature for small radius of horizon is stronger for linear field, but this conclusion is expectable since as we have already mentioned Born-Infeld theory was introduced to modify the electromagnetic field in a way to make it finite at the origin, thus it does not takes place here in the theory with nonminimal derivative coupling apart of the case when $n=3$ .",
"We also pay attention to the case $n=3$ here, namely when $\\beta \\ne 0$ and $\\varepsilon \\ne 0$ for small horizon radius the temperature might change its sign depending on the relation between $\\beta $ and $q$ .",
"We have analyzed the dependences for the temperatures (REF ) and (REF ) for some particular cases of $r_+$ , namely for its extremely large and small values, to understand behaviour of the temperatures better for some intermediate values of $r_+$ we demonstrate it graphically.",
"Figure [REF ] shows this dependence for various values of $\\beta $ , when the other parameters are held fixed (the left graph) and for various values of the cosmological constant $\\Lambda $ (the right graph).",
"We can conclude that the variation of the parameter $\\beta $ affects considerably on the temperature for small horizon radius, whereas for large values of $r_+$ this influence is negligibly small and the behaviour is mainly defined by the AdS-terms in all the cases.",
"Variation of the parameter $\\Lambda $ has substantial influence on the temperature for large $r_+$ , whereas for small $r_+$ its contribution becomes negligibly small.",
"The function $T(r_+)$ might be nonmonotonous, what is better seen on the right graph, this fact might give us some critical behaviour in extended thermodynamic phase space similarly as it is done for charged black holes in the framework of standard General Relativity [80], [81], but this issue will be investigated elsewhere.",
"Figure: Temperature TT as a function of the horizon radius r + r_+ for different values of parameter β\\beta (the left graph) and different values of the cosmological constant Λ\\Lambda (the right one).",
"For all the graphs we have n=3n=3, ε=1\\varepsilon =1, α=0.2\\alpha =0.2, η=0.4\\eta =0.4, q=0.2q=0.2.",
"For the left graph the other parameters are equal to Λ=-8\\Lambda =-8, β=8\\beta =8 (dashed line), β=50\\beta =50 (dotted line) and solid line represents linear field case (β→∞\\beta \\rightarrow \\infty ).",
"For the right graph β=8\\beta =8, and Λ=-2\\Lambda =-2 (solid line), Λ=-4\\Lambda =-4 (dashed line) and Λ=-8\\Lambda =-8 (dotted line)."
],
[
"Wald's approach and entropy of black hole",
"There are several approaches to define entropy of a black hole, some of them take their roots in earlier work of Gibbons and Hawking [82], other approaches are based on Wald's procedure (or method) which can be treated as a generalization of Noether method to derive conserved quantities in gravity [83], [84].",
"It should be pointed out that Wald's procedure applicable to quite general diffeomorphism-invariant theories.",
"It has been applied to numerous black hole solutions in the framework of standard General Relativity as well as its generalizations [53], [54], [55], [85], [86], [87].",
"In Horndeski Gravity the Wald's procedure appears to be a consistent approach to obtain the first law of black hole thermodynamics.",
"We point out here that the scalar potential in Horndeski Gravity has singular behaviour, but the Wald's procedure takes this fact into account [53], [54].",
"We also note that Wald's procedure allowed to derive reasonable and in some sense universal relations for back hole's entropy [53], [54], [55].",
"Here we also utilize Wald's procedure to derive a relation for black hole's entropy and firstly we describe the keypoints of this approach.",
"Assuming that we have a Lagrangian ${\\cal L}$ of a system, we can perform its variation and as a result we write: $\\delta {\\cal L}=e.o.m.+\\sqrt{-g}\\nabla _{\\mu }J^{\\mu },$ where $e.o.m.$ represents the terms which give equations of motion for the system and the term $\\sqrt{-g}\\nabla _{\\mu }J^{\\mu }$ gives rise to the so called surface term in the action integral because of full divergence of the last term, the $J^{\\mu }$ denotes the surface “current”.",
"Using the obtained relation for the current $J^{\\mu }$ one can construct “current” one-form: $J_{(1)}=J_{\\nu }dx^{\\nu }$ and its Hodge dual: $\\Theta _{(n)}=*J_{(1)}$ .",
"Having utilized infinitesimal diffeomorphism given by the vector $\\delta x^{\\mu }=\\xi ^{\\mu }$ one can define the form: $J_{(n)}=\\Theta _{(n)}-i_{\\xi }*{\\cal L}=e.o.m.-d*J_{(2)},$ and here $i_{\\xi }*{\\cal L}$ denotes contraction of the vector field $\\xi ^{\\mu }$ with the dual of the form $*{\\cal L}$ .",
"In case the equations of motion are fulfilled (on-shell condition) we can conclude that the form $J_{(n)}$ is exact, namely $J_{(n)}=dQ_{(n-1)}$ , where $Q_{(n-1)}=-*J_{(2)}$ .",
"The written above relation (REF ) allows to obtain thermodynamic relations, namely the first law of black hole thermodynamics if the infinitesimal diffeomorphism vector $\\xi ^{\\mu }$ is taken to be a Killing vector null at the horizon.",
"It was shown by Wald that the variation of gravitational Hamiltonian can be represented in the form: $\\delta {\\cal H}=\\frac{1}{16\\pi }\\left(\\delta \\int _{c}J_{(n)}-\\int _{c}d(i_{\\xi }\\Theta _{(n)})\\right)=\\frac{1}{16\\pi }\\int _{\\Sigma _{(n-1)}}(\\delta Q-i_{\\xi }\\Theta _{(n)}),$ where $c$ denotes $n$ –dimensional Cauchy surface and $\\Sigma _{(n-1)}$ is its $n-1$ –dimensional boundary which consist of two parts: one on the event horizon and the other at the infinity.",
"The first law of black hole thermodynamics can be obtained from the relation: $\\delta {\\cal H}_{\\infty }=\\delta {\\cal H}_{+},$ we note that the in the latter relation the variation in the left hand side is taken at the infinity and in the right hand side is taken at the events horizon.",
"Using written above relations we can calculate the variation of Hamiltonian and consequently derive the first law.",
"For minimally coupled part of the action with Born-Infeld term we can write: $(\\delta Q-i_{\\xi }\\Theta )_{min}=r^{n-1}\\sqrt{UW}\\left(\\frac{(n-1)}{rW^2}\\delta W+\\frac{2}{UW}\\frac{1}{\\left(1-\\frac{(\\psi ^{\\prime })^2}{\\beta ^2UW}\\right)^{\\frac{3}{2}}}\\left[\\psi ^{\\prime }\\psi \\left(\\frac{\\delta U}{U}+\\frac{\\delta W}{W}\\right)-2\\psi \\delta \\psi ^{\\prime }\\right]-\\frac{\\alpha \\varphi ^{\\prime }}{W}\\delta \\varphi \\right)\\Omega _{(n-1)},$ here $\\Omega _{(n-1)}$ denotes surface $n-1$ –form.",
"For nonminimally coupled part of the action we obtain: $(\\delta Q-i_{\\xi }\\Theta )_{nm}=\\frac{\\eta (n-1)}{2}r^{n-2}\\sqrt{\\frac{U}{W}}\\left(\\frac{(\\varphi ^{\\prime })^2}{2W^2}\\delta W-\\delta \\left(\\frac{(\\varphi ^{\\prime })^2}{W}\\right)+\\frac{2\\alpha r}{(n-1)\\eta }\\varphi ^{\\prime }\\delta \\varphi \\right)\\Omega _{(n-1)}.$ Having combined latter two relations we can obtain total variation: $\\nonumber (\\delta Q-i_{\\xi }\\Theta )_{tot}=r^{n-1}\\sqrt{UW}\\left((n-1)\\left(1+\\frac{\\eta (\\varphi ^{\\prime })^2}{4W}\\right)\\frac{\\delta W}{rW^2}+\\frac{2}{UW}\\frac{1}{\\left(1-\\frac{(\\psi ^{\\prime })^2}{\\beta ^2UW}\\right)^{\\frac{3}{2}}}\\times \\right.\\\\\\left.\\left[\\psi ^{\\prime }\\psi \\left(\\frac{\\delta U}{U}+\\frac{\\delta W}{W}\\right)-2\\psi \\delta \\psi ^{\\prime }\\right]-\\frac{\\eta (n-1)}{2rW}\\delta \\left(\\frac{(\\varphi ^{\\prime })^2}{W}\\right)\\right)\\Omega _{(n-1)}.$ We note that in the limit $\\beta \\rightarrow \\infty $ the given above relation is reduced to the corresponding relation for standard linear Maxwell field [54].",
"Using the written above relation (REF ) we can calculate total variation at the infinity as well as on the horizon.",
"As a result at the infinity we write: $(\\delta Q-i_{\\xi }\\Theta )_{tot}=((n-1)\\delta \\mu +4\\psi _0\\delta q)\\Omega _{(n-1)}.$ The obtained relation is very simple and completely coincides with corresponding relation obtained for linear Maxwell field [54].",
"Taking into account relations for the total charge and gauge potential and performing integration over angular variables we can obtain relation for the variation of the gravitational Hamiltonian at the infinity $\\delta {\\cal H}_{\\infty }$ : $\\delta {\\cal H}_{\\infty }=\\delta M-\\Phi _q\\delta Q$ where $\\delta M$ is the variation of the black hole's mass which can be written in the form: $M=\\frac{(n-1)\\omega _{n-1}}{16\\pi }\\mu $ It should be pointed out that for non-spherical topology of horizon the relation (REF ) should be treated as the variation per unit volume.",
"We would also like to emphasize that obtained relation for the mass (REF ) takes completely the same form as in case of standard General Relativity.",
"Taking the variation of the Hamiltonian at the horizon we arrive at the relation: $\\delta {\\cal H}_{+}=\\frac{(n-1)\\omega _{n-1}}{16\\pi }U^{\\prime }{}(r_+)r^{n-2}_{+}\\delta r_{+}=\\sqrt{U(r_+)W(r_+)}T\\delta \\left(\\frac{{\\cal A}}{4}\\right)=\\left(1+\\frac{\\eta }{4}\\frac{(\\varphi ^{\\prime }{})^2}{W}\\Big |_{r_+}\\right)T\\delta \\left(\\frac{{\\cal A}}{4}\\right).$ where ${\\cal A}=\\omega _{n-1}r^{n-1}_+$ is the horizon area of the black hole.",
"Here we note that variation of the Hamiltonian at the horizon does not include a contribution from the gauge field due to the fact that gauge potential equals to zero at the horizon.",
"Latter relation for the variation of the Hamiltonian at the horizon can't be represented in the form $T\\delta S$ which takes place in the standard General Relativity, in other words the form in the right hand side of the relation (REF ) is not exact, this fact was noted in [54] and a specific procedure was proposed to derive a relation for black hole's entropy.",
"To write the first law of black hole mechanics it was proposed to introduce specific “scalar charge”, related to the scalar field [54].",
"But as it was shown in our earlier works [52], [55] the “scalar charges” can be chosen in different way, and here we take the same form for them as in our previous works and rewrite the latter relation in the form: $\\delta {\\cal H}_{+}=T\\delta S+\\Phi ^{+}_{\\varphi }\\delta Q^{+}_{\\varphi },$ where $S$ is the entropy of the black hole, $Q^{+}_{\\varphi }$ and $\\Phi ^{+}_{\\varphi }$ denote introduced “scalar charge” and related to it potential.",
"These introduced values can be chosen in the form: $S=\\left(1+\\frac{\\eta }{4}\\frac{(\\varphi ^{\\prime }{})^2}{W}\\Big |_{r_+}\\right)\\frac{{\\cal A}}{4},$ $Q^{+}_{\\varphi }=\\omega _{n-1}\\sqrt{1+\\frac{\\eta }{4}\\frac{(\\varphi ^{\\prime }{})^2}{W}\\Big |_{r_+}}, \\quad \\Phi ^{+}_{\\varphi }=-\\frac{{\\cal A}T}{2\\omega _{n-1}}\\sqrt{1+\\frac{\\eta }{4}\\frac{(\\varphi ^{\\prime }{})^2}{W}\\Big |_{r_+}}.$ As we mentioned above the scalar “charge” $Q^{+}_{\\varphi }$ and conjugate potential $\\Psi ^{+}_{\\varphi }$ might be defined in other way, but in our case it allows to derive relation between the temperature and entropy from one side and introduced scalar “charge” and potential from the other one: $\\Phi ^{+}_{\\varphi }Q^{+}_{\\varphi }=-\\frac{{\\cal A}T}{2}\\left(1+\\frac{\\eta }{4}\\frac{(\\varphi ^{\\prime }{})^2}{W}\\Big |_{r_+}\\right)=-2TS$ We point out here that contribution of scalar field charges was examinined for the first time in [89].",
"Using relations (REF ) and corresponding results for the variations at the horizon and at the infinity we can write the first law in the following form: $\\delta M=T\\delta S+\\Phi ^{+}_{\\varphi }\\delta Q^{+}_{\\varphi }+\\Phi _{q}\\delta Q$ The obtained relation (REF ) has very simple form, similar to corresponding relation in standard General Relativity, but nevertheless the introduced definition of the entropy is not supported by some independent way of calculation.",
"It should be pointed out that several attempts to calculate entropy with help of Euclidean methods were made, but there they were related mainly to planar geometry [88] or chargeless case.",
"It should be pointed out here that accurate application of Euclidean action requires properly regularized and renormalized action, this regularization means that we take into account Gibbons-Hawking-York boundary term (REF ) and renormalization or the regularized action should be performed to make the action finite, since we deal with asymptotically AdS solutions, but this issue is a subject of independent investigation and will be performed elsewhere."
],
[
"Conclusions",
"In this work we consider particular case of general Horndeski gravity, namely the theory with nonminimal derivative coupling and we also take into account gauge filed minimally coupled just to gravity sector and given by a Lagrangian of Born-Infeld type.",
"We obtain static solutions which represent black holes.",
"Since the cosmological constant is taken into account it allowed us to consider not only spherically symmetric solution, but also to obtain topological solutions with nonspherical horizon, namely with flat ($\\varepsilon =0$ ) and hyperbolic ($\\varepsilon =-1$ ) ones.",
"In general the structure of the obtained solutions is complicated, but nevertheless they share some common features with black holes' solutions derived in the framework of General Relativity as well as in Horndeski Gravity.",
"Firstly, all the obtained solutions have AdS-like behaviour at large distances, because of the presence of the cosmological constant $\\Lambda $ and due to the influence of the scalar field, but in contrast with the ordinary General Relativity, where the metric at large distances is completely defined by a term with bare cosmological constant $\\Lambda $ in our case we have some effective constant which is defined by the bare one and coupling constants for the scalar field $\\alpha $ and $\\eta $ .",
"The behaviour of the metric for very small distances ($r\\rightarrow 0$ ) depends on the type of topology as well as on the dimension of space.",
"Namely, when $\\varepsilon \\ne 0$ there are three different types of the behaviour of the metric function $U(r)$ , for $n=3$ , $n=4$ and $n\\geqslant 5$ , what is demonstrated by the relations (REF ), (REF ) and (REF ) respectively, whereas for the flat horizon solution ($\\varepsilon =0$ ) the character of the function $U(r)$ for all $n\\geqslant 3$ is the same (REF ).",
"We also stress here that the leading terms for flat geometry ($\\varepsilon =0$ ) as well as nonflat geometry $\\varepsilon \\ne 0$ if $n\\geqslant 4$ is completely defined by the gauge field term and for the particular case $\\varepsilon \\ne 0$ and $n=3$ the dominant term is of Schwarzschild type.",
"In addition we would like to emphasize that the Kretschmann scalar (REF ), which defines the character of singularity at the origin for the flat case $\\varepsilon =0$ and for the nonflat one $\\varepsilon \\ne 0$ if $n\\geqslant 5$ for small distances show the dependence $\\sim 1/r^4$ and does not depend on the charge $q$ , the same situation takes place for linear and power-law fields [55].",
"For $n=4$ and $\\varepsilon \\ne 0$ the Kretshmann scalar has additional peculiarity of a logarithmic character (REF ) and for the case $n=3$ and $\\varepsilon \\ne 0$ due to domination of the Schwarzschild term the Kretshmann scalar has completely different behaviour (REF ).",
"We also obtained and examined the relations for the gauge field and gauge potential and it was shown that for $n=3$ and $\\varepsilon \\ne 0$ the field and potential are nonsingular at the origin, whereas for other considered dimensions and types of geometry they are singular at this point.",
"We point out here that in standard General Relativity with Born-Infeld field where the field and potential are nonsingular at all distances for all dimensions.",
"We have also examined some aspects of black hole's thermodynamics.",
"First of all we have obtained relation for the temperature (REF ) and we show that in the limit $\\beta \\rightarrow \\infty $ we recover the relation (REF ) which was derived in our previous work [55].",
"Careful analysis of the obtained relation shows that for large radius of the horizon $r_+$ the temperature increases almost linearly due to domination of AdS-term in this case.",
"For small radius of horizon the situation is completely different, namely the leading term in this case is related to the gauge field, but it should be pointed out here that in case of finite $\\beta $ the character of dependence $T(r_+)$ is of the type $\\sim \\beta r^{2-n}_+$ , whereas for linear field one arrives at the asymptotic $\\sim r^{3-2n}_+$ this difference can be explained by the fact that Born-Infeld gauge field has more moderate dependence of the $r_+$ than the linear one.",
"To obtain the first law of black hole thermodynamics we have utilized Wald's approach which is applicable to general diffeomorphism-invariant theories.",
"Wald's method is well-posed, but nevertheless the definition of entropy is not an easy task, to introduce the entropy we followed the approach suggested in [54] and used in our work [55] where additional “scalar charges” were introduced.",
"Here we point out that the choice we made to introduce “scalar charge” is not unique, in the paper [54] it was taken in a bit different form, this ambiguity and the fact that we do not have a corresponding “charge” related to the scalar field which appears due to integration of equations of motion makes this final step in Wald's procedure a bit unsatisfactory.",
"To make this step well grounded we should have independent approach to define entropy and as a consequence to write the first law.",
"As it is known the relation for entropy can be derived with help of Euclidean approach.",
"In order to use Euclidean approach one should renormalize the total gravity action (REF ) to make it finite at infinity, we also point out here that several attempts to use Euclidean techniques were made, but they were mainly related to uncharged black hole [42] or black holes with flat topology of horizon [88]."
],
[
"Acknowledgments",
"This work was partly supported by Project FF-83F (No.",
"0119U002203) from the Ministry of Education and Science of Ukraine."
]
] | 2001.03574 |
[
[
"OO-VR: NUMA Friendly Object-Oriented VR Rendering Framework For Future\n NUMA-Based Multi-GPU Systems"
],
[
"Abstract With the strong computation capability, NUMA-based multi-GPU system is a promising candidate to provide sustainable and scalable performance for Virtual Reality.",
"However, the entire multi-GPU system is viewed as a single GPU which ignores the data locality in VR rendering during the workload distribution, leading to tremendous remote memory accesses among GPU models.",
"By conducting comprehensive characterizations on different kinds of parallel rendering frameworks, we observe that distributing the rendering object along with its required data per GPM can reduce the inter-GPM memory accesses.",
"However, this object-level rendering still faces two major challenges in NUMA-based multi-GPU system: (1) the large data locality between the left and right views of the same object and the data sharing among different objects and (2) the unbalanced workloads induced by the software-level distribution and composition mechanisms.",
"To tackle these challenges, we propose object-oriented VR rendering framework (OO-VR) that conducts the software and hardware co-optimization to provide a NUMA friendly solution for VR multi-view rendering in NUMA-based multi-GPU systems.",
"We first propose an object-oriented VR programming model to exploit the data sharing between two views of the same object and group objects into batches based on their texture sharing levels.",
"Then, we design an object aware runtime batch distribution engine and distributed hardware composition unit to achieve the balanced workloads among GPMs.",
"Finally, evaluations on our VR featured simulator show that OO-VR provides 1.58x overall performance improvement and 76% inter-GPM memory traffic reduction over the state-of-the-art multi-GPU systems.",
"In addition, OO-VR provides NUMA friendly performance scalability for the future larger multi-GPU scenarios with ever increasing asymmetric bandwidth between local and remote memory."
],
[
"Introduction",
"With the vast improvements in graphics technology, Virtual Reality (VR) is becoming a potential popular product for major high-tech companies such as Facebook [30], Google [16] and NVIDIA [29].",
"Different from normal PC or mobile graphics applications, VR promises a fully immersive experience to users by directly displaying images in front of users' eyes.",
"Due to the dramatic experience revolution VR brings to users, the global VR market is expected to grow exponentially and generate $30 billion annual revenue by 2022 [34], [24].",
"Despite the growing market penetration, achieving true immersion for VR applications still faces severe performance challenges [20].",
"First, the display image must have a high pixel density as well as a broad field of views which requires a high display resolution.",
"Meanwhile, the high-resolution image must be delivered at an extremely short latency so that users can preserve the continuous illusion of reality.",
"However, the state-of-the-art graphics hardware – the Graphics Processing Units (GPUs) in particular – cannot meet these strict performance requirements [20].",
"Historically, GPUs gain performance improvements through integrating more transistors and scaling up the chip size, but these optimizations on single-GPU system can barely satisfy VR users due to the limited performance boost [33].",
"Multi-GPU system with much stronger computation capability is a promising candidate to provide sustainable and scalable performance for VR applications [33], [21].",
"In recent years, the major GPU verdors combine multiple small GPU models (e.g., GPMs) to build a future multi-GPU system under a single programming model to provide scalable computing resources.",
"They employ high speed inter-GPU links such as NVLINK [28] and AMD Crossfire[3] to achieve fast data transmit among GPMs.",
"The memory system and address mapping in this multi-GPU system are designed as a Non-Uniform Memory Access (NUMA) architecture to achieve 4x storage capacity over single-GPU system.",
"The NUMA-based multi-GPU system employs shared memory space to avoid data duplication and synchronization overheads across the distributed memories [5].",
"In this study, we target the future multi-GPU system because it serves the VR applications more energy-efficiently than distributed multi-GPU system that employs separated memory space, and is becoming a good candidate for future mobile VR applications.",
"Since the entire system is viewed as a single GPU under the single programming model, the VR rendering workloads are sequentially launched and distributed to different GPMs without specific scheduling.",
"Applying this naive single programming model greatly hurts the data locality among rendering workloads and incurs huge inter-GPM memory accesses, which significant constrain the performance of multi-GPU system for VR applications due to the bandwidth asymmetry between the local DRAM and the inter-GPM links.",
"There have been many studies [21], [5], [25], [43] to improve the performance of NUMA-based multi-GPU system by minimizing the remote accesses.",
"However, these solutions are still based on single programming model without considering the special data redundancy in VR rendering, hence, they cannot efficiently solve the performance bottleneck for VR applications.",
"Aiming to reduce the inter-GPM memory accesses, a straightforward method is employing parallel rendering frameworks [7], [19], [13], [14] to split the rendering tasks into multiple parallel sub-tasks under specific software policy before assigning to the multi-GPU system.",
"Since these frameworks are originally designed for distributed multi-GPU system, a knowledge gap still exists on how to leverage parallel rendering programming model to efficiently execute VR applications in NUMA-based multi-GPU system.",
"To bridge this gap, we first investigate three different parallel rendering frameworks (i.e.",
"frame-level, tile-level and object-level).",
"By conducting comprehensive experiments on our VR featured simulator, we find that the object-level rendering framework that distributes the rendering object along with its required data per GPM can convert some remote accesses to local memory accesses.",
"However, this object-level rendering still faces two major challenges in NUMA-based multi-GPU system: (1) a large number of inter-GPM memory accesses because it fails to capture the data locality between left and right view of the same object as well as the data sharing among different objects; (2) the serious workload unbalance among GPMs due to the inefficient software-level distribution and composition mechanisms.",
"To overcome these challenges, we propose object-oriented VR rendering framework (OO-VR) that reduces the inter-GPM memory traffic by exploiting the data locality among objects.",
"Our OOVR framework conducts the software and hardware co-optimizations to provide a NUMA friendly solution for VR multi-view rendering.",
"First, we propose an object-oriented VR programming model that provides a simple software interface for VR applications to exploit the data sharing between the left and right views of the same object.",
"The proposed programming model also automatically groups objects into batches based on their data sharing levels.",
"Then, to combat the limitation of software-level solutions on workload distribution and composition, we design a object aware runtime batch distribution engine in hardware level to balance the rendering workloads among GPMs.",
"We predict the execution time for each batch so that we can pre-allocate the required data of each batch to the local memory to hide long data copy latency.",
"We further design the distributed composition unit in hardware level to fully utilize the rendering output units across all GPMs for best pixel throughput.",
"To summarize, the paper makes following contributions: We investigate the performance of future NUMA-based multi-GPU systems for VR applications, and find that the inter-GPM memory accesses are the major performance bottleneck.",
"We conduct comprehensive characterizations on major parallel rendering frameworks, and observe that the data locality among rendering objects can help to significantly reduce the inter-GPM memory accesses but the state-of-the-art frameworks and multi-GPU systems fail to capture this interesting feature.",
"We propose a software and hardware co-designed Object-Oriented VR (OO-VR) rendering framework that leverages the data locality feature to convert the remote inter-GPM memory accesses to local memory accesses.",
"We further build a VR featured simulator to evaluate our proposed design by rendering VR enabled real-world games with different resolutions.",
"The results show that OO-VR achieves 1.58x performance improvement and 76% inter-GPM memory traffic reduction over the state-of-the-art multi-GPU system.",
"With its nature of NUMA friendly, OO-VR exhibits strong performance scalability and potentially benefits the future larger multi-GPU scenarios with ever increasing asymmetric bandwidth between local and remote memory.",
"In contrast to other traditional graphics applications, the state-of-the-art VR applications employ Head-Mounted-Display (HMD), or VR helmet, to directly present visuals to users' eyes.",
"To display 3D objects in VR, a pair of frames (i.e., stereoscopic frames) are generated for both left and right eyes by projecting the scene onto two 2D plate images.",
"This process is referred as stereo rendering in computer graphics.",
"Figure REF shows an example of such VR projection.",
"The green and yellow boxes represent the rendering process for left and right views, respectively, creating two display images for the HMD.",
"Stereo rendering requires two concurrent rendering process for the two eyes' views, resulting in doubled amount of workload for the VR pipeline.",
"Due to the observation that some objects in the scene (e.g.",
"the robot in Figure REF ) are shared by two eyes, mainstream graphics engines such as NVIDIA and UNITY employ simultaneous multi-projection (SMP) to generate the left and right frames simultaneously through single rendering process [27], [35], [8], [9].",
"This significantly reduces workload redundancy and achieves substantial performance gain.",
"Based on the conventional three-step rendering process (i.e., Geometry Process, Rasterization and Fragment Process) defined by modern graphics application programming interface (API) [2], [1], VR rendering inserts multi-projection process after the geometry process and prior to the Rasterization, shown in Figure REF (a).",
"Thus, when SMP is enabled, VR rendering process is composed of four steps, detailed in Figure REF (b).",
"Basically, VR rendering begins from reading the application-issued vertex from GPU memory.",
"During the geometry process [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 1;, the vertex shader calculates the 3D coordinates of the vertex and assembles them into primitives (i.e.",
"triangles in Figure REF (a)-(1)).",
"After that, the generated triangles pass through the geometry-related shaders which perform clipping, face culling and tessellation to generate extra triangles and remove non-visible triangles.",
"Then, the SMP step [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 2; is responsible for generating multiple projections of a single geometry stream.",
"In other words, GPU executes geometry process only once but produces two positions for each triangle (Figure REF (a)-(2)).",
"These triangles are then streamed into the rasterization stage [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 3; to generate fragments (Figure REF (a)-(3)), each of which is equivalent to a pixel in a 2D image.",
"Finally, the fragment process [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 4; generates pixels by calculating the corresponding fragment attributes to determine their colors and texture (Figure REF (a)-(4)).",
"The output pixels will be written into the frame buffer in GPU memory for displaying."
],
[
"SMP Featured GPU Architectures",
"Traditionally, GPUs are designed as the special-purpose graphics processors for performing modern rendering tasks.",
"Figure REF (c) shows a SMP supported GPU architecture which models the recent NVIDIA Pascal GPUs[27].",
"It consists of several programmable streaming multiprocessors (SMs) [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 1;, some fixed function units such as the GigaThread Engine [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 2;, Raster Engine [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 3;, Polymorph Engine (PME) [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 4;, and Render Output Units (ROPs) [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 5;.",
"Each SM [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 1; is composed of a unified texture/L1 cache (TX/L1 $), several texture units (TXU) and hundreds of shader cores that execute a variety of graphics shaders (e.g., the functions in both geometry and fragment process).",
"The GigaThread Engine [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 2; distributes the rendering workloads among PMEs if there are adequate computing resources.",
"The raster engine [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 3; is a hardware accelerator for rasterization process.",
"Each PME [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 3; conducts input assembler, vertex fetching, and attribute setup.",
"To support multi-view rendering, NVIDIA Pascal architecture integrates an SMP engine into each PME.",
"The SMP engine is capable of processing geometry for two different viewports which are the projection centers for the left and right views.",
"In other words, it duplicates the geometry process from left to right views through changing the projection centers instead of executing the geometry process twice.",
"Finally, the Render Output Units (ROPs) [baseline=(char.base)] shape=circle,fill=black,draw,inner sep=0.5pt] (char) 5; perform anti-aliasing, pixel compression and color output.",
"As Figure REF (c) illustrates, all the SMs and ROPs share a L2 cache and read/write data to the off-chip memory through the memory controllers, each of which is paired to one memory channel.",
"Prior to rendering, GPU memory contents such as framebuffer and texture data are pre-allocated in GPU's off-chip memory.",
"Programmers can manually manage the memory allocation using the graphics APIs such as OpenGL and Direct3D[2], [1].",
"Although recent generations of GPUs have shown capability to deliver good gaming experiences and also gradually evolved to support SMP, it is still difficult for them to satisfy the extremely high demands on rendering throughput from immersive VR applications.",
"The human vision system has both wide field of view (FoV) and incredibly high resolution when perceiving the surrounding world; the requirement for enabling an immersive VR experience is much more stringent than that for PC gaming.",
"Table REF lists the major differences between PC gaming and stereo VR [20].",
"As it demonstrates, stereo VR requires GPU to deliver 116 (58.32$\\times $ 2) Mpixels within 5 ms.",
"Missing the rendering deadline will cause frame drop which significantly damages VR quality.",
"Although the VR vendors today employ frame re-projection technologies such as Asynchronous Time Warp (ATW)[36], [15] to artificially fill in dropped frames, they cannot fundamentally solve the problem of rendering deadline missing due to little consideration on users' perception and interaction.",
"Thus, improving the overall rendering efficiency is still the highest design priority for modern VR-oriented GPUs [6].",
"Table: Differences Between PC Gaming and VRFigure: The Overview of the multi-GPU architecture.",
"Distributed rendering tasks for the same object causes significant remote memory access and data duplication."
],
[
"NUMA-Based Multi-GPU System and Its Performance Bottleneck",
"In recent years, major GPU vendors such as NVIDIA have proposed to integrate multiple easy-to-manufacture GPU chips at package level (i.e., multi-chip design)[5] or at system level[25], [43] using high bandwidth interconnection technologies such as Grand-Reference Signaling (GRS)[32] or NVLinks[28], in order to address future chip density saturation issue.",
"Figure REF shows the overview of the multi-GPU architecture which consists of four GPU models (i.e., GPMs).",
"In terms of compute capability, each GPM is configured to resemble the latest NVIDIA GPU architecture (e.g., Figure REF (c)).",
"Inside each GPM, SMs are connected to the GPM local memory hierarchy including a local memory-side L2 cache and off-chip DRAM, via an XBAR.",
"In the overall multi-chip design (MCM-GPU), XBARs are interconnected through high speed links such as NVLinks to support the communication among different GPMs.",
"This multi-GPU system generally acts as a large single GPU; its memory system and address mapping are designed as a Non-Uniform Memory Access (NUMA) architecture.",
"This design also reduces the programming complexity (e.g., unified programming model similar to CUDA) for GPU developers.",
"Future Multi-GPU System Bottleneck for VR Workloads.",
"As previous works [5], [25] have indicated, data movement among GPMs will become the major obstacle for the continued performance scaling in these future NUMA-based multi-GPU systems.",
"This situation is further exacerbated when executing VR applications caused by the large data sharing among GPMs.",
"Due to the nature of view redundancy in VR applications, the left and right views may include the same object (e.g., the rabbit in Figure REF ) which require the same texture data.",
"However, to effectively utilize the computing resources from all the GPMs in such multi-GPM platforms, the rendering tasks for left and right views will be distributed to different groups or islands of GPMs in a more balanced fashion; each view will then be further broken into smaller pieces and distributed to the individual GPMs of that group.",
"This naive strategy could greatly hurt data locality in the SMP model.",
"For example, if the basic texture data used to describe the rabbit in Figure REF is stored in the local memory of GPM_0, other GPMs need to issue remote memory accesses to acquire this data.",
"Due to the asymmetrical bandwidth between the local DRAM (e.g., 1TB/s) and inter-GPM NVLink (e.g., 64GB/s), the remote memory access will likely become one of the major performance bottlenecks in such multi-GPU system design.",
"More sophisticated parallel rendering frameworks such as OpenGL Multipipe SDK [7], Chromium [19] and Equalizer[13], [14], are designed for distributed environment where they separate memory space and the memory data need to be duplicated in each memory which greatly limits the storage capacity on our NUMA-based multi-GPU systems.",
"Thus, employing them on our architecture requires further investigation and characterization.",
"We will show this study in Section .",
"Figure REF presents the performance of a 4-GPM multi-GPU system as the bandwidth of inter-GPM links is decreased from 1TB/s to 32GB/s (refer to Section for experimental methodology).",
"We can observe that the rendering performance is significantly limited by the bandwidth.",
"On average, applying 128GB/s, 64GB/s and 32GB/s inter-GPM bandwidth results in 22%, 42% and 65% performance degradation compared to the baseline 1TB/s bandwidth, respectively.",
"Although improving the inter-GPM bandwidth is a straightforward method to tackle the problem, it has proven difficult to achieve due to additional silicon cost and power overhead [5].",
"This motivates us to provide software-hardware co-design strategies to enable \"true\" immersive VR experience for future users via significantly reducing the inter-GPM traffic and alleviating the performance bottleneck of executing VR workloads on future multi-GPU platforms.",
"We believe this is the first attempt to co-design at system architecture level for eventually realizing future planet-scale VR.",
"Figure: Normalized performance sensitivity to inter-GPM link bandwidth for a 4-GPM Multi-GPU system.We investigate the performance impact of multi-GPU system for virtual reality by extending ATTILA-sim [10], a cycle-level rasterization-based GPU simulator which covers a wide spectrum of graphics features on modern GPUs.",
"The model of ATTILA-sim is designed upon boxes (a module of functional pipeline) and signals (simulating the interconnect of different components).",
"Because the current ATTILA-sim models an AMD TeraScale2 architecture [17], it is difficult to configure it using the same amount of SMs as NVIDIA Pascal-like architectures [27].",
"To fairly evaluate the design impact, we accordingly scale down other parameters such as the number of ROPs and L2 cache.",
"Similar strategies have been used to study modern graphics architectures in previous works[41], [40], [42].",
"The GPM memory system consists of two level cache hierarchy and a local DRAM.",
"The L1 cache is private to each SM while the L2 cache are shared by all the SMs.",
"Table REF shows the simulation parameters applied in our baseline multi-GPU system.",
"Table: BASELINE CONFIGURATIONIn order to support multi-view VR rendering, we implement the SMP engine in ATTILA-sim based on the state-of-the-art SMP technology [9], [8] which re-projects the triangles in left view to right using updated viewport.",
"Figure REF shows the rendering example of Half-Life 2 after enabling SMP in ATTILA-sim.",
"Our SMP engine first gathers the X coordinate of the display frame which is from -W to +W, where W is a coordinate offset parameter.",
"Then, it duplicates each triangle generated from the geometry process.",
"After that, the SMP engine shifts the viewport of the rendering object by half of W, left or right depending on the eye.",
"The SMP engine can also re-project the triangle based on user-defined viewports for left and right views.",
"Finally, we modify the triangle clipping to prevent the spill over into the opposite eye.",
"We validated the implementation of the SMP engine in ATTILA-sim by comparing the triangle number, fragment number and performance improvement with that from executing VR benchmarks on the state-of-the-art GPUs (e.g.",
"Sponza and San Mangle in NVIDIA VRWork [29]) on NVIDIA GTX 1080 Ti).",
"Specifically, we observe that the added SMP rendering on ATTILA-sim can provide a 27% speed up over the sequential rendering on two views.",
"Figure: The original rendering frame (left) and results with SMP enabled (right).We also model the inter-GPU interconnect as high bandwidth point-to-point NVLinks with 64GB/s bandwidth (one direction).",
"We assume each GPM has 6 ports and each pair of ports is used to connect two GPMs, indicating that the intercommunication between two GPMs will not be interfered by other GPMs.",
"Based on the sensitivity study shown in Figure REF , we configure the inter-GPM link bandwidth as 64GB/s bandwidth.",
"Following the common GPU design, each ROP in our simulation outputs 4 pixels per cycle to the framebuffer.",
"To further alleviate the remote memory access latency on the NUMA-based baseline architecture, we employ the state-of-the-art First-Tough (FT) page placement policy and remote cache scheme [5] to create a fair baseline evaluation environment.",
"Table: BENCHMARKSFigure: Three types of parallel rendering schemes for parallel VR applied on future NUMA-based Multi-GPU systems.Table REF lists the set of graphics benchmarks employed to evaluate our design.",
"This set includes five well-known 3D games, covering different rendering libraries and 3D engines.",
"We also list the original rendering resolution and the number of draw commands for these benchmarks.",
"Two benchmarks (Doom3 and Half-Life 2) from the table are rendered with a range of resolutions (1600$\\times $ 1200, 1280$\\times $ 1024, 640$\\times $ 480) while for other games we adopt 1280x1024 resolution if it is available and supported by the simulator.",
"In order to feature these PC games as VR applications, we modify the ATTILA Common Driver Layer (ACDL) to enable the multi-view rendering.",
"In our experiments, we let all the workloads run to completion to generate the accurate frames on the simulator and gather the average frame latency for each game.",
"Figure: Normalized performance improvement (left) and single frame latency (right) across different benchmarks."
],
[
"Characterizing Parallel Rendering Schemes on Future NUMA-based Multi-GPU Systems",
"Aiming to reduce the NUMA-induced bottlenecks, a straightforward method is to employ parallel rendering schemes in VR applications to distribute a domain of graphics workloads on a targeted computing resource.",
"While some parallel rendering frameworks such as Equalizer [13], [14] and OpenGL Multiple SDK [7] have been used in many cluster-based PC games, the NUMA-based multi-GPU systems face some different challenges when performing parallel rendering.",
"In this section, we perform a detailed analysis using three state-of-the-art parallel rendering schemes (including frame-level, tile-level and object-level parallel rendering) for VR application running on such future NUMA-based multi-GPU architectures, to further understand the design challenges."
],
[
"Alternate Frame Rendering (AFR)",
"Alternate Frame Rendering (AFR), also known as frame-level parallel rendering, executes one rendering process on each GPU in a multi-GPU environment.",
"As Figure REF demonstrates, AFR distributes a sequence of rendering frames along with the required data across different GPMs.",
"AFR is often considered to be a better fit for distributed memory environment since the separate memory spaces make the concurrent rendering of different frames easier to implement[14].",
"To separate our NUMA memory system into unique memory spaces, we leverage the software-level segmented memory allocation to reserve distributed memory segments for each frame.",
"We also employ a simple task scheduler to map the rendering workloads of a frame to a specific GPM.",
"The benefit of this design is to eliminate the inter-GPM commutation.",
"Figure REF shows the performance improvement and single frame latency affected by AFR scheme.",
"The results are normalized to the baseline NUMA-based multi-GPU setup (with 64GB/s NVLink) where the entire system is viewed as a single GPU under the programming model and rendering workloads are directly launched to this system without specific parallel rendering scheduling.",
"On average, AFR improves the performance (i.e., overall frame rate) by 1.67X comparing to the baseline setup.",
"AFR not only eliminates the performance degradation of low bandwidth inter-GPU links, but also increases the rendering throughput by leveraging the SMP feature of the GPM.",
"However, Figure REF (right) also suggests that AFR increases the single frame latency by 59% as a frame is processed by only one GPU.",
"This increased single-frame latency may cause significant motion anomalies, including judder, lagging and sickness in VR system [44], [42] because it highly impacts whether the corresponding display on VR head-gear device can be in sync with the actual motion occurrence.",
"Additionally, we observe that AFR near-linearly increases the memory bandwidth and capacity requirement according to the pre-allocate memory space for each frame.",
"This decreases the maximum system memory capacity which directly limits the rendering resolution, texture details and perceived quality for different VR applications."
],
[
"Tile-Level Split Frame Rendering (SFR)",
"In contrast to AFR, split frame rendering (SFR) tends to reduce the single-frame latency by splitting a single frame into smaller rendering workloads and each GPM only responses to one group of workloads.",
"Figure REF shows a tile-level SFR which splits the rendering frame into several pixel tiles in the screen space and distributes these sets of pixels across different GPMs.",
"This basic method is widely used in cluster-based PC gaming because it requires very low software effort[37].",
"To employ tile-level SFR, we simply leverage the sort-first algorithm to define the tile-window size before the rendering tasks is processed in GPMs.",
"Although this design can effectively reduce single-frame latency, its vertical pixel stripping [37] does distribute left and right views into different GPMs, ignoring the redundancy of the two views.",
"Thus, to enable the SMP under this tile-level SFR, an alternative is to employ a horizontal culling method, shown in Figure REF .",
"It groups the left and right views as a large pixel tile so that the rendering workloads in the left view can by re-projected into the right via the SMP engine to reduce redundancy geometry processing and improve data sharing.",
"Figure: Normalized performance after enabling SFR across different benchmarks."
],
[
"Object-Level Split Frame Rendering",
"Distributing objects among processing units represents a specific type of split rendering frame (SFR).",
"Figure REF shows an example of object-level SFR which is often referred as sort-last rendering [13].",
"In contrast to the traditional vertical and horizontal tile-level SFR, the distribution under object-level SFR begins after the GPU starts the rendering process.",
"During object-level SFR, a root node is selected (e.g., GPM0 in this example) to distribute the rendering objects to other working units (e.g., GPM1, GPM2 and GPM3).",
"Once a worker completes the assembled object, the output color in its local DRAM is sent to the root node to composite the final frame.",
"In this study, we first profile the entire rendering process to get the total number of rendering objects, and then issue them to different GPMs in a round-robin fashion.",
"Note that only one object is executed in each GPM at a time for better data locality.",
"Although this object distribution can also occur during rendering process (e.g., between rasterization and fragment processing [21]), it typical requires to insert additional inter-GPM synchronization which may cause increasing inter-GPM traffic and performance degradation.",
"Thus, we only distribute the objects at the beginning of the rendering pipeline for our experiments.",
"Figure: Total inter-GPM memory traffic across different benchmarks.Figure REF and REF illustrate the performance (i.e., the overall frame rate) impact and inter-GPM memory traffic for different SFR scenarios.",
"The results are normalized to the baseline setup.",
"We have the following observations: (i) The tile-level SFR schemes only slightly improve the rendering performance over the baseline case, e.g., on average 28% and 3% for Tile-level (V) and Tile-level (H), respectively.",
"This is because although processing a small set of pixels via tile-level SFR can improve the data locality within one GPU, the tile-level SFR schemes increase the inter-GPM memory traffic by an average of 50% for the vertical culling (V) and 44% for horizontal culling (H) due to the object overlapping across the tiles.",
"While the horizontal culling (H) fails to capture the data sharing for large objects (e.g., the bridge on the right side of Figure REF ), vertical culling (V) ignores the redundancy between the left and right view.",
"Since when applying SMP-based VR rendering the GPMs do not render the left and right views simultaneously, the large texture data have to be moved frequently across the GPMs.",
"(ii) The object-level SFR outperforms tile-level SFR schemes and achieves an average of 60%, 32% and 57% performance improvement over the baseline, tile-level (V) and tile-level (H), respectively.",
"The speedups are mainly from the inter-GPMs traffic reduction, indicated by Figure REF .",
"By placing the required data in the local DRAM for the rendered objects, Object-level SFR reduces approximately 40% of inter-GPMs traffic compared to the baseline.",
"However, the state-of-the-art object-level SFR can not fully address the NUMA-induced performance bottlenecks for VR execution, because it still executes the objects from the left and right views separately.",
"In other word, it ignores the multi-view redundancy in VR applications which limits its rendering efficiency.",
"Figure: The best-to-worst performance ratio among GPMs in object-level SFR across different workloads.",
"(iii) Additionally, we also observe that the object-level SFR is challenged by low load balance and high composition overhead.",
"Figure REF shows the ratio between the best and the worst performance among different GPMs under the Round-Robin object scheduling policy.",
"Since each object has a variety of graphical properties (e.g., the total amount of triangles, the level of details, the viewport window size, etc), the processing time is typically different for each object.",
"If one GPM is assigned more complex objects than the others, it will take more time to complete the rendering tasks.",
"Since the overall performance of Multi-GPU system is determined by the worst-case GPM processing, low load balance will significantly degrade the overall performance.",
"Meanwhile, the high composition overhead (i.e., assembling all the rendering outputs from different GPMs into a frame) also contributes to the unbalanced execution time.",
"As we mentioned previously, only the root node is employed to distribute and composite rendering tasks in the current object-level SFR.",
"In this case, extra workloads will be issued to the root node while the ROP units of the other GPMs can not be fully utilized during this color output stage, causing bad composition scalability [7].",
"Therefore, we aim to propose software-hardware support to efficiently handle these challenges facing the state-of-the-art object-level SFR in a NUMA-based multi-GPU environment."
],
[
"Object-Oriented VR rendering f-ramework",
"In order to address the performance issues of the object-level SFR applied on future NUMA-based multi-GPU systems, we propose the object-oriented VR rendering framework (OO-VR).",
"The basic design diagram is shown in Figure REF .",
"It consists of several novel components.",
"First, we propose an object-oriented VR programming model at the software layer to support multi-view rendering for the object-level SFR.",
"It also provides an interface to connect the VR applications to the underlying multi-GPU platform.",
"Second, we propose an object-aware runtime distribution engine at the hardware layer to balance the rendering workloads across GPMs.",
"In OO-VR, this distribution engine predicts the rendering time for each object before it is distributed.",
"It replaces the master-slave structure among GPMs so that the actual distribution is only determined by the rendering process.",
"Finally, we design a distributed hardware composition unit to utilize all the ROPs of the GPMs to assemble and update the final frame output from the framebuffer.",
"Due to the NUMA feature, the framebuffer is distributed across all the GPMs instead of only one DRAM partition, so that it can provide 4x output bandwidth of the baseline scenario.",
"We detail each of these components as follows.",
"Figure: Our proposed object-oriented VR rendering framework (OO-VR)."
],
[
"Object-Oriented VR Programming Model",
"The Object-Oriented VR Programming Model extends the conventional object-level SFR as we introduced in Section and uses a similar software structure as today's Equalizer[13], [14] and OpenGL Multipipe SDK (MPK)[7].",
"Figure REF uses an simplified working flow diagram to explain our programming model.",
"In this study, we propose two major components that drive our OO-VR programming model: Object-Oriented Application (OO_Application) to drive the VR multi-view rendering for each object, and Object-Oriented Middleware (OO_Middleware) to reorder objects and regroup the ones that share similar features as a large batch which acts as the smallest scheduling units on the NUMA-based multi-GPU system.",
"The OO_Application provides a software interface (dark blue box) for developers to merge the left and right views of same object as a single rendering tasks.",
"The OO_Application is designed by extending the conventional object-level SFR.",
"For each object, we replace the original viewport which is set during the rendering initialization with two new parameters – viewportL and viewportR, each of which points to one view of the object.",
"In order to enable rendering multi-views at the same time, we apply the built-in openGL extension $GL\\_OVR\\_multiview2$ to set two viewports ID for a single object.",
"After that, each SMP engine integrated in a GPM automatically renders the left and right views to its own positioning using the same texture data.",
"We also design an auto-model to extend the conventional object-level SFR to enable multi-view rendering through generating two fixed viewports for each object via shifting the original viewport along the X coordinate.",
"In this case, only one rendering process needs to be setup for each object.",
"In constrast to the single-path stereo rendering enabled in modern VR SDKs [29], [13], our OO_Application does not decompose the left and right views during the rendering initialization so that it still follows the execution model of the object-level SFR.",
"Figure: Simplified working flow of an Object-Oriented application, middleware and Multi-GPU rendering pipeline.OO_Middleware is the software-hardware bridge to connect the OO_Application and multi-GPU system.",
"It is automatically executed during the application initialization stage to issue a group of the objects to the rendering queue of the multi-GPU system.",
"In the conventional object-level SFR, the objects are scheduled in a master-slave fashion following the programmer-defined order.",
"However, different objects that may share some common texture data are not rendered on the same GPM.",
"As Figure REF illustrates, both \"pillar1\" and \"pillar2\" share the common \"stone\" texture.",
"If they are rendered on different GPMs, the \"stone\" texture may need to be reallocated, increasing remote GPM access.",
"In OO-VR, we leverage OO_Middleware to group objects based on their texture sharing level (TSL) to exploit the data locality across different objects.",
"To implement this, OO_Middeware first picks an object from the head of the queue as the root.",
"It then finds the next independent object of the root and computes the TSL between the two using Equation (REF ).",
"$\\footnotesize TSL = \\sum _{t}^{T} \\left(P_{r}\\left(t\\right) \\cdot P_{n}\\left(t\\right)\\right) / \\sum _{t}^{T}P_{r}\\left(t\\right)$ Where $t$ is the shared texture data between the two objects, $P_{r}(t)$ and $P_{n}(t)$ represent the percentages of $t$ among all the required textures for the root and the target object.",
"TSL represents how many texture data will be shared if we group the target object with the root.",
"If TSL is greater than 0.5, we group them together as a batch and this batch then becomes the new root which consists all textures from the previous iterator and the target object.",
"After this grouping, the OO_Middleware removes the target object from the queue and continues to search for the next object until the total number of triangles within the batch is higher than 4096, or all the objects in the queue have been selected.",
"The triangle number limitation is used to prevent load imbalance from an inflated batch.",
"After this step, this batch is marked as ready and issued to a GPM in the system for rendering.",
"Finally, the OO_Middleware repeats this grouping process for all the objects in the frame until there is no object in the queue.",
"Note that for the objects that have dependency on any of the objects in a batch, we directly merge them to the batch and increase the triangle limitation so that they can follow the programmer-defined rendering order."
],
[
"Object-Aware Runtime Distribution Engine",
"After receiving the batches from the OO_Middleware, the Multi-GPU system needs to distribute them across different GPMs for multi-view rendering.",
"For workload balancing, we propose an object-aware runtime distribution engine at the hardware layer instead of using the software-based distribution method based on master-slave execution used in the conventional object-level SFR.",
"Comparing to the software-level solution which needs to split the tasks before assigning to the multi-GPU system, the hardware engine provides efficient runtime workload distribution by collecting rendering information.",
"Figure REF illustrates the architecture design of the proposed distribution engine.",
"The new hardware architecture is implemented as a micro-controller for the multi-GPU system which responses for predicting the rendering time for each batch, allocating an object to the earliest available GPM, and pre-allocating memory data using the Pre-allocation Units (PAs) in each GPM.",
"Figure: The architecture design of the object-aware runtime distribution engine.Recall the discussion in Section 3 for our evaluation baseline, we employ the first-touch memory mapping policy (FT) [5] to allocate the required data in the local DRAM.",
"Although FT can help reduce inter-GPM memory access, it can also cause performance degradation if the required data is not ready during the rendering process.",
"As a result, we consider to pre-allocate data before objects are being distributed across GPMs.",
"In this case, OO-VR needs to be aware of the runtime information of each GPM to determine which GPM is likely to become idle the earliest.",
"In order to obtain this information, we need to predict approximately how long the current batch will be completed.",
"Equation (REF ) shows a basic way to estimate the rendering time of the current task $X$ , introduced by [39]: $t\\left(X\\right) = RT\\left(g_x, c_x, HW, ST\\right)$ Where ${g_x,c_x}$ is the geometry and texture property of the object $X$ , $HW$ is the hardware information, and $ST$ is the current rendering step (i.e., geometry process, multi-view projection, rasterazition or fragment process) of the object $X$ .",
"While a complex equation can increase the estimation accuracy, it also requires more comprehensive data and increases hardware design complexity and computing overhead.",
"Because the objective of our prediction model is to identify the earliest available GPM instead of accurately predicting each batch's rendering time, we propose a simple linear memorization-based model to estimate the rendering time as Equation (REF ): $t\\left( X \\right) = c_0 \\cdot \\#triangle_x = c_1 \\cdot \\#tv_x + c_2 \\cdot \\#pixel_x$ Where $\\#triangle_x$ , $\\#tv_x$ and $\\#pixel_x$ represent the triangle counts, the number of transformed vertexes and the number of rendered pixels of the current batch, respectively.",
"$c_0$ , $c_1$ and $c_2$ represent the triangle, vertex and pixel rate of the GPM.",
"After building this estimation model, we split the prediction process into two phases: total rendering time estimation and elapsed rendering time prediction.",
"We setup two counters to record the total rendering time and the elapsed rendering time for each GPM.",
"First, we leverage $\\#triangle_x$ (which can be directly acquired from the OO_Application) to predict the total rendering time.",
"During rendering, the distribution engine tracks $\\#tv_x$ and $\\#pixel_x$ from GPMs to calculate the elapsed rendering time.",
"If the $\\#tv_x$ or $\\#pixel_x$ increases by 1, the elapsed counter increases by $c_1$ or $c_2$ , respectively.",
"At the end, by comparing the distance between the two counters from each GPM, we can predict which GPM will become available first.",
"At the beginning of the rendering, the distribution engine uses the first 8 batches to initialize $c_0$ , $c_1$ and $c_2$ .",
"The first 8 batches will be distributed across GPMs under the Round-Robin object scheduling policy and baseline FT memory mapping scheme is also applied to allocate the rendering data.",
"After GPMs complete this round of 8 batches, the total rendering time will be sent back to the distribution engine to calculate $c_0$ , $c_1$ and $c_2$ .",
"Then, starting from the 9th batch, the rendering time predictor is enabled to find the earliest available GPM.",
"After that, the PA Unit pre-allocates the required data to the selected GPMs, and the rendering time predictor updates the predicted total rendering time by increasing the triangle counts.",
"Note that we limit the maximum size of the batch queue to 4 objects to reduce the memory space requirement.",
"Multiple batches could be distributed onto one GPM at the same time.",
"In this case, a PA Unit sequentially fetches the data based on the order of the batch ID.",
"We further observe that even though distribution engine can effectively balance the rendering tasks, it is possible that some large objects may still become the performance bottleneck if all the other batches have been completed.",
"To fully utilize the computing and memory resources of these idle GPMs, we employ a simple fine-grained task mapping mechanism to fairly distribute the rest of the processing units (e.g.",
"triangles in geometry process and fragments in fragment process) to idle GPMs based on their IDs.",
"Meanwhile, the PA units duplicate the required data to the corresponding unused DRAM to eliminate inter-GPMs access for these left-over fine-grained tasks."
],
[
"Distributed Hardware Composition Unit",
"In the conventional object-level SFR, the entire FrameBuffer (FB) is mapped in the DRAM of the master node, and all the rendering outputs will then be transmitted to the master node for the final composition.",
"Although the color outputs can be executed asynchronously with the shader process, a small amount of ROPs in a single GPM limits the pixel rate which impacts the overall rendering performance.",
"Since the NUMA-based multi-GPU system can be considered as a large single GPU, we consider to distribute the composition tasks across all the GPMs which is currently not supported due to the lack of relevant communication mechanism and hardware.",
"For example, shown in Figure REF , we first split the entire FB into 4 partitions using the screen-space coordinate of the final frame.",
"Here we employ the same memory mapping policy as the vertical Tile-level SFR (V).",
"Based on this, we propose the distributed hardware composition unit (DHC) to determine which part of FrameBuffer is used to store what color outputs of the final frame.",
"This design is based on the observation that the color outputs of the final frames only incur a small number of memory access compared to the main rendering phase so that the small amount of remote communication for this phase will not become a performance bottleneck for NUMA-based multi-GPU systems.",
"This is also why vertical culling shown in Figure REF can perform well as the last stage of VR rendering (i.e., after the object-aware runtime distribution for the main rendering phase) since the inter-GPM bandwidth can be effectively utilized by the distributed hardware composition."
],
[
"Overhead Analysis",
"The major hardware components added into the existing multi-GPU system is the object-aware runtime distribution engine, which consists of a rendering time predictor, GPM counters and a batch queue.",
"For the baseline Multi-GPU architecture that we modeled for this work (Table 2), we allocate 64 bits for each counter and 16 bits for each batch ID to store the predicted rendering time.",
"Additionally, to predict the total and elapsed rendering time, twelve 32-bits registers are used to track the triangle counts, the number of transformed vertexes and the number of the rendered pixels for the current batches.",
"In total, we only require 960 bits for storage and several small logic units.",
"We use McPAT[22] to evaluate the area and energy overhead of the added storage and logic units for the distribution engine.",
"The area overhead is 0.59 $mm^2$ under 24nm technology which is 0.18% to modern GPUs (e.g., GTX1080).",
"The power overhead is 0.3W which is 0.16% of TDP to GTX1080."
],
[
"Evaluation",
"We model the object-oriented VR rendering framework (OO-VR) by extending AITTILA-sim [10].",
"To get the object graphical properties (e.g., viewports, number of triangles and texture data), we profile the rendering-traces from our real-game benchmarks as shown in Table REF .",
"Then in ATTILA-sim, we implement the OOVR programming model in its GPUDriver, and the object distribution engine in its command processor, and the distributed hardware composition during the color writing procedure.",
"To evaluate the effectiveness of our proposed OO-VR design, we compare it with several design scenarios: (i) Baseline - the baseline multi-GPU system with single programming model (Section ); (ii) 1TB/s-BW - the baseline system with 1 TB/s inter-GPU link bandwidth; (iii) Object-level - the Object-level SFR which distributes objects among GPMs (Section ); (iv) Frame-level - the AFR which renders entire frame within each GPM; and (v) OO_APP - the proposed object-oriented programming model (Section ).",
"We provide results and detailed analysis of our proposed design on performance, inter-GPU memory traffic, sensitivity study for inter-GPM link bandwidth and the performance scalability over the number of GPMs."
],
[
"Effectiveness On Performance",
"Fig.REF shows the performance results with respect to single frame latency under the five design scenarios.",
"We gather the entire rendering cycles from the beginning to the end for each frame and normalized the performance speedup to baseline case.",
"We show the performance speedup for single frame because it is critical to avoid motion sickness for VR.",
"From the figure, we have several observations.",
"First, without hardware modifications, the OO_APP improves the performance about 99%, 39% an 28% on average comparing to the Baseline, Object-level SFR and 1TB/s-BW, respectively.",
"It combines the two views of the same object and enable the multi-view rendering to share the texture data.",
"In addition, by grouping objects into large batches, it further increases the data locality within one GPM to reduce the inter-GPM memory traffic.",
"However, it still suffers serious workload unbalance.",
"For instance, object-level SFR slightly outperforms OO_APP when executing DM3-1280 and DM3-1600.",
"This is because some batches within these two benchmarks require much longer rendering time than other batches, the software scheduling policy alone in OO_APP can not balance the execution time across GPMs without runtime information.",
"Second, we observe that on average, OO-VR outperforms Baseline, Object-level SFR and OO_APP by 1.58x, 99% and 59%, respectively.",
"With the software and hardware co-design, OO-VR distributes batches based on the predicted rendering time and provides better workload balance than OO_APP.",
"It also increases the pixel rate by fully utilizing the ROPs of all GPMs.",
"We also observe that OO-VR could achieve similar performance as Frame-level parallelism which is considered to provide ideal performance on overall rendering cycles for all frames (as shown in Fig.REF (left)).",
"However, in terms of the single frame latency, Frame-level parallelism suffers 40% slowdown while OO-VR could significantly improve the performance."
],
[
"Effectiveness On Inter-GPU Memory Traffic",
"Reducing inter-GPM memory traffic is another important criteria to justify the effectiveness of OO-VR.",
"Fig.REF shows the impact of OO-VR on inter-GPM memory traffic.",
"Both Baseline and 1TB/s-BW have the same inter-GPM memory traffic, and Frame-Level is processing each frame in one GPM and has near-zero inter-GPM traffic.",
"Moreover, the memory traffic reduction is mainly cause by our software-level design, the inter-GPM traffic is the same under the impact of OO_APP and OO-VR.",
"Therefore, Fig.REF only shows the results for Baseline, Object-Level and OO-VR, and we mainly investigate these three techniques in the following subsections.",
"From the figure, we observe OO-VR can save 76% and 36% inter-GPM memory accesses comparing to the Baseline and Object-level SFR, respectively.",
"This is because OO-VR allocates the required rendering data to the local DRAM of GPMs.",
"The majority inter-GPM memory accesses are contributed by the distributed hardware composition, command transmit and Z-test during fragment process.",
"We observe that the delay caused by these inter-GPM memory accesses can be fully hidden by executing thousands of threads simultaneously in numerous shader cores.",
"In addition, the data transfer via the inter-GPM links also leads to higher power dissipation (e.g.",
"10pj/bit for board or 250pj/bit for nodes based on different integration technologies[5]).",
"By reducing inter-GPM memory traffic, OO-VR also achieves significant energy and cost saving."
],
[
"Sensitivity To Inter-GPM Link Bandwidth",
"Inter-GPU link bandwidth is one of the most important factors in multi-GPU systems.",
"Previous works [5], [25] have shown that increasing the bandwidth of inter-processor link is difficult and requires high fabric cost.",
"To understand how inter-GPM link bandwidth impacts the design choice, we examine the performance gain of OO-VR under a variety of choices on link bandwidth.",
"Fig.REF shows the speedup under different link bandwidth when applying Baseline, Object-level SFR and our proposed OO-VR.",
"In this figure, we normalize the performance to the Baseline with 64GB/s inter-GPM link.",
"We observe that the inter-GPU link bandwidth highly affects the Baseline and Object-level SFR design scenarios.",
"This is because these two designs cannot capture the data locality within the GPM to minimize the inter-GPU memory accesses during rendering.",
"The large amount of shared data across GPMs significantly stalls the rendering performance.",
"In the contrast, OO-VR fairly distributes the rendering workloads into different GPMs and convert numerous remote data to local data.",
"By doing this, it fully utilizes the high-speed local memory bandwidth and is insensitive to the bandwidth of inter-GPM link even the inter-GPM memory accesses are not entirely eliminated.",
"As the local memory bandwidth scales in future GPU design (e.g.",
"High-Bandwidth Memory (HBM)[11]), the performance of the future multi-GPU scenario is more likely to be constrained by inter-GPU memory.",
"In this case, we consider the OO-VR can potentially benefit the future multi-GPU scenario by reducing inter-GPM memory traffic."
],
[
"Scalability of OO-VR",
"Fig.REF shows the average speedup of the Baseline, Object-level SFR and OO-VR as the number of GPMs increases.",
"The results are normalized to single-GPU system.",
"As the figure shows, the Baseline and Object-level SFR suffer limited performance scalability due to the NUMA bottleneck.",
"With 8 GPMs, the Baseline and Object-level SFR only improve the overall performance by 2.08x and 3.47x on average over the single GPU processing.",
"On the other hand, the OO-VR provides scalable performance improvement by distributing independent rendering tasks to each GPM.",
"Hence, with 4 and 8 GPMs, it achieves 3.64x and 6.27x speedup over the single GPU processing, respectively."
],
[
"Related Work",
"Architecture Approach For NUMA Based Multi-GPU System.",
"There have been many works [21], [5], [25] improving the performance for NUMA based multi-GPU system.",
"Some of them[5], [25], [43] introduce architectural optimizations to reduce the inter-GPM memory traffic for GPGPU application while Kim et al.",
"[21] redistribute primitives to each GPM to improve the scalability on performance.",
"However, none of them discusses the data sharing feature of VR application.",
"Our approach exploits the data locality during VR rendering to reduce the inter-GPM memory traffic and achieves scalable performance for multi-GPU system.",
"Parallel Rendering.",
"Currently, PC clusters are broadly used to render high-interactive graphics applications.",
"To drive the cluster, software-level parallel rendering frameworks such as OpenGL Multipipe SDK [7], Chromium [19], Equalizer[13], [14] have been developed.",
"They provides the application programming interface (API) to develop parallel graphics applications for a wide rand of platforms.",
"Their works tend to split the rendering tasks during application development under different configurations.",
"In our work, we propose a software and hardware co-designed object-oriented VR rendering framework for the parallel rendering in NUMA based multi-GPU system.",
"Performance Improvement For Rendering.",
"In order to balance the rendering workloads among multiple GPUs, some studies [26], [18], [12] propose a software-level solution that employs CPU to predict the execution time before rendering to adaptively determine the workload size.",
"However, such software-level method requires a long time to acquire the hardware runtime information from GPUs which causes performance overhead.",
"Our hardware-level scheduler could quickly collect the runtime information and conduct the real-time object distribution to balance the workloads.",
"There are also some works [23], [38] designing NUMA aware algorithms for fast image composition.",
"Instead of implementing a composition kernel in software, we resort to a hardware-level solution that leverages hardware components to distribute the composition tasks across the multi-GPU system to enhance the pixel throughput.",
"Meanwhile, many architecture approaches [4], [41], [40] have been proposed to reduce the memory traffic during rendering.",
"Our work focuses on multi-view VR rendering in multi-GPU system which is orthogonal to these architecture technologies."
],
[
"Conclusions",
"In modern NUMA-based multi-GPU system, the low bandwidth of inter-GPM links significantly limits the performance due to the intensive remote data accesses during the multi-view VR rendering.",
"In this paper, we propose object-oriented VR rendering framework (OO-VR) that converts the remote inter-GPM memory accesses to local memory accesses by exploiting the data locality among objects.",
"First, we characterize the impact of several parallel rendering frameworks on performance and memory traffic in the NUMA-based multi-GPU systems.",
"We observe the high data sharing among some rendering objects but the state-of-the-art rendering framework and multi-GPU system cannot capture this interesting feature to reduce the inter-GPM traffic.",
"Then, we propose an object-oriented VR programming model to combine the two views of the same object and group objects into large batches based on their texture sharing levels.",
"Finally, we design a object aware runtime batch distribution engine and distributed hardware composition unit to achieve the balanced workloads among GPUs and further improve the performance of VR rendering.",
"We evaluate the proposed design using VR featured simulator.",
"The results show that OO-VR improves the overall performance by 1.58x on average and saves inter-GPM memory traffic by 76% over the baseline case.",
"In addition, our sensitivity study proves that OO-VR can potentially benefit the future larger multi-GPU scenario with ever increasing asymmetric bandwidth between local and remote memory."
],
[
"Acknowledgment",
"This research is supported by U.S. DOE Office of Science, Office of Advanced Scientific Computing Research, under the CENATE project (award No.",
"466150), The Pacific Northwest National Laboratory is operated by Battelle for the U.S. Department of Energy under contract DEAC05-76RL01830.",
"This research is partially supported by National Science Foundation grants CCF-1619243, CCF-1537085(CAREER), CCF-1537062."
]
] | 2001.03537 |
[
[
"Superbubbles as an Empirical Characteristic of Directed Networks"
],
[
"Abstract Superbubbles are acyclic induced subgraphs of a digraph with single entrance and exit that naturally arise in the context of genome assembly and the analysis of genome alignments in computational biology.",
"These structures can be computed in linear time and are confined to non-symmetric digraphs.",
"We demonstrate empirically that graph parameters derived from superbubbles provide a convenient means of distinguishing different classes of real-world graphical models, while being largely unrelated to simple, commonly used parameters."
],
[
"Description",
"This document includes all the data analyzed in the main paper.",
"It consists of twelve tables containing all computed graph descriptors and metrics for four different network datasets: the supergenome dataset (Tabs.",
"1–3), the Stanford dataset (Tabs.",
"4–6), the LDBC dataset (Tabs.",
"7–9), and the standard random dataset (Tabs.",
"10–12).",
"The individual datasets are described in the paper.",
"Each of the three tables per dataset contains the undirected graph descriptors, the directed graph descriptors, and the superbubble descriptors, respectively.",
"Values of the undirected graph descriptors for the supergenome dataset.",
"Table: NO_CAPTIONValues of the directed graph descriptors for the supergenome dataset.",
"Table: NO_CAPTIONValues of the superbubble descriptors for the supergenome dataset.",
"Table: NO_CAPTION Values of the undirected graph descriptors for the Stanford dataset.",
"Table: NO_CAPTION Values of the directed graph descriptors for the Stanford dataset.",
"Table: NO_CAPTION Values of the superbubble descriptors for the Stanford dataset.",
"Table: NO_CAPTION Values of the undirected graph descriptors for the LDBC dataset.",
"Table: NO_CAPTION Values of the directed graph descriptors for the LDBC dataset.",
"Table: NO_CAPTION Values of the superbubble descriptors for the LDBC dataset.",
"Table: NO_CAPTION Values of the undirected graph descriptors for the standard random dataset.",
"Table: NO_CAPTION Values of the directed graph descriptors for the standard random dataset.",
"Table: NO_CAPTION Values of the superbubble descriptors for the standard random dataset.",
"Table: NO_CAPTION"
]
] | 2001.03409 |
[
[
"Reconstruction of turbulent flow fields from lidar measurements based on\n large-eddy simulation"
],
[
"Abstract We investigate the reconstruction of a turbulent flow field in the atmospheric boundary layer from a time series of lidar measurements, using Large-Eddy Simulations (LES) and a 4D-Var data assimilation algorithm.",
"This leads to an optimisation problem in which the error between measurements and simulations is minimised over an observation time horizon.",
"To exploit the spatial coherence of the turbulence, we use a quadratic regularisation in the objective function, that is based on the two-point covariance matrix.",
"Moreover, to improve conditioning, and remove continuity constraints, the problem is transformed into a Karhunen-Lo\\`eve basis.",
"For the optimisation, we use a quasi-Newton limited-memory BFGS algorithm combined with an adjoint approach for the gradient.",
"We also consider reconstruction based on a Taylor's frozen turbulence (TFT) model as point of comparison.",
"To evaluate the approach, we construct a series of virtual lidar measurements from a fine-grid LES of a pressure-driven boundary-layer.",
"The reconstruction uses LES on a coarser mesh and smaller domain, and results are compared to the fine-grid reference.",
"Two lidar scanning modes are considered: a classical plan-position-indicator mode, which swipes the lidar beam in a horizontal plane, and a 3D pattern that is based on a Lissajous curve.",
"We find that normalised errors lie between 15% and 25% (error variance normalised by background variance) in the scanning region, and increase to $100\\%$ over a distance that is comparable to the correlation length scale outside this scanning region.",
"Moreover, LES outperforms TFT by 30% to 70% depending on scanning mode and location."
],
[
"Introduction",
"Flow field measurements in the atmospheric boundary layer have relevance for a broad spectrum of applications, ranging from fundamental turbulent boundary layer [18], [35], to analysing turbulent wakes of wind turbines [25], [21], [48] to air quality studies [61], among others.",
"Pulsed lidar sensors allow taking flow field measurements almost simultaneously at multiple locations in space, at frequencies of around one Hertz, leading to a vast amount of measurement data [45].",
"However, lidar sensors are limited to measure a volume averaged line-of-sight wind speed.",
"Several techniques exist for transforming this raw data into velocity vectors.",
"A first category only deals with time-averaged flow statistics, based on the estimation of parameters of an analytical wind speed model [3], [7].",
"In a second approach, the instantaneous velocity is reconstructed but only at discrete measurement points in space using simple algorithms to transform the measured wind speeds to velocities, e.g.",
"assuming horizontal homogeneousity, and stationarity of the velocity field between the measured points in combination with doppler beam swinging lidar scan mode [32], or assuming zero spanwise and vertical velocity in combination with a spinning lidar [51], [49], [38].",
"All the previous approaches ignore the spatial correlations of the turbulent flow field, which are, e.g, known to extend up to $20H$ in neutrally stratified atmospheric boundary layers [16].",
"In three-dimensional variational data assimilation (3D-Var), these correlations are taken into account [30]; a similar method named linear stochastic estimation (LSE) has been simultaneously developed in the turbulence community [1], [2].",
"Recently, this has been applied to microscale flow fields [26], [9], [15].",
"In these last studies, the velocity correlation is modelled using a 3D homogeneous isotropic turbulence spectral correlation tensor.",
"This is often justified in the horizontal directions; however, in boundary layers, the vertical direction shows strong anisotropic behaviour.",
"A method to incorporate the time series of measurements for nonlinear systems have been independently and simultaneously developed in the geoscience and control community and is respectively known as four-dimensional variatiational data assimilation (4D-Var) [28], [31], [27], [55] and nonlinear moving horizon state estimation (MHE) [22].",
"This methodology minimises a linear combination of the mismatch between a time series of real and virtual observations, the model error, and the deviation from the background distribution.",
"The combination of large-eddy simulation (LES) and 4D-Var with lidar data to retrieve turbulent structures was first proposed in [29] conducting a series of twin experiments.",
"Later in a series of papers this methodology was applied to a field measurement campaign, using two different lidars for reconstruction and validation of the methodology [12], [13], [42], [43], [62].",
"To regularise the problem, a Laplacian-based penalty term was used.",
"Continuity was not strictly enforced but added as an additional penalty term.",
"The combination of 4D-Var with a Taylor's frozen turbulence (TFT) model was shown in [47], however, without including spatial regularisation.",
"In the current work, LES-based reconstruction of turbulence from lidar measurements is further investigated for the atmospheric boundary layer, improving on the problem formulation.",
"To this end, regularisation of the problem is based on the background covariance tensor, following a Bayesian inference framework.",
"Moreover, we transform the problem into a Karhunen–Loève basis (or proper orthogonal decomposition (POD) basis), which is constructed from the covariance tensor.",
"This leads to an unconstrained optimisation problem, since the POD basis is by construction divergence free.",
"Moreover, the problem is also optimally conditioned, removing the unfavourable scaling of the condition number of the covariance matrix with growing problem size.",
"By using homogeneity in horizontal directions, which is generally a good approximation for atmospheric boundary layers, the POD basis can be efficiently calculated and stored, since the modes correspond to Fourier modes in horizontal directions.",
"We test the methodology based on virtual lidar measurements in fine-grid reference simulations, using a coarser LES reconstruction grid, as well as a smaller domain.",
"This allows us to perform detailed out-of-sample comparisons of the reconstructed turbulence with the reference solution.",
"The paper is further organised as follows.",
"In § , the lidar observations model is described.",
"§ discusses the state estimation methodology.",
"Subsequently, § introduces the LES and TFT models used for the state estimation.",
"The adjoint-based optimisation methodology is discussed in § , and details of the case setup are introduced in § .",
"Results are presented in § ; conclusions and future outlook in § ."
],
[
"Lidar measurements",
"Lidar sensors measure the velocity component along the laser beam direction.",
"Here, we focus on pulsed lidar sensors, which measure the wind speed at different locations along the beam simultaneously [45].",
"A much used example is the Lockheed Martin WindTracer [26], which is considered in the current work as a reference.",
"This lidar has a moderate range gate width of $\\Delta r={105}{}$ , a pulse length (FWHM) of $\\Delta p_{1/2}={105}{}$ , initial blind zone of $r_0={436}{}$ , and a total of $N_r=100$ range gates, see table REF for a summary.",
"Consider a lidar system mounted at location $x_{\\mathrm {m}}$ (extension to multiple lidar systems is straightforward, but not considered here), and a beam direction $e_{\\mathrm {l}}(t)$ that follows a scanning pattern in time.",
"The lidar measurement locations then correspond with $x_{i}(t) =x_{\\mathrm {m}}+(r_0+\\Delta r(i-1))e_{\\mathrm {l}}(t), ~~~~ i=1\\cdots N_r.$ Due to the finite pulse, range gate width and sampling time, the space–time filtered wind speed around $x_{i}$ is measured, oriented in the direction of the lidar beam.",
"For a single location, we can express [4] $ h_i(u(x,t),t_n) \\triangleq \\frac{1}{T_{\\mathrm {s}}}\\int _{t_{n-1}}^{t_n}\\int _{\\Omega } u(x,t)\\cdot e_{l}(t) \\ \\mathcal {G}_{l}(\\mathsf {Q}(t)(x-x_{i}(t))) \\ \\mathrm {d}x\\mathrm {d}t,$ with $u(x,t)$ the three-dimensional and time-dependent turbulent velocity field, $h_i$ the observation function, $\\mathcal {G}_l$ the lidar filter kernel oriented in the direction of the lidar beam, and $\\mathsf {Q}$ a coordinate transformation between the reference coordinate system $(e_1,e_2,e_3)$ , and a coordinate system aligned with the beam such that $e_{l}=\\mathsf {Q}e_1$ .",
"Since the samples are only collected at discrete time instances, we introduce $t_n = t_0+ nT_{\\mathrm {s}}$ , with $n$ the sample number and $1/T_s=f_s$ the sample frequency.",
"We introduce the vector of observation functions $h_n=[h_{n,1},\\, \\ldots , h_{n,N_r}]^\\top $ , and similarly the vector of observations $y_n \\triangleq [y_{n,1},\\,\\ldots , y_{n,N_r}]^\\top $ , which differ by the vector of measurement errors $w_n$ , so that $y_n = h_n+w_n$ .",
"Note that the subscript $n$ is used to denote the evaluation at $t_n$ , e.g.",
"$h_{n,i}=h_i(u(x,t),t_n)$ .",
"The lidar filter kernel corresponds to a convolution of a box function with width the range gate width $\\Delta r$ , and a Gaussian filter kernel with width $\\Delta p$ [4], [32].",
"Using $x=[x_1,\\,x_2,\\,x_3]^\\top $ , this corresponds to $ \\mathcal {G}_l(x)& = \\frac{1}{2\\Delta p_{1/2}}\\left[\\mathrm {erf}\\left(\\frac{x_1 + \\Delta r/2}{\\Delta p_{1/e}}\\right)-\\mathrm {erf}\\left( \\frac{x_1 - \\Delta r/2}{\\Delta p_{1/e}}\\right)\\right]\\delta (x_2)\\delta (x_3),$ with $\\delta $ the Dirac delta function and $\\Delta p_{1/e}= \\Delta p_{1/2}/(2\\sqrt{\\log 2})$ the $1/e$ pulse width.",
"We note that in the direction perpendicular to the beam, a Gaussian filter could be used, with a filter width that corresponds to the beam width.",
"The maximum beam width occurs at $R_{\\mathrm {max}}$ and can be roughly estimated by $\\lambda R/d={23}{}$ [17], with $\\lambda $ the wavelength of the lidar and $d$ the beam waist.",
"Practical LES grid resolutions for ABL simulations typically range from $2-{60}{}$ , such that for simplicity, the Gauss kernel in perpendicular directions is approximated by a Dirac delta function.",
"Table: Summary of the lidar parameters of a Lockheed Martin WindTracer ."
],
[
"General methodology",
"Given an approximate state equation, the 4D-Var algorithm provides a best estimate of the time-dependent state given a series of measurements.",
"Here, we introduce the method, mainly following ideas from [23], [14], [31], adopting it to the context of lidar measurements and LES-based state models and turbulent flows in the atmospheric boundary layer.",
"Consider the (exact) space–time velocity field $u(x,t)$ , and an approximate state equation (e.g.",
"the LES equations).",
"We then have $\\frac{\\partial u}{\\partial t} &= f(u,p) + v, \\\\u(x,t_0) &= u_0(x), $ where $f$ is a shorthand notation for the momentum balance in the approximate flow model, $v(x,t)$ is the model mismatch, and $p$ are additional parameters in the model setup that are known, and do not need to be estimated (cf.",
"§REF , and § for more details).",
"Furthermore, $u_0(x)$ is the initial condition at time $t_0$ .",
"We presume that the state is divergence free (using the Boussinesq approximation for ABL flows), so that $\\nabla \\cdot u_0=0$ , and also $\\nabla \\cdot v=0$ .",
"Further practical details on the state equation are discussed in § .",
"Given a measurement series $\\mathsf {Y}\\triangleq [y_{1}, \\,\\dots , y_{N_s}]$ , a related series of states $\\mathsf {U} \\triangleq [u_{0},\\,\\dots , u_{N_s}]$ is considered.",
"Since both the measurements and the state equation contain errors ($w_{n}$ and $v(x,t)$ respectively) that are unknown random variables, the distributions $P(\\mathsf {Y})$ and $P(\\mathsf {U})$ can be introduced.",
"A popular approach for finding a good estimate for the state $u(x,t)$ ($t_0\\le t \\le t_f$ ) is to find the state $u$ that maximises the conditional probability distribution $P(\\mathsf {U}|\\mathsf {Y})$ , i.e.",
"given the measurement series $\\mathsf {Y}$ .",
"Applying Bayes' rule yields $P(\\mathsf {U}|\\mathsf {Y}) = P(\\mathsf {Y}|\\mathsf {U})P(\\mathsf {U})/P(\\mathsf {Y})$ .",
"However, finding the state $u(x,t)$ that maximises this probability, leads to an optimisation problem over the full space–time state space, that is very high dimensional.",
"Moreover, expressing $P(\\mathsf {U}) = P(u_0,\\ldots ,u_{N_s})$ requires knowledge of the space–time correlation function of the bias $v(x,t)$ , which is usually not straightforward.",
"Therefore, in an alternative approach, the modeled space–time velocity field $\\widetilde{u}(x,t)$ is considered, following from $\\frac{\\partial \\widetilde{u}}{\\partial t} &= f(\\widetilde{u},p), \\\\\\widetilde{u}(x,t_0) &= u_0(x).",
"$ Since this is a deterministic equation, we also introduce the solution operator $\\mathcal {M}_{t}(u_0(x))\\triangleq \\widetilde{u}(x,t)$ .",
"Further, since $h$ is a linear function, $y_n & = h(\\mathcal {M}_{t}(u_0(x)),t_n)+ h(\\epsilon (x,t),t_n) +w_n, $ with $\\epsilon \\triangleq u - \\widetilde{u} = u - \\mathcal {M}_{t}(u_0)$ .",
"Note that for linear systems, the probability distribution of $\\epsilon $ is simply a linear transformation of the distribution of $v$ .",
"However, since this distribution is not known, we simply consider the distribution of $\\zeta _n \\triangleq h(\\epsilon (x,t),t_n) +w_n$ , with random variables $\\zeta _n$ that are independent and Gaussian distributed with same variance $\\gamma ^2$ , which is further tuned during the setup of the optimisation problem (cf.",
"below).",
"We then consider the state that maximises the conditional probability $P(\\widetilde{\\mathsf {U}}|\\mathsf {Y})$ .",
"Elaborating leads to $P(\\widetilde{\\mathsf {U}}|\\mathsf {Y}) = P(\\widetilde{\\mathsf {U}}) P(\\mathsf {Y}|\\widetilde{\\mathsf {U}})/P(\\mathsf {Y}) \\sim P(u_0) \\prod _{n} P(y_n|\\mathcal {M}_{t_n}(u_0))$ .",
"Thus, using the assumed distribution for $\\zeta $ $P(\\widetilde{\\mathsf {U}}|\\mathsf {Y}) \\sim P(u_0) \\prod _{n=1}^{N_{\\mathrm {s}}} \\exp \\left(\\frac{\\Vert y_n-h_n(\\mathcal {M}_{t}(u_0), t_n) \\Vert ^2}{2\\gamma ^2}\\right).$ In order to further elaborate $P(u_0)$ , and avoid mathematical technicalities related to probability distributions over infinite dimensional spaces, it is assumed here that this distribution is finite-dimensional or can be approximated by a finte-dimensional random process.",
"For instance, given the mean of the velocity field $U(x)=\\langle u(x) \\rangle $ , and its two-point covariance ${\\mathsf {B}}_{ij}(x,\\breve{x})= \\langle u_i^{\\prime }(x) u_j^{\\prime }(\\breve{x}) \\rangle $ (with $u^{\\prime }=u-U$ ), a truncated Karhunen–Loève decomposition can be used to arrive at a finite dimensional representation.",
"This leads to [5] $ u&\\approx U(x) + \\sum _{k=1}^{N_{\\mathrm {m}}}a_{k}(t) \\lambda _{k}^{1/2}\\psi _{k}(x)=U + \\Psi \\Lambda ^{1/2} a,$ where $a=[a_1, \\,\\ldots , a_{N_{\\mathrm {m}}}]^\\top $ are now uncorrelated random variables with unit variance, and where $\\lambda _k$ are the eigenvalues of ${\\mathsf {B}}_{ij}$ (ordered such that $\\lambda _1\\ge \\lambda _2\\ge \\cdots $ ), with $\\Lambda = \\mathrm {diag}(\\lambda _1, \\,\\ldots , \\lambda _{N_{\\mathrm {m}}})$ and $\\psi _k$ are the eigenvectors of ${\\mathsf {B}}_{ij}$ with $\\Psi = [\\psi _1, \\,\\ldots , \\psi _{N_{\\mathrm {m}}}]$ .",
"They are obtained by solving the following Fredholm eigenvalue problem $ \\frac{1}{|\\Omega |}\\int _{\\Omega }\\mathsf {B}(x,\\breve{x})\\psi _{k}(\\breve{x}) \\ \\mathrm {d}\\breve{x} = \\lambda _{k} \\psi _{k}(x),$ where the eigenfunctions are normalised such that $\\Vert \\psi _{k}(x)\\Vert = 1$ .",
"Finally, using (REF ), this leads to $P(u_0)\\sim P(a)$ .",
"Moreover, although velocity fluctuations in turbulent boundary layers are known to deviate from a Gaussian distribution [34], it is a valid first order approximation, and thus, $P(u_0)\\sim P(a) \\sim \\exp (\\frac{1}{2}\\Vert a \\Vert ^2)$ .",
"Bringing all above assumptions together, and formulating the optimisation in terms of $\\log [P(\\widetilde{\\mathsf {U}}|\\mathsf {Y})]$ , which does not change the optimum, leads to [23] $&\\underset{a}{\\mathrm {minimise}} &&J(a) = \\frac{1}{2}\\Vert a\\Vert ^2 + \\frac{1}{2\\gamma ^2} \\sum _{n=1}^{N_{\\mathrm {m}}} \\left\\Vert y_n-h\\left(\\mathcal {M}_{t}(U + \\Psi \\Lambda ^{1/2} a), t_n\\right) \\right\\Vert ^2.",
"$ We note that the value of $\\gamma ^2$ is at this point unknown – further selection is discussed in § REF .",
"Above problem can be interpreted as minimizing the model–measurement mismatch, given a regularisation $\\frac{1}{2}\\Vert a\\Vert ^2$ , that is particularly important in areas where no measurement information is given.",
"The formulation based on a Karhunen–Loève decomposition as discussed above, leads to a regularisation that is optimally conditioned.",
"If the velocity field $u_0$ were to be used directly as optimisation variable, the regularisation would typically take the form $\\frac{1}{2}\\Vert u_0\\Vert ^2_{\\mathsf {B}^{-1}}$ , with a condition number that corresponds to $\\lambda _1/\\lambda _N$ (given $N$ degrees of freedom in the finite-dimensional representation of $u_0$ )."
],
[
"Formulation of an efficient Karhunen–Loève basis",
"In order to use the approach proposed in §REF , a Karhunen–Loève basis for the wind-field distribution is required.",
"This can in principle be based on an ensemble of all possible wind fields that occur in the ABL assembled over many years.",
"However, acquiring the covariance tensor for this would be nontrivial, and the resulting basis may require many modes for an accurate parameterisation of all possible states.",
"Therefore, we envisage a different approach, in which the covariance tensor is parametrised depending on a number of background parameters $p$ (e.g.",
"wind direction, friction velocity, surface roughness, Obukhov length, Rossby number, etc.).",
"These parameters are presumed to change slowly in time, and are either given, or estimated using a different overarching algorithm.",
"In the current manuscript, we focus in particular on neutral conditions, and simplify the approach to that of estimating the wind field in a neutral pressure driven boundary layer in equilibrium (the extension to Ekman layers should be straightforward, but is not considered here).",
"For this case, the relevant background parameters are given by $p=[u_*, z_0, H, \\theta ]$ , with $u_*$ the friction velocity, $z_0$ the surface roughness, $H$ the boundary layer height, and $\\theta $ the mean wind direction.",
"The mean-velocity profile in is then given by [46] $U(x) = \\frac{u_*}{\\kappa }\\left(\\log \\left(\\frac{z}{z_0}\\right) + F\\left(\\frac{z}{H}\\right)\\right)\\mathsf {Q}_3e_1,$ with $z\\triangleq x_3$ and $F(z/H)$ an outer-layer velocity deficit function, that in the classical picture of inner–outer separation, can be uniquely determined from a single simulation.",
"Further, $\\mathsf {Q}_3(\\theta )$ is a rotation matrix around around the wall normal direction that reorients the coordinate axis in the horizontal plane to a system with main wind direction $\\theta $ .",
"Similarly, using outer-layer scaling arguments [57], and horizontal homogeneity, the covariance tensor can be expressed as $\\mathsf {B}(x,\\breve{x}) = u_*^2 \\ \\mathsf {Q}_3 \\mathsf {B}^+\\left(\\frac{x_1-\\breve{x}_1}{H},\\frac{x_2-\\breve{x}_2}{H},\\frac{x_3}{H},\\frac{\\breve{x}_3}{H}\\right)\\mathsf {Q}_3^\\top ,$ with $\\mathsf {B}^+$ the covariance normalised by friction velocity and boundary layer height, obtained in a system with main wind direction oriented in the $x_1$ direction.",
"We note that as a result of outer-layer similarity, $B$ will not depend on the surface roughness $z_0$ [54].",
"It is easily shown that for horizontal homogeneous directions, as encountered approximately in atmospheric boundary layer flows, eigenfunctions simply correspond to Fourier modes [5], such that $\\psi _{k,m}(x) = \\widehat{\\psi }_{k,m}(x_3)\\exp (\\mathrm {i}(k_1x_1+k_2x_2))$ .",
"Inserting this expression in (REF ) and integrating over $x_1$ and $x_2$ gives $\\frac{1}{H}\\int _{0}^{H}\\widehat{\\mathsf {B}}(k,x_3,\\breve{x}_3)\\widehat{\\psi }_{k,m}(\\breve{x}_3)\\ \\mathrm {d}\\breve{x}_3 = \\widehat{\\lambda }_{k, m} \\widehat{\\psi }_{k,m}(x_3), $ normalised such that $\\Vert \\widehat{\\psi }_{k,m}\\Vert =1$ , where $\\widehat{{\\mathsf {B}}}_{ij}(k,x_3,\\breve{x}_3) = \\langle \\widehat{u}_i(k,x_3)\\widehat{u}_j^*(k,\\breve{x}_3) \\rangle $ is the horizontal Fourier transform of the correlation tensor, with $k=[k_1,k_2]^{\\top }$ .",
"In order to further elaborate, consider velocity fields that are discretised in space on a domain $L_1\\times L_2 \\times H$ .",
"Consistent with our LES code (cf.",
"§), we consider $N_1 \\times N_2 \\times N_3$ grid points that are uniformly distributed, and a grid that is staggered in the vertical direction for the $u_3$ component, leading to $N_1N_2(3N_3-1)$ degrees of freedom.",
"Thus, wavenumbers in horizontal directions are integer multiples of $k_1^*\\triangleq 2/L_1$ , $k_2^*\\triangleq 2/L_2$ , with cutoff the wave number corresponding to $k_c\\triangleq [/\\Delta _1, /\\Delta _2]^\\top $ , with $\\Delta _1=L_1/N_1$ , and $\\Delta _2=L_2/N_2$ .",
"Given this setup, (REF ) requires the solution of $\\frac{1}{4}N_1N_2$ eigenvalue problems (factor $\\frac{1}{4}$ because of symmetries), each of size $(3N_3-1)\\times (3N_3-1)$ .",
"This allows an easier construction of higher dimensional basis, compared to, e.g., the snapshot POD method [52], where the rank is limited to the amount of samples used for the determination of $\\widehat{\\mathsf {B}}$ .",
"Expanding the velocity field using the POD eigenfunctions leads to $ u^{\\prime }&=\\sum _{k,m}c_{k,m}\\widehat{\\lambda }^{1/2}_{k, m}\\psi _{k,m} \\\\&=\\sum _{m}c_{0, m}\\widehat{\\lambda }^{1/2}_{k, m}\\psi _{0,m} + 2\\sum _{k^+,m}\\widehat{\\lambda }^{1/2}_{k, m}\\left(\\operatorname{Re}(c_{k, m})\\operatorname{Re}(\\psi _{k,m}) -\\operatorname{Im}(c_{k, m})\\operatorname{Im}(\\psi _{k,m})\\right), \\nonumber $ where in the second step the conjugate symmetry of $\\widehat{\\mathsf {B}}$ is used.",
"This transforms the complex modes $\\psi _{k,m}$ to an equivalent set of real orthogonal modes, $\\operatorname{Re}(\\psi _{k,m})$ and $\\operatorname{Im}(\\psi _{k,m})$ , by adding together positive and corresponding negative wave numbers.",
"In this way conjugate symmetry constraints in the optimisation problem are avoided.",
"The wavenumber $k^+$ is chosen such that all coefficients are only used once, here we use $k^+= k\\mid (k_1>0~\\text{or}~k_1=0, k_2>0)$ .",
"For $k=0$ the modes are real, and therefore the complex coefficients are omitted.",
"Grouping all coefficients, eigenvalues and modes gives $\\widehat{a} &= \\begin{bmatrix}c_{0,1} & \\cdots & \\sqrt{2}\\operatorname{Re}(c_{k_c,3N_z})& \\sqrt{2}\\operatorname{Im}(c_{k_c,3N_z})\\end{bmatrix}, \\\\\\widehat{\\Lambda } &= \\mathrm {diag}\\begin{pmatrix}\\lambda _{0,1} & \\cdots &\\lambda _{k_c,3N_z}& \\lambda _{k_c,3N_z}\\end{pmatrix}, \\\\\\widehat{\\Psi }&=\\begin{bmatrix}\\psi _{0,1}& \\cdots & \\sqrt{2}\\operatorname{Re}(\\psi _{k_c,3N_z})& -\\sqrt{2}\\operatorname{Im}(\\psi _{k_c,3N_z})\\end{bmatrix},$ where the scaling factor $\\sqrt{2}$ is added such that $\\Vert \\widehat{\\Psi }_k \\Vert =1$ , and $\\langle \\widehat{a}_k^2 \\rangle =1$ .",
"Converting to a basis using the $N_{\\mathrm {m}}$ most energetic modes can be done by $a &= S\\mathsf {P}\\widehat{a}, \\\\\\Lambda &= \\mathsf {S}\\mathsf {P}\\widehat{\\Lambda }\\mathsf {P}^\\top \\mathsf {S}^\\top , \\\\\\Psi &= \\widehat{\\Psi }\\mathsf {P}^\\top \\mathsf {S}^\\top ,$ where $\\mathsf {P}$ represents a permutation matrix ordering the eigenvalue in descending order, and $\\mathsf {S} = \\begin{bmatrix} \\mathsf {I} & 0 \\end{bmatrix}$ is a selection matrix that removes all modes with order higher than $ N_{\\mathrm {m}}$ , and $\\mathsf {I}$ is an identity matrix of size $N_{\\mathrm {m}}\\times N_{\\mathrm {m}}$ .",
"We note that it is required to select $N_m\\le N_1N_2(2N_3-1)+1$ .",
"This results from the fact that the velocity field is solenoidal.",
"In the discretised LES system (cf.",
"next section), there are $N_1N_2N_3-1$ independent continuity constraints, and this leads to a subspace of $N_1N_2N_3-1$ modes in the discrete POD basis that are orthogonal to the solenoidal subspace, with eigenvalue zero."
],
[
"Large-eddy simulations",
"The main state-space model that we consider in the current work is based on large-eddy simulations of a neutrally stable pressure driven boundary layer.",
"We presume knowledge of the setup parameters $p=[u_*, z_0, H, \\theta ]$ , and simply orient the simulation domain such that $e_1=e_\\theta $ .",
"The governing equations then correspond to $\\frac{ \\partial \\widetilde{u}}{\\partial t} &= -\\widetilde{u}\\cdot \\nabla \\widetilde{u} + u_*^2 H e_1 - \\frac{1}{\\rho } \\nabla \\widetilde{p} + \\nabla \\cdot \\tau _{\\mathrm {SGS}}, \\\\0 &=\\nabla \\cdot \\widetilde{u} , $ where $\\widetilde{u}$ is the filtered velocity field, $-\\rho u_*^2 H e_1$ is the mean background pressure gradient, and $\\widetilde{p}$ is the remaining filtered pressure fluctuation.",
"For the subgrid-scale model $\\tau _{\\mathrm {SGS}}$ we use the Smagorinsky model [53] with Smagorinsky coefficient $C_s=0.14$ in combination with wall-damping [33] with $n=1$ .",
"In horizontal directions, we consider a computational domain that is sufficiently large for boundary conditions not to influence the solution in the estimation region of interest, so that simple periodic boundary conditions can be used, and boundary fields in space do not need to be estimated.",
"In the vertical direction, nonpermeable slip boundary conditions are used both along top and bottom wall.",
"Along the bottom wall, this is supplemented with a classical wall model [39], following the implementation proposed by [10]; see also [36] for further details.",
"All state-space simulations are performed using our in-house LES code SP-Wind [37], [41], in which above model is implemented using a pseudospectral discretisation in the horizontal directions, and a fourth-order energy conserving scheme in the vertical directions [60].",
"For the time integration we use a fourth-order Runge–Kutta method, where the time step is fixed, approximately corresponding to a CFL number of 0.4."
],
[
"Taylor's-frozen turbulence model",
"As an alternative simpler reference, we also consider the Taylor's frozen turbulence (TFT) model [56] as a state-space model.",
"It corresponds to $\\widetilde{u}(x,t)=\\mathcal {M}_{t}(u_0(x))=u_0(x-U_{\\infty }e_1(t-t_0))$ , with $U_{\\infty }$ a characteristic convection speed, or equivalently in differential form $\\frac{\\partial \\widetilde{u}}{\\partial t} = -U_{\\infty }e_1\\cdot \\nabla \\widetilde{u}.$ For $U_{\\infty }$ , we use the lidar mount-height velocity, which is simply estimated as $U_{\\infty }/u_*= \\kappa ^{-1}\\log (z_{\\mathrm {m}}/z_{0})$ .",
"The model is solved using the same time integration and spatial discretisation scheme as the LES code.",
"Only horizontal boundary conditions are necessary, for which we also use periodicity."
],
[
"Optimisation methodology and adjoint equations",
"For the optimisation problem (REF ) we use the L-BFGS algorithm from [11], which is a quasi-Newton algorithm suitable for large scale problems.",
"The step length is determined by the Moré–Thuente line-search algorithm [40].",
"Due to the vast amount of optimisation variables, finite difference gradient calculation is computationally not feasible, and instead a continuous adjoint approach is applied.",
"The sensitivity of the cost function is given by $\\frac{\\partial J}{\\partial a_i}= a_i -\\lambda _i^{1/2}\\int _{\\Omega }\\psi _i(x)\\cdot \\xi (x,t_0)\\mathrm {d}x,$ where $\\xi $ is the adjoint variable (see Appendix $\\ref {sec:adjoint_deriv}$ for a derivation), which follows from solving the adjoint equations (cf.",
"below).",
"In this way, the gradient can be computed to all the variables at the cost of one extra simulation, which has a similar form and computational cost as the forward equations.",
"The derivation of the continuous adjoint equations is quite standard, and not repeated here (see [6], [19] for respectively a DNS derivation, and a derivation of the additional LES specific terms).",
"The adjoint equations can be summarised as $-\\frac{\\partial \\xi }{\\partial t} &= \\widetilde{u}\\cdot \\nabla \\xi +\\xi \\cdot \\nabla \\widetilde{u}+\\nabla \\cdot \\tau ^*_{SGS} + \\frac{1}{\\rho }\\nabla \\pi + \\sum _{i=1}^{N_{\\mathrm {r}}}f_{i}, \\\\\\nabla \\cdot \\xi &= 0.$ Here $\\xi $ and $\\pi $ are respectively the adjoint velocity and pressure variable, $\\tau ^*_{SGS}$ is the adjoint subgrid-scale model, and $f_{i}$ is the adjoint forcing term connected to measurement $i$ (see Appendix $\\ref {sec:adjoint_deriv}$ for a derivation), and given by $ f_{i} = \\frac{1}{\\gamma ^2T_{\\mathrm {s}}}\\sum _{n=1}^{N_{\\mathrm {s}}}(y_{n,i}-h_{n,i}) \\, \\mathcal {G}_{\\mathrm {l}}\\!\\left(\\mathsf {Q}(t)(x-x_{i}(t))\\right)e_{\\mathrm {l}}(t) \\, \\mathrm {H}\\!\\left(\\frac{T_{\\mathrm {s}}}{2}-\\left|t - t_{n-\\frac{1}{2}}\\right|\\right),$ where $\\mathrm {H}$ is the Heaviside function.",
"The adjoint equations need to be integrated backwards in time.",
"The starting condition for the adjoint variables specified at the end of the horizon is simply $\\xi (x,t_f) = 0$ (see Appendix REF for details).",
"For the integration we use a fourth-order discrete adjoint Runge–Kutta scheme [20], which is consistent with our forward time integration.",
"The spatial boundary conditions correspond to periodicity in the horizontal directions, and impermeability for the bottom and top of the domain in combination with respectively an adjoint wall model at the bottom.",
"Details are found in [19].",
"The adjoint TFT equations are very similar to the adjoint LES-equations, i.e.",
"$-\\frac{\\partial \\xi }{\\partial t} &= U_{\\infty }\\cdot \\nabla \\xi + \\sum _{i=1}^{N_{\\mathrm {r}}}f_{i}.$ These equations also have to be solved backwards in time with initial condition $\\xi _0 = 0$ and periodic boundary conditions in space.",
"The forcing term $f_{i}$ is the same as the one used by the adjoint LES (REF )."
],
[
"Simulation setup",
"In the current work, a reference simulation is used to take virtual lidar measurements.",
"Afterwards the reconstruction of the flow field is performed on a different domain with a coarser mesh.",
"Both domains are schematically represented on figure REF , and the details are summarised in table REF and further discussed below.",
"All simulations use a boundary layer height of $H={1}{}$ and a surface roughness of $z_{0}={0.1}{}$ , which is a common overland value, and a friction number of $u_* = {0.5}{^{-1}}$ , which leads to a wind speed of around ${8}{^{-1}}$ at ${100}{}$ .",
"For the reference simulation we use a relatively fine grid resolution of $0.015H \\times 0.015H\\times 0.005H$ , combined with a long domain $30H \\times 5.4H\\times H$ to avoid spurious periodic correlations.",
"In addition, the boundary conditions of the reference domain are shifted over $d_s=H$ to ensure statistically homogeneous inflow, and avoid locking of large-scale motions [41].",
"The fringe region is located between $15.5H$ and $14.5H$ upstream of the lidar mount, the distance between recycling and fringe region is chosen as $3H$ , this is also visualised on figure REF .",
"A spinup period of $50H/u_*$ is used, to ensure a statistical steady state has been reached, before virtual lidar measurements are taken.",
"The optimisation domain is smaller than the reference domain, but should at least encompass the region of influence of the time series of lidar measurements.",
"Based on the lidar setup discussed in §REF and the assimilation time horizon of $0.1H/u_*$ (see §REF ), suitable domain dimensions are found to be $18H \\times 5.4H\\times H$ .",
"Reconstruction of velocity scales smaller than the lidar filter size is not possible, and therefore adding these scales is not improving the reconstruction, and needlessly increases the computational complexity.",
"Therefore the grid is chosen to be $0.05H \\times 0.05H\\times 0.0167H$ .",
"Note that by using a different grid resolution for reference and reconstruction, a bias is introduced in the state space model, although admittedly, the expected bias with respect to real measurements can be quite different (the use of real data is however not in the scope of the current study).",
"The two-point covariance matrix $B$ is found to be sensitive to the grid resolution and is therefore computed on the same grid used for the reference simulation.",
"The domain is chosen as $18H \\times 5.4H\\times H$ – a tradeoff between accuracy and reasonable computational cost, caused by the long time averaging that is necessary to get sufficient statistical convergence (see further).",
"We note that the largest flow structures in an ABL, i.e., streamwise streaks, are of the same scale as our domain.",
"Therefore, we find that the far correlations of the streamwise velocity component are influenced by the periodic boundary conditions.",
"This effect is briefly discussed in §REF .",
"The simulation is spun up over a period of $50H/u_*$ , and subsequentially sampled every $0.01H/u_*$ over a time horizon of $100H/u_*$ , leading to $10^4$ samples.",
"Note that the equations are reflection symmetric with respect to a streamwise-vertical plane, and therefore, we add the mirrored samples as well.",
"Finally, the two-point covariance tensor needs to be converted from the correlation to the optimisation domain.",
"The domain dimensions are the same, such that only a restriction to the coarser grid is needed.",
"For the horizontal and vertical directions, we use a cutoff filter and linear interpolation respectively.",
"To assure that the POD modes obtained from the restricted covariance matrix are solenoidal, the restricted matrix is projected on the solenoidal subspace of the coarser grid by means of a Helmholtz decomposition."
],
[
"Lidar setup",
"We demonstrate the methodology using two scanning modes.",
"To keep the two trajectories comparable, we use the same scanning period of $T_{\\mathrm {p}}=0.1H/u_*={200}{}$ .",
"The lidar mount is located at $x_{\\mathrm {m}}=[0, 0, 0.1H]$ for both cases.",
"In a first case study, we set the virtual lidar in plan position indicator (PPI) scanning mode with zero elevation angle, thus tracking a horizontal sweeping trajectory.",
"The direction of the beam is given by $e_{\\mathrm {l}}(t) = \\mathsf {Q}_3(\\phi (t))e_1$ , such that $\\mathsf {Q}(t)=\\mathsf {Q}_3(t)$ .",
"The azimuthal angle $\\phi $ is given by $\\phi (t) = \\Delta \\phi \\text{ Triag}(t/T_{\\mathrm {p}}) + $ , where $\\mathrm {Triag}(t)\\triangleq 1/\\, \\mathrm {sin}^{-1}(\\mathrm {sin}(2t))$ is the triangle wave function with unit amplitude and period, such that the lidar has a constant azimuthal angular velocity of $|\\partial \\phi /\\partial t|=2\\Delta \\phi /T_{\\mathrm {p}}$ .",
"For the azimuthal range $\\Delta \\phi $ we take $\\Delta \\phi = 2\\sin ^{-1}(2H/R_{\\mathrm {max}})$ (see figure REF ).",
"In a second case study, we study a 3D scanning pattern.",
"First we define a parametric curve $l(t)$ , where we use $l(t)=[1, Asin(\\omega _2 t -\\delta ),B(sin(\\omega _3t)+1)]$ , a special case of a 3D Lissajous curve, in which $A$ and $B$ respectively control the horizontal and vertical extent of the trajectory, and the ratio of angular speeds $\\omega _3/\\omega _2$ , and phase $\\delta $ control the shape of the curve.",
"The construction is such that the lidar does not scan lower than the lidar mount height.",
"The lidar direction is given by $e_{\\mathrm {l}}(t) = \\mathsf {Q}_3()l(t)/||l(t)||$ .",
"For this study we respectively take $A=\\tan (\\frac{1}{2}\\Delta \\phi )$ , which is easily verified to lead to the same spanwise extent as the sweeping lidar, and $B=\\tan \\Delta \\theta $ with $\\Delta \\theta = \\sin ^{-1}(0.8H/R_{\\mathrm {max}})$ , which gives a maximum scanning altitude of $0.9H$ .",
"Further, $\\omega _2=4/T_{\\mathrm {p}}$ , $\\omega _3=3/2\\omega _2$ , and $\\delta =/2$ .",
"Table: Summary of the setup parameters of the different simulations."
],
[
"Optimisation setup",
"The time horizon of the optimisation $T\\triangleq t_f- t_0$ is chosen equal to a single scanning period of the lidar $T_{\\mathrm {p}}$ .",
"Very long horizons lead to very large gradients due to the chaotic behaviour of turbulence.",
"Moreover, since we don't explicitly include model errors, and optimally match $\\widetilde{u}$ to the measurements, $\\widetilde{u}$ and $u$ will diverge to the point where new measurements do not contribute to the reconstruction.",
"All modes with positive eigenvalues are taken into the POD basis, such that we optimise over the full space of solenoidal velocity fields.",
"The amount of modes corresponds to $N_{\\mathrm {m}} = N_1N_2(2N_3-1)+1$ , which is obtained by substracting the number of independent continuity constraints $N_1N_2N_3-1$ from the total degrees of freedom $N_1N_2(3N_3-1)$ .",
"In table REF the most important optimisation parameters are summarised.",
"For the optimisation, we use 8 Hessian correction pairs for the L-BFGS-B method.",
"For the Wolfe condition parameters we take $c_1=10^{-4}$ and $c_2=0.9$ , which are standard values selected for quasi-Newton methods [44].",
"In the following sections we elaborate further on the convergence and stopping criteria for the optimisation, and on the tuning of the variance $\\gamma ^2$ of to the model–measurement uncertainty."
],
[
"Adjoint gradient calculation and optimisation convergence",
"In this section we discuss general properties of the optimisation.",
"As a test case we use the PPI-scanning mode in combination with the LES reconstruction model.",
"We use $\\gamma ^2=10^{-4}$ for the model–measurement uncertainty (this is further elaborated in §REF ).",
"First of all, the streamwise component of the adjoint field is given in figure REF .",
"The field gives a representation of the sensitivity of the cost function to a local perturbation in velocity and the propagated effect through the Navier–Stokes equations.",
"The adjoint field is clearly seen to originate from the lidar measurement locations, due to the forcing term of the adjoint equations being a convolution of the mismatch between the observed and simulated lidar measurements with the lidar filter kernel.",
"The adjoint velocity field propagates upstream due to the reverse sign of the convection term.",
"Note that for a large part of the domain, the adjoint field remains (almost) zero, which indicates that flow information in this region does not influence the measurements.",
"The accuracy of the adjoint gradient is also verified by comparing against a finite-difference evaluation for a few selected perturbation directions.",
"We find that the relative error of the gradient remains smaller than $0.1\\%$ for the selected cases.",
"More details are provided in appendix .",
"Figure: The right and left figure respectively represent the reduction of the cost function and gradient as a function of the amount of PDE evaluations.",
"The orange, green and blue dots respectively represent the J o J_{\\mathrm {o}}, J b J_{\\mathrm {b}} contribution of the cost function and the whole cost function J=J o +J b J= J_{\\mathrm {o}} + J_{\\mathrm {b}}.All optimisations are started from an initial guess $a = 0$ .",
"The convergence history of the PPI case with $\\gamma ^2=10^{-4}$ is shown in figure REF .",
"The left panel of the figure shows the evolution of the cost function as a function of the number of PDE evaluations (LES or adjoint) during the optimisation.",
"For sake of analysis, we split the cost function (see (REF )) into two parts, i.e.",
"$J= J_{\\mathrm {b}} + J_{\\mathrm {o}} $ , with $J_{\\mathrm {b}} = \\frac{1}{2}\\Vert a\\Vert ^2, ~~~J_{\\mathrm {o}} = \\frac{1}{2\\gamma ^2} \\sum _{n=1}^{N_{\\mathrm {m}}} \\left\\Vert y_n-h\\left(\\mathcal {M}_{t}(U + \\Psi \\Lambda ^{1/2} a), t_n\\right) \\right\\Vert ^2.$ The first represents the background variability (of the initial condition), and acts as a regularisation term, while the second represents the variability of the observation–model mismatch.",
"Both are shown in the convergence history in figure REF as well.",
"Each outer optimisation iteration requires a simulation of the forward and adjoint equations, and an additional line search in case the Wolfe conditions are not satisfied.",
"The latter is found to only happen in 15 occasions, so that the number of total PDE simulations (shown in the horizontal axes in figure REF ) is by good approximation twice the number of iterations.",
"The right panel of figure REF shows the $L_2$ -norm of the gradient vector $\\nabla _{a} J$ .",
"Since the optimisation problem is unconstrained, a (local) optimum will correspond with $\\nabla _{a}J=0$ .",
"Given the accuracy of the gradient, which is found to be $\\textit {O}(10^{-3})$ (see appendix ), we see that the gradient converges up to a relative value of $2\\times 10^{-3}$ ."
],
[
"Tuning of the combined model–measurement uncertainty",
"In this section the unknown variance $\\gamma ^2$ , introduced in §REF , is tuned.",
"The regularisation term of the cost function $J_{\\mathrm {b}}$ controls the complexity of the solution, while $\\gamma ^2 J_{\\mathrm {o}}$ represents the differences between the simulated and observed measurements.",
"By controlling the parameter $\\gamma ^2$ , the relative trust in the model–measurement accuracy is adjusted.",
"For the tuning, we consider the PPI scanning mode combined with the LES model.",
"In figure REF we show the Pareto front of $J_{\\mathrm {b}}$ versus $\\gamma ^2J_{\\mathrm {o}}$ , for different values of $\\gamma ^2$ .",
"It is clearly seen that below $\\gamma ^2=10^{-4}$ there is almost no decrease in $\\gamma ^2J_{\\mathrm {o}}$ while the complexity of the solution increases significantly, which is an indication of overfitting.",
"Therefore we use $\\gamma ^2=10^{-4}$ .",
"This value is also used for the TFT model and for the Lissajous scanning mode.",
"Figure: Pareto front for the optimisation using the LES model, between the the regularisation part of the cost function J b J_{\\mathrm {b}} and the model–observation mismatch γ 2 J o \\gamma ^2J_{\\mathrm {o}}.",
"The annotations at the markers denote different variances γ 2 \\gamma ^2 of the combined model and measurement uncertainty."
],
[
"Covariance matrix and POD modes",
"We start by showing the structure of the covariance tensor.",
"To this end, we visualise the correlation tensor, which is defined as ${\\mathsf {C}}_{ij}(x,x^{\\prime })=\\frac{{\\mathsf {B}}_{ij}(x,x^{\\prime })}{({\\mathsf {B}}_{ij}(x,x){\\mathsf {B}}_{ij}(x^{\\prime },x^{\\prime }))^{1/2}},$ and is an often-used nondimensionalised version of the covariance tensor [50], [24]).",
"Figure REF visualises different cross-sections of the diagonal components (${\\mathsf {C}}_{11}$ , ${\\mathsf {C}}_{22}$ , ${\\mathsf {C}}_{33}$ ) of the correlation tensor for reference point $x^{\\prime }=[0,0,\\frac{1}{4}L_3]$ .",
"Figure REF (a) visualises ${\\mathsf {C}}_{11}$ , which consists of a central inclined positive lobe laterally surrounded by two negative correlated lobes.",
"The streamwise velocity component of the tensor ${\\mathsf {C}}_{11}(x,x^{\\prime })$ , has been studied in [16], but almost exclusively for 1D streamwise and spanwise variations of $x-x^{\\prime }$ .",
"In this work, a similar decay of $C_{11}(x,x^{\\prime })$ was found for the streamwise direction of ${\\mathsf {C}}_{11}(x,x \\pm 5He_1)=0.1$ .",
"Note that in [16] at around $10H$ a zero crossing of the correlation was found and beyond this region anti-correlation, is found which stretches to at least to $30H$ .",
"However, our domain size is too small to observe this phenomenon.",
"Moreover, an influence of periodic boundary conditions exists and leads to an overestimation of the correlation near the edges of the domain.",
"The correlation of the spanwise velocity component ${\\mathsf {C}}_{22}$ has a main lobe, which has a steeper inclination angle compared to ${\\mathsf {C}}_{11}$ , and has vertically situated anti-correlated side lobes.",
"Finally, the vertical velocity component correlation ${\\mathsf {C}}_{33}$ has a relatively narrow main lobe in the spanwise direction.",
"In contrast with the other components, no significant anti-correlated lobes are found.",
"Figure $\\ref {fig:eigenvector}$ visualises the most energetic complex eigenfunctions $\\widehat{\\Psi }_{k,1}(x_3)$ for the first 5 streamwise $k_1$ and spanwise $k_2$ wave numbers.",
"Each eigenfunction can by an inverse Fourier transform be transformed to a 3D solenoidal vector field."
],
[
"Turbulent flow field reconstruction",
"We now discuss the reconstruction of velocity fields from lidar observations.",
"We start by looking at qualitative comparisons between the reference field and reconstructed velocity fields using an LES model.",
"Subsequently, errors are quantified in more details, and comparison with reconstructions using a TFT model are also included.",
"In figure REF , a comparison between the reconstructed velocity fluctuations $\\widetilde{u}^{\\prime }$ and the reference velocity fluctuations $u^{\\prime }$ are visualised for the PPI lidar scanning mode.",
"Cross-sections of the streamwise velocity are shown at the start ($t=t_0$ ) and end ($t=t_f$ ) of the observation time window.",
"For the horizontal cross-sections (figure REF b), in general a good correspondence between $\\widetilde{u}^{\\prime }$ and $u^{\\prime }$ is found within the scanned area.",
"We also observe that for the initial and final field, fluctuations in an area upstream and downstream of the scanning region, are also well reconstructed.",
"This is explained by convective transport of flow information out of the measurement area during the measurements.",
"Beyond this convective transport of flow information, additional upstream information is provided due to the relatively long $u_1$ correlations in streamwise direction.",
"For the vertical cross-section (figure REF a), direct flow field observations are only available at $z=0.1H$ .",
"Due to the lack of mean transport in the vertical direction, additional information is only available due to the regularisation term, and the spatial coherence introduced by the LES model.",
"Nevertheless, it is seen that the large scale motions are reasonably well constructed in the vertical plane.",
"Figure REF shows a comparison between reconstructed $h_n$ and reference $y_n$ measurements.",
"We note that the optimisation problem effectively minimises this difference.",
"Thus, as may be expected, the trends are well represented by the reconstructed measurements.",
"The lidar measurement signal as a function of time for a fixed range gate number (figure REF b) transforms from a relatively smooth signal close to the sensor location to a more irregular further away due to the larger distance covered by the range gate.",
"Next, we focus on the reconstructed velocity field in the lidar plane, but at a moment in time when the lidar is pointing elsewhere.",
"To this end, figure REF compares the reconstructed velocity field to the reference field along two lines in the the streamwise and spanwise direction respectively, at time $t=t_0+ T/4$ .",
"At this moment, the lidar is in a spanwise extremum position.",
"It is observed that the streamwise velocity component $u_1$ is well constructed, except for the smallest scales.",
"This is expected, since the lidar observations themselves are spatially filtered, while the LES reconstruction mesh is also coarser than the reference.",
"Looking at the streamwise and wall-normal components, the quality of the reconstruction appears somewhat lower, though the main trends are still recovered.",
"However, when quantifying the errors in more detail (see below), we find that errors in absolute value are of the same order of magnitude for the three components, indicating that the reconstruction is of similar quality for all three directions, even though the lidar only measures along it's line of sight, which is dominantly oriented along the $u_1$ direction.",
"Figure: The top, middle and bottom figure present the streamwise, spanwise and wall-normal velocity component at t=t i +T/4t = t_i + T/4.",
"The left and right hand side respectively show the velocity component along a line in the streamwise direction through the lidar mount, i.e.",
"(x 1 ,0,0.1H)(x_1,0,0.1H) and in the spanwise direction at mount height, and 8H8H upstream of the mount point, i.e.",
"(-8H,x 2 ,0.1H)(-8H,x_2,0.1H).In Figure REF the velocity field reconstruction for the Lissajous scanning mode is shown.",
"Three spanwise–vertical planes are shown at $t=t_0$ comparing reconstructed to reference velocity field.",
"The dashed line shows the trajectory of the intersect of these planes with the lidar beam during the complete time horizon.",
"The further upstream the plane, the more spatially extended this trajectory is.",
"As further quantified below, quality of reconstruction along the height of the boundary layer is a lot better is for this scanning pattern, since direct measurements are available up to $0.9H$ .",
"Figure: Cross-section at x 1 =-10Hx_1 = -10H, x 1 =-5Hx_1 = -5H and x 1 =0x_1 = 0 of the streamwise component of the velocity field at t=t 0 t=t_0.",
"The left and right hand side respectively show the reconstructed velocity field u ˜ 1 \\widetilde{u}_1 and reference velocity field u 1 u_1.",
"The dashed line denotes the intersection of the lidar beam with the respective plane for the complete time horizon.Figure: (left) Normalised error variances as function of height.",
"(right) Evolution of error variances over the time window, evaluated at height z=0.1Hz=0.1H.",
"(top) Streamwise velocity component; (middle) spanwise velocity component; (bottom) vertical velocity component.",
"(⋯\\cdots ): error with respect to unfiltered reference; (—): error with respect to filtered reference.",
"(▵\\triangle ): PPI scanning mode; (∘\\circ ): Lissajous scanning mode.",
"(Blue): reconstruction based on LES, (Orange): reconstruction based on TFT, (Black, □\\square ): background variance.In order to quantify errors in more detail, we define the error on the streamwise component as $\\epsilon _1(x,t) = I_c^F \\circ \\widetilde{u}_1(x,t)-u_{1}(x,t)$ , where $\\widetilde{u}_1$ is the reconstructed velocity, $I_c^F$ is a coarse-to-fine interpolation operator (for which we simply use linear interpolation), and $u_{1}$ the fine-grid reference velocity.",
"We introduce the error variance averaged over a horizontal region of interest $\\Gamma $ as $\\mathrm {Var}[\\epsilon _1(z,t)] = \\langle (\\epsilon _1)^2\\rangle _{\\Gamma }$ .",
"To this end, we select a horizontal region that is located inside the lidar observation area.",
"To avoid boundary effects on the error, we omit regions that are less than one convective distance $U_{\\infty }T$ away from the scanning boundaries, and use $\\Gamma (z)=\\mathsf {Q}_3()[r\\cos (\\phi ) + U_\\infty T, r\\sin (\\phi ) ,z]$ with $r\\in [R_0 ,R_{\\mathrm {max}}-2U_\\infty T]$ , $\\phi \\in [-\\Delta \\phi /2, \\Delta \\phi /2]$ (see also §REF ).",
"Similar to $\\mathrm {Var}[\\epsilon _1(z,t)]$ , the error variances for the velocity components in spanwise and wall-normal direction are also introduced.",
"Finally, since lidars effectively measure a filtered velocity field, we also construct errors based on a filtered reference velocity field.",
"To this end, we filter the fine-grid LES field using a stream-wise cutoff filter with $k_{\\mathrm {c}} = /(\\Delta p^2 + \\Delta R^2)^{1/2}$ , with $\\Delta p^2= 3/(2\\log 2) \\Delta p_{1/2}^2$ roughly approximating the lidar filter kernel (REF ).",
"An overview of the normalised error variance is shown in figure REF for reconstruction with an LES model, and as point of comparison also for reconstruction with the TFT model.",
"Errors based on unfiltered and filtered reference fields are shown, and results for both the PPI and Lissajous scanning modes are included.",
"As an additional reference, we also show, per component in figure REF , the background variance $\\langle u_{i}^{\\prime 2}\\rangle _{\\Gamma }$ , which is the error obtained by predicting with the mean velocity profile.",
"This is the best estimate without or far away from any measurements.",
"First of all, it is observed in figure REF that both scanning trajectories give good results at lidar mount height, though the PPI mode slightly outperforms the Lissajous mode, with a variance of the $u_{1}$ component that is on average only $15\\%$ of the background variance, compared to $25\\%$ for the Lissajous mode.",
"The best results are obtained in the middle of the assimilation window, and progressively increases towards the bounds.",
"When looking at higher altitudes, the Lissajous mode clearly outperforms the PPI mode, with an average normalized error variance of $25\\%$ in the region $z$ from $0.1$ to $0.9$ , compared to $55\\%$ for the PPI mode.",
"Moreover, in the lidar scanning region, the variance of errors in $u_{1}$ is of the same order than those of errors in $u_{2}$ , and $u_{3}$ (in absolute value).",
"This indicates that the estimation distributes the errors evenly in all direction.",
"However, since in boundary layers, the background variance of $u^{\\prime }_{1}$ is much larger than that of $u^{\\prime }_{2}$ , and $u^{\\prime }_{3}$ , the relative error of the reconstruction in the $x_1$ direction is much lower.",
"This explains, the better matching of the $u^{\\prime }_{1}$ signal in figure REF (which is simply larger in amplitude than the other components).",
"Finally, it is further also found that LES consistently outperforms TFT, for example for the PPI scanning mode at lidar mount heigth, the error variance is $24\\%$ an increase by $60\\%$ compared to the LES.",
"The maximum error of the TFT model in the scanning region is $35\\%$ , which is an increase by $70\\%$ compared to the LES case."
],
[
"Conclusion",
"In the current study we investigated reconstructing turbulent flow field from lidar measurements by using a 4D-Var approach in combination with a LES model.",
"The problem was regularised using the background covariance tensor, and reformulated using a POD basis.",
"This allowed the elimination of the continuity constraint, and also led to a better conditioned formulation.",
"Moreover, we used horizontal homogeneity of boundary layers to efficiently construct and represent the POD basis.",
"In order to test the methodology, we constructed virtual lidar measurements from a fine-grid pressure-driven boundary layer, and reconstructed the turbulent flow field using LES on a coarser mesh.",
"This allowed for a detailed error analysis.",
"Different lidar scanning modes were investigated, and a comparison with a TFT model was also included.",
"Overall, LES based reconstruction was quite effective.",
"Inside the general lidar scanning region, we found that errors in the streamwise velocity fluctuation lie between $15\\%$ and $25\\%$ (error variance normalised by background variance).",
"Moreover, LES outperformed TFT by $30\\%$ to $70\\%$ .",
"In spanwise and wall-normal directions, the reconstruction quality was the same in absolute value, but worse in relative value, since in turbulent boundary layers the streamwise background variability is substantially higher than spanwise and wall-normal variability.",
"In the current work, we studied a pressure driven boundary layer as a proxy for a neutral atmospheric boundary layer.",
"In reality, effects related to the Ekman spiral (changing wind direction with height), and the capping inversion and free-atmosphere stratification can play an important role in boundary-layer height velocity profile.",
"Moreover, near-wall stratification has strong effects on turbulence, requiring the use of correlation tensors that depend on the stratification regime.",
"These are important directions for future research.",
"A further working assumption in the current work, was the use of horizontal homogeneity for the construction of a POD basis in combination with an outer-scaling argument for scaling independent from surface roughness.",
"In reality effects of inhomogeneity in terrain can have an effect on the outer region of the ABL, and further research needs to investigate whether an imperfect (based on homogeneity assumptions), but efficient POD basis remains effective when terrain features (e.g.",
"wind turbines, buildings) are introduced.",
"Moreover, the explicit inclusion of model uncertainties [58], may also further improve the methodology."
],
[
"Derivation and validation of the adjoint gradient",
"In this appendix, we derive the adjoint equations for the calculation of the gradient of the cost function $J(a)$ to the control variables $a$ .",
"For the derivation we use a Lagrangian approach [8], similar to the approach by [19] to which we refer for further details.",
"To this end, we first reformulate the optimisation problem (REF ) by removing the explicit solution operator $\\mathcal {M}_{t}$ , and instead explictly adding the state space constraints, leading to $&\\underset{a, \\widetilde{u}, p}{\\mathrm {minimise}} &&\\mathcal {J}(a, \\widetilde{u}) = \\frac{1}{2}\\Vert a\\Vert ^2 + \\frac{1}{2\\gamma ^2} \\sum _{n=1}^{N_{\\mathrm {m}}} \\left\\Vert y_n-h\\left(u(t), t_n\\right) \\right\\Vert ^2, \\nonumber \\\\&\\text{subject to} && \\frac{ \\partial \\widetilde{u}}{\\partial t} +\\widetilde{u}\\cdot \\nabla \\widetilde{u}-u_*^2 H e_1+\\frac{1}{\\rho } \\nabla p- \\nabla \\cdot \\tau _{\\mathrm {SGS}} = 0 \\nonumber \\\\& && \\nabla \\cdot \\widetilde{u}= 0 \\nonumber \\\\& && \\widetilde{u}(x, t_0)-\\Psi \\Lambda a = 0.$ We note that by construction,$J(a)=\\mathcal {J}(a,\\mathcal {M}_{t}(u_0))$ .",
"For ease of notation the state variables and the adjoint variables are grouped together and respectively given by $q=[\\widetilde{u}, p]^\\top $ and $q^*=[\\xi ,\\pi , \\chi ]^\\top $ .",
"In an analogous way we group together the state space constraints $\\mathcal {B}(a,q)=[\\mathcal {N}^{\\mathrm {m}},\\mathcal {N}^{\\mathrm {c}},\\mathcal {C}]^\\top $ , which are respectively the momentum, continuity equations and the constraint for the initial condition.",
"The Lagrangian of above problem is now defined as ${L}(a, q,q^*) \\triangleq \\mathcal {J}(a,q) + (q^*,\\mathcal {B}(a,q))$ and is given by ${L}(a, q, q^*) &=\\underbrace{\\frac{1}{2}||a||^2 + \\frac{1}{2\\gamma ^2}\\sum _{n=1}^{N_{\\mathrm {s}}}||y_n-h_n || }_{\\mathcal {J}} \\nonumber \\\\& + \\underbrace{\\int _{t_0}^{t_f}\\int _{\\Omega }\\xi \\cdot \\left(\\frac{ \\partial \\widetilde{u}}{\\partial t} +\\widetilde{u}\\cdot \\nabla \\widetilde{u}-u_*^2 H e_1 +\\frac{1}{\\rho } \\nabla p- \\nabla \\cdot \\tau _{\\mathrm {SGS}}\\right)\\ \\mathrm {d}x\\mathrm {d}t}_{(\\xi ,\\mathcal {N}^{\\mathrm {m}})}\\nonumber \\\\&+ \\underbrace{\\int _{t_0}^{t_f}\\int _{\\Omega }\\pi \\left(\\nabla \\cdot \\widetilde{u}\\right)\\ \\mathrm {d}x\\mathrm {d}t}_{(\\pi ,\\mathcal {N}^{\\mathrm {c}})}+\\underbrace{\\int _{\\Omega } \\chi \\cdot \\left(\\widetilde{u}(x,t_0)-\\Psi \\Lambda a\\right)\\ \\mathrm {d}x}_{(\\chi ,\\mathcal {C})}.$ It can be shown [59] that, if the adjoint variables are chosen such that ${L}_{q}(\\delta q) = 0$ and the state space constraints are satisfied $\\mathcal {B}(a,q)=0$ , then $J_{a}(\\delta a)={L}_{a}(\\delta a)$ .",
"Here we use the Riesz representation theorem to relate gradients to derivatives [8], for example for the cost function this gives $J_{a}(\\delta a) \\triangleq \\left.",
"\\frac{\\mathrm {d}}{\\mathrm {d}\\alpha }J(a+ \\alpha \\delta a)\\right|_{\\alpha =0}= \\left(\\nabla _{a} J, \\delta a\\right).$ Further elaborating ${L}_{q}(\\delta q) = 0$ gives ${L}_{q}(\\delta q) = &\\left(\\frac{\\partial \\mathcal {J}}{\\partial \\widetilde{u}},\\delta \\widetilde{u}\\right)+\\left(\\xi , \\frac{\\partial \\mathcal {N}^{\\mathrm {\\mathrm {m}}}}{\\partial \\widetilde{u}}\\delta \\widetilde{u}\\right)+\\left(\\xi ,\\frac{\\partial \\mathcal {N}^{\\mathrm {\\mathrm {m}}}}{\\partial p} \\delta p\\right) \\nonumber \\\\+&\\left(\\pi ,\\frac{\\partial \\mathcal {N}^{\\mathrm {\\mathrm {c}}}}{\\partial \\widetilde{u}}\\delta \\widetilde{u}\\right)+ \\left(\\chi ,\\frac{\\partial \\mathcal {C}}{\\partial \\widetilde{u}}\\delta \\widetilde{u}\\right) = 0,$ where the terms that are trivially zero are left out.",
"Partial integration of the terms $(\\xi ,[\\partial \\mathcal {N}^{\\mathrm {m}}/\\partial \\widetilde{u}]\\delta \\widetilde{u})$ and $(\\pi ,[\\partial \\mathcal {N}^{\\mathrm {c}}/\\partial {\\widetilde{u}}]\\delta \\widetilde{u})$ , and $(\\xi ,[\\partial \\mathcal {N}^{\\mathrm {\\mathrm {m}}}/\\partial p]\\delta p)$ respectively lead to the unforced adjoint momentum equation and adjoint continuity equation.",
"This is a standard derivation: for the DNS-equations it can be found in [6], the additional LES terms are derived in [19].",
"The remaining nonzero terms will be discussed in the subsequent sections."
],
[
"Adjoint contribution of the observations",
"Linearisation of the cost function to $\\widetilde{u}$ gives $\\left(\\frac{\\partial \\mathcal {J}}{\\partial \\widetilde{u}},\\delta \\widetilde{u}\\right)&=\\int _{t_0}^{t_f}\\int _{\\Omega }\\frac{1}{\\gamma ^2T_{\\mathrm {s}}}\\sum _{i=1}^{N_{\\mathrm {r}}}\\sum _{n=1}^{N_{\\mathrm {s}}}(y_{n,i}-h_{n,i}) \\nonumber \\\\& \\qquad \\qquad \\times \\mathcal {G}_{\\mathrm {l}}\\left(\\mathsf {Q}(t)(x-x_{i}(t))\\right)e_{\\mathrm {l}}(t) \\, \\mathrm {H}\\!\\left(\\frac{T_{\\mathrm {s}}}{2}-\\left|t - t_{n-\\frac{1}{2}}\\right|\\right)\\cdot \\delta \\widetilde{u}\\, \\mathrm {d}x\\mathrm {d}t\\nonumber \\\\&=\\left(\\sum _{i=1}^{N_{\\mathrm {r}}}f_{i},\\delta \\widetilde{u}\\right),$ which leads to the forcing term $f_i$ for each lidar measurement in the adjoint momentum equations ()."
],
[
"Adjoint contribution of POD constraints",
"Linearisation of $(\\chi ,\\mathcal {C})$ to $\\widetilde{u}$ gives $\\left(\\chi ,\\frac{\\partial \\mathcal {C}}{\\partial \\widetilde{u}}\\delta \\widetilde{u}\\right) = \\int _{\\Omega } \\chi \\cdot \\delta \\widetilde{u}(x,t_0)\\ \\mathrm {d}x.$ The only other contribution at $t_0$ comes from the term $[\\partial \\delta \\widetilde{u}/\\partial t,\\xi ]$ from the linearised momentum equation, and the combination of both needs to reduce to zero.",
"This can be further elaborated by partial integration $ \\int _{t_0}^{t_f}\\int _{\\Omega } \\xi \\cdot \\frac{\\partial \\delta \\widetilde{u}}{\\partial t}\\ \\mathrm {d}{x}\\mathrm {d}t = \\left.\\int _{\\Omega } \\xi \\cdot \\delta \\widetilde{u}\\ \\mathrm {d}{x}\\right|_{t_0}^{t_f}-\\int _{t_0}^{t_f}\\int _{\\Omega } \\frac{\\partial \\xi }{\\partial t}\\cdot \\delta \\widetilde{u}\\ \\mathrm {d}{x}\\mathrm {d}t.$ The second term contributes to the adjoint momentum equations.",
"The first term evaluated at $t_f$ can be eliminated if $\\xi (x,t_f)=0$ is used as starting condition for the adjoint equations.",
"The first term evaluated at $t_0$ combined with (REF ) can be eliminated if $\\chi (x)=\\xi (x,t_0)$ ."
],
[
"Derivation of the adjoint gradient",
"The linearisation of the reduced cost function $J$ to the control variables $a$ , is given by (provided ${L}_{q}(\\delta q) = 0$ and $\\mathcal {B}(a,q)=0$ ) $J_{a}(\\delta a)={L}_{a}(\\delta a) = a^\\top \\delta a - \\int _{\\Omega } \\chi \\cdot \\Psi \\Lambda ^{1/2}\\delta a\\ \\mathrm {d}x.$ The partial derivative to the $i$ -th mode is readily identified as $\\partial J/\\partial a_i = a_i - \\lambda _i^{1/2}\\int _{\\Omega }\\xi (x,t_0)\\cdot \\psi _i \\ \\mathrm {d}x$ (for which $\\chi (x)=\\xi (x,t_0)$ was also used)."
],
[
"Adjoint gradient validation",
"In this appendix we compare the adjoint gradient, with finite differences.",
"The finite difference approximation of the Gâteaux derivative is given by $\\left(\\nabla J,\\delta a\\right) \\approx \\frac{J(a+\\alpha \\delta a)-J(a)}{\\alpha }, $ where $\\alpha $ is a factor which controls the step size.",
"$\\alpha =10^{-6}$ is found as a tradeoff between nonlinear effects $\\sim \\alpha ^2$ and the finite precision arithmetic by which the calculations are performed, starting to dominate at very small $\\alpha $ .",
"The gradient is validated for both a laminar and turbulent initial velocity field $\\widetilde{u}$ of the reconstruction model.",
"Since validating for all the components of the gradient $\\nabla J$ would be to time consuming, we use the steepest descent direction $\\delta a = \\nabla J/ \\Vert \\nabla J\\Vert $ obtained from the adjoint approach for the evaluation of the finite-difference method (REF ), and compare results only for this direction.",
"The results are shown in table REF for the PPI scanning mode, and are similar for the Lissajous scanning mode.",
"The relative precision for the gradient is around $10^{-3}$ for the LES and $10^{-8}$ for the TFT model.",
"The difference is explained by the differences in spatial discretisation errors, which are significantly larger for the nonlinear terms of the LES, compared to the TFT.",
"Single components of the gradient have also been used, with similar results but are not further reported here.",
"Table: Comparison of the adjoint and finite difference gradient, for LES and TFT models, using different initial states.",
"`L' and `T' are respectively abbreviations for laminar and turbulent initial flow conditions."
]
] | 2001.03469 |
[
[
"On the computational content of Zorn's lemma"
],
[
"Abstract We give a computational interpretation to an abstract instance of Zorn's lemma formulated as a wellfoundedness principle in the language of arithmetic in all finite types.",
"This is achieved through G\\\"odel's functional interpretation, and requires the introduction of a novel form of recursion over non-wellfounded partial orders whose existence in the model of total continuous functionals is proven using domain theoretic techniques.",
"We show that a realizer for the functional interpretation of open induction over the lexicographic ordering on sequences follows as a simple application of our main results."
],
[
"*"
],
[
"*"
],
[
"We give a computational interpretation to an abstract instance of Zorn's lemma formulated as a wellfoundedness principle in the language of arithmetic in all finite types.",
"This is achieved through Gödel's functional interpretation, and requires the introduction of a novel form of recursion over non-wellfounded partial orders whose existence in the model of total continuous functionals is proven using domain theoretic techniques.",
"We show that a realizer for the functional interpretation of open induction over the lexicographic ordering on sequences follows as a simple application of our main results.",
"Keywords.",
"Zorn's lemma, Gödel's functional interpretation, domain theory, continuous functionals, higher-order computability"
],
[
"Introduction",
"The correspondence between proofs and programs is one of the most fundamental ideas in computer science.",
"Initially connecting intuitionistic logic with the typed lambda calculus, it has since been extended to incorporate a wide range of theories and programming languages.",
"A challenging problem in this area is to give a computational interpretation to the axiom of choice in the setting of classical logic.",
"A number of ingenious solutions have been proposed, ranging from Spector's fundamental consistency proof of classical analysis using bar recursion [25] to more modern approaches, which include the Berardi-Bezem-Coquand functional [2], optimal strategies in sequential games [9], and Krivine's `quote' and `clock' [16].",
"In this paper, we introduce both a new form of recursion and a new computational interpretation of a choice axiom.",
"In contrast to the aforementioned works, which all focus on variants of countable choice, we give a direct computational interpretation to an axiomatic formulation of Zorn's lemma.",
"Our work is closest in spirit to Berger's realizability interpretation of open induction on the lexicographic ordering via open recursion [4] - an idea which was later transferred to the setting of Gödel's functional interpretation in [22].",
"However, a crucial difference here is that we do not work with a concrete order, but a general parametrised variant of Zorn's lemma, from which induction on the lexicographic ordering can be considered a special case.",
"After formulating an axiomatic version of Zorn's lemma in the language of Peano arithmetic in all finite types, we study related forms of recursion on chain bounded partial orders.",
"In particular, we introduce a new recursive scheme based on the notion of a `truncation', and give precise domain theoretic conditions under which the resulting fixpoint in the partial continuous functionals is total (Theorem REF ).",
"We then demonstrate that we can use our new form of recursion to solve the functional interpretation of our variant of Zorn's lemma.",
"Our approach completely separates the issues of correctness (that our program does what it's supposed to do) with that of totality (that our program is well-defined).",
"The main correctness result (Theorem REF ) is extremely general, and its proof short and direct, suggesting that our realizing terms are natural in a fundamental way.",
"To establish totality we make use of our earlier domain theoretic results, and again provide conditions which ensure that our computational interpretation is satisfied in the continuous functionals.",
"We conclude with a concrete example which ties everything together, demonstrating that the functional interpretation of open induction over the lexicographic ordering can be given as a special case of our general result.",
"This work aims to achieve several things.",
"Our new recursive schemes on chain bounded partial orders form a contribution to higher-order computability theory, which we believe is of interest in its own right.",
"The subsequent computational interpretation of Zorn's lemma is a new result in proof theory, which we hope will lead to novel applications in future work.",
"Finally, through our general and abstract setting we provide some fresh insights into known computational interpretations of variants of the axiom of choice, particularly open recursion [4] and Spector's original bar recursion [25]."
],
[
"Preliminaries",
"We begin by presenting some essential background material.",
"Gödel's functional interpretation, which only appears from Section onwards, will be introduced later."
],
[
"Zorn's lemma",
"Zorn's lemma is central to this article, and features not only as a proof technique but also in the guise of an axiomatic principle.",
"In what follows, $<$ will always denote a strict partial order, and $\\le $ its reflexive closure.",
"Definition 3.1 We call a partially ordered set $(S,<)$ chain bounded if every nonempty chain $\\gamma \\subseteq S$ (i.e.",
"nonempty totally ordered subset of $S$ ) has an upper bound in $S$ , that is an element $u\\in S$ such that $x\\le u$ for all $x\\in \\gamma $ .",
"Theorem 3.2 (Zorn's lemma) Let $(S,<)$ be a nonempty partially ordered set which is chain bounded.",
"Then $S$ contains at least one maximal element, that is an element $x\\in S$ such that $\\lnot (x<y)$ for all $y\\in S$ .",
"The following well-known application of Zorn's lemma will form a running illustration throughout the paper: Example 3.3 Let $R$ be some nontrivial ring with unity, and define $(S,\\subset )$ to be the set of all proper ideals of $R$ partially ordered by the strict subset relation.",
"Then $S$ is nonempty since $\\lbrace 0\\rbrace \\in S$ , and is also chain bounded since for any nonempty chain $\\gamma $ , the set $\\bigcup _{x\\in \\gamma } x$ is also a proper ideal of $R$ and thus an element of $S$ .",
"Therefore by Zorn's lemma, $S$ has a maximal element, or in other words, $R$ has a maximal ideal.",
"Our ability to apply Zorn's lemma to establish the existence of maximal ideals relies crucially on the fact that the upper bound $\\bigcup _{x\\in \\gamma } x$ is also a proper ideal.",
"This in turn is due to the fact that $x$ being a proper ideal is a `piecewise' property, in that it can be reduced to an infinite conjunction ranging over finite pieces of information about $x$ .",
"We now make this intuition precise, leading to a modification of Zorn's lemma (Theorem REF ) close in spirit to open induction as studied by Raoult [24].",
"This will form the basis of our syntactic version of Zorn's lemma presented in Section .",
"Definition 3.4 An approximation function on the set $X$ relative to some sets $D$ and $U$ is taken to be a mapping $[{\\cdot }]_{(\\cdot )}:X\\times D\\rightarrow U$ , where the sets $D$ and $U$ will play the following intuitive roles: $D$ is an index set of `sizes', $U$ is a set of `approximations' of elements of $X$ .",
"We call $[{x}]_{d}\\in U$ the approximation of $x$ of size $d$ .",
"Definition 3.5 We say that $(X,<)$ is chain bounded with respect to the approximation function $[{\\cdot }]_{}:X\\times D\\rightarrow U$ if any nonempty chain $\\gamma \\subseteq X$ has an upper bound $\\tilde{\\gamma }\\in X$ satisfying the additional property that for all $d\\in D$ there is some $x\\in \\gamma $ such that $[{\\tilde{\\gamma }}]_{d}=[{x}]_{d}$ .",
"Example 3.6 Let $(2^R,\\subset )$ be the powerset of some set $R$ , and $D$ the set of all finite subsets of $R$ .",
"Let $U:=\\lbrace f:d\\rightarrow \\lbrace 0,1\\rbrace \\; | \\; d\\in D\\rbrace $ and define $[{x}]_{d}:d\\rightarrow \\lbrace 0,1\\rbrace $ by $[{x}]_{d}(a)=1\\Leftrightarrow a\\in x.$ Then $(2^R,\\subset )$ is chain bounded with respect to $[{\\cdot }]_{}$ .",
"To see this, given a chain $\\gamma $ let $\\tilde{\\gamma }:=\\bigcup _{x\\in \\gamma } x$ and suppose that $a\\in \\tilde{\\gamma }$ .",
"Then there must be some $x_a\\in \\gamma $ such that $a\\in x_a$ .",
"For $d\\in D$ define $x:=\\max _{\\subset }\\lbrace x_a\\; | \\; a\\in d\\cap \\tilde{\\gamma }\\rbrace \\in \\gamma $ , and note that $x$ is well defined since $\\gamma $ is totally ordered.",
"Now, if $[{\\tilde{\\gamma }}]_{d}(a)=1$ then $a\\in d\\cap \\tilde{\\gamma }$ and thus $a\\in x$ , and so $[{x}]_{d}(a)=1$ .",
"On the other hand, if $[{\\tilde{\\gamma }}]_{d}=0$ then $a\\notin \\tilde{\\gamma }$ and so $a\\notin x$ (since $a\\in x$ trivially implies $a\\in \\tilde{\\gamma }$ , hence $[{x}]_{d}(a)=0$ .",
"Therefore $[{\\tilde{\\gamma }}]_{d}=[{x}]_{d}$ .",
"Definition 3.7 We call a predicate $P(x)$ on $X$ piecewise with respect to the approximation function $[{\\cdot }]_{}:X\\times D\\rightarrow U$ if $P(x)\\Leftrightarrow (\\forall d\\in D) Q([{x}]_{d})$ for some predicate $Q(u)$ on $U$ .",
"Theorem 3.8 Let $(X,<)$ be a partially ordered set which is chain bounded w.r.t.",
"the approximation function $[{\\cdot }]_{}:X\\times D\\rightarrow U$ , and $P(x)$ a predicate on $X$ which is piecewise w.r.t the same function.",
"Then whenever $P(x)$ holds for some $x\\in X$ , there exists $y\\in X$ such that $P(y)$ holds but $\\lnot P(z)$ whenever $y<z$ .",
"Let $S:=\\lbrace x\\in X\\; | \\; P(x)\\rbrace $ , and take some nonempty chain $\\gamma \\subseteq S$ .",
"Our first step is to show that $\\tilde{\\gamma }\\in S$ , from which it follows that $(S,<)$ is chain bounded.",
"Since $P(x)\\Leftrightarrow (\\forall d\\in D) Q([{x}]_{d})$ for some predicate $Q(u)$ , it suffices to show that $Q([{\\tilde{\\gamma }}]_{d})$ for all $d\\in D$ .",
"But using that for any $d$ there exists some $x\\in \\gamma \\subseteq S$ with $[{\\tilde{\\gamma }}]_{d}=[{x}]_{d}$ we're done, since $Q([{x}]_{d})$ follows from $P(x)$ .",
"Now, suppose that $P(x)$ holds for some $x\\in X$ , and thus $S$ is nonempty.",
"By Zorn's lemma, $S$ contains a maximal element $y$ .",
"We clearly have $P(y)$ , and if $y<z$ then $z\\notin S$ and thus $\\lnot P(z)$ .",
"Example 3.9 Let $(2^R,\\subset )$ be the powerset of some nontrivial ring $R$ , with $[{x}]_{d}$ defined as in Example REF , and let $P(x)$ be denote the predicate `$x$ is a proper ideal of $R$ '.",
"Then this is a piecewise predicate w.r.t.",
"$[{\\cdot }]_{}$ , since each condition of being a proper ideal can be formulated in a piecewise way.",
"For instance, $0\\in x$ is equivalent to $\\forall d\\; (0\\in d\\Rightarrow [{x}]_{d}(0)=1)$ and analogously for $1\\notin x$ .",
"Similarly, closure of $x$ under addition can be formulated in a piecewise way as $\\begin{aligned}\\forall d,r,r^{\\prime }\\; (\\lbrace r,r^{\\prime },r+r^{\\prime }\\rbrace \\subseteq d& \\wedge [{x}]_{d}(r)=[{x}]_{d}(r^{\\prime })=1\\\\&\\Rightarrow [{x}]_{d}(r+r^{\\prime })=1)\\end{aligned}$ and analogously for closure under left and right sided multiplication.",
"Therefore the existence of a maximal ideal also follows from Theorem REF above.",
"Note that since in $r,r^{\\prime }$ above are always elements of the finite set $d$ , $\\forall r,r^{\\prime }$ can be treated as a bounded quantifier, and so `$x$ is a proper ideal of $R$ ' is piecewise even with respect to some quantifier-free $Q(u)$ ."
],
[
"Formal theories of arithmetic",
"In the remainder of this article, our definitions and results typically take place in one of the following settings: Within a formal theory of arithmetic in higher-types (syntactic); Within a type structure of continuous functionals, either the total or partial (semantic).",
"We now outline both of these settings in turn.",
"Our basic formal system will be the standard theories of Peano (resp.",
"Heyting) arithmetic in all finite types $\\mathrm {PA}^\\omega $ ($\\mathrm {HA}^\\omega $ ).",
"For us, the finite types $\\bf {T}$ will be generated by the following grammar: $\\rho ,\\tau ::=\\mathsf {B}\\; | \\; \\mathsf {N}\\; | \\; \\rho \\times \\tau \\; | \\; \\rho ^\\ast \\; | \\; \\rho \\rightarrow \\tau $ These represent base types for booleans $\\mathsf {B}$ and natural numbers $\\mathsf {N}$ , and in addition to the usual function type $\\rho \\rightarrow \\tau $ include cartesian products $\\rho \\times \\tau $ and finite sequence types $\\rho ^\\ast $ as primitives.",
"Note that alternatively, we could work over a minimal type structure $\\mathsf {N}\\; | \\; \\rho \\rightarrow \\tau $ and code up products and finite sequences as derived constructions.",
"For full definitions of $\\mathrm {PA}^\\omega $ resp.",
"$\\mathrm {HA}^\\omega $ the reader is directed to e.g.",
"[1], [14], [26], bearing in mind that officially we would need to extend the canonical theories presented there with additional constants and axioms for dealing with cartesian products and list operations, which is nevertheless entirely standard (for details see e.g.",
"[26] and [27]).",
"Terms of $\\mathrm {PA}^\\omega $ resp.",
"$\\mathrm {HA}^\\omega $ are those of Gödel's System T (with product and sequence types).",
"We denote by $0_\\rho :\\rho $ a canonical zero object of type $\\rho $ .",
"Formulas of $\\mathrm {PA}^\\omega $ (resp.",
"$\\mathrm {HA}^\\omega $ ) include atomic formulas $=_\\mathsf {B}$ and $=_\\mathsf {N}$ for equality at base types, and are built using the usual logical connectives, together with quantifiers for each type.",
"Axioms and rules include those of full classical (resp.",
"intuitionistic) logic, non-logical axioms for the constants symbols together with equality axioms and the axiom of induction.",
"Equality at higher types is defined inductively e.g.",
"$f=_{\\rho \\rightarrow \\tau } g:=\\forall x^\\rho (fx=gx)$ , and we include axioms for extensionality, so that our formulation of $\\mathrm {PA}^\\omega $ corresponds to the fully extensional $\\mathrm {E\\mbox{-}PA}^\\omega $ of [14].",
"The canonical models for $\\mathrm {PA}^\\omega $ include the type structures of all set-theoretic functionals $\\mathcal {S}^\\omega $ together with total continuous functional $\\mathcal {C}^\\omega $ .",
"However, the majority of recursive schemes which have been used to interpret the axiom of choice (including essentially all known variants of bar recursion) are no longer satisfiable in $\\mathcal {S}^\\omega $ , and instead have $\\mathcal {C}^\\omega $ as their canonical model.",
"In the remainder of this section, we outline some key facts about this model."
],
[
"The continuous functionals in all finite types",
"In one sentence, the type structure $\\mathcal {C}^\\omega $ of continuous functionals consists of functionals which only require a finite piece of information about their input to compute a finite piece of information about their output.",
"Over the years, they have turned out to form an elegant and robust class of functionals, and in particular are the standard model for bar recursive extensions of the primitive recursive functionals.",
"There are various ways of characterising the continuous functionals, dating back to Kleene [13] (whose construction was based on associates) and Kreisel [15] (who instead used formal neighbourhoods).",
"However, here we follow the domain theoretic approach of Ershov [8], who demonstrated that the continuous functionals can be constructed as the extensional collapse of the total objects in the type structure ${\\mathcal {P}}^\\omega $ of partial continuous functionals.",
"This in particular provides us with a simple method for showing that our new recursive schemes are satisfied in $\\mathcal {C}^\\omega $ , namely proving that the corresponding fixpoints in ${\\mathcal {P}}^\\omega $ represent total objects.",
"For accomprehensive account of all this, the reader is encouraged to consult [18] or the recent book [17].",
"Here we provide no more than a brief overview of the relevant theory.",
"For each finite type $\\sigma $ , we define the domain $P_\\sigma $ of partial continuous functionals of that type as follows: $P_\\mathsf {B}:=\\mathbb {B}_\\bot $ and $P_\\mathsf {N}:=\\mathbb {N}_\\bot $ where $\\mathbb {B}_\\bot $ resp.",
"$\\mathbb {N}_\\bot $ are the usual flat domains of booleans and natural numbers, $P_{\\rho \\times \\tau }:=P_\\rho \\times P_\\tau $ , $P_{\\sigma ^\\ast }:=\\lbrace [ {x_0,\\ldots ,x_{n-1}} ]\\; | \\; n\\in \\mathbb {N}\\mbox{ and }x_i\\in P_\\sigma \\rbrace \\cup \\lbrace \\bot \\rbrace $ and finally $P_{\\rho \\rightarrow \\tau }:=[P_\\rho \\rightarrow P_\\tau ]$ where $[D\\rightarrow E]$ denotes the domain of all functions between $X$ and $Y$ which are continuous in the domain theoretic sense (i.e.",
"are monotone and preserve lubs of chains).",
"We write ${\\mathcal {P}}^\\omega :=\\lbrace P_\\sigma \\rbrace _{\\sigma \\in \\bf {T}}$ for this type structure of partial continuous functionals.",
"For each type $\\sigma $ , we define the set $T_\\sigma \\subset P_\\sigma $ of total objects in the usual way as $T_\\mathsf {B}:=\\mathbb {B}$ and $T_\\mathsf {N}:=\\mathbb {N}$ , $T_{\\rho \\times \\tau }:=T_\\rho \\times T_\\tau $ , $T_{\\sigma ^\\ast }:=\\lbrace [ {x_0,\\ldots ,x_{n-1}} ]\\; | \\; n\\in \\mathbb {N}\\mbox{ and }x_i\\in T_\\sigma \\rbrace $ and finally $T_{\\rho \\rightarrow \\tau }:=\\lbrace f\\in P_{\\rho \\rightarrow \\tau }\\; : \\; \\forall x(x\\in T_\\sigma \\Rightarrow fx\\in T_\\tau )\\rbrace $ .",
"Furthermore, we define an equivalence relation $\\approx _\\sigma $ on $T_\\sigma $ to equate total objects that agree on total inputs: $x\\approx _\\mathsf {B}y$ iff $x=y$ and similarly for $\\approx _\\mathsf {N}$ , $(x,x^{\\prime })\\approx _{\\rho \\times \\tau } (y,y^{\\prime })$ iff $x\\approx _\\rho y$ and $y\\approx _\\tau y^{\\prime }$ , $[ {x_0,\\ldots ,x_{n-1}} ]\\approx _{\\sigma ^\\ast } [ {y_0,\\ldots ,y_{m-1}} ]$ iff $n=m$ and $x_i\\approx _{\\sigma } y_i$ for all $i<n$ and finally $f\\approx _{\\rho \\rightarrow \\tau } g$ iff $fx\\approx _\\tau gx$ for all $x\\in T_\\rho $ .",
"It turns out that all total objects are hereditarily extensional, in the sense that if $f\\in T_{\\rho \\rightarrow \\tau }$ and $x\\approx _\\rho y$ then $fx\\approx _\\tau fy$ , and therefore the extensional collapse $C_\\sigma :=T_\\sigma /\\approx _\\sigma $ of the total objects constitutes a hierarchy $\\mathcal {C}^\\omega :=\\lbrace C_\\sigma \\rbrace _{\\sigma \\in \\bf {T}}$ of functionals in its own right.",
"We call this hierarchy the total continuous functionals, and as shown by Ershov, $\\mathcal {C}^\\omega $ is in fact isomorphic to the constructions of Kleene and Kreisel.",
"It is well known that $\\mathcal {C}^\\omega $ is a model of $\\mathrm {PA}^\\omega $ , and so in particular, any closed term $e:\\sigma $ of System T has a canonical interpretation $e_C\\in C_\\sigma $ , which can in turn be represented by some element $e_P\\in T_\\sigma $ of the corresponding equivalence class in ${\\mathcal {P}}^\\omega $ .",
"Suppose now that we extend System T with some new constant symbol $\\Phi :\\sigma $ which satisfies a recursive defining axiom $(\\ast ) \\ \\ \\ \\Phi (x_1,\\ldots ,x_n)= r(\\Phi ,x_1,\\ldots ,x_n)$ where $r$ is a closed term of System T. We can equivalently express $(\\ast )$ as $\\Phi =e(\\Phi )$ for $e:\\sigma \\rightarrow \\sigma $ defined by $e(f):=\\lambda x_1,\\ldots ,x_n.r(f,x_1,\\ldots ,x_n).$ Now, since $e$ is primitive recursive, it has a total representation $e_P\\in T_{\\sigma \\rightarrow \\sigma }\\subset [P_\\sigma \\rightarrow P_\\sigma ]$ , and it is a basic fact of domain theory that $\\Phi $ can be given an interpretation $\\Phi _P$ in ${\\mathcal {P}}^\\omega $ as a least fixed point of $e_P$ i.e.",
"$\\Phi _P:=\\bigsqcup _{n\\in \\mathbb {N}} e_P^n(\\bot _\\sigma )$ satisfies $\\Phi _P=e_P(\\Phi _P)$ .",
"If we can now show that $\\Phi _P$ is in fact total, in other words that $\\Phi _P(x_1,\\ldots ,x_n)$ is total for all total inputs $x_1,\\ldots ,x_n$ , then defining $\\Phi _C:=[\\Phi _P]_{\\approx _{\\sigma }}\\in C_\\sigma $ we have $\\Phi _C=[\\Phi _P]_{\\approx _\\sigma }=[e_P(\\Phi _P)]_{\\approx _\\sigma }=[e_P]_{\\approx _{\\sigma \\rightarrow \\sigma }}[\\Phi _P]_{\\approx _\\sigma }=e_C(\\Phi _C)$ and therefore the object $\\Phi _C$ satisfies the defining axiom $(\\ast )$ in $\\mathcal {C}^\\omega $ .",
"In other words, $\\mathcal {C}^\\omega $ is a model of the theory $\\mathrm {PA}^\\omega +\\Phi $ , where by the latter we mean the extension of $\\mathrm {PA}^\\omega $ with the new constant $\\Phi $ and axiom $(\\ast )$ .",
"In short, in order to show that the extension of System T with some new form of recursion $\\Phi $ is satisfied in $\\mathcal {C}^\\omega $ , it suffices to show that the natural interpretation of $\\Phi $ as a fixpoint in ${\\mathcal {P}}^\\omega $ is total.",
"This approach has been widely used in the past to show that various forms of strong recursion arising from the axiom of choice have $\\mathcal {C}^\\omega $ as a model (see e.g.",
"[4] or [5]), and will be fundamental for us as well in Section .",
"In addition to showing that extensions of System T have a model, we must also confirm that they represent programs, in the sense that any object of type $\\mathsf {N}$ can be effectively reduced to a numeral.",
"This follows by appealing to Plotkin's adequacy theorem [20]: We observe that terms of System T plus our new recursor $\\Phi $ can be viewed as terms in PCF (recursion being dealt with by using the fixpoint combinator), which in addition inherit the usual call-by-value reduction semantics, with the defining axiom $(\\ast )$ being interpreted as a rewrite rule.",
"By showing that $\\Phi $ represents a total object in the semantics of PCF within ${\\mathcal {P}}^\\omega $ , it follows that any closed term $e:\\mathsf {N}$ in our extended calculus is denoted by some natural number i.e.",
"$[e]\\in \\mathbb {N}$ , and by the adequacy theorem $e$ must then reduce to the numeral $\\underline{n}$ .",
"Remark.",
"In order to avoid burdening ourselves with too many subscripts, in the remainder of this paper we use the same notation for $e:\\sigma $ in $\\mathrm {PA}^\\omega $ , its canonical interpretation $e\\in C_\\sigma $ and some suitable representation $e\\in P_\\sigma $ , rather than laboriously writing $e_C$ resp.",
"$e_P$ whenever we are working in continuous models.",
"Where there is any ambiguity, we make absolutely clear which system we are working in, and in the case of $e_P$ for primitive recursive $e$ we write explicitly how $e$ can be represented as a partial object unless this is obvious."
],
[
"A syntactic formulation of Zorn's lemma",
"In this short section, we present a general axiomatic formulation of Zorn's lemma.",
"This will be based on Theorem REF , and is close in spirit to the axiom of open induction as studied in [4].",
"Like open induction, our axiom is of course weaker than the full statement of Zorn's lemma.",
"Nevertheless, as we will see in Section , it in fact generalises open induction, and so in particular can be used to formalize highly non-trivial proofs such as Nash-Williams' minimal bad-sequence construction (cf.",
"[4], [22]).",
"To be more specific, our axiom schema will take the shape of a maximum principle of the form $\\exists x P(x)\\rightarrow \\exists y (P(y)\\wedge \\forall z>y\\lnot P(z))$ where $P(x)$ will range over formulas which are piecewise in the sense of Definition REF and $<$ denotes some chain bounded partial order.",
"However, our precise formulation of the axiom will be within the language of $\\mathrm {PA}^\\omega $ , and therefore both the notion of a piecewise formula and the relation $<$ need to be represented in a suitable way.",
"Remark.",
"From now all we annotate important definitions and results with the theory or model in which they take place, which will usually be some extension of $\\mathrm {PA}^\\omega $ resp.",
"$\\mathrm {HA}^\\omega $ or one of $\\mathcal {C}^\\omega $ or ${\\mathcal {P}}^\\omega $ .",
"Definition 4.1 ($\\mathrm {PA}^\\omega $ /$\\mathrm {HA}^\\omega $ ) Suppose that $[{\\cdot }]_{(\\cdot )}:\\sigma \\times \\delta \\rightarrow \\nu $ is a closed term of System T, and $Q(u^\\nu )$ is a formula in the language of $\\mathrm {PA}^\\omega $ /$\\mathrm {HA}^\\omega $ .",
"Then we say that the formula $P(x^\\sigma ):\\equiv \\forall d^\\delta \\; Q([{x}]_{d})$ is piecewise w.r.t.",
"$[{\\cdot }]_{}$ .",
"Now, while it is too restrictive to demand that $<$ be represented by some primitive recursive functional $\\sigma \\times \\sigma \\rightarrow \\mathsf {B}$ , for all applications we are interested in it suffices that $<$ can be expressed as a $\\Sigma _1$ formula as follows: $x<y:\\equiv \\exists a^\\rho (y=_\\sigma x\\oplus a\\wedge x\\prec a)$ where now $\\oplus :\\sigma \\times \\rho \\rightarrow \\sigma $ and $\\prec :\\sigma \\times \\rho \\rightarrow \\mathsf {B}$ are closed terms of System T for some type $\\rho $ (we use $x\\oplus a$ to denote $\\oplus (x,a)$ and $x\\prec a$ to denote $\\prec (x,a)=1$ , and similarly $a\\succ x$ to denote $x\\prec a$ ).",
"Definition 4.2 ($\\mathrm {PA}^\\omega $ /$\\mathrm {HA}^\\omega $ ) Let $[{\\cdot }]_{(\\cdot )}:\\sigma \\times \\delta \\rightarrow \\nu $ , $\\oplus :\\sigma \\times \\rho \\rightarrow \\sigma $ and $\\prec :\\sigma \\times \\rho \\rightarrow \\mathsf {B}$ be closed terms of System T. The axiom schema $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ is given by $\\begin{aligned}\\exists x^\\sigma \\forall d^\\delta Q([{x}]_{d})\\rightarrow &\\exists y^\\sigma (\\forall d\\; Q([{y}]_{d})\\\\&\\wedge \\forall a\\succ y\\;\\exists d\\; \\lnot Q([{y\\oplus a}]_{d}))\\end{aligned}$ where $Q(u^\\nu )$ ranges over arbitrary formulas of $\\mathrm {PA}^\\omega $ (and does not contain $x,y,a,d$ free).",
"Note that our axiomatic formulation no longer mentions a main ordering $<$ , which is instead induced by $\\oplus $ and $\\prec $ .",
"Note also that chain boundedness of $<$ is not formulated as a part of the axiom itself, and as such, validity of $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ in some given type structure will depend on the interpretation of $<$ being chain bounded in that model.",
"We could of course seek to incorporate chain boundedness into the syntactic definition of Zorn's lemma and give a computational interpretation to the axiom as a whole.",
"This would lead to a fascinating but extremely complex computational problem which would steer us in a quite different direction to the current article, and so we leave this to future work (cf.",
"Section ).",
"We now illustrate our new principle by continuing our example from Section REF , whose computational content has already been studied in in [23].",
"Example 4.3 Let $\\sigma :=\\mathsf {N}\\rightarrow \\mathsf {B}$ , $\\delta :=\\mathsf {N}$ , $\\nu :=\\mathsf {B}^\\ast $ and $\\rho :=\\mathsf {N}\\times (\\mathsf {N}\\rightarrow \\mathsf {B})$ and define $\\begin{aligned}[{x}]_{d}&:=[ {x(0),\\ldots ,x(d-1)} ]\\\\x\\oplus (n,y)&:=x\\cup y\\\\x\\prec (n,y)&:= \\overline{x(n)}\\cdot y(n)\\end{aligned}$ where $\\overline{b}$ represents the negation of the boolean $b$ and $(x\\cup y)(n):=\\mbox{$1$ if $x(n)=1$ or $y(n)=1$ else $0$}.$ These are all clearly definable as closed terms of System T, and in this case $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ is equivalent to $\\begin{aligned}\\exists x P(x)\\rightarrow &\\exists y(P(y)\\\\&\\wedge \\forall (n,z)(\\overline{y(n)}=z(n)=1\\rightarrow \\lnot P(y\\cup z)))\\end{aligned}$ for $P(x):\\equiv \\forall d\\; Q([ {x(0),\\ldots ,x(d-1)} ])$ .",
"Here we can imagine objects $x:\\mathsf {N}\\rightarrow \\mathsf {B}$ as characteristic functions for subsets of the natural numbers.",
"Moreover, given some countable ring $R$ whose elements can be coded up as natural numbers and whose operations $+_R$ and $\\cdot _R$ represented as primitive recursive functions $\\mathsf {N}\\times \\mathsf {N}\\rightarrow \\mathsf {N}$ , the existence of a maximal ideal in $R$ would be provable in $\\mathrm {PA}^\\omega +\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ .",
"We do not give full details of this (an outline of the formalisation can be found in [23]).",
"Instead we simply sketch why both $\\mathcal {S}^\\omega $ and $\\mathcal {C}^\\omega $ satisfy $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ and are thus models of $\\mathrm {PA}^\\omega +\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ .",
"Working in $\\mathcal {C}^\\omega $ (the same argument is also valid for $\\mathcal {S}^\\omega $ ) we apply Theorem REF for $X:=C_{\\mathsf {N}\\rightarrow \\mathsf {B}}\\cong \\mathbb {B}^\\mathbb {N}$ which via the identification of sets with their characteristic function is isomorphic to the powerset of $\\mathbb {N}$ , together with the proper subset relation, observing that $x\\subset y\\Leftrightarrow \\exists (n,z)\\in \\mathbb {N}\\times \\mathbb {B}^\\mathbb {N}(y=x\\cup z\\wedge n\\notin x\\wedge n\\in z)$ where the right hand side is just the interpretation of the formula $\\exists (n,z)(y=x\\oplus (n,z)\\wedge y\\prec (n,z))$ in $\\mathcal {C}^\\omega $ .",
"Clearly $(X,\\subset )$ is chain bounded w.r.t.",
"$[{x}]_{d}:=[ {x(0),\\ldots ,x(d-1)} ]$ using a simplified version of the argument in Example REF .",
"Therefore for any formula $Q([ {u(0),\\ldots ,u(k-1)} ])$ in $\\mathcal {C}^\\omega $ on finite sequences of natural numbers the resulting formula $P(x):\\equiv \\forall d\\; Q([ {x(0),\\ldots ,x(d-1)} ])$ on $X$ in $\\mathcal {C}^\\omega $ is piecewise w.r.t.",
"$[{\\cdot }]_{}$ , and thus by Theorem REF whenever $\\exists x P(x)$ is satisfied there exists some $y\\in X$ such that $P(y)$ , and also $\\lnot P(z)$ whenever $y<z$ (or alternatively $\\forall (n,z)(n\\notin x\\wedge n\\in z\\Rightarrow \\lnot P(y\\cup z)$ ).",
"Thus $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ is valid in $\\mathcal {C}^\\omega $ ."
],
[
"Recursion over chain bounded partial orders",
"We now come to our first main contribution, in which we study modes of recursion over chain bounded partial orders that form an analogue to the axiom $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ .",
"A precise connection between a restricted form of $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ and our second mode of recursion will be presented in Section , but the results of this section are more general, and we consider them to be of interest in their own right.",
"As such, this section could be read as a short, self-contained study in which we explore different recursion schemes over orderings induced by the parameters $(\\oplus ,\\prec )$ .",
"Totality of our recursors will be justified using a variant of Theorem REF , and the two main modes of recursion considered here will primarily differ in how we achieve `piecewise-ness' of the totality predicate.",
"The first, which we characterise as `simple' recursion, uses a sequential continuity principle but is valid only for discrete output types, whereas the second, which we call `controlled' recursion, is total for arbitrary output type but uses an auxiliary parameter in the recursor itself to ensure wellfoundedness.",
"For the remainder of this section, we fix types $\\sigma ,\\rho ,\\delta $ and $\\nu $ , together with closed terms $[{\\cdot }]_{}:\\sigma \\times \\delta \\rightarrow \\nu $ , $\\oplus :\\sigma \\times \\rho \\rightarrow \\sigma $ and $\\prec :\\sigma \\times \\rho \\rightarrow \\mathsf {B}$ of System T, which are analogous to those in Section .",
"For definitions and results below which take place in the model ${\\mathcal {P}}^\\omega $ , note that $[{\\cdot }]_{}\\in T_{\\sigma \\times \\delta \\rightarrow \\nu }$ denotes some canonical representation of the corresponding term of System T as a total continuous functional, and similarly for $\\oplus \\in T_{\\sigma \\times \\rho \\rightarrow \\sigma }$ and $\\prec \\in T_{\\sigma \\times \\rho \\rightarrow \\mathsf {B}}$ (cf.",
"Section REF )."
],
[
"Simple recursion over $(\\oplus ,\\prec )$",
"The first recursion scheme we consider is represented by the constant $\\Phi ^\\theta _{\\oplus ,\\prec }$ equipped with defining equation $\\Phi fx=_\\theta fx(\\lambda a \\; .",
"\\; \\Phi f(x\\oplus a)\\mbox{ if $a\\succ x$ else $0_\\theta $})$ where $f:\\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\theta $ and $x:\\sigma $ , and we recall that $0_\\theta $ is a canonical zero term of type $\\theta $ .",
"Note that in the defining equation we suppressed the parameters on $\\Phi $ - and we will continue to do this whenever there is no risk of ambiguity.",
"In what follows, it will be helpful to use the abbreviation $\\Phi _{f,x}:=\\lambda a \\; .",
"\\; \\Phi f(x\\oplus a)\\mbox{ if $a\\succ x$ else $0_\\theta $}$ so that the defining equation can then be expressed as $\\Phi fx=fx\\Phi _{f,x}.$ Definition 5.1 (${\\mathcal {P}}^\\omega $ ) Let $L\\subseteq T_\\sigma $ .",
"We say that a functional $\\psi \\in P_{\\sigma \\rightarrow \\theta }$ is piecewise continuous with respect to $[{\\cdot }]_{}$ and $L$ , if for any $x\\in L$ such that $\\psi x\\in T_\\theta $ there exists some $d\\in T_\\delta $ such that $\\forall y\\in T_\\sigma ([{x}]_{d}=[{y}]_{d}\\Rightarrow \\psi y\\in T_\\theta ).$ Definition 5.2 (${\\mathcal {P}}^\\omega $ ) A partial order $<$ on $T_\\sigma $ is compatible with $(\\oplus ,\\prec )$ if $x<x\\oplus a$ for any $(x,a)\\in T_{\\sigma \\times \\rho }$ with $x\\prec a$ (i.e.",
"$\\prec (x,a)=_\\mathbb {B}1$ ).",
"The next definition is a slight adaptation of Definition REF , where now we require $\\tilde{\\gamma }$ to be an element of some subset $L$ of the main partial order.",
"Definition 5.3 (${\\mathcal {P}}^\\omega $ ) A partial order $<$ on $T_\\sigma $ is chain bounded with respect to $[{\\cdot }]_{}$ and $L\\subseteq T_\\sigma $ if every nonempty chain $\\gamma \\subseteq T_\\sigma $ has an upper bound $\\tilde{\\gamma }\\in L$ satisfying the property that for any $d\\in T_\\delta $ there exists some $x \\in \\gamma $ such that $[{\\tilde{\\gamma }}]_{d}=[{x}]_{d}$ .",
"We now come to our first totality result.",
"This establishes a condition on inputs $f$ which ensures totality of $\\Phi f$ .",
"As we will see, in certain natural situations we can use this to show that $\\Phi f$ is total for any total $f$ , and thus $\\Phi $ itself is total.",
"However, our result is more general as it also allows us to establish totality of $\\Phi f$ in cases where $\\Phi $ may not be.",
"Theorem 5.4 (${\\mathcal {P}}^\\omega $ ) Let $\\Phi $ denote the least fixed point of the primitive recursive defining equation (REF ), and suppose that there exist $<$ on $T_\\sigma $ and $L\\subseteq T_\\sigma $ such that $<$ is compatible with $(\\oplus ,\\prec )$ and chain bounded w.r.t.",
"$[{\\cdot }]_{}$ and $L$ .",
"Let $f\\in T_{\\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\theta }$ .",
"Then whenever $\\Phi f\\in P_{\\sigma \\rightarrow \\theta }$ is piecewise continuous w.r.t.",
"$[{\\cdot }]_{}$ and $L$ , it follows that $\\Phi f\\in T_{\\sigma \\rightarrow \\theta }$ .",
"By Zorn's lemma.",
"Suppose that $\\Phi f$ is piecewise continuous, and consider the set $S\\subseteq T_\\sigma $ given by $S:=\\lbrace x\\in T_\\sigma \\; | \\; \\Phi fx\\notin T_\\theta \\rbrace .$ We first show that $(S,<)$ is chain bounded in the usual sense.",
"Taking some nonempty chain $\\gamma \\subseteq S$ , by chain boundedness in the sense of Definition REF this has some upper bound $\\tilde{\\gamma }\\in L$ .",
"Suppose for contradiction that $\\Phi f\\tilde{\\gamma }\\in T_\\theta $ .",
"By piecewise continuity of $\\Phi f$ there exists some $d\\in T_\\delta $ such that $[{\\tilde{\\gamma }}]_{d}=[{y}]_{d}$ implies $\\Phi fy\\in T_\\theta $ for any $y\\in T_\\sigma $ .",
"But then there exists some $x\\in \\gamma $ with $[{\\tilde{\\gamma }}]_{d}= [{x}]_{d}$ and thus $\\Phi fx\\in T_\\theta $ , contradicting $x\\in S$ .",
"Therefore $\\Phi f\\tilde{\\gamma }\\notin T_\\theta $ and thus $\\tilde{\\gamma }\\in S$ .",
"To prove the main result, suppose for contradiction that $\\Phi f\\notin T_{\\sigma \\rightarrow \\theta }$ , which implies that $S\\ne \\emptyset $ .",
"Then by Zorn's lemma, $S$ has some maximal element $x$ .",
"But for any $a\\in T_\\rho $ with $x\\prec a$ we have $x<x\\oplus a$ by compatibility, and thus $\\Phi _{f,x}(a)=\\Phi f(x\\oplus a)\\in T_\\theta $ .",
"It follows that $\\Phi _{f,x}\\in T_{\\rho \\rightarrow \\theta }$ , since in the other case $\\lnot (x\\prec a)$ we have $\\Phi _{f,x}(a)=0_\\theta \\in T_\\theta $ .",
"But then by totality of $f$ we have $\\Phi fx=fx\\Phi _{f,x}\\in T_\\theta $ , contradicting $x\\in S$ .",
"Therefore $S=\\emptyset $ and so $\\Phi f\\in T_{\\sigma \\rightarrow \\theta }$ .",
"The technique we have used in this proof is a generalisation of the proof of Theorem 0.3 from [3], which uses Zorn's lemma to show that the so-called Berardi-Bezem-Coquand functional defined in [2] is total.",
"We now give a concrete example of how the result can be applied, but first we state and prove a sequential continuity lemma (cf.",
"also [3]), which will also be useful in later sections.",
"Lemma 5.5 (${\\mathcal {P}}^\\omega $ ) Let $\\theta $ be a discrete type i.e.",
"one which does not contain function types.",
"Suppose that $\\psi \\in P_{(\\mathsf {N}\\rightarrow \\sigma )\\rightarrow \\theta }$ where $\\sigma $ is some arbitrary type, that $x\\in T_{\\mathsf {N}\\rightarrow \\sigma }$ satisfies $x(\\bot )=\\bot _\\sigma $ and that $\\psi x\\in T_\\theta $ .",
"Then there is some $d\\in \\mathbb {N}$ such that for any $y\\in P_{\\mathsf {N}\\rightarrow \\sigma }$ , whenever $x(i)=y(i)$ for all $i<d$ then $\\psi x=\\psi y$ .",
"We use a simple adaptation of the proof of Lemma 0.1 of [3].",
"Since $T_\\theta $ is open in the Scott topology whenever $\\theta $ is discrete, there is some compact $x_0\\sqsubseteq x$ such that $\\psi x_0\\in T_\\theta $ , and since $\\psi x_0\\sqsubseteq \\psi x$ we must in fact have $\\psi x_0=\\psi x$ (that $y\\sqsubseteq z$ implies $y=z$ for $y\\in T_\\theta $ is evidently true for $\\theta =\\mathbb {N}_\\bot $ or $\\theta =\\mathbb {B}_\\bot $ , and holds for arbitrary discrete $\\theta $ by induction over its structure).",
"Now, since $x_0$ is compact (i.e.",
"contains only a finite amount of information) there is some $d\\in \\mathbb {N}$ such that $x_0(i)=\\bot _\\sigma $ for all $i\\ge d$ .",
"Suppose now that $y\\in P_{\\mathsf {N}\\rightarrow \\sigma }$ satisfies $x(i)=y(i)$ for all $i<d$ .",
"We claim that $x_0\\sqsubseteq y$ .",
"To see this, note that for $i<d$ we have $x_0(i)\\sqsubseteq x(i)=y(i)$ , for $i\\ge d$ we have $x_0(i)=\\bot _\\sigma \\sqsubseteq y(i)$ , and for $i=\\bot $ since $x(\\bot )=\\bot _\\sigma $ we must also have $x_0(\\bot )=\\bot _\\sigma \\sqsubseteq y(\\bot )$ .",
"Therefore $\\psi x_0\\sqsubseteq \\psi y$ and since $\\psi x_0\\in T_\\theta $ we must have $\\psi y=\\psi x_0=\\psi x$ .",
"Example 5.6 Let $[{\\cdot }]_{}$ , $\\oplus $ and $\\prec $ be the obvious total representatives of the primitive recursive functions defined in Example REF i.e.",
"extensions that are defined also on non-total input, for example $\\begin{aligned}[{x}]_{d}&:=[ {x(0),\\ldots ,x(d-1)} ]\\mbox{ for $d\\in \\mathbb {N}$ else $\\bot $}\\end{aligned}$ and so on.",
"We observe that $T_\\sigma =T_{\\mathsf {N}\\rightarrow \\mathsf {B}}$ is the set of all functions $x:\\mathbb {N}_\\bot \\rightarrow \\mathbb {B}_\\bot $ which are monotone (in the domain theoretic sense) and satisfy $x(n)\\in \\mathbb {B}$ whenever $n\\in \\mathbb {N}$ .",
"We define $L\\subset T_{\\mathsf {N}\\rightarrow \\mathsf {B}}$ to consist of those functions which are strict, in that they satisfy in addition $x(\\bot )=\\bot $ .",
"Now suppose that $\\theta $ is discrete.",
"Then any function $\\psi \\in P_{(\\mathsf {N}\\rightarrow \\mathsf {B})\\rightarrow \\theta }$ is piecewise continuous w.r.t.",
"$[{\\cdot }]_{}$ and $L$ .",
"To see this, take any strict $x$ such that $\\psi x\\in T_\\theta $ .",
"Then by Lemma REF there exists some $d\\in \\mathbb {N}$ such that for any $y\\in P_{\\mathsf {N}\\rightarrow \\mathsf {B}}$ (and so in particular $y\\in T_{\\mathsf {N}\\rightarrow \\mathsf {B}}$ ) we have $\\psi y=\\psi x\\in T_\\theta $ whenever $[{x}]_{d}=[ {x(0),\\ldots ,x(d-1)} ]=[ {y(0),\\ldots ,y(d-1)} ]=[{y}]_{d}$ .",
"Next define $<$ on $T_{\\mathsf {N}\\rightarrow \\mathsf {B}}$ by $x<y$ iff $x(i)=1\\Rightarrow y(i)=1$ for all $i\\in \\mathbb {N}$ and there exists at least one $j\\in \\mathbb {N}$ with $x(j)=0$ and $y(j)=1$ .",
"Then $<$ is compatible with $(\\oplus ,\\prec )$ , and moreover, for any nonempty chain $\\gamma \\subseteq T_{\\mathsf {N}\\rightarrow \\mathsf {B}}$ define $\\tilde{\\gamma }\\in L$ by $\\tilde{\\gamma }(n):={\\left\\lbrace \\begin{array}{ll}1 & \\mbox{if $x(n)=1$ for some $x\\in \\gamma $}\\\\ 0 & \\mbox{otherwise}\\end{array}\\right.",
"}$ and $\\tilde{\\gamma }(\\bot )=\\bot $ .",
"Then clearly $x\\le \\tilde{\\gamma }$ for all $x\\in \\gamma $ , and moreover for any $d\\in \\mathbb {N}=T_{\\mathsf {N}}$ , by a variant of the argument in Example REF we have $[{\\tilde{\\gamma }}]_{d}=[{x}]_{d}$ for some $x\\in \\gamma $ .",
"Thus $<$ is chain bounded w.r.t.",
"$[{\\cdot }]_{}$ and $L$ .",
"Now, let $\\Phi $ denote the least fixed point in ${\\mathcal {P}}^\\omega $ of $\\Phi fx=_\\theta fx(\\lambda (n,y)\\; .",
"\\; \\Phi f(x\\cup y)\\mbox{ if $\\overline{x(n)}\\cdot y(n)=1$ else $0_\\theta $}).$ By Theorem REF , taking any total $f$ , since $\\Phi f\\in P_{(\\mathsf {N}\\rightarrow \\mathsf {B})\\rightarrow \\theta }$ is automatically piecewise continuous, we have that $\\Phi f$ is total, and therefore $\\Phi $ is a total object in ${\\mathcal {P}}^\\omega $ .",
"This implies that $\\mathcal {C}^\\omega \\models \\mathrm {PA}^\\omega +\\Phi $ for the extension of $\\mathrm {PA}^\\omega $ with some new constant satisfying the defining axiom (REF )."
],
[
"On non-discrete output types",
"In Example REF we have essentially shown that a simple variant of `update induction' in the sense of [4] is total.",
"In fact, with a slight modification of the above proof we would be able to reprove totality of update induction in its general form.",
"However, in this paper we are primarily interested in forms of recursion on chain bounded partial orders which do not correspond to the simple recursive scheme (REF ).",
"The reason for this is that in order to establish totality of $\\Phi f$ for any total $f$ , we are typically required to restrict the complexity of the output type $\\theta $ to being discrete, so that something along the lines of Lemma REF applies.",
"As we will see, this is a problem for the functional interpretation.",
"Before we go on, we illustrate why extending $\\mathrm {PA}^\\omega /\\mathrm {HA}^\\omega $ with (REF ) for non-discrete types does not even result in a consistent theory!",
"Let us set $\\theta :=\\mathsf {N}\\rightarrow \\mathsf {N}$ and define $f$ by $fxp:=\\lambda n\\; .",
"\\; 1+p(n,\\lbrace n\\rbrace )(n+1)$ where we identify the set $\\lbrace n\\rbrace $ with its characteristic function of type $\\mathsf {N}\\rightarrow \\mathsf {B}$ .",
"Then defining $k:=\\Phi f(\\emptyset )0:\\mathsf {N}$ we have $\\begin{aligned}k&=1+\\Phi f({\\lbrace 0\\rbrace })(1)=2+\\Phi f({\\lbrace 0,1\\rbrace })(2)\\\\&=\\ldots =k+1+\\Phi f({\\lbrace 0,\\ldots ,k\\rbrace })(k+1)>k\\end{aligned}$ which is inconsistent with $\\mathrm {PA}^\\omega /\\mathrm {HA}^\\omega $ .",
"The key point at which the argument from Example REF fails is that Lemma REF is no longer valid for $\\theta :=\\mathsf {N}\\rightarrow \\mathsf {N}$ : if $\\psi x$ is a function then it can in general query an infinite part of $x$ .",
"To overcome this we could restrict our attention to those $f$ such that $\\Phi f$ is piecewise continuous and thus total for non-discrete output type: For example, let $fxp:=\\lambda n<\\underline{N}\\; .",
"\\; 1+p(n,\\lbrace n\\rbrace )(n+1)$ for some numeral $\\underline{N}:\\mathsf {N}$ , so that $p$ is only queried finitely many times.",
"Then working in ${\\mathcal {P}}^\\omega $ , for total $x$ we would have $\\Phi fx=\\lambda n\\; .",
"\\; 1+{\\left\\lbrace \\begin{array}{ll}\\Phi f(x\\cup \\lbrace n\\rbrace )(n+1) & \\mbox{if $x(n)=1\\wedge n<N$}\\\\ 0 & \\mbox{otherwise} \\end{array}\\right.",
"}$ and so a point of continuity for $\\Phi fx$ could be taken to be the maximum of all points of continuity of the functions $\\lambda y.\\Phi f(y\\cup \\lbrace n\\rbrace )(n+1)$ for $n<N$ and at point $y:=x$ .",
"We now propose cleaner way of extending (REF ) to non discrete output types.",
"Instead of restricting $f$ , we add a new parameter $\\omega $ which controls the recursion directly."
],
[
"Controlled recursion over $(\\oplus ,\\prec )$",
"We modify the scheme (REF ), resulting in a slightly more elaborate mode of recursion in which the continuity behaviour is controlled by some auxiliary functional $\\omega $ .",
"Define the constant $\\Psi ^\\theta _{\\oplus ,\\prec }$ (from now on omitting the parameters) by $\\begin{aligned}&\\Psi \\omega fx=_\\theta \\\\&f\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}(\\lambda a\\; .",
"\\; \\Psi \\omega f(\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\oplus a)\\mbox{ if $a\\succ \\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}$ else $0_\\theta $})\\end{aligned}$ where $f:\\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\theta $ and $\\omega :\\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\sigma $ and $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}$ is defined by $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}:=_\\sigma \\omega x(\\lambda a\\; .",
"\\; \\Psi \\omega f(x\\oplus a)\\mbox{ if $a\\succ x$ else $0_\\theta $}).$ Observe that $\\Psi $ is still defined as the fixed point of a simple closed term of $\\mathrm {PA}^\\omega $ , and as it will turn out, this modified scheme will allow us to admit output of arbitrary type level.",
"Moreover, we will show later that by instantiating $\\omega $ by a suitable closed term of $\\mathrm {PA}^\\omega $ , we can use this recursive scheme to define a realizer for the functional interpretation of our axiomatic form of Zorn's lemma.",
"As before, we use the abbreviation $\\Psi _{\\omega ,f,x}:=\\lambda a\\; .",
"\\; \\Psi \\omega f(x\\oplus a)\\mbox{ if $a\\succ x$ else $0_\\theta $}$ so that the defining equation (REF ) now becomes $\\Psi \\omega fx=f\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\Psi _{\\omega ,f,\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}}$ for $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}:=\\omega x\\Psi _{\\omega ,f,x}$ .",
"We now give a totality theorem analogous to Theorem REF , but with the notion of piecewise continuity replaced by a slightly more subtle property.",
"Definition 5.7 (${\\mathcal {P}}^\\omega $ ) We say that a pair of functionals $\\psi \\in P_{\\sigma \\rightarrow \\sigma }$ and $\\phi \\in P_{\\sigma \\rightarrow \\theta }$ form a truncation with respect to $[{\\cdot }]_{}$ , $L\\subseteq T_\\sigma $ and some partial order $<$ on $T_\\sigma $ if the following two conditions are satisfied: For any $x,y\\in T_\\sigma $ , if $\\psi x\\in T_\\sigma $ and $\\psi x<y$ then $x<y$ .",
"For any $x\\in L$ such that $\\psi x\\in T_\\sigma $ and $\\phi (\\psi x)\\in T_\\theta $ there exists some $d\\in T_\\delta $ such that $\\forall y\\in T_\\sigma ([{x}]_{d}=[{y}]_{d}\\Rightarrow \\psi x=\\psi y).$ Example 5.8 Continuing from Example REF with $\\sigma :=\\mathsf {N}\\rightarrow \\mathsf {B}$ but $\\theta $ now arbitrary, for any $N\\in \\mathbb {N}$ the continuous functional $\\psi _N\\in T_{\\sigma \\rightarrow \\sigma }$ defined by $\\psi _Nx(n):={\\left\\lbrace \\begin{array}{ll}x(n) & \\mbox{if $n<N$}\\\\ 1 & \\mbox{otherwise}\\end{array}\\right.",
"}$ and $\\psi _Nx(\\bot )=\\bot $ forms a truncation with any other functional $\\phi \\in P_{\\sigma \\rightarrow \\theta }$ w.r.t $[{\\cdot }]_{}$ , $L$ and $<$ .",
"To see this, observe that for any strict $x$ , if $y\\in T_\\sigma $ satisfies $[ {x(0),\\ldots ,x(N-1)} ]=[ {y(0),\\ldots ,y(N-1)} ]$ then $\\psi _Nx=\\psi _Ny$ and so $\\psi _N$ satisfies condition (REF ) of being a truncation for $d:=N$ .",
"For condition (REF ), if $\\psi _Nx<y$ this means that $x(i)=1\\Rightarrow y(i)=1$ for all $i<N$ , $y(i)=1$ for all $i\\ge N$ , and $x(j)=0$ and $y(j)=1$ for some $j<N$ , from which it follows easily that $x<y$ .",
"On the other hand, the continuous functional $\\phi _N\\in T_{\\sigma \\rightarrow \\sigma }$ defined by $\\phi _Nx(n):={\\left\\lbrace \\begin{array}{ll}x(n) & \\mbox{if $n<N$}\\\\ 0 & \\mbox{otherwise}\\end{array}\\right.",
"}$ and $\\phi _Nx(\\bot )=\\bot $ satisfies condition (REF ) of being a truncation, but not condition (REF ), since for $x$ representing the characteristic function of the singleton set $\\lbrace N\\rbrace $ we have $\\phi _Nx<x$ but not $x<x$ .",
"Finally, the identity functional $\\iota \\in T_{\\sigma \\rightarrow \\sigma }$ clearly satisfies condition (REF ) of being a truncation, but condition (REF ) fails for e.g.",
"$\\phi $ also the identity function, since for some arbitrary $x\\in L$ there is no $d\\in \\mathbb {N}$ such that for any total $y$ , $[ {x(0),\\ldots ,x(d-1)} ]=[ {y(0),\\ldots ,y(d-1)} ]$ implies $x(n)=y(n)$ for all $n\\in \\mathbb {N}$ .",
"Theorem 5.9 (${\\mathcal {P}}^\\omega $ ) Let $\\Psi $ denote the least fixed point of the primitive recursive defining equation (REF ), and suppose that there exist $<$ on $T_\\sigma $ and $L\\subseteq T_\\sigma $ such that $<$ is compatible with $(\\oplus ,\\prec )$ and chain bounded w.r.t.",
"$[{\\cdot }]_{}$ and $L$ .",
"Let $f\\in T_{\\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\theta }$ and $\\omega \\in T_{\\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\sigma }$ .",
"Then whenever $\\lbrace \\cdot \\rbrace ^{\\Psi }_{\\omega ,f}\\in P_{\\sigma \\rightarrow \\sigma }$ and $\\lambda x\\; .",
"\\; fx\\Psi _{\\omega ,f,x}\\in P_{\\sigma \\rightarrow \\theta }$ form a truncation w.r.t.",
"$[{\\cdot }]_{}$ , $L$ and $<$ , it follows that $\\Psi \\omega f\\in T_{\\sigma \\rightarrow \\theta }$ .",
"We again appeal to Zorn's lemma, but this time on the set $S\\subseteq T_\\sigma $ given by $S:=\\lbrace x\\in T_\\sigma \\; | \\; \\mbox{either $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\notin T_\\sigma $ or $\\Psi \\omega fx\\notin T_\\theta $}\\rbrace .$ To show that $(S,<)$ is chain bounded in the usual sense, take some nonempty chain $\\gamma \\subseteq S$ and consider its upper bound $\\tilde{\\gamma }\\in L$ in the sense of Definition REF .",
"As before, we want to show that $\\tilde{\\gamma }\\in S$ , so we assume for contradiction that this is not the case, which means that both $\\lbrace \\tilde{\\gamma }\\rbrace ^{\\Psi }_{\\omega ,g}\\in T_\\sigma $ and $\\Psi \\omega f\\tilde{\\gamma }\\in T_\\theta $ .",
"But then by Definition REF (REF ) - observing that $\\lbrace \\tilde{\\gamma }\\rbrace ^{\\Psi }_{\\omega ,g}$ and $f\\lbrace \\tilde{\\gamma }\\rbrace ^{\\Psi }_{\\omega ,g}\\Psi _{\\omega ,f,\\lbrace \\tilde{\\gamma }\\rbrace ^{\\Psi }_{\\omega ,g}}=\\Psi \\omega f \\tilde{\\gamma }$ are both total - there exists some $d\\in T_\\delta $ such that $\\lbrace \\tilde{\\gamma }\\rbrace ^{\\Psi }_{\\omega ,f}=\\lbrace y\\rbrace ^{\\Psi }_{\\omega ,f}$ for any $y\\in T_\\sigma $ satisfying $[{\\tilde{\\gamma }}]_{d}=[{y}]_{d}$ .",
"But since by Definition REF there exists some $x\\in \\gamma $ such that $[{\\tilde{\\gamma }}]_{d}=[{x}]_{d}$ it therefore follows that $\\lbrace \\tilde{\\gamma }\\rbrace ^{\\Psi }_{\\omega ,f}=\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}$ and thus $\\Psi \\omega f\\tilde{\\gamma }=f\\lbrace \\tilde{\\gamma }\\rbrace ^{\\Psi }_{\\omega ,f}\\Psi _{\\omega ,f,\\lbrace \\tilde{\\gamma }\\rbrace ^{\\Psi }_{\\omega ,f}}=f\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\Psi _{\\omega ,f,\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}}=\\Psi \\omega fx$ which imply that $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\in T_\\sigma $ and $\\Psi \\omega fx\\in T_\\theta $ and thus $x\\notin S$ , a contradiction.",
"Thus $\\tilde{\\gamma }\\in S$ and $S$ is chain bounded.",
"We now suppose that the conclusion of the main result is false, which means that there exists some $x\\in T_\\sigma $ such that $\\Psi \\omega fx\\notin T_\\theta $ , and so in particular $x\\in S$ and thus $S$ is nonempty.",
"By Zorn's lemma, $S$ contains a maximal element $x$ .",
"We now show that $x\\notin S$ , a contradiction.",
"Since $x$ is maximal, for any $x<y$ we must have $\\lbrace y\\rbrace ^{\\Psi }_{\\omega ,f}\\in T_\\sigma $ and $\\Psi \\omega fy\\in T_\\theta $ .",
"We first show that $\\Psi _{\\omega ,f,x}$ is total: For any $a\\in T_\\rho $ , either $x\\succ a$ and so $\\Psi _{\\omega ,f,x}a=0_\\theta \\in T_\\theta $ , or $x\\succ a$ and thus by compatibility we have $x<x\\oplus a$ and therefore $\\Psi _{\\omega ,f,x}a=\\Psi \\omega f(x\\oplus a)\\in T_\\theta $ .",
"But then since $\\omega $ , $x$ and $\\Psi _{\\omega ,f,x}$ are all total, it follows that $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}=\\omega x\\Psi _{\\omega ,f,x}\\in T_\\sigma $ .",
"We now show that $\\Psi _{\\omega ,f,\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}}$ is total: For $a\\in T_\\rho $ , either $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\succ a$ and so $\\Psi _{\\omega ,g,\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}}a=0_\\theta \\in T_\\theta $ , or $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\succ a$ and thus by compatibility we have $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}<\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\oplus a$ .",
"But now using condition (REF ) of $\\lbrace \\cdot \\rbrace ^{\\Psi }_{\\omega ,f}$ forming a truncation, we have $x<\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\oplus a$ and thus $\\Psi \\omega f(\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\oplus a)\\in T_\\theta $ .",
"Now, since $f$ , $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}$ and $\\Psi _{\\omega ,f,\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}}$ are all total, it follows that $\\Psi \\omega fx=f\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}\\Psi _{\\omega ,f,\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}}\\in T_{\\theta }$ .",
"We have therefore proven that if $x$ is maximal, then both $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}$ and $\\Psi \\omega fx$ are total and so $x\\notin S$ , contradicting that $S$ has a maximal element.",
"Therefore $S=\\emptyset $ and so $\\Psi \\omega fx\\in T_\\theta $ for any $x\\in T_\\sigma $ and we have shown totality of $\\Psi \\omega f$ .",
"Example 5.10 We now consider Example REF from the perspective of controlled recursion, using a truncation similar to that given in Example REF above.",
"Let us extend the language of $\\mathrm {PA}^\\omega $ with a new constant $\\Omega $ with defining equation $\\Omega nfx=f\\langle {x}\\rangle _{n}(\\lambda a\\; .",
"\\; \\Omega nf(\\langle {x}\\rangle _{n}\\oplus a)\\mbox{ if $a\\succ \\langle {x}\\rangle _{n}$ else $0$})$ where $n:\\mathsf {N}$ , $f:\\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\theta $ and $\\langle {x}\\rangle _{n}$ is defined by $\\langle {x}\\rangle _{n}:=_{\\mathsf {N}\\rightarrow \\mathsf {B}}\\lambda i\\; .",
"\\; x(i)\\mbox{ if $i<n$ else 1}.$ Then $\\Omega $ is definable as $\\Omega nfx:=\\Psi (c n)fx$ where $\\Psi $ satisfies (REF ) and $c:\\mathsf {N}\\rightarrow \\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\sigma $ is the primitive recursive functional defined by $cnxp:=\\lambda i\\; .",
"\\; x(i)\\mbox{ if $i<n$ else 1}.$ Working from now on in ${\\mathcal {P}}^\\omega $ , for each $n\\in \\mathbb {N}$ we can interpret $\\Omega n$ as being a least fixed point of the equation (REF ) for $\\omega $ instantiated as the total representation in ${\\mathcal {P}}^\\omega $ of $cn$ as above.",
"We apply Theorem REF to show that $\\Omega n$ is total.",
"The compatibility and chain boundedness requirements are the same as in Theorem REF , and so in our setting have already been dealt with in Example REF .",
"Take $f\\in T_{\\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\theta }$ with $\\omega :=cn\\in T_{\\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\sigma }$ .",
"To see that $\\lbrace \\cdot \\rbrace ^{\\Psi }_{\\omega ,f}$ and $\\lambda x\\; .",
"\\; fx\\Psi _{\\omega ,f,x}$ form a truncation, we use a similar argument to Example REF .",
"We can assume that $\\omega =cn$ is interpreted in ${\\mathcal {P}}^\\omega $ as $\\omega xp(i)={\\left\\lbrace \\begin{array}{ll}x(i) & \\mbox{if $i<n$}\\\\ 1 & \\mbox{otherwise}\\end{array}\\right.",
"}$ and $\\omega xp(\\bot )=\\bot $ , and so for any strict $x$ , if $y\\in T_\\sigma $ satisfies $[ {x(0),\\ldots ,x(n-1)} ]=[ {y(0),\\ldots ,y(n-1)} ]$ then $\\omega xp=\\omega yp=\\omega yq$ for any functionals $p,q\\in P_{\\rho \\rightarrow \\theta }$ , and so in particular $\\lbrace x\\rbrace ^{\\Psi }_{\\omega ,f}=\\omega x\\Psi _{\\omega ,f,x}=\\omega y\\Psi _{\\omega ,f,y}=\\lbrace y\\rbrace ^{\\Psi }_{\\omega ,f}$ .",
"This establishes property (REF ), and property (REF ) follows analogously to Example REF .",
"Therefore $\\Psi \\omega f=\\Psi (cn)f\\in T_{\\sigma \\rightarrow \\theta }$ for arbitrary $n\\in \\mathbb {N}$ and $f\\in T_{\\sigma \\rightarrow (\\rho \\rightarrow \\theta )\\rightarrow \\theta }$ , which implies that the object $\\Omega $ defined by $\\Omega nfx:=\\Psi (cn)fx$ is total and satisfies the equation (REF ) in ${\\mathcal {P}}^\\omega $ .",
"Therefore $\\Omega $ also has an interpretation in $\\mathcal {C}^\\omega $ , i.e.",
"$\\mathcal {C}^\\omega \\models \\mathrm {PA}^\\omega +\\Omega $ .",
"Note that no conditions were imposed on $\\theta $ , and so totality of $\\Omega $ also holds for non-discrete $\\theta $ .",
"The functional defined in (REF ) is rather strongly controlled by $cn$ (we claim in fact that $\\Omega $ is definable as a term of System T).",
"This deliberately simplistic example was chosen simply to illustrate Theorem REF .",
"Nevertheless, later we will require a much more subtle truncation for realizing the functional interpretation of lexicographic induction, which certainly does lead us beyond the realm of primitive recursion."
],
[
"The functional interpretation of Zorn's lemma",
"In the last section we introduced two general variants of recursion over chain bounded partial orders.",
"We will now show that our controlled variant is well suited for solving the functional interpretation of our axiomatic formulation of Zorn's lemma from Section .",
"We begin by recalling some essential facts about the functional interpretation.",
"Full details can be found in [1] or [14].",
"For those readers not familiar with the functional interpretation, we aim to at least present, in a self contained manner, the concrete computational problem we need to solve.",
"This alone should suffice in order to understand later sections.",
"Such a reader is advised to skip directly ahead to Section REF (perhaps skimming through Section REF on the way)."
],
[
"An overview of the functional interpretation",
"So that we can formally state a higher-type variant of Gödel's soundness theorem for the functional interpretation, we need to recall the so-called weakly extensional variant $\\mathrm {WE\\mbox{-}PA}^\\omega $ of $\\mathrm {PA}^\\omega $ , which is obtained from the latter by simply replacing the axiom of extensionality with a quantifier-free rule form (see [14] for details, though this is not necessary to understand what follows).",
"This is because the interpretation is unable to deal with the axiom of extensionality and thus cannot be applied directly to $\\mathrm {PA}^\\omega $ (see [1] or [14]).",
"The functional interpretation assigns to each formula $A$ of $\\mathrm {WE\\mbox{-}PA}^\\omega $ a new formula $|{A}|^{x}_{y}$ where now $x$ and $y$ are (possibly empty) tuples of variables of some finite type.",
"The precise definition is by induction over the logical structure of $A$ , and is given in Figure 1, where in the interpretation of disjunction, $i$ is an object of natural number type and $P\\vee _i Q$ denotes $(i= 0\\rightarrow P)\\wedge (i\\ne 0\\rightarrow Q)$ .",
"The basic functional interpretation applies only to intuitionistic theories.",
"In order to deal with classical logic, we need to combine the interpretation with some variant of the negative translation $A\\mapsto A^N$ as an initial step.",
"We do not give any further details, but simply state the main soundness theorem for classical arithmetic.",
"In the following, $\\mathrm {QF\\mbox{-}AC}$ denotes the axiom of quantifier-free choice i.e.",
"the schema $\\forall x^\\rho \\exists y^\\sigma \\; A_0(x,y)\\rightarrow \\exists f^{\\rho \\rightarrow \\sigma }\\forall x\\; A_0(x,fx)$ where $\\rho $ and $\\sigma $ are arbitrary types and $A_0(x,y)$ ranges over quantifier-free formulas.",
"Figure: The functional interpretationTheorem 6.1 (cf.",
"Theorem 10.7 of [14], but essentially due to Gödel [11]) Let $A$ be a formula in the language of $\\mathrm {WE\\mbox{-}PA}^\\omega $ .",
"Then whenever $\\mathrm {WE\\mbox{-}PA}^\\omega +\\mathrm {QF\\mbox{-}AC}\\vdash A$ we can extract a term $t$ of $\\mathrm {WE\\mbox{-}HA}^\\omega $ whose free variables are the same as those of $A$ , and such that $\\mathrm {HA}^\\omega \\vdash \\forall y|{A^N}|^{t}_{y}.$ Remark.",
"Note that $\\forall y|{A^N}|^{t}_{y}$ is provable even in a quantifier-free fragment of $\\mathrm {WE\\mbox{-}HA}^\\omega $ , the intuitionistic variant of $\\mathrm {WE\\mbox{-}PA}^\\omega $ .",
"The main result in the remainder of this section is to extend Theorem REF above to include our formulation of Zorn's lemma.",
"Generally speaking, in order to expand the soundness theorem to incorporate extensions of $\\mathrm {WE\\mbox{-}PA}^\\omega +\\mathrm {QF\\mbox{-}AC}$ with new axioms $X$ , it suffices to provide a new recursive scheme $\\Omega $ such that the functional interpretation of $X^N$ has a solution in $\\mathrm {HA}^\\omega +\\Omega $ .",
"A classical example of this is with $X$ as the axiom of countable choice, and $\\Omega $ the scheme of bar recursion in all finite types (cf.",
"[1] or [14]).",
"Here on the other hand, we set $X$ to be our syntactic formulation of Zorn's lemma, and $\\Omega $ a functional definable from our scheme of controlled recursion from Section REF ."
],
[
"The functional interpretation of $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$",
"We now outline how the combination of the functional interpretation with the negative interpretation acts on the axiom $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ as given in Definition REF , subject to the additional restriction that $Q(u)$ ranges over quantifier-free formulas of $\\mathrm {WE\\mbox{-}PA}^\\omega $ (similar restrictions can be found in [4] and [7] in the context of open induction).",
"This restriction still allows us to deal with most concrete examples we are interested in (including the existence of maximal ideals in countable commutative rings in Example REF and also Higman's lemma, which we will discuss later), but simplifies the interpretation considerably (though we conjecture that in many cases, and in particular for concrete example considered in Section , a solution for general $Q(u)$ can be reduced to that of quantifier-free $Q(u)$ , subject to modification of the parameters $[{}]_{},\\oplus ,\\prec $ ).",
"In what follows, we make use of the fact that the quantifier-free formulas of $\\mathrm {WE\\mbox{-}PA}^\\omega $ are decidable, in the sense that whenever $A_0(x_1,\\ldots ,x_n)$ is quantifier-free with free variables $x_1,\\ldots ,x_n$ there is a closed term $t_A$ of System T so that $\\mathrm {HA}^\\omega \\vdash \\forall x_1,\\ldots ,x_n(t_Ax_1\\ldots x_n=1\\leftrightarrow A_0(x_1,\\ldots ,x_n))$ .",
"This also means that the functional interpretation essentially interprets quantifier-free formulas as themselves.",
"Let us now fix closed terms $[{}]_{},\\oplus ,\\prec $ and consider $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ as given in Definition REF , but where now $Q(u)$ is assumed to be quantifier-free.",
"There are several variants of the negative translation which can be applied.",
"Applying standard variant due to Kuroda, as used in [14], and using a few standard intuitionistic laws together with Markov's principle (all of which can be interpreted by the intuitionistic functional interpretation), it suffices to solve the functional interpretation of $\\begin{aligned}\\exists x^\\sigma \\forall d^\\delta Q([{x}]_{d})\\rightarrow &\\lnot \\lnot \\exists y^\\sigma (\\forall d\\; Q([{y}]_{d})\\\\&\\wedge \\forall a\\succ y\\;\\exists d\\; \\lnot Q([{y\\oplus a}]_{d})).\\end{aligned}$ We must now apply the rules of Figure 1 to (REF ), which we do step by step.",
"We first observe that the inner part of the conclusion of (REF ) within the double negations is translated to $\\exists y^\\sigma , h^{\\rho \\rightarrow \\delta }\\forall d, a(Q([{y}]_{d})\\wedge (a\\succ y\\rightarrow \\lnot Q([{y\\oplus a}]_{ha})).$ Therefore the double negated conclusion is partially interpreted (i.e.",
"before the final instance of the $\\forall $ -rule) as $\\forall F,G\\; \\exists y,h(Q([{y}]_{Fyh})\\wedge (Gyh\\succ y\\rightarrow \\lnot Q([{y\\oplus Gyh}]_{h(Gyh)}))$ where here $F:\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\delta $ and $G:\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\rho $ .",
"Therefore, interpreting the main implication, in order to solve the functional interpretation of (REF ) we must produce three terms $r,s,t$ which take as input $x,F,G$ and have output types $\\delta ,\\sigma $ and $\\rho \\rightarrow \\sigma $ respectively, and satisfy $Q([{x}]_{r})\\rightarrow Q([{s}]_{Fst})\\wedge (Gst\\succ s\\rightarrow \\lnot Q([{s\\oplus Gst}]_{t(Gst)}))$ where for readability we suppress the input parameters, so that $r$ should actually read $rxFG$ throughout, and similarly for $s$ and $t$ .",
"Though (REF ) looks complicated, it can be given a fairly intuitive characterisation as follows.",
"The original axiomatic formulation of Zorn's lemma is equivalent (using $\\mathrm {QF\\mbox{-}AC}$ ) to the statement that given some $x^\\sigma $ satisfying $\\forall dQ([{x}]_{d})$ we can find some $y$ also satisfying $\\forall d Q([{y}]_{d})$ together with an $h:\\rho \\rightarrow \\delta $ witnessing maximality of $y$ in the sense that $\\lnot Q([{y\\oplus a}]_{ha})$ for any $a\\succ x$ .",
"On the other hand, the computational interpretation of Zorn's lemma given as (REF ) says that for any $x^\\sigma $ together with `counterexample functionals' $F,G$ we can produce elements $s$ and $t$ (in terms of $x,F,G$ ), where $s$ approximates our maximal element $y$ in the sense that it satisfies $Q([{s}]_{d})$ not for all $d$ but just for $d:=Fst$ , while $t$ approximates $h$ in the sense that it satisfies $\\lnot Q([{s\\oplus a}]_{t(a)}$ not for all $a\\succ s$ but just for $a:=Gst$ whenever $Gst\\succ s$ .",
"Indeed, this can be seen as a slightly more intricate version of Kreisel's no-counterexample interpretation, and the relationship between $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ and (REF ) is similar to the relationship between Cauchy convergence and `metastability' (see [14])."
],
[
"Solving the functional interpretation of $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$",
"From this point onwards, we no longer need to deal directly with the functional interpretation.",
"Rather, our focus is on solving the functional interpretation of $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ as given in (REF ).",
"To be more precise, we will construct realizing terms $r$ , $s$ and $t$ which each take as input $x:\\sigma $ , $F:\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\delta $ and $G:\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\rho $ and satisfy, $Q([{x}]_{r})\\rightarrow Q([{s}]_{Fst})\\wedge C(G,s,t)$ for any input, where $C(G,y,h)$ abbreviates the formula $C(G,y,h):\\equiv Gyh\\succ y\\rightarrow \\lnot Q([{y\\oplus Gyh}]_{h(Gyh)}).$ Interestingly, we do not require $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ in order to verify our realizing terms.",
"Instead, we work in $\\mathrm {HA}^\\omega $ extended with two recursively defined constants together with a simple universal axiom which we label `relevant part'.",
"That this formal theory has a model is a separate question, which we discuss after presenting our main result (Theorem REF ).",
"Definition 6.2 ($\\mathrm {HA}^\\omega $ ) Let $t_C$ denote the term of System T satisfying $t_CGyh=1\\leftrightarrow C(G,y,h)$ , which exists since $\\prec $ is decidable and $Q(u)$ is quantifier-free.",
"For the remainder of this section, we fix some closed term $e:(\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\delta )\\rightarrow (\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\sigma )$ of System T, so that all definitions and results that follows are parametrised by $e$ .",
"Definition 6.3 ($\\mathrm {HA}^\\omega $ ) Define the new constant $\\Omega _e:(\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\delta )\\rightarrow \\sigma \\rightarrow \\delta $ by $\\Omega _eFx=F\\langle {x}\\rangle ^{\\Omega _e}_{F}(\\lambda a\\; .",
"\\; \\Omega _eF(\\langle {x}\\rangle ^{\\Omega _e}_{F}\\oplus a)\\mbox{ if $a\\succ \\langle {x}\\rangle ^{\\Omega _e}_{F}$ else $0_\\delta $})$ where $\\langle {x}\\rangle ^{\\Omega _e}_{F}$ is shorthand for $\\langle {x}\\rangle ^{\\Omega _e}_{F}:=eFx(\\lambda a\\;.",
"\\; \\Omega _eF(x\\oplus a)\\mbox{ if $a\\succ x$ else $0_\\delta $})$ Furthermore, we use the abbreviation $\\Omega _{e,F,x}:=\\lambda a\\; .",
"\\; \\Omega _eF(x\\oplus a)\\mbox{ if $a\\succ x$ else $0_\\delta $}$ so that (REF ) can be expressed as $\\Omega _e Fx=F\\langle {x}\\rangle ^{\\Omega _e}_{F}\\Omega _{e,F,\\langle {x}\\rangle ^{\\Omega _e}_{F}}$ for $\\langle {x}\\rangle ^{\\Omega _e}_{F}=eFx\\Omega _{e,F,x}$ .",
"Definition 6.4 ($\\mathrm {HA}^\\omega +\\Omega _e$ ) We define in the language of $\\mathrm {HA}^\\omega +\\Omega _e$ the `relevant part' axiom for $\\Omega _e$ as $\\mathrm {RP}_e\\; : \\; \\forall x,F([{x}]_{\\Omega _eFx}=[{\\langle {x}\\rangle ^{\\Omega _e}_{F}}]_{\\Omega _eFx}).$ Intuitively, the relevant part axiom says that if we take the approximation of $x$ of size $\\Omega _eFx$ , then actually this approximation has no more information than that of the truncated version $\\langle {x}\\rangle ^{\\Omega _e}_{F}$ of $x$ , and so the latter already contains the `relevant part' of this approximation.",
"We will see a natural example of an $e$ which satisfies this axiom in Section .",
"Definition 6.5 ($\\mathrm {HA}^\\omega +\\Omega _e$ ) Define the constant $\\Gamma _e:(\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\delta )\\rightarrow (\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\rho )\\rightarrow \\sigma \\rightarrow \\sigma ^\\ast $ in the language of $\\mathrm {HA}^\\omega +\\Omega _e$ by $\\Gamma _eFGx:=y::{\\left\\lbrace \\begin{array}{ll}[ {} ] & \\mbox{if $t_CGy\\Omega _{e,F,y}=1$}\\\\ \\Gamma _eFG(y\\oplus Gy\\Omega _{e,F,y}) & \\mbox{otherwise}\\end{array}\\right.",
"}$ for $y:=\\langle {x}\\rangle ^{\\Omega _e}_{F}$ , where here $y::l$ denotes the appending of $y$ to the front of the list $l$ .",
"i.e.",
"$y::[l_1,\\ldots ,l_{j-1}]:=[y,l_1,\\ldots ,l_{j-1}]$ .",
"Theorem 6.6 ($\\mathrm {HA}^\\omega +\\Omega _e+\\Gamma _e+\\mathrm {RP}_e$ ) Define terms $r$ , $s$ and $t$ as follows: $\\begin{aligned}rxFG&:=_\\delta \\Omega _{e}Fx\\\\sxFG&:=_\\sigma \\mathrm {tail}(\\Gamma _eFGx)\\\\txFG&:=_{\\rho \\rightarrow \\delta }\\Omega _{e,F,\\mathrm {tail}(\\Gamma _eFGx)}\\end{aligned}$ where $\\mathrm {tail}(l)$ denotes the last element of the list $l$ (and $\\mathrm {tail}([])=0_\\sigma $ ).",
"Then provably in $\\mathrm {HA}^\\omega +\\Omega _e+\\Gamma _e+\\mathrm {RP}_e$ we have $\\forall x,F,G(Q([{x}]_{r})\\rightarrow Q([{s}]_{Fst})\\wedge C(G,s,t))$ where in the above formula we write just $r$ instead of $rxFG$ , and similarly for $s$ and $t$ .",
"Fixing $F$ and $G$ , we prove by induction on $n$ that $\\begin{aligned}\\forall x(|\\Gamma _eFGx|=n&\\wedge Q([{x}]_{rx})\\\\&\\rightarrow Q([{sx}]_{F(sx)(tx)})\\wedge C(G,sx,tx))\\end{aligned}$ where here $rx$ is shorthand for $rxFG$ (i.e.",
"the parameter $x$ is now explicitly written since it varies in the induction).",
"Since $|\\Gamma _eFGx|\\ge 1$ , our base case is $n=1$ which means that $t_CGy\\Omega _{e,F,y}=1$ and $\\Gamma _eFGx=[ {y} ]$ for $y:=\\langle {x}\\rangle ^{\\Omega _e}_{F}$ .",
"But this implies that $sx=y$ and $tx=\\Omega _{e,F,y}$ , and thus in particular $C(G,sx,tx)$ holds.",
"Next, we observe that $[{x}]_{rx}=[{x}]_{\\Omega _eFx}\\stackrel{(a)}{=}[{sx}]_{\\Omega _eFx}\\stackrel{(b)}{=}[{sx}]_{F(sx)(tx)}$ where (a) follows from $\\mathrm {RP}_e$ and the definitions of $rx$ and $sx$ , while for (b) we use that $\\Omega _eFx=Fy\\Omega _{e,F,y}=F(sx)(tx).$ Thus from $Q([{x}]_{rx})$ we can infer $Q([{sx}]_{F(sx)(tx)})$ , which establishes (REF ) for $n=1$ .",
"For the induction step, suppose that $|\\Gamma _eFGx|=n+1$ , which implies that $t_CGy\\Omega _{e,F,y}=0$ .",
"Setting $y:=\\langle {x}\\rangle ^{\\Omega _e}_{F}$ as before, and in addition $a:=Gy\\Omega _{e,F,y}$ , by unwinding definitions it follows from $\\lnot C(G,y,\\Omega _{e,F,y})$ that $a\\succ y$ and thus $\\Omega _{e,F,y}(a)=\\Omega _eF(y\\oplus a)$ , $Q([{y\\oplus a}]_{\\Omega _{e,F,y}(a)})$ and thus $Q([{y\\oplus a}]_{\\Omega _eF(y\\oplus a)})$ by (i).",
"Now since $\\Gamma _eFGx=y::\\Gamma _eFG(y\\oplus a)$ and thus $|\\Gamma _eFG(y\\oplus a)|=n$ , we can apply the induction hypothesis for $x^{\\prime }:=y\\oplus a$ .",
"Since $rx^{\\prime }=\\Omega _eFx^{\\prime }=\\Omega _eF(y\\oplus a)$ it follows from (ii) that $Q([{x^{\\prime }}]_{rx^{\\prime }})$ and therefore we have $Q([{sx^{\\prime }}]_{F(sx^{\\prime })(tx^{\\prime })})$ and $C(G,sx^{\\prime },tx^{\\prime })$ .",
"But since $sx=\\mathrm {tail}(y::\\Gamma _eFG(y\\oplus a))=\\mathrm {tail}(\\Gamma _eFG(x^{\\prime }))=sx^{\\prime }$ and similarly $tx=tx^{\\prime }$ , it follows that $Q([{sx}]_{F(sx)(tx)})\\wedge C(G,sx,tx)$ , which establishes (REF ) for $n^{\\prime }=n+1$ .",
"This completes the induction, and (REF ) follows by taking some arbitrary $F,G,x$ and letting $n:=|\\Gamma _eFGx|$ in (REF ).",
"The above result which solves the functional interpretation of $\\mathrm {ZL}_{[{}]_{},\\oplus ,\\prec }$ is valid for arbitrary $e$ .",
"However, it is only useful if the theory $\\mathrm {HA}^\\omega +\\Omega _e+\\Gamma _e+\\mathrm {RP}_e$ has a reasonable model.",
"The final results of this section establish some conditions by which both $\\Omega _e$ and $\\Gamma _e$ give rise to total objects and hence exist in $\\mathcal {C}^\\omega $ .",
"An example of a setting where $\\mathrm {RP}_e$ is also valid in $\\mathcal {C}^\\omega $ is given in Section .",
"Theorem 6.7 (${\\mathcal {P}}^\\omega $ ) Let $\\Omega _e$ denote a fixed point of the primitive recursive defining equation (REF ) - where the closed primitive recursive term $e$ is interpreted as some total object in ${\\mathcal {P}}^\\omega $ - and suppose that there exist $<$ and $L$ such that $<$ is compatible with $(\\oplus ,\\prec )$ and chain bounded w.r.t.",
"$[{\\cdot }]_{}$ and $L$ .",
"Suppose in addition that $\\langle {\\cdot }\\rangle ^{\\Omega _e}_{F}\\in P_{\\sigma \\rightarrow \\sigma }$ and $\\lambda x\\; .",
"\\; Fx\\Omega _{e,F,x}\\in P_{\\sigma \\rightarrow \\delta }$ form a truncation w.r.t.",
"$[{\\cdot }]_{}$ , $L$ and $<$ for any total $F$ .",
"Then $\\Omega _e$ is total.",
"This is a simple adaptation of Theorem REF , taking $\\Omega _e:=\\lambda F\\; .",
"\\; \\Psi (eF)F$ .",
"If $F\\in T_{\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\delta }$ then also $eF\\in T_{\\sigma \\rightarrow (\\rho \\rightarrow \\delta )\\rightarrow \\sigma }$ by totality of $e$ , and thus whenever $\\lbrace \\cdot \\rbrace ^{\\Psi }_{eF,F}$ and $\\lambda x\\; .",
"\\; Fx\\Psi _{eF,F,x}$ form a truncation w.r.t.",
"$[{\\cdot }]_{}$ , $L$ and $<$ then $\\Omega _eF=\\Psi (eF)F\\in T_{\\sigma \\rightarrow \\delta }$ .",
"But the truncation condition is exactly that given as the statement of this theorem, and if this holds for arbitrary total $F$ then $\\Omega _e$ is also total.",
"Theorem 6.8 (${\\mathcal {P}}^\\omega $ ) Let $\\Gamma _e$ denote a fixed point of the defining equation (REF ).",
"Under the assumptions of Theorem REF , $\\Gamma _e$ is total.",
"We can define $\\Gamma _e:=\\lambda F,G\\; .",
"\\; \\Psi (\\omega F)(fFG)$ where $\\omega $ and $f$ are total representations in ${\\mathcal {P}}^\\omega $ of the following functionals definable in $\\mathrm {HA}^\\omega +\\Omega _e$ : $\\begin{aligned}\\omega Fxp&:=_\\sigma \\langle {x}\\rangle ^{\\Omega _e}_{F}\\\\fFGxp&:=_{\\sigma ^\\ast }x::{\\left\\lbrace \\begin{array}{ll}[] & \\mbox{if $t_CGx\\Omega _{e,F,x}$}\\\\ p(Gx\\Omega _{e,F,x}) & \\mbox{otherwise}\\end{array}\\right.",
"}\\end{aligned}$ where here $p:\\rho \\rightarrow \\sigma ^\\ast $ (note that totality of $\\omega $ and $f$ follows from totality of primitive recursive functionals plus totality of $\\Omega _e$ as established in Theorem REF above).",
"To see that $\\Gamma _e$ satisfies (REF ) is just a case of unwinding the definitions.",
"Now, if $F$ and $G$ are total it follows that $\\omega F$ and $fFG$ are also total, and so by Theorem REF , $\\Gamma _eFG=\\Psi (\\omega F)(fFG)$ is total if we can show that $\\lbrace \\cdot \\rbrace ^{\\Psi }_{\\omega F,fFG}$ and $\\lambda x.",
"(fFG)x\\Psi _{\\omega F,fFG,x}$ form a truncation.",
"But $\\lbrace x\\rbrace ^{\\Psi }_{\\omega F,fFG}=\\langle {x}\\rangle ^{\\Omega _e}_{F}$ , and so this follows from the assumption that $\\langle {\\cdot }\\rangle ^{\\Omega _e}_{F}$ and $\\lambda x.Fx\\Omega _{e,F,x}$ form a truncation.",
"Formally, if $\\langle {x}\\rangle ^{\\Omega _e}_{F}$ is total for $x\\in L$ (which it always is by totality of $\\Omega _e$ ), then since in addition $Fy\\Omega _{e,F,y}$ is total for $y:=\\langle {x}\\rangle ^{\\Omega _e}_{F}$ then $\\langle {\\cdot }\\rangle ^{\\Omega _e}_{F}$ has a point of continuity $d$ for $x$ .",
"Condition (REF ) follows trivially.",
"Therefore we have shown that $\\Gamma _e$ is total.",
"Remark.",
"Our use of controlled recursion means that there are no type level restrictions on the output types $\\Omega _eFx:\\delta $ or $\\Gamma _eFGx:\\sigma ^\\ast $ .",
"This not only permits a greater degree of generality but is essential even for simple applications: In Example REF , $\\sigma :=\\mathsf {N}\\rightarrow \\mathsf {B}$ and thus $\\sigma ^\\ast $ is a higher type."
],
[
"Application: The lexicographic ordering",
"We conclude the paper by showing how our parametrised results can now be implemented in the special case of induction over the lexicographic ordering on sequences.",
"This constitutes a direct counterpart to open induction as presented in [4], and is closely related to the recursive scheme introduced in [22] for extracting a witness from the proof of Higman's lemma.",
"Definition 7.1 ($\\mathrm {HA}^\\omega $ ) Let $\\theta $ be some arbitrary type, and suppose that $\\lhd :\\theta \\times \\theta \\rightarrow \\mathsf {B}$ is a decidable relation on $\\theta $ such that induction over $\\lhd $ is provable in $\\mathrm {HA}^\\omega $ .",
"Setting $\\sigma :=\\mathsf {N}\\rightarrow \\theta $ , $\\delta :=\\mathsf {N}$ , $\\rho :=\\mathsf {N}\\times (\\mathsf {N}\\rightarrow \\theta )$ and $\\nu :=\\theta ^\\ast $ , define $\\begin{aligned}[{x}]_{n}&:=[ {x(0),\\ldots ,x(n-1)} ]\\\\x\\oplus (n,y)&:=[{x}]_{n}\\; @ \\;y\\\\(n,y)\\succ x &:=y(n)\\lhd x(n)\\end{aligned}$ where $([ {x(0),\\ldots ,x(n-1)} ]\\; @ \\;y)(i):=x(i)$ if $i<n$ and $y(i)$ otherwise.",
"We define $\\mathrm {LEX}_{\\rhd }$ to be the principle $\\mathrm {ZL}_{[{\\cdot }]_{},\\oplus ,\\prec }$ for the parameters given above i.e.",
"$\\begin{aligned}\\exists x\\forall d Q([{x}]_{d})&\\rightarrow \\exists y(\\forall d Q([{y}]_{d})\\\\&\\wedge \\forall (n,z)(z(n)\\lhd y(n)\\rightarrow \\exists d\\lnot Q([{[{y}]_{n}\\; @ \\;z}]_{d})).\\end{aligned}$ Our axiom $\\mathrm {LEX}_\\rhd $ is essentially the contrapositive of open induction as presented in [4], and as such the theory $\\mathrm {WE\\mbox{-}PA}^\\omega +\\mathrm {QF\\mbox{-}AC}+\\mathrm {LEX}_\\rhd $ (for various instantiations of $\\rhd $ ) is capable not only of formalizing large parts of mathematical analysis but also giving direct formalizations of minimal bad sequence arguments common in the theory of well quasi orderings.",
"We now show how it can be given a direct computational interpretation using the theory developed so far.",
"Lemma 7.2 (${\\mathcal {P}}^\\omega $ ) Define $L\\subset T_{\\sigma }$ to be the set of all strict total objects i.e.",
"those satisfying $x(\\bot )=\\bot _\\theta $ (recall that $\\sigma =\\mathsf {N}\\rightarrow \\theta $ ), and let the partial order $<$ on $T_{\\sigma }$ by defined by $y>x:\\Leftrightarrow \\exists n\\in \\mathbb {N}([{y}]_{n}=_{\\mathbb {N}^\\ast }[{x}]_{n}\\wedge y(n)\\lhd x(n))$ where here $\\lhd $ is now interpreted as a total functional $T_{\\theta \\times \\theta \\rightarrow \\mathsf {B}}$ .",
"In other words, $y>x$ if it is lexicographically smaller than $x$ w.r.t.",
"$\\lhd $ .",
"Then $<$ is compatible with $(\\oplus ,\\prec )$ and chain bounded w.r.t.",
"$[{\\cdot }]_{}$ and $L$ .",
"Compatibility is clear, while chain boundedness follows easily using a standard construction for the lexicographic ordering.",
"Take some nonempty chain $\\gamma \\subset T_\\sigma $ and inductively define the sequence of total objects $u_k\\in T_\\theta $ for $k\\in \\mathbb {N}$ by taking $u_k$ to be the $\\lhd $ -minimal element of the set $S_k:=\\lbrace x(k)\\; | \\; x\\in \\gamma \\mbox{ and $(\\forall i<k)(x(i)=u_i)$}\\rbrace \\subseteq T_\\theta .$ Note that $S_k$ are nonempty by induction on $k$ , and $u_k$ is well-defined since the $\\lhd $ -minimum principle is provable from induction over $\\lhd $ , which is provable in $\\mathrm {HA}^\\omega $ and thus satisfied by the total elements $T_\\theta $ .",
"Now define $\\tilde{\\gamma }(k):=u_k$ for $k\\in \\mathbb {N}$ and $\\tilde{\\gamma }(\\bot )=\\bot $ , which is clearly an element of $L\\subset T_{\\mathsf {N}\\rightarrow \\theta }$ .",
"It follows by definition that for any $d\\in \\mathbb {N}$ there exists some $x\\in \\gamma $ with $[{x}]_{d}=[ {u_0,\\ldots ,u_{d-1}} ]=[{\\tilde{\\gamma }}]_{d}$ .",
"To see that $\\tilde{\\gamma }$ is an upper bound, take some $x\\in \\gamma $ and assume that $x\\ne \\tilde{\\gamma }$ .",
"Let $k\\in \\mathbb {N}$ be the least with $x(k)\\ne \\tilde{\\gamma }(k)=u_k$ .",
"Then by definition of $u_k$ there is some $y\\in \\gamma $ with $[{y}]_{k}=[ {u_0,\\ldots ,u_{k-1}} ]=[{x}]_{k}$ and $y(k)=u_k$ .",
"Since $<$ is a total order on $\\gamma $ we must have either $x<y$ or $y<x$ , and since $x(k)\\ne y(k)$ this means that either $x(k)\\lhd y(k)$ or $y(k)\\lhd x(k)$ .",
"But by minimality of $u_k=y(k)$ we must have $\\tilde{\\gamma }(k)=y(k)\\lhd x(k)$ and thus $\\tilde{\\gamma }>x$ .",
"This proves that $x\\le \\tilde{\\gamma }$ for any $x\\in \\gamma $ .",
"Our next step is to define a suitable closed term $e$ of $\\mathrm {HA}^\\omega $ which not only induces a truncation in the sense of Theorem REF but also satisfies $\\mathrm {RP}_e$ in the total continuous functionals.",
"For this, we introduce a powerful idea that is already implicit in Spector's fundamental bar recursive interpretation of the axiom of countable choice [25], and has been studied in more detail in [19].",
"From now on we make the fairly harmless assumption that the canonical object $0_\\theta $ is minimal w.r.t to $\\lhd $ (this could in theory be circumvented but having it makes what follows slightly simpler).",
"Definition 7.3 ($\\mathrm {HA}^\\omega $ ) For $x:\\sigma $ and $n:\\mathsf {N}$ let $\\overline{{x},{n}}:=[{x}]_{n}\\; @ \\;(\\lambda i.0_\\theta ):\\sigma ,$ and define the primitive recursive functional $\\eta :(\\sigma \\rightarrow \\mathsf {N})\\rightarrow \\sigma \\rightarrow \\sigma $ by $\\eta \\phi xk:=_\\theta {\\left\\lbrace \\begin{array}{ll}0_\\theta & \\mbox{if $(\\exists i\\le k)(\\phi (\\overline{{x},{i}})<i)$}\\\\ x(k) & \\mbox{if $(\\forall i\\le k)(\\phi (\\overline{{x},{i}})\\ge i)$}\\end{array}\\right.",
"}$ where we note that the bounded quantifiers can be represented as bounded search terms in System T. Lemma 7.4 (${\\mathcal {P}}^\\omega $ ) Let us represent $\\eta $ in ${\\mathcal {P}}^\\omega $ by the total continuous functional $\\eta \\phi xk:={\\left\\lbrace \\begin{array}{ll}0_\\theta & \\mbox{if $(\\exists i\\le k)(\\forall j\\le i(\\phi (\\overline{{x},{j}})\\in \\mathbb {N})\\wedge \\phi (\\overline{{x},{i}})<i)$}\\\\ x(k) & \\mbox{if $(\\forall i\\le k)(\\phi (\\overline{{x},{i}})\\in \\mathbb {N}\\wedge \\phi (\\overline{{x},{i}})\\ge i)$}\\\\ \\bot & \\mbox{otherwise}\\end{array}\\right.",
"}$ with $\\eta \\phi x\\bot =\\bot $ .This is a standard domain theoretic interpretation of $\\eta $ , where the bounded search terminates with 0 for the first $i\\le k$ it finds with $\\phi (\\overline{{x},{i}})<i$ , and returns $\\bot $ if $\\phi (\\overline{{x},{i}})$ is undefined for any $i$ that is queried.",
"Then for any $\\phi \\in P_{\\sigma \\rightarrow \\mathsf {N}}$ , the functionals $\\eta \\phi \\in P_{\\sigma \\rightarrow \\sigma }$ and $\\phi $ form a truncation w.r.t.",
"$[{\\cdot }]_{}$ , $L$ and $<$ .",
"Part (REF ) is simple: Suppose that $x,y,\\eta \\phi x\\in T_\\sigma $ and $\\eta \\phi x<y$ so that there exists some $n\\in \\mathbb {N}$ with $[{y}]_{n}=[{\\eta \\phi x}]_{n}$ and $y(n)\\lhd \\eta \\phi x(n)$ .",
"Since we cannot have $y(n)\\lhd 0_\\theta $ by our minimality assumption, we must have $\\eta \\phi x(n)=x(n)$ .",
"But then by definition of $\\eta $ it follows that $\\eta \\phi x(k)=x(k)$ for all $k<n$ , and thus $[{y}]_{n}=[{x}]_{n}$ and so $x<y$ .",
"For part (REF ), let us now assume that $x\\in L$ with $\\eta \\phi x\\in T_\\sigma $ and $\\phi (\\eta \\phi x)\\in \\mathbb {N}$ .",
"We first show that there exists some $n\\in \\mathbb {N}$ with $\\phi (\\overline{{x},{n}})<n$ .",
"Suppose for contradiction that for all $i\\in \\mathbb {N}$ we have either $\\phi (\\overline{{x},{i}})=\\bot $ or $\\phi (\\overline{{x},{i}})\\ge i$ .",
"The first possibility is ruled out since if $\\phi (\\overline{{x},{i}})=\\bot $ then $\\eta \\phi xi=\\bot $ contradicting totality of $\\eta \\phi x$ .",
"But this means that $\\eta \\phi x=x$ (since also $\\eta \\phi \\bot =\\bot =x(\\bot )$ ).",
"But then $\\phi (\\eta \\phi x)=\\phi x\\in \\mathbb {N}$ and so by Lemma REF there exists some $d\\in \\mathbb {N}$ such that $\\phi x=\\phi y$ whenever $x(i)=y(i)$ for all $i<d$ .",
"Now set $N:=\\max \\lbrace \\phi x+1,d\\rbrace $ and consider $y:=\\overline{{x},{N}}$ .",
"Then $x(i)=y(i)$ for all $i<N$ and so also for all $i<d$ , which implies that $\\phi (\\overline{{x},{N}})=\\phi x<\\phi x+1\\le N$ a contradiction.",
"Therefore we have shown there exists some $n\\in \\mathbb {N}$ with $\\phi (\\overline{{x},{n}})<n$ , from which it follows that $\\eta \\phi x=\\overline{{x},{m}}$ for the least such $m\\in \\mathbb {N}$ with this property (again, $\\phi (\\overline{{x},{j}})\\in \\mathbb {N}$ for all $j\\le m$ by totality of $\\eta \\phi x$ ).",
"Let us now suppose that $y\\in P_\\sigma $ satisfies $[{x}]_{m}=[{y}]_{m}$ .",
"Then for $k<m$ , since $\\phi (\\overline{{y},{i}})=\\phi (\\overline{{x},{i}})\\ge i$ for all $i\\le k$ it follows that $\\eta \\phi yk=y(k)=x(k)$ , and if $k\\ge m$ , since $\\phi (\\overline{{y},{m}})=\\phi (\\overline{{x},{m}})<m$ it follows that $\\eta \\phi yk=0$ , and thus $\\eta \\phi y=\\overline{{x},{m}}=\\eta \\phi x$ .",
"Lemma 7.5 (${\\mathcal {P}}^\\omega $ ) Let $\\Omega _e$ be a fixed point of the equation (REF ) as in Theorem REF , where now $e$ is defined by $eFxp:=\\eta (\\lambda y.Fy(p|_y))x$ for $\\eta $ as in Definition REF (resp.",
"Lemma REF ) and $p|_y(n,z):=p(z)\\mbox{ if $z(n)\\lhd y(n)$ else 0}.$ Then $\\langle {\\cdot }\\rangle ^{\\Omega _e}_{F}\\in P_{\\sigma \\rightarrow \\sigma }$ and $\\lambda x.Fx\\Omega _{e,F,x}\\in P_{\\sigma \\rightarrow \\mathsf {N}}$ form a truncation w.r.t.",
"$[{\\cdot }]_{}$ , $L$ and $<$ for any $F$ .",
"We first observe that $\\langle {x}\\rangle ^{\\Omega _e}_{F}=eFx\\Omega _{e,F,x}=\\eta (\\lambda y.Fy(\\Omega _{e,F,x}|_y))x.$ We now argue that for any $i\\in \\mathbb {N}$ we have $\\Omega _{e,F,x}|_{\\overline{{x},{i}}}=\\Omega _{e,F,\\overline{{x},{i}}}.$ For this we only need to check arguments $(n,y)$ which satisfy $(n,y)\\succ \\overline{{x},{i}}$ i.e.",
"$y(n)\\lhd (\\overline{{x},{i}})(n)$ .",
"But by minimality of $0_\\theta $ this is only possible if $n<i$ and $y(n)\\lhd x(n)$ , in which case $\\begin{aligned}\\Omega _{e,F,x}|_{\\overline{{x},{i}}}(n,y)&=\\Omega _{e,F,x}(n,y)=\\Omega _eF([{x}]_{n}\\; @ \\;y)\\\\&=\\Omega _eF([{\\overline{{x},{i}}}]_{n}\\; @ \\;y)=\\Omega _{e,F,\\overline{{x},{i}}}(n,y).\\end{aligned}$ Since $\\eta \\phi x$ only depends on $\\phi $ for arguments of the form $\\overline{{x},{i}}$ , it follows that $\\langle {x}\\rangle ^{\\Omega _e}_{F}=\\eta \\phi _{F,\\Omega } x\\mbox{ \\ \\ for \\ \\ }\\phi _{F,\\Omega }:=\\lambda y.Fy\\Omega _{e,F,y}.$ But for any $F$ , by Lemma REF applied to $\\phi :=\\phi _{F,\\Omega }$ as defined above, we have that $\\eta \\phi _{F,\\Omega }$ and $\\phi _{F,\\Omega }$ form a truncation w.r.t.",
"$[{\\cdot }]_{}$ , $L$ and $<$ , and the result follows.",
"Corollary 7.6 (${\\mathcal {P}}^\\omega $ ) Let $\\Omega _e$ and $\\Gamma _e$ be fixed points of the equations (REF ) and (REF ) respectively, for $e$ be as defined in Lemma REF .",
"Then $\\Omega _e$ and $\\Gamma _e$ are total, and thus $\\mathcal {C}^\\omega \\models \\mathrm {HA}^\\omega +\\Omega _e+\\Gamma _e$ .",
"Directly from Lemmas REF and REF together with Theorems REF and Theorem REF .",
"Lemma 7.7 $\\mathrm {RP}_e$ is valid in $\\mathcal {C}^\\omega $ for $e$ as in Lemma REF .",
"The argument in the proof of Lemma REF that $\\langle {\\cdot }\\rangle ^{\\Omega _e}_{F}=\\eta \\phi _{F,\\Omega }$ for $\\phi _{F,\\Omega }:=\\lambda y.Fy\\Omega _{e,F,y}$ is also valid in $\\mathcal {C}^\\omega $ , and a simpler version of the argument in the proof of Lemma REF verifies that there is some $n\\in \\mathbb {N}$ such that $\\phi _{F,\\Omega }(\\overline{{x},{n}})<n$ , and moreover $\\langle {x}\\rangle ^{\\Omega _e}_{F}=\\eta \\phi _{F,\\Omega }x=\\overline{{x},{m}}$ where $m\\in \\mathbb {N}$ is the least satisfying this property.",
"But since $\\phi _{F,\\Omega }(\\overline{{x},{m}})=\\phi _{F,\\Omega }(\\langle {x}\\rangle ^{\\Omega _e}_{F})=\\Omega _eFx$ and thus $\\Omega _eFx<m$ , it follows that $[{x}]_{\\Omega _eFx}=[{\\overline{{x},{m}}}]_{\\Omega _eFx}=[{\\langle {x}\\rangle ^{\\Omega _e}_{F}}]_{\\Omega _eFx}$ and so $\\mathrm {RP}_e$ is satisfied.",
"Theorem 7.8 For any type $\\theta $ and relation on $\\lhd $ such that induction over $\\lhd $ is provable in $\\mathrm {HA}^\\omega $ , the functional interpretation of (the negative translation of) $\\mathrm {LEX}_\\lhd $ can be solved by a term in $\\mathrm {HA}^\\omega +\\Omega _e+\\Gamma _e$ , provably in $\\mathrm {HA}^\\omega +\\Omega _e+\\Gamma _e+\\mathrm {RP}_e$ , for any closed term $e$ of System T. Moreover, defining $e$ as in Lemma REF , we have $\\mathcal {C}^\\omega \\models \\mathrm {HA}^\\omega +\\Omega _e+\\Gamma _e+\\mathrm {RP}_e$ .",
"The first claim follows directly from Theorem REF , and the second from Corollary REF and Lemma REF ."
],
[
"Conclusion and open questions",
"In this paper, we explored various notions of recursion over chain bounded partial orders, and gave a general theorem on solving the functional interpretation of an axiomatic, parametrised form of Zorn's lemma.",
"We intend this work to be taken as the starting point for a number of much broader research questions in both proof theory and computability theory, which we hope to pursue in the future.",
"These include the following: Can particular instances of $\\Phi $ and $\\Psi $ as in Section be connected to known forms of strong recursion, particularly variants of bar recursion?",
"We conjecture, for example, that $\\Omega _e$ and $\\Gamma _e$ as given in Section are definable using Spector's variant of bar recursion, using ideas from [21].",
"Are more general results along the lines of [6], [10], [19], [21] possible?",
"The relationship between our simple and controlled recursors has many parallels to that between modified bar recursion and Spector's variant.",
"It was shown in [6] that the former in fact defines the latter over System T. Under certain conditions, can we show that our simple recursor actually defines the controlled variant?",
"It was also shown in [6] that Spector's bar recursion is S1-S9 computable in $\\mathcal {C}^\\omega $ , but modified bar recursion is not.",
"Does an analogous result hold in our setting?",
"Can we formulate Theorems REF and REF so that they apply to non-continuous models, such as the majorizable functionals [12]?",
"What other applications of our abstract computational interpretation of Zorn's lemma are possible?",
"Are there cases where a sensible choice of the parameters could lead to a more concise formalisation of a well-known proof, and consequently a more natural and efficient extracted program?",
"In the other direction, can our framework be applied to give a computational interpretation to instances of Zorn's lemma stronger than even countable dependent choice?",
"If we were modify our formulation of Zorn's lemma so that chain boundedness is given as part of the syntactic definition, rather than being implicitly dealt with in some model, how would we then solve its functional interpretation?",
"Acknowledgements.",
"The author is grateful to the anonymous referees for their valuable corrections and suggestions."
]
] | 2001.03540 |
[
[
"A Non-Equilibrium Approach to Photon Emission from the Late Stages of\n Relativistic Heavy-Ion Collisions"
],
[
"Abstract Cross sections for photon production in hadronic scattering processes have been calculated according to an effective chiral field theory.",
"For $\\pi + \\rho \\to \\pi + \\gamma$ and $\\pi + \\pi \\to \\rho + \\gamma$ processes, these cross sections have been implemented into a novel hadronic transport approach (SMASH), which is suitable for collisions at low and intermediate energies.",
"The implementation is verified by systematically comparing the thermal photon rate to theoretical expectations.",
"The photon rates we obtain are compared to previous works, where scattering processes mediated by $\\omega$ mesons are found to contribute significantly to the total photon production.",
"Finally, the impact of considering the finite width of the $\\rho$ meson is investigated, and a significant enhancement of photon production in the low-energy region is observed.",
"This work is the first step towards a consistent treatment of photon emission in hybrid hydrodynamics+transport approaches.",
"The quantification of the importance of the hadronic stage for the resolution of the direct photon flow puzzle is a next step and can be applied to identify equilibrium and non-equilibrium effects in the hadronic afterburner."
],
[
"Introduction",
"Photons are direct and unique probes in heavy-ion collisions.",
"They escape the fireball unaffected owing to their purely electromagnetic coupling and are produced in all stages of the collision.",
"Hence, they draw a time-integrated picture of the entire evolution, carrying properties from the medium to the detector.",
"It is therefore essential to fully understand the production mechanisms and properties of photons in each stage of the collision.",
"This work focuses on the late stages, where the fireball is believed to have expanded and cooled down sufficiently to find quarks and gluons confined again into hadrons.",
"For this purpose, a non-equilibrium approach for photon emission from hadronic interactions is presented."
],
[
"Model Description",
"The photon production framework introduced in the following is based on the SMASH (Simulating Many Accelerated Strongly-interacting Hadrons) hadronic transport approach [1], [2].",
"It is designed for the description of heavy-ion collisions at low and intermediate energies.",
"Furthermore it was successfully applied as an hadronic afterburner [3], [4] which allows for a non-equilibrium study of the late stages of relativistic heavy-ion collisions.",
"In SMASH, photon production in binary, hadronic scattering processes is implemented based on an effective chiral field theory with mesonic degrees of freedom [5].",
"Among those are pseudoscalar mesons, vector mesons, axial vector mesons and the photon.",
"The corresponding Lagrangian reads $\\mathcal {L} = & \\ \\dfrac{1}{8} \\ F_{\\pi }^2 \\ Tr\\left(D_{\\mu }UD^{\\mu }U^{\\dagger }\\right) \\ + \\ \\dfrac{1}{8} \\ F_{\\pi }^2 \\ Tr\\left(M\\left(U+U^{\\dagger }-2\\right)\\right) - \\ \\dfrac{1}{2} \\ Tr \\left(F_{\\mu \\nu }^{L}F^{L \\mu \\nu } \\ + \\ F_{\\mu \\nu }^{R} \\ F^{R \\mu \\nu } \\right) + \\\\&\\ m_0^2 \\ Tr\\left(A_{\\mu }^LA^{L\\mu } + A_{\\mu }^RA^{R\\mu } \\right) \\ + \\ \\gamma \\ Tr \\left(F_{\\mu \\nu }^{L}U F^{R \\mu \\nu } U^{\\dagger }\\right) \\ - \\ i \\xi \\ Tr \\left(D_{\\mu }UD_{\\nu }U^{\\dagger }F^{L\\mu \\nu } + D_{\\mu }U^{\\dagger }D_{\\nu }UF^{R\\mu \\nu } \\right) - \\\\&\\ \\dfrac{2em_V^2}{\\tilde{g}} B_{\\mu } \\ Tr \\Bigl (Q\\tilde{V}^{\\mu }\\Bigr ) \\ - \\ \\dfrac{1}{4}\\left(\\partial _{\\mu } B^{\\nu } - \\partial _{\\nu } B^{\\mu }\\right)^2 + \\ \\dfrac{2 e^2 m_0^2}{g_0^2} B_{\\mu }B^{\\mu } Tr\\left(Q^2\\right)+ \\ g_{VV\\phi } \\ \\varepsilon _{\\mu \\nu \\alpha \\beta } \\ Tr \\Big [\\partial ^{\\mu } V^{\\nu } \\partial ^{\\alpha } V^{\\beta } \\phi \\Big ]$ and contains the couplings and interaction of the aforementioned particles.",
"For further details about the underlying theoretical framework the reader is referred to [6].",
"Based on the properties enclosed in Equation (REF ), it is possible to derive the Feynman rules, matrix elements and cross sections for the photon producing scattering processes.",
"We limit our explorations to processes of the kind $\\ \\pi + \\pi \\rightarrow \\rho + \\gamma \\ $ and $\\ \\pi + \\rho \\rightarrow \\pi + \\gamma \\ $ as those provide the leading contributions.",
"The specific scattering processes considered are collected in Table REF .",
"As in [6], they are grouped into processes mediated by ($\\pi $ , $\\rho $ , $a_1$ ) mesons (left) and those mediated by the $\\omega $ meson (right).",
"It is evident, that processes (e) and (g) as well as (d) and (h) differ only with respect to their mediating particles but have identical initial and final states.",
"Those processes are first treated separately, to allow for a straight-forward validation of the presented framework.",
"Once validated, these processes are incoherently added while accounting for their respective form factors, as described in [5].",
"It should be noted that within the afore presented framework, the $\\rho $ meson is assumed to be a stable particle with vanishing width.",
"It is however known that in reality, $\\Gamma _\\rho $ = 0.149 GeV [7], such that a more realistic description is necessary.",
"An attempt is made to extend the presented framework to also describe broad $\\rho $ mesons.",
"The SMASH resonance treatment is employed and the $\\rho $ meson mass is sampled from a Breit-Wigner distribution.",
"The initial or final state $\\rho $ masses enter the computation of the photon cross sections directly, higher-order corrections are however not introduced.",
"The disagreement between the $\\rho $ meson mass in the loops and the in-/outgoing $\\rho $ meson masses introduces a systematic error which is found to be smaller than 11% in the configurations tested.",
"Further details are explained in [5].",
"Table: Photon production processes by mediating particles.",
"Processes mediated by (π\\pi , ρ\\rho , a 1 a_1) mesons are displayed on the left, those mediated by the ω\\omega meson on the right."
],
[
"Results",
"The cross sections [8] as well as the photon framework are subject to a proof of concept in Figure REF .",
"The thermal photon rate as obtained from SMASH (lines) is compared to the corresponding semi-analytical expectation values (bands) for each of the processes listed in Table REF .",
"The photon rates are extracted in an infinite matter simulation with SMASH at a temperature of 150 MeV.",
"Error bands of the theoretical expectation are due to uncertainties in the temperature extraction of the medium.",
"Within errors, a perfect agreement can be observed between the photon rates resulting from SMASH and those from the semi-analytical benchmark; hence verifying the presented framework.",
"Figure: Thermal photon rates extracted from SMASH (lines) as compared to semi-analytical theoretical expectations (band) at T = 150 MeV for processes (a)-(g).",
"(π\\pi , ρ\\rho , a 1 a_1) mediated processes are displayed on the left, ω\\omega mediated processes on the right.",
"Form factors are not applied and the ρ\\rho meson is treated as a stable particle.To assess the significance of introducing additional degrees of freedom, a comparison with previous works [9], as similarly done in [10], is performed.",
"In contrast to the above presented framework, the degrees of freedom therein are limited to $\\pi $ , $\\rho $ , $\\eta $ mesons and the photon, thus lacking processes mediated by $\\omega $ and $a_1$ mesons.",
"In Figure REF a comparison of the photon rates obtained from SMASH (including form factor corrections and having combined processes (e) and (g) as well as (d) and (h)) to parametrizations of the photon rates from [9], [11] is shown.",
"It can be observed that the photon rates determined within [9] yield smaller contributions than SMASH for $E \\lesssim 0.4$ GeV and vice versa.",
"Especially for high photon energies, this discrepancy becomes more pronounced.",
"It can be deduced that for a realistic description of mesonic photon production, $\\omega $ and $a_1$ mediated processes should not be neglected.",
"Similar observations have already been made in [12].",
"As mentioned above, the presented framework is capable of describing stable $\\rho $ mesons only but can, under certain assumptions, be extended to finite-width $\\rho $ mesons.",
"The effect on the thermal photon rate of treating the $\\rho $ meson as a resonance is shown in Figure REF , where the dashed lines correspond to $\\Gamma _\\rho $ = 0 GeV and the solid lines to $\\Gamma _\\rho $ = 0.149 GeV.",
"Only a minor effect is found for $\\pi + \\pi \\rightarrow \\rho + \\gamma $ processes, whereas there is a significant increase of photon production for $\\pi + \\rho \\rightarrow \\pi + \\gamma $ processes especially in the low- and mid-energy regime.",
"The latter can be explained by the reduced kinematic threshold for the occurrence of such a scattering process since $\\rho $ mesons below the pole mass are also accessible once the finite width is taken into consideration.",
"Whether or not this effect is visible in final particle spectra of simulated heavy-ion collisions is however yet to be assessed as it highly depends on the energy of the produced photons.",
"Figure: Comparison of thermal photon rates for Γ ρ \\Gamma _\\rho = 0 GeV (dashed) and Γ ρ \\Gamma _\\rho = 0.149 GeV (solid) for π+π→ρ+γ\\pi + \\pi \\rightarrow \\rho + \\gamma (orange) and π+ρ→π+γ\\pi + \\rho \\rightarrow \\pi + \\gamma (blue) processes at T = 150 MeV.",
"Form factors are included."
],
[
"Conclusion and Outlook",
"Cross sections for the production of photons in hadronic scattering processes have been derived from an effective chiral field theory with mesonic degrees of freedom and implemented into the SMASH transport model.",
"A proof of concept has been successfully performed by means of a systematic comparison to semi-analytically calculated theoretical expectation values for the thermal photon rate.",
"Comparisons to previous works describing photon production in hadronic matter have demonstrated the importance of including $\\omega $ and $a_1$ mediated production processes.",
"An extension of the presented framework to finite-width $\\rho $ mesons has further shown a significant increase of photon production in $\\pi + \\rho \\rightarrow \\pi + \\gamma $ scattering processes in the low- and intermediate-energy range.",
"This work is a first step to consistently treat photon emission in hybrid hydrodynamics+transport approaches.",
"The 3+1D viscous hydrodynamics code MUSIC [13], [14] and SMASH [1], [2] are particularly well suited for such an approach as both rely on the same underlying field theory for photon production in the hadronic phase.",
"Not only can it be applied to describe a heavy-ion collision at RHIC/LHC energies, but also to benchmark the importance of a non-equilibrium treatment in the late stages of relativistic heavy-ion collisions as compared to a pure hydrodynamic evolution.",
"To this end, the incorporation of bremsstrahlung contributions is essential and constitutes a next step.",
"Such a hybrid model provides a great opportunity to identify and study non-equilibrium effects in the afterburner and to assess the importance of the late stages to contribute to the resolution of the direct photon flow puzzle [15].",
"Acknowledgements This project was supported by the DAAD funded by BMBF with Project-ID 57314610.",
"A.S. acknowledges support by the Stiftung Polytechnische Gesellschaft Frankfurt am Main.",
"C.G.",
"is supported in part by the Natural Sciences and Engineering Research Council of Canada.",
"H.E.",
"and J.M.T.R.",
"acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) through the Grant No.",
"CRC-TR 211 “Strong-interaction matter under extreme conditions”.",
"J.M.T.R.",
"was further supported by the DFG through Grant No.",
"411563442.",
"Computational resources have been provided by the Center for Scientific Computing (CSC) at the Goethe- University of Frankfurt and the GreenCube at GSI."
]
] | 2001.03378 |
[
[
"Multi-layer Optimizations for End-to-End Data Analytics"
],
[
"Abstract We consider the problem of training machine learning models over multi-relational data.",
"The mainstream approach is to first construct the training dataset using a feature extraction query over input database and then use a statistical software package of choice to train the model.",
"In this paper we introduce Iterative Functional Aggregate Queries (IFAQ), a framework that realizes an alternative approach.",
"IFAQ treats the feature extraction query and the learning task as one program given in the IFAQ's domain-specific language, which captures a subset of Python commonly used in Jupyter notebooks for rapid prototyping of machine learning applications.",
"The program is subject to several layers of IFAQ optimizations, such as algebraic transformations, loop transformations, schema specialization, data layout optimizations, and finally compilation into efficient low-level C++ code specialized for the given workload and data.",
"We show that a Scala implementation of IFAQ can outperform mlpack, Scikit, and TensorFlow by several orders of magnitude for linear regression and regression tree models over several relational datasets."
],
[
"Introduction",
"The mainstream approach in supervised machine learning over relational data is to specify the construction of the data matrix followed by the learning task in scripting languages such as Python, MATLAB, Julia, or R using software environments such as Jupyter notebook.",
"These environments call libraries for query processing, e.g., Pandas or SparkSQL , or database systems, e.g., PostegreSQL .",
"The materialized training dataset then becomes the input to a statistical package, e.g., mlpack , scikit-learn , TensorFlow , and PyTorch ) that learns the model.",
"There are clear advantages to this approach: it is easy to use, allows for quick prototyping, and does not require substantial programming knowledge.",
"Although it uses libraries that provide efficient implementations for their functions (usually by binding to efficient low-level C code), they miss optimization opportunities for the end-to-end relational learning pipeline.",
"In particular, they fail to exploit the relational structure of the underlying data, which was removed by the materialization of the training dataset.",
"The effect is that the efficiency on one machine is often severely limited, with common deployments having to rely on expensive and energy-consuming clusters of machines to perform the learning task.",
"The thesis of this paper is that this performance limitation can be overcome by systematically optimizing the end-to-end relational learning pipeline as a whole.",
"We introduce Iterative Functional Aggregate Queries, or IFAQ for short, a framework that is designed to uniformly process and automatically optimize the various phases of the relational learning pipeline.",
"IFAQ takes as input a program that fully specifies both the data matrix construction and the model training in a dynamically-typed language called D-IFAQ.",
"This language captures a fragment of scripting languages such as Python that is used for rapid prototyping of machine learning models.",
"This is possible thanks to the iterative and collection-oriented constructs provided by D-IFAQ.",
"An IFAQ program is optimized by multiple compilation layers, whose optimizations are inspired by techniques developed by the data management, programming languages, and high-performance computing communities.",
"IFAQ performs automatic memoization, which identifies code fragments that can be expressed as batches of aggregate queries.",
"This optimization further enables non-trivial loop-invariant code motion opportunities.",
"The program is then compiled down to a statically-typed language, called S-IFAQ, which benefits from further optimization stages, including loop transformations such as loop fusion and code motion.",
"IFAQ investigates optimization opportunities at the interface between the data matrix construction and learning steps and interleaves the code for both steps by pushing the data-intensive computation from the second step past the joins of the first step, inspired by recent query processing techniques , .",
"The outcome is a highly optimized code with no separation between query processing and machine learning.",
"Finally, IFAQ compiles the optimized program into low-level C++ code that further exploits data-layout optimizations.",
"IFAQ can outperform equivalent solutions by orders of magnitude.",
"The contributions of this paper are as follows: [leftmargin=.1in] Section introduces the IFAQ framework, which comprises languages and compiler optimizations that are designed for efficient end-to-end relational learning pipelines.",
"As proof of concept, Section demonstrates IFAQ for two popular models: linear regression and regression tree.",
"Section shows how to systematically optimize an IFAQ program in several stages.",
"Section benchmarks IFAQ, mlpack, TensorFlow, and scikit-learn.",
"It shows that IFAQ can train linear regression and regression tree models over two real datasets orders-of-magnitude faster than its competitors.",
"In particular, IFAQ learns the models faster than it takes the competitors to materialize the training dataset.",
"Section further shows the performance impact of individual optimization layers.",
"Figure: A relational learning task is expressed in D-IFAQ, transformed into an optimized S-IFAQ expression, and compiled to efficient C++ code."
],
[
"Overview",
"Figure REF depicts the IFAQ workflow.",
"Data scientists use software environments such as Jupyter Notebook to specify relational learning programs.",
"In our setting, such programs are written in a dynamically-typed language called D-IFAQ and subject to high-level optimizations (Section REF ).",
"Given the schema information of the dataset, the optimized program is translated into a statically-typed language called S-IFAQ (Section REF ).",
"If there are type errors, they are reported to the user.",
"IFAQ performs several optimizations inspired by database query processing techniques (Section REF ).",
"Finally, it synthesizes appropriate data layouts, resulting in efficient low-level C/C++ code (Section REF )."
],
[
"IFAQ Core Language",
"The grammar of the IFAQ core language is given in Figure REF .",
"This functional language support the following data types.",
"The first category consists of numeric types as well as categorical types (e.g., boolean values, string values, and other custom enum types).",
"Furthermore, for categorical types the other alternative is to one-hot encode them, the type of which is represented as $\\mathbb {R}^{\\text{$n$}}_{\\text{$T_1$}}$.",
"This type represents an array of $n$ real numbers each one corresponding to a particular value in the domain of $T_1$ .",
"In the one-hot encoding of a value of type $T_1$ , only the $i^{th}$ value is 1 and the rest are zero.",
"However, a value of type denoting the one-hot encoding of $\\mathbb {R}^{\\text{$n$}}_{\\text{$T_1$}}$, can take arbitrary real numbers at each position.",
"The second category consists of record types, which are similar to structs in C; the record values contain various fields of possibly different data types.",
"The partial records, where some fields have no values are referred to as variants.",
"Figure: Grammar of IFAQ core language.The final category consists of collection data types such as (ordered) sets and dictionaries.",
"Database relations are represented as dictionaries (mapping elements to their multiplicities).",
"In D-IFAQ, the elements of collections can have different types.",
"However, in S-IFAQ the elements of collections should have the same data type.",
"In order to distinguish the variables with collection data type with other types of variables, we denote them as x and $x$ , respectively.",
"The top-level program consists of several initialization expressions, followed by a loop.",
"This enables IFAQ to express iterative algorithms such as gradient descent optimization algorithms.",
"The rest of the constructs of this language are as follows: [label=()] various binary and unary operations on both scalar and collection data structures,More specifically ring-based operations.",
"let binding for assigning the value of an expression to a variable, conditional expressions $\\sum \\nolimits _{x \\in \\text{$\\text{\\text{e}}_{1}$}}\\text{$\\text{\\text{e}}_{2}$}$ : the summation operator in order to iterate over the elements of a collection ($\\text{\\text{e}}_{1}$) and perform a stateful computation using a particular addition operator (e.g., numerical addition, set union, bag union),This operator could be every monoid-based operator.",
"Thus, even computing the minimum of two numbers is an addition operator.",
"$\\operatornamewithlimits{{{\\lambda }}}\\nolimits _{x \\in \\text{$\\text{\\text{e}}_{1}$}}\\text{$\\text{\\text{e}}_{2}$}$ : constructing a dictionary where the key domain is $\\text{\\text{e}}_{1}$ and the value for each key $x$ is constructed using $\\text{\\text{e}}_{2}$, constructing sets ([[e]]) and dictionaries ({{e $\\rightarrow $e}}) given a list of elements, dom(e): getting the domain of a dictionary, $\\text{\\text{e}}_{0}$($\\text{\\text{e}}_{1}$): retrieving the associatiated value to the given key in a dictionary, constructing records ($\\texttt {\\lbrace }\\overrightarrow{x=e}\\texttt {\\rbrace }$ ) and variants ($\\texttt {<}x=e\\texttt {>}$ ), statically (e.f) and dynamically (e[f]) assessing the field of a record or a variant.",
"Applications As proof of concept, we show in this section how linear regression and regression tree models can be expressed in D-IFAQ.",
"We assume that the model is learned over a training dataset Q with features $a_1,\\ldots , a_n$ and label $a_{n+1}$ , which is the output of a query over a database with relations $\\mathbf {R}_1, \\ldots , \\mathbf {R}_m$ .",
"Further applications that can benefit from IFAQ include supervised learning problems (e.g., logistic regression, factorization machines, and support vector machines), as well as unsupervised learning problems (e.g., k-means clustering, principal component analysis, and learning Bayesian networks via Chow-Liu trees).",
"IFAQ further supports different optimization algorithms, e.g., batch or stochastic gradient descent, coordinate descent, and alternating minimization.",
"Linear Regression: A linear regression model is given by: $LR(x) &= \\sum \\nolimits _{f \\in \\textbf {F}} \\mathbf {\\theta }(f) \\text{$*$} x[f]$ where $x$ is a record with fields $\\textbf {F} = \\lbrace a_1,\\ldots , a_n\\rbrace $ , and $\\mathbf {\\theta }$ is a dictionary with these fields as its keys.",
"The $\\mathbf {\\theta }$ dictionary defines the parameters of the model, and the output of $LR(x)$ is a prediction for the label given input $x$ .",
"Linear regression models require categorical features to be one-hot encoded.",
"Thus, for a categorical feature $a_i$ , the field $x[a_i]$ encodes a vector of one indicator value for each value in the domain of $a_i$ (Section REF ).",
"The corresponding parameter element $\\mathbf {\\theta }(a_i)$ encode a vector of parameters which is of the same dimension as $x[a_i]$ .",
"We assume without loss of generality that $x[a_1]$ only takes value 1 and then $\\mathbf {\\theta }(a_1)$ is the intercept of the model.",
"Given the training dataset Q, the error of fitting $LR$ to Q is given by the sum of the least squares loss function:It is common to penalize $J(\\mathbf {\\theta })$ with a regularization term.",
"IFAQ supports regularization as well, but we omit it from the examples to avoid clutter.",
"$J(\\mathbf {\\theta })&= \\frac{1}{2|\\textbf {Q}|} \\sum _{x \\in \\texttt {dom(}\\textbf {Q}\\texttt {)}} \\big (\\sum _{f \\in \\textbf {F}} \\mathbf {\\theta }(f)\\text{$*$} x[f] - x[a_{n+1}]\\big )^2$ We minimize $J(\\mathbf {\\theta })$ using batch gradient descent (BGD) optimization, which repeatedly updates each parameter $\\mathbf {\\theta }(f)$ by learning rate $\\alpha $ in the direction of the partial derivative of $J(\\mathbf {\\theta })$ w.r.t.",
"$\\mathbf {\\theta }(f)$ until convergence.",
"This is represented in the following D-IFAQ program (Q is expressed in IFAQ as a query over a multi-relational database): $&\\texttt {let }\\textbf {F} = \\texttt {[\\!",
"[}a_1,a_2,\\ldots ,a_n\\texttt {]\\!",
"]}\\text{ }\\texttt {in }\\\\&\\mathbf {\\theta } \\leftarrow \\mathbf {\\theta }_0\\\\&\\texttt {while(}\\texttt { not converged }{}\\texttt {)\\;\\lbrace }\\\\&\\quad \\mathbf {\\theta } = \\text{$\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\lambda $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}${\\sbox {\\scriptstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}}{\\sbox {\\scriptscriptstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}$ f1 F$$ ((f1) - |Q|$\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\Sigma $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\Sigma $}}}$ x dom(Q)$$ Q(x) *($\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\Sigma $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\Sigma $}}}$ f2 F$$ (f2) * x[f2] - x[an+1])* x[f1]) } This program is inefficient, because it computes for each BGD iteration all pairwise products $x[f_1] * x[f_2]$ .",
"These products, however, are invariant, and can be computed once outside the while loop.",
"This is the main high-level optimization performed by IFAQ for this case (Section REF ).",
"The rewriting would gather a collection of these products, which can be computed once as database aggregate queries.",
"The following BGD iterations can then be computed directly over these aggregates, and do not require a pass over $\\textbf {Q}$ .",
"The collection of these aggregates represents the non-centered covariance matrix (or covar matrix for short).",
"The aggregates can then be further optimized using ideas from the database query processing, in particular they can be computed directly over the input database and without materializing $\\textbf {Q}$ (Section REF ).",
"We next provide a running example which is used to explain the optimization rewritings in the following sections.",
"In retail forecasting scenarios, the goal is to learn a model which can predict future sales for items at different stores.",
"We assume the retailer has a database with three relations: Sales(item, store, units), StoRes(store, city), Items(item, price).",
"The goal is to learn a model that predicts $u$ with the features $\\textbf {F} = \\lbrace i, s, c, p\\rbrace $ .",
"The training dataset is given by the result of the join of the three relations: $\\textbf {Q}=\\textbf {S}\\bowtie \\textbf {R}\\bowtie \\textbf {I}$ .",
"We learn the model with BGD using the D-IFAQ program above.",
"To avoid clutter, we focus on the core computation of the program and we make two simplifications: (1) we assume $\\frac{\\alpha }{|\\textbf {Q}|} = 1$ , and (2) we hide the term for $x[a_{n+1}]$ .",
"Then the inner-loop expression is given by: $&\\mathbf {\\theta } = \\text{$\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\lambda $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}${\\sbox {\\scriptstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}}{\\sbox {\\scriptscriptstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}$ f1 F$$ ( (f1) - $\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\Sigma $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\Sigma $}}}$ x dom(Q)$$ Q(x) * ($\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\Sigma $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\Sigma $}}}$ f2 F$$ (f2) *x[f2]) * x[f1]) where $\\textbf {F} = \\texttt {[\\!",
"[}\\texttt {`}i\\texttt {`}, \\texttt {`}s\\texttt {`}, \\texttt {`}c\\texttt {`}, \\texttt {`}p\\texttt {`}\\texttt {]\\!",
"]}$ .",
"Figure: Transformation steps for an expression written in IFAQ.Decision Tree.",
"We next consider learning decision trees using the CART algorithm .",
"With our optimizations, we can learn both classification and regression trees efficiently.",
"In the following, we focus on regression trees.",
"A regression tree model is a (binary) tree with inner nodes representing conditional control statements to model decisions and their consequences.",
"Given an element $x$ in the dataset Q, the condition for a feature $f$ is of the form $x[f] \\text{ op } t$ , which is denoted as $c(f, \\text{op}, t)$ here.",
"For categorical features (e.g., city), $t$ may be a set of categories and $\\text{op}$ denotes inclusion.",
"For continuous features (e.g., price), $t$ is a real number and $\\text{op}$ is inequality.",
"Leaf nodes represent predictions for the label.",
"For regression trees, the prediction is the average of the label values in the fragment of the training dataset that satisfies all control statements on the root to leaf path.",
"For a given node $N$ , let $\\delta $ encode the conjunction of all conditions along the path from the root to $N$ .",
"The CART algorithm recursively seeks for the condition that minimises the following optimization problem for a given cost function: $\\min _{f \\in \\mathbf {F}} \\min _{t \\in \\mathbf {T}_f} \\texttt {cost}(\\textbf {Q}, \\delta \\wedge c(f, \\text{op}, t)) + \\texttt {cost}(\\textbf {Q}, \\delta \\wedge c(f, \\text{!op}, t)).$ where !op denotes the negation of op, $\\mathbf {T}_f$ is the set of all possible thresholds for $f$ .",
"Once this condition is found, a new node with condition $c(f, \\text{op}, t)$ is constructed and the algorithm recursively constructs the next node for each child.",
"For regression trees, the cost is given by the variance, which is represented as the following D-IFAQ expression: $\\texttt {cost}(\\textbf {Q}, \\delta ^{\\prime })=&\\sum \\limits _{x\\in \\texttt {dom(}\\textbf {Q}\\texttt {)}}\\textbf {Q}(x)\\text{$*$} {}x[a_{n+1}]^2\\text{$*$} {}\\delta ^{\\prime }-\\\\&\\frac{1}{\\sum \\limits _{x\\in \\texttt {dom(}\\textbf {Q}\\texttt {)}}\\textbf {Q}(x)\\text{$*$} {}\\delta ^{\\prime }}(\\sum \\limits _{x\\in \\texttt {dom(}\\textbf {Q}\\texttt {)}}\\textbf {Q}(x)\\text{$*$} x[a_{n+1}]\\text{$*$} {}\\delta ^{\\prime })^2$ In this formula, $\\delta ^{\\prime }$ can be $\\delta \\wedge c(f, \\text{op}, t)$ or $\\delta \\wedge c(f, \\text{!op}, t)$ .",
"In contrast to linear regression, all aggregates depend on node-specific information ($\\delta ^{\\prime }$ ).",
"It is thus not possible to hoist and compute them only once for all recursions of the CART algorithm.",
"Nevertheless, all other optimizations presented in the next section are applicable.",
"Optimizations This section details the IFAQ optimizations.",
"Figure REF overviews the transformations applied at different stages.",
"High-Level Optimizations The high-level optimizations applied to the D-IFAQ expressions are: normalization, loop scheduling, factorization, static memoization, and loop-invariant code motion.",
"The first two transformations are preprocessing steps for factorization.",
"The impact of static memoization becomes positive once it is combined with loop-invariant code motion.",
"Normalization.",
"This transformation brings the expressions into a normalized form.",
"Similar to other algebraic-based systems (e.g., logical circuits) the normal form is sum of products.",
"Thus, this transformation applies distributivity and pushes the products inside summation (Figure REF ).",
"The inner-loop of our running example: Table: Acknowledgments"
],
[
"Applications",
"As proof of concept, we show in this section how linear regression and regression tree models can be expressed in D-IFAQ.",
"We assume that the model is learned over a training dataset Q with features $a_1,\\ldots , a_n$ and label $a_{n+1}$ , which is the output of a query over a database with relations $\\mathbf {R}_1, \\ldots , \\mathbf {R}_m$ .",
"Further applications that can benefit from IFAQ include supervised learning problems (e.g., logistic regression, factorization machines, and support vector machines), as well as unsupervised learning problems (e.g., k-means clustering, principal component analysis, and learning Bayesian networks via Chow-Liu trees).",
"IFAQ further supports different optimization algorithms, e.g., batch or stochastic gradient descent, coordinate descent, and alternating minimization.",
"Linear Regression: A linear regression model is given by: $LR(x) &= \\sum \\nolimits _{f \\in \\textbf {F}} \\mathbf {\\theta }(f) \\text{$*$} x[f]$ where $x$ is a record with fields $\\textbf {F} = \\lbrace a_1,\\ldots , a_n\\rbrace $ , and $\\mathbf {\\theta }$ is a dictionary with these fields as its keys.",
"The $\\mathbf {\\theta }$ dictionary defines the parameters of the model, and the output of $LR(x)$ is a prediction for the label given input $x$ .",
"Linear regression models require categorical features to be one-hot encoded.",
"Thus, for a categorical feature $a_i$ , the field $x[a_i]$ encodes a vector of one indicator value for each value in the domain of $a_i$ (Section REF ).",
"The corresponding parameter element $\\mathbf {\\theta }(a_i)$ encode a vector of parameters which is of the same dimension as $x[a_i]$ .",
"We assume without loss of generality that $x[a_1]$ only takes value 1 and then $\\mathbf {\\theta }(a_1)$ is the intercept of the model.",
"Given the training dataset Q, the error of fitting $LR$ to Q is given by the sum of the least squares loss function:It is common to penalize $J(\\mathbf {\\theta })$ with a regularization term.",
"IFAQ supports regularization as well, but we omit it from the examples to avoid clutter.",
"$J(\\mathbf {\\theta })&= \\frac{1}{2|\\textbf {Q}|} \\sum _{x \\in \\texttt {dom(}\\textbf {Q}\\texttt {)}} \\big (\\sum _{f \\in \\textbf {F}} \\mathbf {\\theta }(f)\\text{$*$} x[f] - x[a_{n+1}]\\big )^2$ We minimize $J(\\mathbf {\\theta })$ using batch gradient descent (BGD) optimization, which repeatedly updates each parameter $\\mathbf {\\theta }(f)$ by learning rate $\\alpha $ in the direction of the partial derivative of $J(\\mathbf {\\theta })$ w.r.t.",
"$\\mathbf {\\theta }(f)$ until convergence.",
"This is represented in the following D-IFAQ program (Q is expressed in IFAQ as a query over a multi-relational database): $&\\texttt {let }\\textbf {F} = \\texttt {[\\!",
"[}a_1,a_2,\\ldots ,a_n\\texttt {]\\!",
"]}\\text{ }\\texttt {in }\\\\&\\mathbf {\\theta } \\leftarrow \\mathbf {\\theta }_0\\\\&\\texttt {while(}\\texttt { not converged }{}\\texttt {)\\;\\lbrace }\\\\&\\quad \\mathbf {\\theta } = \\text{$\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\lambda $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}${\\sbox {\\scriptstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}}{\\sbox {\\scriptscriptstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}$ f1 F$$ ((f1) - |Q|$\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\Sigma $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\Sigma $}}}$ x dom(Q)$$ Q(x) *($\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\Sigma $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\Sigma $}}}$ f2 F$$ (f2) * x[f2] - x[an+1])* x[f1]) } This program is inefficient, because it computes for each BGD iteration all pairwise products $x[f_1] * x[f_2]$ .",
"These products, however, are invariant, and can be computed once outside the while loop.",
"This is the main high-level optimization performed by IFAQ for this case (Section REF ).",
"The rewriting would gather a collection of these products, which can be computed once as database aggregate queries.",
"The following BGD iterations can then be computed directly over these aggregates, and do not require a pass over $\\textbf {Q}$ .",
"The collection of these aggregates represents the non-centered covariance matrix (or covar matrix for short).",
"The aggregates can then be further optimized using ideas from the database query processing, in particular they can be computed directly over the input database and without materializing $\\textbf {Q}$ (Section REF ).",
"We next provide a running example which is used to explain the optimization rewritings in the following sections.",
"In retail forecasting scenarios, the goal is to learn a model which can predict future sales for items at different stores.",
"We assume the retailer has a database with three relations: Sales(item, store, units), StoRes(store, city), Items(item, price).",
"The goal is to learn a model that predicts $u$ with the features $\\textbf {F} = \\lbrace i, s, c, p\\rbrace $ .",
"The training dataset is given by the result of the join of the three relations: $\\textbf {Q}=\\textbf {S}\\bowtie \\textbf {R}\\bowtie \\textbf {I}$ .",
"We learn the model with BGD using the D-IFAQ program above.",
"To avoid clutter, we focus on the core computation of the program and we make two simplifications: (1) we assume $\\frac{\\alpha }{|\\textbf {Q}|} = 1$ , and (2) we hide the term for $x[a_{n+1}]$ .",
"Then the inner-loop expression is given by: $&\\mathbf {\\theta } = \\text{$\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\lambda $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}${\\sbox {\\scriptstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}}{\\sbox {\\scriptscriptstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\lambda $}}}$ f1 F$$ ( (f1) - $\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\Sigma $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\Sigma $}}}$ x dom(Q)$$ Q(x) * ($\\mathop {\\vphantom{\\sum }\\mathchoice{\\sbox {\\displaystyle \\sum }\\hbox{{.9}{!",
"}{$\\m@th \\Sigma $}}}{}{}{}}{\\sbox {\\textstyle \\sum }\\hbox{{}{!",
"}{$\\m@th \\Sigma $}}}$ f2 F$$ (f2) *x[f2]) * x[f1]) where $\\textbf {F} = \\texttt {[\\!",
"[}\\texttt {`}i\\texttt {`}, \\texttt {`}s\\texttt {`}, \\texttt {`}c\\texttt {`}, \\texttt {`}p\\texttt {`}\\texttt {]\\!",
"]}$ .",
"Figure: Transformation steps for an expression written in IFAQ.Decision Tree.",
"We next consider learning decision trees using the CART algorithm .",
"With our optimizations, we can learn both classification and regression trees efficiently.",
"In the following, we focus on regression trees.",
"A regression tree model is a (binary) tree with inner nodes representing conditional control statements to model decisions and their consequences.",
"Given an element $x$ in the dataset Q, the condition for a feature $f$ is of the form $x[f] \\text{ op } t$ , which is denoted as $c(f, \\text{op}, t)$ here.",
"For categorical features (e.g., city), $t$ may be a set of categories and $\\text{op}$ denotes inclusion.",
"For continuous features (e.g., price), $t$ is a real number and $\\text{op}$ is inequality.",
"Leaf nodes represent predictions for the label.",
"For regression trees, the prediction is the average of the label values in the fragment of the training dataset that satisfies all control statements on the root to leaf path.",
"For a given node $N$ , let $\\delta $ encode the conjunction of all conditions along the path from the root to $N$ .",
"The CART algorithm recursively seeks for the condition that minimises the following optimization problem for a given cost function: $\\min _{f \\in \\mathbf {F}} \\min _{t \\in \\mathbf {T}_f} \\texttt {cost}(\\textbf {Q}, \\delta \\wedge c(f, \\text{op}, t)) + \\texttt {cost}(\\textbf {Q}, \\delta \\wedge c(f, \\text{!op}, t)).$ where !op denotes the negation of op, $\\mathbf {T}_f$ is the set of all possible thresholds for $f$ .",
"Once this condition is found, a new node with condition $c(f, \\text{op}, t)$ is constructed and the algorithm recursively constructs the next node for each child.",
"For regression trees, the cost is given by the variance, which is represented as the following D-IFAQ expression: $\\texttt {cost}(\\textbf {Q}, \\delta ^{\\prime })=&\\sum \\limits _{x\\in \\texttt {dom(}\\textbf {Q}\\texttt {)}}\\textbf {Q}(x)\\text{$*$} {}x[a_{n+1}]^2\\text{$*$} {}\\delta ^{\\prime }-\\\\&\\frac{1}{\\sum \\limits _{x\\in \\texttt {dom(}\\textbf {Q}\\texttt {)}}\\textbf {Q}(x)\\text{$*$} {}\\delta ^{\\prime }}(\\sum \\limits _{x\\in \\texttt {dom(}\\textbf {Q}\\texttt {)}}\\textbf {Q}(x)\\text{$*$} x[a_{n+1}]\\text{$*$} {}\\delta ^{\\prime })^2$ In this formula, $\\delta ^{\\prime }$ can be $\\delta \\wedge c(f, \\text{op}, t)$ or $\\delta \\wedge c(f, \\text{!op}, t)$ .",
"In contrast to linear regression, all aggregates depend on node-specific information ($\\delta ^{\\prime }$ ).",
"It is thus not possible to hoist and compute them only once for all recursions of the CART algorithm.",
"Nevertheless, all other optimizations presented in the next section are applicable."
],
[
"Optimizations",
"This section details the IFAQ optimizations.",
"Figure REF overviews the transformations applied at different stages."
],
[
"High-Level Optimizations",
"The high-level optimizations applied to the D-IFAQ expressions are: normalization, loop scheduling, factorization, static memoization, and loop-invariant code motion.",
"The first two transformations are preprocessing steps for factorization.",
"The impact of static memoization becomes positive once it is combined with loop-invariant code motion.",
"Normalization.",
"This transformation brings the expressions into a normalized form.",
"Similar to other algebraic-based systems (e.g., logical circuits) the normal form is sum of products.",
"Thus, this transformation applies distributivity and pushes the products inside summation (Figure REF ).",
"The inner-loop of our running example: Table: Acknowledgments"
]
] | 2001.03541 |
[
[
"Chromospheric Heating by Acoustic Waves Compared to Radiative Cooling:\n II -- Revised Grid of Models"
],
[
"Abstract Acoustic and magnetoacoustic waves are considered to be possible agents of chromospheric heating.",
"We present a comparison of deposited acoustic energy flux with total integrated radiative losses in the middle chromosphere of the quiet Sun and a weak plage.",
"The comparison is based on a consistent set of high-resolution observations acquired by the IBIS instrument in the Ca II 854.2 nm line.",
"The deposited acoustic-flux energy is derived from Doppler velocities observed in the line core and a set of 1737 non-LTE 1D hydrostatic semi-empirical models, which also provide the radiative losses.",
"The models are obtained by scaling the temperature and column mass of five initial models VAL B-F to get the best fit of synthetic to observed profiles.",
"We find that the deposited acoustic-flux energy in the quiet-Sun chromosphere balances 30-50 % of the energy released by radiation.",
"In the plage, it contributes by 50-60 % in locations with vertical magnetic field and 70-90 % in regions where the magnetic field is inclined more than 50 degrees to the solar surface normal."
],
[
"Introduction",
"The energy released by radiation from the solar chromosphere is mostly concentrated in the strong lines of Ca 2, Mg2, and hydrogen Lyman-$\\alpha $ as well as in the H$^-$ continuum [24].",
"It is characterized by net radiative cooling rates (radiative losses).",
"The estimate of the total radiative losses integrated over the height of the chromosphere is 4300 W m$^{-2}$ in the quiet Sun [1] and it is by a factor of 2 to 4 higher in active regions [25].",
"This radiative cooling must be balanced by heating processes that deliver energy to the chromosphere.",
"The dominant process heating the upper layers of the solar atmosphere is still being intensively debated.",
"There are many agents considered by various studies [9], which could be grouped into two fundamentally different classes: (1) Mechanisms that are usually connected with heat releases during the reconnection processes in the magnetic field and (2) mechanisms that are associated with the deposit of energy by various kinds of waves.",
"In the present work, we focus on the latter class.",
"Waves are generated near the solar surface by turbulent motions of the convective plasma and propagate upwards, where they can dissipate a considerable part of their energy in the chromosphere.",
"In the magnetized solar atmosphere, there exist three principal types of MHD waves: Alfvén waves as well as fast and slow magnetoacoustic waves [10], [11].",
"These waves with magnetic component are considered extremely important.",
"Low-frequency waves may represent a significant source of energy, related to the so-called magnetic portals [8] or to magnetoacoustic-gravity waves [7].",
"These waves are generally not allowed to propagate higher into the atmosphere because their frequency does not exceed the expected photospheric cutoff frequency 5.2 mHz [2].",
"However, in regions where the photospheric magnetic field is inclined with respect to the gravity vector, the cutoff frequency can be lowered by means of the ramp effect [22].",
"This allows waves with frequencies far below 5.2 mHz, which would otherwise be trapped in the photosphere, to propagate into the upper atmosphere.",
"Recently, [15] used Solar Dynamics Observatory [14] data to discuss thoroughly the relations between magnetic field properties and the propagation of acoustic waves.",
"Sobotka16Paper I [20] studied the hypothesis of chromospheric heating by acoustic and magnetoacoustic waves.",
"This work was based on 70-minute long observations of the Ca2 854.2 nm near-infrared line with the Interferometric Bidimensional Spectrometer [5] at the Dunn Solar Telescope.",
"Sequences of Dopplergrams were used to compute energy flux carried by (magneto)acoustic waves and a simplistic grid of seven semi-empirical non-LTE (that is, with departures from the local thermodynamic equilibrium, LTE) hydrostatic models of the atmosphere to estimate the radiative losses.",
"Although the propagation and dissipation of waves in the chromosphere is generally a time-dependent process, 1D static models are still a reasonable tool to represent time-averaged physical conditions in long-lived structures of the solar atmosphere [6].",
"In Sobotka16 we found that a significant portion of the radiative losses could be replenished by the dissipative acoustic flux generated by the $p$ -modes, which were converted to magnetoacoustic modes in the inclined magnetic field.",
"We have shown that there was a correlation between the estimate of the acoustic flux dissipation in the region and the estimate of the radiative losses.",
"That provided an indication that there indeed was a contribution of the acoustic waves to the heating of the chromosphere, which should be considered.",
"A weak point of that work was a scarce grid of only seven atmospheric models, which were used to calculate the radiative cooling rates as well as the acoustic fluxes.",
"In this paper we present an analysis of the same observations as in Sobotka16, applying a much finer grid of models obtained by up- and downscaling the frequently used semi-empirical models of [24], to make sure that the comparison of deposited acoustic flux and radiative losses is more conclusive."
],
[
"Observations and data analysis",
"The target, a slowly decaying bipolar active region NOAA 11005 located at 25.2 N and 10.0 W (heliocentric angle $\\vartheta = 23$ ) was observed with IBIS on 2008 October 15 from 16:34 to 17:44 UT.",
"A pore surrounded by a superpenumbra [21] was present in the leading part of the region.",
"A weak chromospheric plage with equal polarity was located near the pore (see Figure 1 in Sobotka16).",
"The IBIS data set was described in detail in [19], [21], and Sobotka16.",
"Two spectral lines, Ca2 854.2 nm (intensity only) and Fe1 617.33 nm (full Stokes vector) were observed simultaneously.",
"The relevant parameters for both spectral lines are summarized in Table REF .",
"Table: Parameters of the IBIS data setThe data-processing procedures were described thoroughly by [19] and [21].",
"We concentrated on the region of interest (ROI) 137 $\\times $ 397 (82 $\\times $ 238 pixels) that included the plage eastward of the pore and a quiet-Sun region.",
"The location of ROI in the whole field of view is shown in Figure 1 of Sobotka16.",
"The following observables were obtained from the IBIS data set (see Sobotka16 for details): (1) The magnetic-field vector in the photosphere, retrieved from the full-Stokes spectral scan of Fe1 617.33 nm taken at 17:10 UT by means of the Stokes inversion code based on response functions [16].",
"Because the acoustic flux in the chromosphere is very sensitive to the inclination of magnetic field to the solar surface normal, we revise the calculations of Sobotka16 to avoid uncertainties in regions with weak fields, where the Stokes $Q$ , $U$ , and $V$ signals are dominated by noise.",
"In such cases, the code returns an unreliable line-of-sight (LOS) inclination near 90 (transversal direction), which can be removed by a mask that for the fields weaker than 350 G sets to zero all inclination angles in the range $\\pm 5$ from the transversal direction.",
"We update the method of removing the 180 ambiguity of the LOS azimuth, using the AMBIG code [12].",
"In the revised calculations, we also removed a bug in the transformation from the LOS reference frame to the local reference frame (LRF) that led to an overestimation of magnetic inclination in the plage.",
"We use routines from the AZAM code [13] for the transformation from the LOS reference frame to LRF.",
"(2) Time-dependent line-of-sight velocities, measured in the inner wings of the Ca2 854.2 nm core at $\\pm 18$ pm and in the line center.",
"According to [4], the inner wings are formed at approximately 900–1000 km above the optical depth $\\tau _{500} = 1$ and the line centre at 1400–1500 km.",
"These velocities are used to calculate power spectra of oscillations in two different layers of the chromosphere.",
"(3) Mean profiles of the Ca2 854.2 nm line, obtained by time-averaging over the 70-minute observing period of the observed profiles with removed Doppler shifts.",
"They are used to find the most appropriate semi-empirical models at each location in ROI.",
"The cadence of scans (52 s) and the length of the time-series sets the maximum detectable frequency of oscillations to 9.6 mHz and the frequency resolution to 0.24 mHz, suitable to analyze low-frequency waves.",
"Power spectra of the line-of-sight velocities were calculated using the standard Fourier analysis and the results were discussed in [21].",
"The acoustic energy fluxes at the heights of 900 km and 1500 km were estimated following [3].",
"The method consists in an integration over frequencies of the product of gas density, spectral power density, and group velocity of energy transport.",
"The frequency integration range spans between the acoustic cutoff frequency and the maximum detected frequency—see Sobotka16 for details.",
"The quantities depend on the gas density and pressure at the given height, which are taken from the model atmosphere.",
"The acoustic cutoff frequency is proportional to the cosine of the magnetic field inclination to the solar surface normal.",
"Taking into account that the acoustic flux estimated at 900 km approximates the incoming energy flux that may partly dissipate in the atmosphere, while the acoustic flux at 1500 km corresponds to the energy flux that has passed without dissipation, then the deposited acoustic flux in the chromospheric layers between 900 km and 1500 km is the difference between these two fluxes.",
"The corresponding net radiative cooling rates are integrated in the same range of heights."
],
[
"Grid of chromospheric models",
"The grid of models has been constructed keeping the number of free parameters as low as possible.",
"We scale the existing set of semi-empirical 1D hydrostatic models VAL A–F, which describe the solar atmosphere from intranetwork to bright network features, by changing independently the temperature and column mass stratifications.",
"Adopting the column mass as an independent variable makes it possible to conserve the condition of hydrostatic equilibrium in the scaled models.",
"Figure: Temperature stratifications of models VAL A–F normalized to VAL C(solid lines) in the range hh from 2100 km to 250 km.Dashed lines show the approximation by fourth-order polynomials.The original VAL models specify the column mass $m$ , optical depth $\\tau _{500}$ at 500 nm, temperature $T$ , microturbulent velocity $v_{\\rm t}$ , hydrogen density $n_{\\rm H}$ , electron density $n_{\\rm e}$ , total pressure $P_{\\rm tot}$ , gas pressure to total pressure ratio $P_{\\rm g}/P_{\\rm tot}$ , and density $\\rho $ in 52 geometrical heights $h$ ranging from $-75$ km to 2400–2700 km ($h = 0$ is at $\\tau _{500} = 1$ ).",
"The microturbulence stratification is practically equal in all models A–F at the heights from $-75$ km to 2000 km, that is, in the whole photosphere and chromosphere, and it is not changed in scaled models.",
"To improve numerical accuracy of the following computations, we resampled the models to 103 heights by means of the linear interpolation.",
"The scaling of these initial models is done in two steps: (1) $m$ and $T$ stratifications are changed and (2) the other model quantities are re-calculated.",
"Figure: Linear relations of the fourth-order polynomial coefficients a 1-4 a_{1-4}of the temperature scaling.",
"Diamonds show values correspondingto the models VAL A–F.We can find an adequate way to change $m$ and $T$ by adopting, for example, the VAL C model as the initial one and reproducing the stratifications of all the other VAL models.",
"In general, the scaling algorithm must be able to reproduce all the stratifications using any VAL model as the initial one.",
"The following solution was adopted: The initial column mass stratification $m_0(h)$ is changed for $h \\ge 450$ km ($m < 0.1$ g cm$^{-2}$ ) whereas it is kept unchanged in deeper layers.",
"The changes are controlled by scaling parameter $p_{\\rm m}$ , so that the scaled column mass is $m_x &=& b_x m_{0x}, \\ {\\rm where} \\\\b_x &=& p_{\\rm m} \\ {\\rm for} \\ h \\ge 2000 \\, {\\rm km}, \\\\b_x &=& p_{\\rm m} + (1-p_{\\rm m})\\, x/(x_{450}-x_{2000}) \\\\& & {\\rm for} \\ 2000 \\, {\\rm km} > h \\ge 450 \\, {\\rm km},$ and $x$ is the sampling (row) index in the model table.",
"In upper layers, from the maximum height down to $h \\approx 2000$ km, $b_x$ is simply a multiplicative factor equal to $p_{\\rm m}$ .",
"In deeper layers ($h < 2000$ km), this factor linearly decreases or increases to reach unity at $h \\approx 450$ km.",
"The initial temperature stratification is changed in the range of heights from approximately 250 km to 2100 km; the upper and lower parts remain equal to the initial model.",
"To derive the scaling parameter $p_{\\rm T}$ , we normalize the temperature stratifications $T_x$ of all VAL models to VAL C: $A_x = T_x /T_{x \\, {\\rm VAL\\, C}})$ .",
"These normalized stratifications can be approximated by a fourth-order polynomial in the form $A^\\prime (x) = 1 + \\sum _{i=1}^{4} a_i x^i$ (Figure REF ).",
"The coefficients $a_i$ are mutually dependent and $a_{2-4}$ can be expressed by multiples of $a_1$ (Figure REF ), particularly, $a_{i = 2,3,4} = a_1 q_i$ , where $q_i = [-7.90 \\cdot 10^{-2}, 2.21 \\cdot 10^{-3}, -2.03 \\cdot 10^{-5}]$ .",
"The values of $q_i$ are practically independent of the initial model.",
"Thus, $a_1$ can be used as the temperature scaling parameter, $p_{\\rm T} = a_1$ .",
"The scaled temperature stratification (in the range 250–2100 km) is then calculated as $ T_x = T_{0x} (1 + p_{\\rm T} x + p_{\\rm T} \\sum _{i=2}^{4} q_i x^i), $ where $T_{0x}$ are the initial temperatures.",
"Figure: Temperature versus column mass plots of the VAL A–F models scaled fromthe model C. Red and black lines show the scaling results and the originalVAL models, respectively.",
"The symbols denote sampling of the models.For each of the initial models VAL we calculated a grid of 2806 scaled models using a combination of the parameters $p_{\\rm m} = [0.5,\\, 0.6,\\, ...,\\, 4.9,\\, 5.0]$ and $p_{\\rm T} = [-0.030,\\, -0.029,\\, ...,\\, 0.029,\\, 0.030]$ .",
"In Figure REF we show that the models VAL A–F can be reproduced by scaling the model C, applying the sets of parameters $p_{\\rm m} = [0.5,\\, 0.7,\\, 1.0,\\, 1.3,\\, 1.6,\\, 2.0]$ and $p_{\\rm T} = [-0.021,\\, -0.010,\\, 0.000,\\, 0.006,\\, 0.012,\\, 0.020]$ , respectively.",
"In total, we have 16836 models in the grid parametrized by $p_{\\rm m}$ , $p_{\\rm T}$ , and the initial model selection.",
"Figure: Maps of (a) Ca2 854.2 nm line-core intensity, (b) selection of initial models,(c) temperature, (d) electron density, (e) density, and (f) gas pressure.",
"The maps (c–f) areretrieved from resulting scaled models at h=900h = 900 km.The models, which have been obtained by varying the temperature structure on the column-mass scale, are used as the input atmospheres in the non-LTE radiative-transfer codes MALI.",
"MALI stands for the Multi-level Accelerated Lambda Iterations technique with preconditioning of the statistical-equlibrium equations according to [17], [18].",
"The density structure is obtained from the hydrostatic equilibrium for a given temperature structure and the radiative transfer equation is solved in a semi-infinite atmosphere subject to standard boundary conditions.",
"First we solve the hydrogen problem using a 5-level plus continuum atomic model.",
"To obtain the gas density, we add the helium but we neglect the helium ionization in computing the electron density within the atmosphere.",
"For hydrogen resonance lines Lyman-$\\alpha $ and Lyman-$\\beta $ we consider the standard angle-averaged partial frequency redistribution (PRD) for the scattering part of the source function and all other lines are treated in the complete redistribution.",
"The microturbulence, taken from the initial VAL model, is consistently included in the hydrostatic equilibrium as the turbulent pressure and it enters also in the line broadening calculations.",
"The resulting electron densities are then used, together with the temperature structure, in the Ca2 version of the MALI code, which solves the non-LTE problem for a 5-level plus continuum model of Ca2–Ca3 ions.",
"This governs the Ca2 H and K resonance lines, the Ca2 infrared triplet lines and the five continua.",
"Again, PRD is used for both resonance lines H and K. We perform detailed synthesis of the Ca2 854.2 nm line, using all relevant broadening mechanisms.",
"Due to uncertainties in the Van der Waals damping parameter, we adjust it in order to better fit the wings of this line.",
"From the grid of models we compute the synthetic line profiles of the 854.2 nm line and use them to find the best fit to the observed profiles (see Section ) at all positions in ROI.",
"The energy losses due to radiation are characterized by the net radiative cooling rates or simply the net radiative losses.",
"For selected best-fit models, at each position in ROI and model height, we finally compute the net radiative cooling rates due to hydrogen, Ca2, and Mg2.",
"These are the main contributors in the solar chromosphere as demonstrated in [24].",
"The Mg2 version of our MALI code uses a 5-level plus continuum Mg2–Mg3 atomic model and treats the two resonance lines Mg2 h and k and the triplet lines.",
"Both resonance lines are again computed using the PRD approach.",
"Contrary to optically-thin losses from the solar corona, the chromospheric losses in the above listed lines must be computed by solving the complex non-LTE radiative transfer problem, because these lines are optically thick and there is no simple and sufficientlly precise prescription for an easy estimate of such losses.",
"The resulting net radiative losses are then integrated along the atmospheric height in the range from 900 km to 1500 km, which corresponds to the range where the measured acoustic energy flux is dissipated (see Section ).",
"At such heights, the losses are dominated by Ca2 lines with some contribution of Mg2 lines h and k and hydrogen continua.",
"A strong coolant is the hydrogen Lyman-$\\alpha $ line, but this is formed higher in the atmosphere close to the transition zone and thus does not enter our integration over the heights of interest."
],
[
"Deposited acoustic flux compared to radiative cooling",
"The initial models VAL B–F were assigned to different areas in ROI in accordance with the brightness of the Ca2 854.2 nm line core (Figures REF a and b).",
"Then, a scaled model from the grid, which provided the best match of the synthetic $I_\\lambda ^{\\rm syn}$ to the local mean observed profile $I_\\lambda ^{\\rm obs}$ of the line, was assigned to each of 19516 positions in ROI.",
"The models have been found by minimizing the merit function $ \\chi ^2({\\rm model}) = \\sum _\\lambda [I_\\lambda ^{\\rm syn}({\\rm model}) - I_\\lambda ^{\\rm obs}]^2.",
"$ In total, 1737 different models were used.",
"The retrieved values of model $T$ , $n_{\\rm e}$ , $\\rho $ , and $P_{\\rm g}$ at the geometrical height $h = 900$ km are shown in Figures REF c–f.",
"Because each position in ROI is characterized by a 1D model unrelated to its surroundings, the maps of $\\rho $ , and $P_{\\rm g}$ have a noisy appearance.",
"The local differences, however, are small—for example, the standard deviation of gas pressure in the “noisy” subfield 6$<x<$ 11, 30$<y<$ 40 is 0.5 Pa (9 % of the mean value).",
"Examples of typical temperature stratifications in quiet-Sun and plage are depicted in the left panel of Figure REF .",
"The inset image of the Ca2 854.2 nm line core intensity shows positions of the examples: the quiet-Sun temperatures (black curves) correspond to the bottom white line, while those in the plage are taken along the top white line and the colors of the temperature curves change from dark red to orange moving from left to right.",
"Figure: Scatter plots of total integrated radiative losses versus deposited acoustic fluxin the region of interest (19516 points).",
"The acoustic fluxes are calculated using (a) theoriginal magnetic inclination angles and (b) the corrected ones.Solid lines show average (red) and median (green) values together with red dashed linesof ±1σ\\pm 1\\sigma .",
"The straight dashed line represents the full balance of radiativelosses by acoustic flux deposit.The total net radiative cooling rates (a sum of the Ca2, Mg2, and H contributions) are calculated for each model in ROI and integrated over the height range 900–1500 km.",
"The right panel of Figure REF shows examples of the height dependence of cooling rates in the quiet Sun and plage at the same positions as of the temperature stratifications.",
"In the plage, the total cooling rates have a maximum at $h = 1000$ –1100 km, which corresponds to maximum cooling rates of Ca2 (the Mg2 contribution is by an order lower).",
"A secondary peak, caused by enhanced contribution of H and H$^-$ continua in hot and dense atmospheres, appears around $h = 800$ km in the brightest part of the plage (orange curves).",
"It does not enter our integration range.",
"The map of the total integrated radiative losses is depicted in Figure REF a.",
"We calculate the deposited acoustic fluxes (see Section ) using two different sets of magnetic field inclinations to estimate the influence of uncertainties in the inclination angle: (i) Original inclination angles retrieved from the inversion, including the unreliable values near 90 in weak-field or non-magnetic regions.",
"The resulting map of the deposited acoustic flux is shown in Figure REF b.",
"(ii) Corrected inclination angles, where the unreliable values are set to zero by means of the mask described in Section .",
"The corresponding deposited acoustic flux map is shown in in Figure REF c. The coefficient of spatial correlation between the total integrated radiative losses and the deposited acoustic flux is 0.47 in the case of (i) and 0.50 in the case of (ii).",
"Scatter plots of total integrated radiative losses $L$ versus the acoustic energy flux $\\Delta F_{\\rm ac}$ deposited in the chromosphere between $h = 900$ and 1500 km are shown in Figure REF a and b for the original inclination angles and the corrected ones, respectively.",
"The red solid line represents mean values of $\\Delta F_{\\rm ac}$ that fall into 200 W m$^{-2}$ wide bins of the $L$ histogram.",
"The bins must contain at least 100 points to calculate the mean value.",
"Likewise, the green line represents median values.",
"The red dashed lines delimit the $\\pm 1\\sigma $ range that characterizes the scatter of individual points in each bin.",
"When the unreliable horizontal magnetic inclination is included (Figure REF a), the deposited acoustic fluxes in the quiet-Sun area ($L < 5000$ W m$^{-2}$ ) almost double those calculated using the corrected inclination values (Figure REF b).",
"This is not realistic because of too many points where $\\Delta F_{\\rm ac} > L$ .",
"The increase of $\\Delta F_{\\rm ac}$ is much smaller in the plage, where the magnetic inclination is determined reliably.",
"In the further discussion, we shall use the deposited acoustic fluxes calculated using the corrected inclination angles.",
"For most of the points (88 %) in the plot (Figure REF b), $L > \\Delta F_{\\rm ac}$ , that is, the deposited acoustic flux is insufficient to balance the radiative losses and maintain the (semi-empirical) temperature at corresponding positions in ROI.",
"The scatter of $\\Delta F_{\\rm ac}$ versus $L$ is large, so that we have to express the contribution of the deposited acoustic flux to the radiative losses statistically, using the ratios $\\overline{\\Delta F_{\\rm ac}} / \\overline{L}$ and ${\\rm median} (\\Delta F_{\\rm ac}) / \\overline{L}$ .",
"Their values start at 0.3 (median 0.2) for $L < 3500$ W m$^{-2}$ and increase to 0.5 (median 0.3) for $3500 < L <5000$ W m$^{-2}$ in the quiet area.",
"They reach 0.7 (median 0.6) with a peak of 0.9 for $5000 < L <6500$ W m$^{-2}$ at the periphery of the plage, where the magnetic inclination is large, and drop to 0.5 (median 0.4) for $L >6500$ W m$^{-2}$ in the brightest part of the plage with the nearly vertical magnetic field.",
"The points with $L < \\Delta F_{\\rm ac}$ will be discussed in Section .",
"Compared to the results of of Sobotka16, the median values of $\\Delta F_{\\rm ac}$ are consistent excepting the brightest parts of the plage ($L > 6500$ W m$^{-2}$ ), where the magnetic inclination was overestimated.",
"Moreover, the revision of the magnetic inclination map resulted in an increase of $\\Delta F_{\\rm ac}$ at the plage periphery ($L \\simeq 6000$ W m$^{-2}$ ).",
"Other differences can be explained by improved values of the gas density, to which the acoustic fluxes are directly proportional.",
"The present set of 1737 atmospheric models provides a more realistic density distribution than the simple grid of only seven models."
],
[
"Discussion and conclusions",
"A quantitative comparison of deposited acoustic energy flux with total integrated radiative losses in the middle chromosphere of the quiet Sun and a weak plage is made.",
"The comparison is based on a consistent set of high-resolution observations acquired by the IBIS instrument in the line Ca2 854.2 nm.",
"The deposited acoustic flux is derived from Doppler velocities observed in the line core and a set of 1737 non-LTE 1D hydrostatic semi-empirical models, which are also used for the calculation of radiative losses.",
"The models are obtained by scaling the temperature and column mass of five initial models VAL B–F [24] to get the best fit of synthetic to time-averaged observed profiles of the Ca2 854.2 nm line.",
"The fit quality $\\chi ^2_{\\rm min}$ changes with the position in ROI.",
"We define the bad-fit areas where $\\chi ^2_{\\rm min}$ is larger than $2\\sigma $ of its statistical distribution.",
"Several of them, where the observed profiles are broader than the synthetic ones, coincide with regions of enhanced magnetic field in the pore and plage.",
"Contours that outline the bad-fit areas together with the map of magnetic-field strength are shown in Figure REF a.",
"The Zeeman broadening of the Ca2 line with Landé factor $g = 1.1$ probably does not play a major role.",
"Rather, this effect might be explained by the fact that our 1D semi-empirical models do not account for the complex 3D distribution of thermodynamic and magnetic parameters in such areas.",
"The bad-fit areas represent 12.5 % of the data set and a mask is made to remove the affected points from the scatter plot of $L$ versus $\\Delta F_{\\rm ac}$ .",
"This removal alters the plot shown in Figure REF b and the derived statistical values only in the part of $L > 6500$ W m$^{-2}$ , corresponding to the brightest plage region, where $\\overline{\\Delta F_{\\rm ac}}$ becomes larger by the factor of 1.3 on average and its contribution to the total integrated radiative losses $\\overline{\\Delta F_{\\rm ac}} / \\overline{L}$ increases from 0.5 to 0.6.",
"Figure: (a) Map of magnetic field strength (in logarithmic scale) with contours of areaswhere the fit of synthetic to observed Ca2 line profiles is worse than 2σ2\\sigma .",
"(b) Areas of radiative losses smaller than the deposited acoustic flux (white)together with the bad-fit areas (black).In 12 % points of our data set, the deposited acoustic energy flux is larger than the total integrated radiative losses ($L < \\Delta F_{\\rm ac}$ ).",
"Regions formed by these points are shown in Figure REF b together with the bad-fit ones in white and black colors, respectively.",
"The $L < \\Delta F_{\\rm ac}$ regions appear at locations where the magnetic inclination is larger than 60 and temporal variations of Doppler velocity and intensity of the Ca2 854.2 nm core are strong (cf.",
"Figure 1 and its animation in Sobotka16): around the border between the pore's superpenumbra and plage (subfield 0$<x<$ 10, 15$<y<$ 25) and partly at the periphery of the plage.",
"We have shown that the deposited acoustic-flux energy in the quiet-Sun chromosphere balances 30–50 % of the energy released by radiative losses.",
"The energy carried by (magneto)acoustic waves in the plage supplies 50–60 % of the radiated energy at locations with vertical magnetic field and 70–90 % in regions where the magnetic field is inclined more than 50.",
"These values are statistical averages of results with a large individual scatter and they are based on one observation of a single small solar area.",
"They are also critically sensitive to the correct determination of the magnetic field inclination, particularly in the quiet-Sun region.",
"We also have to note that the area considered as quiet Sun in our ROI is close to the plage and it falls within the extended canopy region of the plage's magnetic field (see Figure 1 in Sobotka16).",
"The effect of “magnetic shadows”, which are related to the elevated magnetic field forming the Ca2 fibrils [23], reduces the oscillatory power in this region, as seen from the power maps in Figure 7 of [21].",
"Consequently, the deposited acoustic flux in our “quiet area” may be lower than that in quiet regions far from the plage.",
"Additional studies of different quiet and active regions in various chromospheric lines together with precise measurements of magnetic field inclination are needed to obtain more general and conclusive results.",
"ACKNOWLEDGMENTS We thank the anonymous reviewer for valuable comments and suggestions.",
"This work was supported by the Czech Science Foundation and Deutsche Forschungsgemeinschaft under the common grant 18-08097J – DE 787/5-1 and the institutional support RVO:67985815 of the Czech Academy of Sciences.",
"The Dunn Solar Telescope was run by the National Solar Observatory (NSO), which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA), for the National Science Foundation.",
"IBIS has been built by INAF/Osservatorio Astrofisico di Arcetri with contributions from the Universities of Firenze and Roma “Tor Vergata”, NSO, and the Italian Ministries of Research (MIUR) and Foreign Affairs (MAE).",
"The observations at IBIS were supported by the Rome Tor Vergata “Innovative techniques and technologies for the study of the solar magnetism” grant funded by MIUR."
]
] | 2001.03413 |
[
[
"Pseudorapidity dependence of anisotropic flow in heavy ion collisions\n with ALICE"
],
[
"Abstract The pseudorapidity dependence of flow coefficients, $v_n$, is presented over a large range in pseudorapidity ($-3.5 \\leq \\eta \\leq 5$) in Pb$-$Pb collisions at 5.02 TeV.",
"The flow coefficients are measured with 2- and 4-particle cumulants with $|\\Delta \\eta | > 0$ and $|\\Delta \\eta | > 2$.",
"The difference between $v_2\\{2,|\\Delta \\eta|>0\\}$ and $v_2\\{2,|\\Delta \\eta|>2\\}$ implies changes in either the de-correlation of flow vectors or non-flow effects between the two methods.",
"The AMPT model shows a qualitative agreement with the measurements."
],
[
"Introduction",
"At the LHC, a hot and dense medium is created in Pb$-$ Pb collisions.",
"It is commonly referred to as the Quark-Gluon-Plasma (QGP).",
"In heavy-ion collisions, the rapid expansion of the system causes the particles to be boosted by a common velocity field.",
"Spatial anisotropies arise from the overlap of the two heavy ions or fluctuations in the initial state.",
"The spatial anisotropies cause anisotropy in the momentum space of the created particles through the evolution of the system.",
"The azimuthal flow can be described by the coefficients $v_n$ of a Fourier series decomposition of emitted particles [1], $\\frac{\\mathrm {d}N}{\\mathrm {d} \\varphi } \\propto \\frac{1}{2 \\pi } [1 + 2 \\sum ^\\infty _{n=1} v_n \\cos (n[\\varphi - \\Psi _n])]$ with $v_n = \\langle \\cos (n [\\varphi - \\Psi _n]) \\rangle .$ The pseudorapidity dependence of the flow coefficients can give information about the shear viscosity ($\\eta /s$ ) of the system, e.g., temperature dependence of $\\eta /s$ [2].",
"At forward rapidities, there is a shorter lifetime in the QGP phase, which can lead to hadronic viscosity to play a more significant role.",
"In [3], [4], [5] measurements of $v_2$ was presented in the central regions of pseudorapidity for Pb-Pb at 5.02 and 2.76 TeV at the LHC.",
"Measurements of $v_n$ were extended to forward rapidities in Pb$-$ Pb collisions at 2.76 TeV [6] and in Au$-$ Au collisions at 130 GeV [7]."
],
[
"Experimental Setup",
"The data used in the analysis is collected with the ALICE experiment.",
"The Time Projection Chamber (TPC) is used for tracking of particles in the pseudorapidity region $-1.1 < \\eta <1.1 $ .",
"The Inner Tracking System (ITS) is used in combination with the TPC for tracking and is also used for triggering.",
"At large pseudorapidities the Forward Multiplicity Detector (FMD) is placed in both the forward ($1.7 < \\eta < 5.0$ ) and backward direction ($-3.5 < \\eta < -1.7$ ).",
"The VZERO scintillators are also placed at large rapidities ($-3.7 < \\eta < -1.7$ and $2.8 < \\eta < 5.1$ ) and are used for triggering and determination of centrality."
],
[
"Methods",
"The flow coefficients $v_n$ are calculated using the Generic Framework [8].",
"In the analysis two methods with sub-events were used.",
"Using the TPC as the reference region, means that there is $|\\Delta \\eta | > 0$ in the TPC and $|\\Delta \\eta | > 2$ in the FMD (Figure REF left).",
"If there are no de-correlation of the symmetry planes as a function of $\\eta $ and no breaking of factorization, then the calculation for differential flow $v_n^{\\prime }$ in the sub-event $B$ is $v_n^{^{\\prime }B} = \\frac{\\langle v_n^{^{\\prime }B} v_n^C \\rangle }{\\sqrt{\\langle v_n^B v_n^C \\rangle } },$ where $v_n^B$ and $v_n^C$ is reference flow in region $B$ and $C$ .",
"The reference region is always chosen to be opposite in $\\eta $ to the differential region.",
"E.g.",
"particles of interest in region $A$ are correlated with the reference particles from region $C$ .",
"If one use the FMD as reference region instead, there is $\\Delta \\eta > 2$ in the TPC and $\\Delta \\eta > 4$ in the FMD (Figure REF right).",
"In this case for e.g.",
"$v_n^{\\prime }$ in region $B$ the calculation is, $v_n^{^{\\prime }B} = \\frac{\\langle v_n^{^{\\prime }B} v_n^D \\rangle }{\\sqrt{\\langle v_n^A v_n^D \\rangle } }$ Figure: The two methods used for calculating v n (η)v_n(\\eta ).",
"pp is the differential vector and QQ is the reference vector."
],
[
"Results",
"Figure REF shows the two-particle cumulant results with TPC as reference region.",
"There is a clear ordering of the flow coefficients, $v_2 > v_3 > v_4$ and $v_n$ is increasing from central to semi-central collisions.",
"$v_3$ and $v_4$ have a much weaker centrality dependence than $v_2$ , which indicates that the higher orders are mostly driven by initial state fluctuations.",
"The $v_n$ measurements are overlayed with simulation results from AMPT with string melting [9].",
"The AMPT simulation describes the data qualitatively but not quantitatively.",
"In Figure REF , the results with a large $\\eta $ -gap with the FMD as reference region are shown, and the 4-particle cumulant results.",
"The difference in the magnitude between $v_2\\lbrace 2\\rbrace $ and $v_2\\lbrace 4\\rbrace $ is from flow fluctuations.",
"The similar shape of $v_2\\lbrace 2\\rbrace $ and $v_2\\lbrace 4\\rbrace $ could imply that $v_2\\lbrace 2\\rbrace $ is without non-flow, since 4-particle cumulants are known to suppress non-flow effects.",
"In the middle pseudorapidity region, there is a clear difference compared to Figure REF by the flat shape.",
"The flat shape implies that there are some effects that a small $\\eta $ -gap might not be able to suppress, e.g., non-flow and factorization effects.",
"The AMPT model, also presented for comparison, again, does not describe the data quantitatively.",
"Figure: Measurements of v n {2}v_n\\lbrace 2\\rbrace using the TPC as reference region.",
"The boxes are the systematical uncertainties (divided into correlated and uncorrelated uncertainties) and the statistical uncertainties are mostly smaller than the marker size."
],
[
"Summary",
"In this proceeding, we present the pseudorapidity dependence of flow coefficients covering a wide range in pseudorapidity in Pb$-$ Pb collisions at 5.02 TeV.",
"The results show that $v_n$ measured with a large $\\eta $ -gap and 4-particle cumulant has a flatter shape than the 2-particle cumulant with a small $\\eta $ -gap.",
"The change of the shape of $v_2\\lbrace 2\\rbrace $ when using either long-range or short-range correlations implies changes in either de-correlation of flow vectors or non-flow effects between the two methods.",
"The measurements of $v_3$ and $v_4$ do not have a strong centrality dependence and are therefore mostly driven by fluctuations.",
"AMPT has a qualitative agreement with presented measurements, but further improvements are needed.",
"In the future, 3+1D hydrodynamic calculations might be compared to constrain initial state models and study the longitudinal dynamics of the hot and dense matter."
]
] | 2001.03407 |
[
[
"Gamma-ray emission in alpha-particle reactions with C, Mg, Si, Fe"
],
[
"Abstract Cross sections for the strongest gamma-ray emission lines produced in alpha-particle reactions with C, Mg, Si, Fe have been measured in the range E_alpha = 50 - 90 MeV at the center for proton therapy at the Helmholtz-Zentrum Berlin.",
"Data for more than 60 different gamma-ray lines were determined, with particular efforts for lines that are in cross section compilations/evaluations with astrophysical purpose, and where data exist at lower projectile energies.",
"The data are compared with predictions of a modern nuclear reaction code and cross-section curves of the latest evaluation for gamma-ray line emission in accelerated-particle interactions in solar flares."
],
[
"Introduction",
"Cross sections for the emission of nuclear $\\gamma $ -ray lines are a basic ingrediant for the analysis and interpretation of observations in the low-energy $\\gamma $ -ray band of astrophysical sites with an important population of accelerated ions.",
"The latest compilation of Murphy et al.",
"[1], aimed mainly at de-excitation lines produced in accelerated-particle interactions in solar flares, features 248 cross-section curves for $\\gamma $ -ray lines emitted in proton, $^3$ He and $\\alpha $ -particle reactions with abundant isotopes in the solar atmosphere.",
"The curves are typically given from reaction threshold to several hundred MeV per nucleon, which is the important energy range for the accelerated particle populations in solar flares.",
"For most of the curves, however, data exist only at proton energies below about 25 MeV and for $\\alpha $ -particles below about 10 MeV per nucleon and the extrapolation to higher energies relied on calculations with the nuclear reaction code TALYS [2].",
"Of particular importance are the most prominent lines that often stand out in observed spectra from strong solar flares.",
"They can be used to deduce ambient abundances and the energetic-particle composition and energy spectrum [3], [4].",
"These lines are from the deexcitation of the first or second excited state in the most abundant species: $^{12}$ C, $^{14}$ N, $^{16}$ O, $^{20}$ Ne, $^{24}$ Mg, $^{28}$ Si, $^{56}$ Fe.",
"The cross section curves for the emission of these lines in proton and $\\alpha $ -particle interactions with these nuclei typically show a broad maximum around 15-20 MeV, due to inelastic scattering reactions, reaching the several hundred mb range.",
"This is followed by an nearly exponential fall-off to higher energies for proton reactions while a second, very broad maximum around 60 MeV is predicted for $\\alpha $ -particle reactions.",
"This maximum may be explained by fusion-evaporation reactions like ($\\alpha $ ,dnp) or ($\\alpha $ ,2n2p), where the residual nucleus is identical to the target nucleus.",
"This second cross section maximum is important in solar flares with relatively high accelerated $\\alpha $ -particle to proton ratios $\\alpha $ /p and hard energy spectra.",
"However, there are practically no experimental data available for the emission of $\\gamma $ -ray lines in $\\alpha $ -particle reactions above $E_{\\alpha }$ = 40 MeV.",
"We therefore decided to measure cross sections for the emission of strong lines in $\\alpha $ -particle reactions around the predicted second cross section maximum.",
"In the first experiment, we concentrated on the elements C, O, Mg and Fe.",
"The chosen energy range, $E_{\\alpha }$ = 50 - 90 MeV also continues to higher energies the data ($E_{\\alpha }$$\\le $ 40 MeV), obtained in previous experiments at the Orsay tandem-van-de-Graaff accelerator [5], [6]."
],
[
"Experiment",
"The experiments have been done at the center for proton therapy of the Helmholtz-Zentrum Berlin in two campaigns in 2015 and 2016.",
"Pulsed beams of $\\alpha $ particles with $E_{\\alpha }$ = 50 MeV in 2015 and $E_{\\alpha }$ = 60, 75 and 90 MeV in 2016 have been produced using the Van-de-Graaff injector and the K-132 separated sector cyclotron of the Helmholtz center.",
"They were directed onto self-supporting target foils of C, Mg, Si and Fe inside a spherical chamber of 15 cm diameter made of stainless steel and stopped in a thick copper block about 4 m downstream of the target chamber.",
"The target chamber was equipped with two glass windows perpendicular to the beam direction, where one of them served for optical monitoring of the beam spot through the use of an alumina target.",
"In both campaigns, 4 detectors, one LaBr3:Ce scintillation detector and three HP-Ge detectors in a wide angular range and at large distances of typically 50-70 cm from the target, have been used for the detection of the emitted $\\gamma $ rays.",
"The pulsed beams were produced with relatively low repetition rates of $\\sim $ 80 kHz to $\\sim $ 1 MHz and a small pulse width well below 1 ns.",
"This allowed an efficient suppression of background, in particular radiation from the beam stop and interactions of secondary neutrons in the detectors, but also radiation from activated material and natural radioactivity.",
"Beam intensities were of the order of 0.5-3 nA, resulting in typical detector count rates of a few thousand counts per second.",
"The beam current was repeatedly measured in two well-shielded Faraday cups sitting just after the beam extraction from the cyclotron and about 1 m upstream of the targets.",
"Values of the two Faraday cups agreed to better than 5%.",
"In 2016, a third cup was installed at the beam stop, allowing a constant monitoring of the beam intensity.",
"The efficiency calibration of the detectors has been done with standard radioactive sources, $^{133}$ Ba, $^{137}$ Cs, $^{60}$ Co and $^{152}$ Eu for energies below about 1.4 MeV and extensive Geant simulations to extrapolate the detector efficiencies to higher energies and also to obtain precise detector response functions.",
"Figure: Timing channel (T2) versus energy channel (E2) of events in the LaBr 3 _3:Ce detector recorded during the irradiation of the C target with a 90-MeV α\\alpha -particle beam.The detector signals were treated with standard NIM modules, providing a signal for the deposited energy and another signal for the timing of the event.",
"The timing signal was defined as the time difference between the detector signal and the beam pulse signal, both determined with constant fraction discriminators fed into a time-to-amplitude converter NIM module.",
"The energy and timing data of the 4 detectors were then digitized and recorded in event-by-event mode and a time stamp of 100 ns resolution using an acquisition system of FAST ComTec [7], providing also the dead times of the 4 energy and 4 timing channels.",
"Figure 1 shows a zoom in the timing vs. energy plane of data recorded by the LaBr$_3$ :Ce detector in the irradiation of the C target with a 90-MeV $\\alpha $ -particle beam.",
"The slightly inclined, horizontal branch at T2 $\\sim $ 4190 shows the prompt $\\gamma $ rays induced by beam interactions in the target.",
"At E2 $\\sim $ 4200-5400 is the 4.4-MeV complex including the full-energy and escape lines, some narrow lines can also be seen below E2 $\\sim $ 1500.",
"A broader horizontal band inside T2 $\\sim $ 4100-4160 with several narrow lines below E2 $\\sim $ 2400 can also be recognized.",
"These events, arriving about 4 ns after the prompt target $\\gamma $ rays, have been identified to come from interactions of secondary particles, mostly neutrons probably, inside the stainless-steel walls of the reaction chamber.",
"Events arriving still later are from interactions in further away beam tubes and supporting structures and from secondary neutrons interacting in the LaBr$_3$ :Ce crystal.",
"Beam-induced $\\gamma $ rays from interactions in the beam stop are in this histogram around T2 $\\sim $ 2700 (not shown).",
"Figure: Compton-subtracted spectrum of the HP-Ge detector at 90o to the beam direction from irradiation of the Mg target with 75-MeV α\\alpha particles."
],
[
"Data analysis and results",
"Spectra of prompt $\\gamma $ rays from the target were obtained by a selection of events within a narrow band around the visible horizontal branchs in the energy-timing plane from prompt target $\\gamma $ rays as shown in Figure 1.",
"With the LaBr$_3$ :Ce detectors, events from interactions of secondary particles could be completely excluded by the selection, while for the HP-Ge detectors, with a time resolution not better than 3-4 ns, not all events from secondary-particle interactions in the target chamber walls could be separated.",
"Their contributions, essentially lines from the first few excited states of $^{56}$ Fe and $^{52}$ Cr, could however be estimated from their time profile in the LaBr$_3$ :Ce detector.",
"This was important for cross-section determinations with the Fe target.",
"Integrals for narrow $\\gamma $ -ray lines could be directly taken from these spectra, while for broader lines and line complexes, Compton-subtracted spectra were used.",
"These spectra were obtained by an unfolding procedure made possible by dedicated measurements with radioactive sources and extensive Geant simulations for the detector response functions in the two experimental setups.",
"The accuracy of the Compton subtraction from about 8 MeV down to the 511-keV line was estimated to be better than 20%.",
"An example of Compton-subtracted spectrum is shown in Figure 2.",
"Some of the most important lines for each target, broad lines (like $E_{\\gamma }$ = 4439 keV in $^{12}$ C) or narrow ones (e.g.",
"$E_{\\gamma }$ = 1368 and 1779 keV in $^{24}$ Mg and $^{28}$ Si, respectively) were merged with other lines in a complex structure and needed line-shape calculations to determine their intensity.",
"Examples for the 1779-keV line and the 4439-keV line are shown in Figures 3, 4.",
"The uncertainties related to these decompositions is added to the statistical and other systematic uncertainties like the detector efficiencies.",
"In cases of lines very close in energy, like the 4439-keV line of $^{12}$ C and the 4445-keV line of $^{11}$ B, the sum has been used for the line cross section determination.",
"Figure: Symbols: Compton-subtracted spectrum around the 4.4-MeV line complex of the HP-Ge detector at 116o to the beam direction during irradiation of the C target with 50-MeV α\\alpha particles.",
"The shape of the 4439-keV line of 12 ^{12}C has been obtained using coupled-channels calculations for inelastic scattering as in .Differential cross section data from the 3 HP-Ge detectors have been obtained for about 60 different $\\gamma $ -ray lines, and for most of them at all $\\alpha $ -particle energies.",
"With the LaBr$_3$ :Ce detectors, however, owing to their lower energy resolution, no cross section data could be obtained for some lines in complex structures.",
"Finally, the $\\gamma $ -ray emission cross sections have been obtained by Legendre-polynomial fits of the 3 or 4 detector data.",
"To the resulting cross-section uncertainty from the fit were added the target thickness and beam charge uncertainties.",
"Cross section curves for the strongest line of each target, here from the deexcitation of the first excited level in the major isotopes $^{12}$ C, $^{24}$ Mg, $^{28}$ Si and $^{56}$ Fe are shown on Figure 5.",
"The data show effectively clearly a second cross section bump for the 1.78-MeV $^{28}$ Si and 0.85-MeV $^{56}$ Fe lines, approximately at the energies predicted by the compilation [1].",
"The latter, adjusted to the data at lower energies, that were obtained at the tandem-Van-de-Graaff accelerators of Washington [9], [10] and Orsay [5], however, overestimates significantly the measured cross sections.",
"For the 1.37-MeV line of $^{24}$ Mg, there is a hint for a weak second cross section bump, in any case significantly less pronounced than predicted by the compilation.",
"Its prediction for the cross section curve of the 4.44-MeV line of $^{12}$ C agrees reasonably with the present data.",
"Figure: Gamma-ray line emission cross sections for α\\alpha -particle reactions with, from left to right and from upper to lower: C, Mg, Si and Fe.",
"Red filled circles are the present data, blue filled squares are from Ref.",
", green triangles are from Ref.",
"for C and from Ref.",
"for Mg, Si and Fe.",
"The cyan curves are the cross section excitation functions from the cross section compilation .Similar disagreements can also be seen for cross sections of the other strong lines, albeit with a slightly better agreement for the lines from the Fe target compared to the other targets.",
"We also did calculations with the nuclear reaction code Talys [2] with default parameters for the nuclear structure and potentials.",
"For the lines from the deexcitation of the first few levels of the major target isotopes, they generally agree at energies below $E_{\\alpha }$ $\\sim $ 30 MeV with the data at the 30% level or better, but disagree with the present data up to factors of 2-3.",
"Here, a comprehensive study of structure parameters and nuclear reaction potentials in Talys as it has been done in [5], [6] could certainly improve significantly the agreement with data and allow an accurate description of the total $\\gamma $ -ray emission in $\\alpha $ -particle reactions and an extrapolation to higher projectile energies."
],
[
"Conclusions",
"Cross section data for the emission of about 60 $\\gamma $ -ray lines could be determined for $\\alpha $ -particle reactions with targets of C, Mg, Si and Fe at $E_{\\alpha }$ = 50, 60, 75 and 90 MeV.",
"For the strongest lines from each target, there is now with the present data a complete coverage of experimental data from threshold to 90 MeV.",
"Furthermore, cross sections for many new lines could be added, where no data were available from previous experiments.",
"Comparison of cross section curves of the latest cross section compilation and nuclear reaction code calculations show significant disagreements with the new data.",
"This underlines the importance and necessity of experimental work at particle accelerators for the establishment of an accurate cross section data base in a wide projectile energy range that is needed for applications in astrophysics, but also for other applications with energetic particles like hadrontherapy.",
"New accelerator centers open possibilities for future measurements in astrophysics and nuclear physices [11]."
],
[
"Acknowledgments",
"We like to thank the staff at the section for proton therapy of the Helmholtz-Zentrum Berlin for their strong involvement to provide the high-quality beams in the time periods between the cancer therapy sessions and their generous and efficient help in the planning and setup of the experiments."
]
] | 2001.03600 |
[
[
"Seismic horizon detection with neural networks"
],
[
"Abstract Over the last few years, Convolutional Neural Networks (CNNs) were successfully adopted in numerous domains to solve various image-related tasks, ranging from simple classification to fine borders annotation.",
"Tracking seismic horizons is no different, and there are a lot of papers proposing the usage of such models to avoid time-consuming hand-picking.",
"Unfortunately, most of them are (i) either trained on synthetic data, which can't fully represent the complexity of subterranean structures, (ii) trained and tested on the same cube, or (iii) lack reproducibility and precise descriptions of the model-building process.",
"With all that in mind, the main contribution of this paper is an open-sourced research of applying binary segmentation approach to the task of horizon detection on multiple real seismic cubes with a focus on inter-cube generalization of the predictive model."
],
[
"Introduction",
"High-quality tracking of subterranean reflections is of utmost importance to the task of seismic interpretation: it influences all of the subsequent processing steps, including velocity model building.",
"Such picking, on the other hand, requires a lot of time of even very qualified experts, and, consequently, a lot of automatic methods were proposed.",
"Recent works [5], [4] propose to use deep learning based approaches.",
"Unfortunately, most of the proposed researches focuses on single cube: it is split into train/validation sets, and used to both train the model and evaluate its quality.",
"This approach somewhat reflects the activity of seismic specialist: for example, if the train set consists only of multiple slices, uniformly cut from the cube, and the model is used to <<interpolate>> labeled horizon on the rest of the data.",
"On the other hand, splitting the cube into train/validation sets along one plane is unfair: due to slow changes of information inside the volume, we can't use this validation data to evaluate model performance.",
"Moreover, we want to use our approach on completely new seismic cubes: to this end, our train set must consist of multiple cubes.",
"Training such models is paired with more obstacles due to varying equipment and settings of shooting, and this raises the bar for the amount of information needed to create a model that can work well on unseen data even higher.",
"Note that we can't use synthetic data to enlarge our dataset: at the moment, we can't plausibly replicate complex subterranean structures as we are unable to reliably simulate time-lasting processes of earth shaping.",
"This work is organized as follows: first of all, we describe our data at hand both qualitatively and quantitatively; then we rigorously define task of horizon detection and our approach to it; after that, we strictly define our research on inter-cube generalization and report its results; finally, we discuss shortcomings of the task formulation, as well as propose new method of tracking seismic reflection."
],
[
"Previous work",
"We rely on our framework SeismiQB, which was presented in [CITE].",
"To briefly recap, it allows to store seismic data in a fast data format together with attached horizons (either hand-labeled or any other), cut crops of desired shape from it and apply various transformations to the values.",
"This library also allows to easily define even the most sophisticated neural networks with just a few lines of code, and there are also implementations for various popular architectures ranging from simple ResNet [3] and UNet-like [6] models to state-of-the-art EfficientNet [8] and DeepLab [1].",
"Table REF provides detailed information about four of the (anonymized) cubes in our dataset.",
"Images REF , REF illustrate completely different inner structure of seismic data, which is a consequence of both varying equipment of shooting and different subterranean patterns in distant locations.",
"We can see, that Cube 3 is very plain and almost linear, while Cube 4 contains sharp turns and fissures, so we expect varying performance of the neural networks on them.",
"Each of the cubes is paired with a number (from four to nine) of hand-labeled horizons which have diverse nature: some of them track the most noticeable reflections, some of them mark boundaries between interlayers, few of them follow fissures and cracks.",
"Worse yet, they track different phase: some of them are at maximas of amplitudes, some of them are at minimas, while most of human-picked horizons have varying phase over the domain.",
"Quality evaluation of tracked reflections must be further explored: training models on bad data is obviously detrimental to its predictive ability.",
"Information about the amount of labeled horizons for each cube is also presented in Table REF .",
"Table: Results for models that are trained and tested on the same cubeFigure: NO_CAPTION SeismiQB's capability to transform values in the cube is crucial: as we know, cubes contain excessive information with the relation to the task of tracking reflections.",
"Thus, we can use a wealth of functions to transform the space of $(x, y, t)$ to the space of $(x, y, \\omega )$ with $\\omega $ being an arbitrary hand-developed feature like frequency, sign, phase, etc, in order to represent information in a more suitable way.",
"This new data can be used as a complementary one, or replace original image entirely.",
"In section [REF SECTION] we investigate some of the transforms and their influence on the quality of the predictive model."
],
[
"Formal task description",
"There are multiple approaches to solve the task in hand: we can cast it as a regression problem, or as binary or multiclass segmentation.",
"In this paper, we use the binary segmentation method.",
"We use $N_x, N_y, N_t$ as sizes of crop along each respective dimension, and $n$ is reserved for the number of horizons.",
"Despite every crop being a 3D entity cut from the original cube, we don't use 3D (volumetric) convolutions due to no metric gains and prohibitive computational costs of such neural networks.",
"Consequently, our models perceive input data as 2D entity with channels, and uses 2D (spatial) convolutions to process it.",
"In order to cast the task into a binary segmentation one, model should predict 3D array of the same shape, as the input data, with 1's corresponding to horizons and 0's to background.",
"Thus, CNN should learn mapping from $\\mathcal {R}^{N_x \\times N_y \\times N_t}$ on itself, which is commonly achieved by an encoder-decoder architectures.",
"We evaluated multiple models in such fashion, ranging from simple UNet [6] to the state-of-the-art DeepLab [1].",
"We also want our model to look at the image from a side view: to this end, we use any of the spatial axis (cross-line or inline) as the channel one.",
"Due to heavy class imbalance (volume of horizons is extremely small compared to the volume of the whole image), we use Dice coefficient as the loss function.",
"Note that we can artificially increase the volume of horizon class by thickening the labeled surface: that helps model to distinguish classes.",
"On the other hand, that hampers its ability to finely locate reflections, which is undesirable, and it also makes some of the closer horizon to overlap with one another, which is absolutely unacceptable."
],
[
"Train protocol",
"We train models on batches of 64 randomly generated crops for 1000 iterations by Adam optimizer with default parameters, augmented by inverse-time learning rate decay schedule, which takes approximately one hour to finish.",
"Crops are scaled to a $[0, 1]$ range via min-max scaling in order to have the same range of values in every cube.",
"Also, at train time we augment crops with various distortions: additive and multiplicative noises, affine and perspective transformations, cutout [2] and elastic transform [7].",
"This process is aimed at making model robust to imperfections of the data and to make training examples more diverse.",
"The shape of the input crops deserves special attention.",
"First observation is that cube changes very slowly along its spatial dimensions: therefore, there is no need to make it big in both cross-lines and inlines, and keeping one of them big enough should suffice.",
"Secondly, shape can either be fixed during the whole time of training and inference or dynamically change at training time.",
"We found that randomly generating crop shape from tenth of the cube to its half helps inter-cube generalization, but takes much more time for training (two or three times longer) and prone to unstable results."
],
[
"Train setups",
"In order to exhaustively study inter-cube generalization, we've prepared multiple combinations of train and validation setups.",
"That allows us to thoroughly examine the problem and make definitive conclusions about the obtained results.",
"To evaluate the quality of the predicted horizons, we use multiple metrics, including the percentage of covered area (compared to the ground truth), mean error ($l_1$ -difference) and percentage of area of predicted horizon inside 5-ms window of the hand-labeled one.",
"Figure: NO_CAPTIONFirst of all, we train and test neural network on the same cube.",
"We use only every 200-th inline during train, totaling in no more than 15 slides for each individual cube as a train data.",
"This setting corresponds to a situation, where seismic specialist labels sought-for horizon only on every 200-th slide and wants its automatic prolongation on the whole cube.",
"Due to slow change of the cube along its spatial axis, we anticipate good performance in this task.",
"Results, presented in Table REF , are consistent with our expectations: predicted horizons cover most of the needed area while staying close to the hand-labeled ones.",
"Qualitative estimation shows that prediction makes sense from a geological point of view: it follows the same phase, having discontinuities only in the locations of fissures or sharp faults.",
"Table: Results for models that are trained and tested on the same cubeModel, trained on one cube, hardly works on the unseen data.",
"That can be attributed to varying subterranean structure, as well as to the unequal values inside individual cubes: despite scaling values to a $[0, 1]$ range, the exact distribution can diverse.",
"Thus, we need to train the model on multiple cubes in order to work in other regions.",
"Next, we make model learn from two cubes with validation of the results on the third.",
"Note that, due to small size, we exclude Cube 2 from this part of research: it is just not big enough.",
"All in all, we trained three models, with results shown in Table REF .",
"We can easily spot a glaring problem here: predictions cover only a fraction of the spatial size.",
"Manual check allows us to conclude that the model works somewhat well (detected horizons are in 5ms window) in places where the texture of the validation data is similar to the learned one, and in order to generalize better we must provide more diverse seismic information at train time.",
"Our last model, trained on three cubes and tested on Cube 2, confirms this statement, achieving both good coverage and mean error results, presented in Table REF .",
"An example of a model working on unseen data can be seen at Image REF .",
"Table: Results for models that are trained on two cubes and tested on the unseen dataTable: Results for model that is trained on three cubes and tested on the unseen data"
],
[
"Discussion",
"Despite accomplishing acceptable results, there is a major issue.",
"The task in hand is ill-defined: it is unclear, which of the horizons do we want to detect, how many of them, with which rules of tracking (e.g.",
"on which phase the horizon should be).",
"Moreover, we have virtually no control at the model's behavior on unseen data: we can't choose which horizon to track.",
"That contradicts with the very nature of the problem: in most situations, we want to know the location of a particular chosen horizon.",
"There is a simple way of attacking this problem: just get more train labels, one way or the other.",
"That would require significant human resources, yet would not solve the issue completely: we would make model to detect more horizons, but that does not eliminate situations where we need to track the one, not present in predictions.",
"This is a simple, but a rather unscalable approach.",
"A better way is to communicate prior information about horizon location to the model, at the same time making it follow only one reflection at a time.",
"Doing so requires significant enhancement of our framework, but appears to be the best way to tackle the issue.",
"Introducing a mechanism of hand-selecting horizon to track would both make results more interpretable and appealing."
],
[
"Conclusion",
"In this paper, we systematically study one of the approaches to seismic horizon detection.",
"With multiple carefully designed settings we identify both situations where deep learning based methods can work well, as well as harder tasks that are yet to be successfully tackled by neural networks; in each of the settings we show both quantitive and qualitative results.",
"We also demonstrate crucial problems with the task itself and propose to move away from unsupervised horizon detection (in a sense that we don't control locations of the predicted horizons) to a supervised one."
]
] | 2001.03390 |
[
[
"Average radial structures of gas convection in the solar granulation"
],
[
"Abstract Gas convection is observed in the solar photosphere as the granulation, i.e., having highly time-dependent cellular patterns, consisting of numerous bright cells called granules and dark surrounding-channels called intergranular lanes.",
"Many efforts have been made to characterize the granulation, which may be used as an energy source for various types of dynamical phenomena.",
"Although the horizontal gas flow dynamics in intergranular lanes may play a vital role, but they are poorly understood.",
"This is because the Doppler signals can be obtained only at the solar limb, where the signals are severely degraded by a foreshortening effect.",
"To reduce such a degradation, we use Hinode's spectroscopic data, which are free from a seeing-induced image degradation, and improve its image quality by correcting for straylight in the instruments.",
"The dataset continuously covers from the solar disk to the limb, providing a multidirectional line-of-sight (LOS) diagnosis against the granulation.",
"The obtained LOS flow-field variation across the disk indicates a horizontal flow speed of 1.8-2.4 km/s.",
"We also derive the spatial distribution of the horizontal flow speed, which is 1.6 km/s in granules and 1.8 km/s in intergranular lanes, and where the maximum speed is inside intergranular lanes.",
"This result newly suggests the following sequence of horizontal flow: A hot rising gas parcel is strongly accelerated from the granular center, even beyond the transition from the granules to the intergranular lanes, resulting in the fastest speed inside the intergranular lanes, and the gas may also experience decelerations in the intergranular lane."
],
[
"Introduction",
"Solar granules are bright cells in the solar photosphere, surrounded by dark channels called intergranular lanes [66].",
"A granule is a manifestation of an overshooting of a hot upflow from the convectively unstable subsurface layers into the stable photosphere [73].",
"The excess pressure above the upflow pushes the gas toward sideways.",
"Radiative cooling, along with an associated increase in density, makes the material less buoyant, and the material begins to descend.",
"A dark intergranular feature is formed at the location of the downflow.",
"The enormous amount of the kinetic energy deposited in the granulation is transformed and may be used to drive various types of photospheric phenomena, e.g., exploding granules [69], [54], [23], [18], supersonic flows [64], [44], [9], [51], and reversed granulation [58], [57], [31].",
"In addition, the granulation plays a role of an MHD wave driver, serving as the energy transfer in a non-thermal manner from the photosphere to the upper layers, the chromosphere and the corona (see [50] for a review).",
"Thus, the characterization of flow fields is remarkably beneficial for comprehending not only the granulation itself but also the other convection-driven phenomena.",
"Our observational knowledge regarding the horizontal flow field still remains poor, compared to a vertical component of the flows, simply because its derivation is tricky.",
"The Doppler signals for a horizontal component can be measured only near the solar limb [16], [7], [4], [5].",
"[6] first confirmed the observational existence of a horizontal outflow inside the granules, and detected the weaker amplitude of a horizontal flow (0.25 km/s) than that of an upflow (0.40 km/s).",
"[27], [41], and [45] found a twofold larger amplitude in the horizontal flows than that in the vertical ones.",
"These authors reported an amplitude of a horizontal flow, which largely depends on the geometrical height, e.g., roughly 2.0 km/s, 0.7 km/s, and 0.6 km/s, respectively, at a geometrical height of approximately 100 km.",
"[24] and [56] attempted to depict the height structure of the vertical- and horizontal-flow fields, where the latter value is deduced from the vertical flow amplitude in accordance with the mass flux balance, i.e., div($\\rho v$ ) = 0, where $\\rho $ is the gas density and $v$ is the gas flow velocity.",
"It should be emphasized that any solar limb observations are vulnerable to a spatial degradation owing to the foreshortening effect.",
"This effect causes an image degradation because the object's length along the LOS becomes shorter than the actual length as the observer's angle inclines toward the plane of the solar surface.",
"Its spatial smearing effect is crucial even with the highest spatial resolution currently available.",
"A feature tracking technique has been developed and extensively used as an alternative approach over the last decades.",
"This technique calculates the cross-correlation within two or more successive images, often observed at the center of the solar disk center, and measures the relative displacement of the moving features (mostly granules).",
"The resulting speeds of the horizontal flows are found to be 0.8 - 1.3 km/s for the root mean square (RMS) [10], [40], [28], [38], [12].",
"By contrast, [3] developed a totally different approach using a machine learning to find an immanent relation between the image intensity and horizontal gas flows in a numerically synthesized photosphere.",
"It provides similar results to that of the feature tracking technique.",
"However, the feature tracking technique should be used with caution for a purpose of deriving the velocity fields at a spatial scale smaller than or even comparable to a granule, since this technique can characterize only distinctive shapes in an image (i.e., the rounded shape of a granule).",
"Several authors investigated a limitation of the technique, by comparing the real horizontal velocity fields in the granulation synthesized through a numerical simulation with the output retrieved by the technique: a poor agreement was particularly in intergranular lanes [35] or at a small spatial scale ($<$ 1 [Mm]) [37], and the retrieved flow speed in RMS is smaller than the real speed by a factor of 3 [75] or the kinetic energy of a horizontal flow is underestimated by a factor of 5 to 6 [79].",
"To characterize the horizontal flow field at a small spatial scale down to a granule, we will return to Doppler analysis at the limb observation, by using the spectral data currently available with a good spatial resolution.",
"The space-borne telescope onboard the Hinode spacecraft [30] is suited to meeting this requirement.",
"Its spatial resolution is the highest among those used in the previous studies that characterize the Doppler velocity fields over the center-to-limb.",
"The space-borne telescope benefits from providing an almost constant data quality by eliminating time-dependent degradation such as an atmospheric seeing.",
"Moreover, we attempted to minimize the image degradation owing to the instrumental stray light occurring in the Hinode's optical configuration, e.g., primary mirror with obscuration of the secondary mirror; the stray light degrades the image quality owing to the light property of diffraction, which distributes the incoming light over several pixels in the camera [60].",
"[13] investigated how much stray light in the Hinode's optical configuration degrades the image contrast of the granulation, and found that this degradation reduces the contrast by almost half.",
"Recently, several authors have attempted to remove out the stray light, by applying deconvolution processing, which restores the real image under a lack of the image degradation.",
"The restored data shows an almost twofold larger contrast in intensity [39], [78] and in the convective velocity fields [49].",
"Our study adopts a deconvolution technique in Hinode's spectral data, eliminating the major concern in a limb analysis.",
"Our objective is to derive the amplitude of the horizontal flow and its spatial distribution in the solar granulation.",
"The remainder of this paper is organized as follows.",
"In Section 2, we describe the observations as well as our approach to calculating the horizontal flow fields.",
"In Section 3, we report our observational results.",
"In Section 4, we discuss the averaged nature of the flow structures in the granulation.",
"Finally, in Section 5, we summarize our findings."
],
[
"Observations",
"The observation data in this work were taken by a spectropolarimeter ($SP$ ; [34]) used by the Solar Optical telescope (SOT) on board the Hinode spacecraft [71], [68], [61], [25].",
"The Hinode/SP observed quiet regions, recording the Stokes $I$ , $Q$ , $U$ , and $V$ profiles of the Fe i 630.15 and 630.25 nm spectral lines with a spectral sampling of 21.5 mÅ.",
"The spatial pixel sizes along the slit and scanning directions are approximately 0$$ 16 and 0$$ 15, respectively, with a spatial resolution of 0$$ 3.",
"The integration time was 1.6 s per slit position.",
"The slit orientation is fixed in the solar N–S direction.",
"For the disk center observation, the observation was executed on August 25, 2009, between 08:01 and 09:59 UT.",
"The FOV of $45\\times 609$ was repeatedly scanned 117 times with 30 slit positions per scan with an average cadence of 62 s. For the center-to-limb observation, the SOT team members, including the authors, conducted raster scans from the solar disk center to the north limb, and thus the data did not incur a Doppler shift induced by the solar rotation (Table REF ).",
"Portions of the $\\mu $ ranges are overlapped with respect to each dataset such that the wide coverage of $\\mu $ from 0.11 to 0.96 is continuous.",
"The center-to-limb data employed the same observational setup (e.g., pixel sampling and integration time) as the disk center data, although its slit-length is replaced by a longer one (81$$ 2).",
"The observation durations of these datasets typically reach 1 hour, which is considerably longer than the typical lifetime of the granulation (6 min; [22]) and the oscillatory motions (5 min; [14]).",
"The SP data were calibrated using the standard routine SP_PREP in the Solar SoftWare package ([33]); this procedure performs i) dark field correction, ii) flat-field correction, iv) the correction of curved spectral line, iv) the removal of periodic wavelength and spatial shifts in the period of the spacecraft, and vi) the calibration of intensity variation along the SP slit caused by the thermal deformation of the instrumental optics.",
"Finally, a systematic error of the absolute velocity (i.e., the offset of the absolute wavelength position) was estimated in the same manner as [48] such that the line core of our average spectral profile, integrated broadly over space and time, matches that in a well-calibrated spectral catalogue [2], falling within a systematic error of 0.18 km/s.",
"This study focuses on normal convection in hydrodynamics, without disturbance by magnetic elements.",
"The dataset covers a quiet region but also unavoidably covers magnetized atmosphere.",
"To see how much the atmosphere is magnetized, we computed the polarization profile (Stokes $V$ ) averaged over the entire FOV in each dataset.",
"The degree of polarization is described by the peak values of the profiles, of which the highest value reaches 0.15$\\%$ relative to the continuum intensity, and the other profiles range between 0.01$\\%$ and 0.06$\\%$ .",
"We conjecture that such a tiny contamination by a magnetized atmosphere should not significantly affect the statistical results."
],
[
"Spatial deconvolution",
"To correct for the instrumental stray light contamination, the spectral data undergo a deconvolution processing (see [26] for an overview).",
"The processing used is the same algorithm as that developed in [49], which is based on the Richardson-Lucy algorithm [53], [36], incorporating a regularization term designated to suppress the noise amplification [15].",
"This algorithm requires information of a point-spread-function (PSF) for the employed instrument, provided here by [13], who modeled the PSF including the major factors of image degradation induced by the optical configuration of the SOT."
],
[
"How to derive the LOS flow field",
" We processed the deconvolved spectral profiles into velocity fields through a bisector analysis [17], the detailed setup of which is described in [48], [49].",
"A bisector gives a wavelength position dividing the spectral line in two equal width, and its placement from the original wavelength is attributed to the Doppler shift at various intensity levels.",
"Computing the bisectors of the Fe i 630.15 nm spectral line, we obtained the LOS velocity fields at six bisector levels, defined here by $I/I_{0}$ =0.70, 0.65, 0.60, 0.55, 0.50, 0.45, where $I$ is intensity level at each spatial pixel, and $I_{0}$ is the continuum intensity averaged over the disk center.",
"Note that this $I_{0}$ definition is also adopted for the non-disk center observations, aiming to provide the same temperature layer sampling irrespective of the heliocentric angle.",
"The physical meaning of such an approach is that any observed spectral profiles are normalized with the absolute intensity, i.e., rather, the bisector levels are defined using the data number (DN) as the unit.",
"Thus, our bisector analysis aims to sample the identical temperature layer even at a different heliocentric angle under the following rough approximation; any observed light with a certain intensity originates from the corresponding temperature layer through the Planck function, valid within the optical thick regime and in the first-order approximation of the temperature stratification along the optical depth, which is moderately sufficient for the solar photosphere [20].",
"The stratification of the photospheric temperature stratification is nearly invariant with respect to the solar latitude, namely, as large as an enhancement of 2-3 K at the pole [52], corresponding to a geometrical height difference of a few km.",
"Consequently, the bisector intensity levels sampled at the absolute DN reflect the same temperature layer, irrespective of the heliocentric angle.",
"Intensity levels of 0.70-0.45 are related to the Planck function temperature, through the average quiet Sun model of [76], then proving the corresponding geometrical heights of roughly from 50 km to 140 km.",
"The calculated geometrical height should be accepted when the temperature stratification holds the plane-parallel structure although the actual photosphere does not, as the corrugation of the iso-optical depth surface is estimated to 30 km/ in rms [65].",
"The intensity range of $I/I_{0}$ =0.70 to 0.45 is included between the line core and continuum even at an off disk-center, except at the extreme limb.",
"Fig.REF confirmed this fact, showing spectral profiles normalized with the disk center continuum intensity at several different heliocentric angles.",
"Spectral profiles separately averaged in granules ($I_{c}>$ 0) and intergranular lanes ($I_{c}<$ 0) are illustrated on the top and bottom panels, respectively, where $I_{c}$ is the average continuum intensity at each heliocentric angle.",
"One noticeable concern in our dataset, intermittently observed from 2007 to 2017, is an instrumental throughput degradation over time; [34] reported a decreasing trend of approximately 17 % in five years.",
"To minimize the throughput difference among our dataset, by referring to irradiance scan data regularly executed every month from the beginning of the Hinode launch, we normalize the DN in each dataset to be identical with this reference data at the closest timing (TableREF ).",
"Fig.REF demonstrates the validity of such correction, showing smooth variation of the continuum intensity across the center-to-limb.",
"Our elaborate bisector processing mentioned above provides a reliable sampling of the same temperature layer across the solar disk."
],
[
"Removal of the 5-min oscillation",
"The derived velocity field includes another-origin (not the convection-origin), namely, the 5-min oscillation.",
"This oscillation is an assembly of numerous numbers of eigenmodes generated with the solar spherical stratification [32], and typically fluctuates 0.3-0.4 km/s in RMS in the photospheric velocity fields [72].",
"A subsonic filter [70], [57], [40] is a well-established technique for extracting the convective velocity alone from a time series of velocity map by filtering out the 5-min oscillations.",
"Our study adopted the same setup as described in [48], [49].",
"Each dataset from the disk center to the limb undergoes the above filtering processing separately.",
"In all the created power spectra in the time-spatial frequency domain, the 5-min oscillatory power seems to be located in the identical frequency domain among the dataset, whereas the power decreases toward the limb presumably owing to the predominant vertical fluctuations [67], [59]."
],
[
"Deriving the horizontal flow",
"We introduce how to determine $v_{h, std}$ (horizontal flow speed in standard deviation) from the LOS amplitude variations over the disk-center to limb.",
"Any LOS flow fields, except for when observed at the disk center, consist of components vertical and horizontal to the solar surface.",
"We assume that the granulation has no latitude-dependence and that flows in the vertical and horizontal directions are uncorrelated.",
"Under this assumption, one of the horizontal components $v_{x}$ (defined by the spectrograph slit direction projected on the solar surface plane) can be related to the other observable physical quantities through the following equation: $v_{los, std}^2 (\\mu )=\\mu ^{2} v_{z, std}^{2}+(1-\\mu ^{2}) v_{x, std}^{2},$ where $v_{los, std}$ is the observed convective velocity in the standard deviation along the LOS, $v_{z, std}$ stands for vertical motion obtained at the disk center, $v_{x, std}$ stands for the horizontal motions we aim to derive, and $\\mu $ is the heliocentric angular distance represented conventionally as $\\mu =\\textrm {cos} \\ \\theta $ ; here, $\\theta $ is a heliocentric angle.",
"Past studies [27], [41] have also applied the above equation but with the standard deviations replaced by the RMS.",
"Their purpose is to derive $v_{x}$ averaged over granules and intergranular lanes, where the RMS is also valid as long as the average value of the vertical flow speed holds at zero.",
"Our interest is to derive $v_{x}$ in the granules and intergranular lanes separately, which should be derived through the standard deviation form, because each average vertical flow is non-zero as the granules tend to have an upflow and the intergranular lanes have a downflow.",
"The fitting process provides a reasonable value of $v_{x, std}$ , which minimizes the dispersion of $v_{los, std}$ versus the resulting value of the right-side of the equation.",
"Note that $v_{los, std}$ reflects only one of the two directions along the horizontal plane, although the gas flow along its perpendicular direction actually exists.",
"In symmetric horizontal gas flows, the actual amplitude should be given through $v_{h, std}= \\sqrt{2} v_{x, std}.$"
],
[
"LOS velocity field",
"First, we report the obtained the center-to-limb variation of $v_{los}$ .",
"Fig.REF shows the arbitrary chosen snapshots of the continuum intensity (top panels) and the co-spatial $v_{los}$ at a bisector level of 0.45 (bottom panels), at a different $\\mu $ of 1.0, 0.8, 0.6, and 0.4 from the left to right panels.",
"The contrast of $v_{los}$ is the smallest at $\\mu $ =1.0 and the largest at $\\mu $ =0.6, as shown in their RMS values of 0.79 and 1.09 km/s, respectively.",
"As the heliocentric angular distance $\\mu $ decreases (toward the limb), the foreshortening effect conspicuously appears: granular and intergranular shapes become compressed along the north-south direction while keeping their length along the east-west direction, resulting in elliptical shapes in their appearance.",
"In the $v_{los}$ maps, we can find two characteristics along the south-north and east-west directions, respectively.",
"Regarding the south-north direction, the peak of the blue-shifted signal does not coincide with the center of the granule; the front (south) side of the granules generally displays strong blue-shifted signals, whereas the rear (north) side generally shows weak blue-shifted or occasionally red-shifted, signals (Fig.REF ).",
"This trend is particularly seen in the slices sandwiched by the two arrows facing each other, indicated by a and b, which upon closer inspection are displayed in the left panels of Fig.REF .",
"In the granules, the peak of the blue-shifted signal is offset toward the front side (i.e., the south side) from that of the continuum intensity, as shown in 0.1-0.9 Mm and 1.4-2.0 Mm of sample a, and more clearly seen in 0.4-1.6 Mm of sample b.",
"This asymmetry between the south and north sides of the granule is in good agreement with a granular radial flow; when viewed from the inclined observer's angle, the south side reflects a horizontal flow approaching toward the observer (i.e., blue-shifted), while the north side shows a horizontal flow moving away from the observer (i.e., red-shifted).",
"For the other characteristic, the existence of a non-radial flow is implied by the spatial relation between $v_{los}$ along the east-west direction and the granulation morphology.",
"The dark intergranular lane, sandwiched by adjoining granules along the east-west direction, is filled with a blue-shifted signal.",
"Conspicuous examples of such intergranular lanes are seen in the slices indicated by the arrows A and B in Fig.REF , the plots of which are shown in 0.4-1.3 Mm of sample A and 1.5-2.0 Mm of sample B.",
"This trend can be explained by the granular non-radial flow, because $v_{los}$ on the west and east sides are not aligned in the radial direction.",
"Our finding of radial flow is not surprising if the gas convection holds the mass conservation.",
"The other finding does not come from a natural consequence of the gas convection, indicating a non-radial flow can develop strongly enough to be observed.",
"By contrast, we cannot clearly see the opposite trends: namely, for a slice of intergranular lane cutting through the south-north direction, no clear trend of weak red-shifted signals at the front side or strong red-shifted signals at the rear side, and for a slice cutting through the east-west direction, no red-shifted signal associated with intergranular lanes sandwiched by adjoining granules.",
"We attribute the lack of a clear indication of these opposite trends to still the remaining lack of spatial resolution because low-intensity signals in intergranular lanes are strongly diluted with higher-intensity signals from granules and are preferentially lost."
],
[
"Amplitude of horizontal flow field",
"Fig.REF describes $v_{los, std}$ variation over the center-to-limb at a bisector level of 0.55.",
"In this Figure, we calculates all the standard deviation values along the slit direction with a constant spacing of 0$$ 16, and arranges the values in order of heliocentric angular distance.",
"The standard deviation is calculated from pixels covering whole the repeatedly scanned duration for each dataset (typically more than 1 hour) and covering 2 Mm within the center part of the scanned range, since pixels close to the edge are uncertainly deconvolved as they need information outside the FOV.",
"Each of the pixel coverages correspond to 2 and 10 samples of a typical extent and lifetime of a granule, respectively.",
"Fig.REF represents an arc-shape variation of $v_{los, std}$ over the center-to-limb; the amplitude starts from 0.82 km/s at $\\mu =$ 1.0 and monotonically increases toward the limb, reaching 1.22 at $\\mu =$ 0.6, and decreases at $\\mu <$ 0.6.",
"This arc-shape variation is a net consequence of the two competing factors: The decreasing factor of $v_{los, std}$ toward the limb is due to the foreshortening effect, whereas the increasing factor is due to the faster horizontal flow than the vertical flow.",
"Although this increasing factor affects at the limb ($\\mu <$ 0.6), the decreasing factor there becomes effective enough to overcome the increasing trend.",
"For a reasonable estimation of $v_{h, std}$ , we have to make a proper selection of which heliocentric angles should be adopted into the fitting (Eq.REF ).",
"This fitting function should be adopted to samples at as many heliocentric angles as possible but simultaneously under the minimal foreshortening degradation.",
"Accordingly, we adopted four different coverages of the heliocentric angles, i.e., $\\mu $ = 0.6-1.0, 0.7-1.0, 0.8-1.0, and 0.9-1.0, one example of which is shown with the gray-solid curve by adapting $v_{x, std}$ =1.38 km/s (or $v_{h, std}$ =1.95 km/s) into the fitting.",
"The reason for eliminating the remaining angles (i.e., $\\mu <$ 0.6) is that Eq.REF does not model the decreasing factor owing to the foreshortening degradation.",
"Table REF summarizes the amplitudes resulting from the fitting with a combination of the six bisector levels and the four fitted ranges.",
"When only heliocentric angles close to the disk center (e.g., $\\mu $ =0.9-1.0) are adopted to the fitting, the estimated $v_{x, std}$ becomes higher than those with a wide coverage of the heliocentric angles (e.g., $\\mu $ =0.6-1.0).",
"This is due to a less prominent foreshortening degradation close to the disk center.",
"Whereas different fitted ranges result in a different value of $v_{h, std}$ , those at a bisector level of 0.70 are roughly 1.8-2.4 km/s, because the maximum value (2.43 km/s) is derived with a fitted range of $\\mu $ =0.9-1.0 and the minimum value (1.93 km/s) is derived with a fitted range of $\\mu $ = 0.6-1.0.",
"This dependence on the fitting range induces a complexity in strictly determining the value of $v_{h, std}$ .",
"Instead, we determine the value of $v_{h, std}$ by averaging over four outputs with four fitting ranges, which are listed in the right-most column in Table REF .",
"For greater simplicity, the horizontal flow speed, averaging over all the bisector levels from 0.45 to 0.70 is 2.10 km/s.",
"We make a crude estimation of the error of $v_{hor, std}$ derived through the fitting.",
"Referring to a standard deviation of 0.39 km/s in one of the value $v_{los, std}$ in Fig.REF , we distributed the random fluctuation of 0.39 km/s on the $v_{los, std}$ variation over $\\mu $ =0.60-1.00 and searched for the best fitted $v_{hor, std}$ for a hundred of time, finding the standard error is 0.04 km/s."
],
[
"Horizontal flow fields in granules and intergranular lanes",
" The standard deviation of $v_{los}$ , when viewed from an inclined heliocentric angle, can be attributed to the amplitude of $v_{h}$ as its flowing direction is randomly distributed.",
"Fig.REF shows scatter plots between $I_{c}$ and $v_{los}$ .",
"The left panel ($\\mu =$ 1.00) presents the relation of a granular upflow and intergranular downflow with a somewhat small scatter.",
"On the right panel ($\\mu $ =0.60), the aforementioned relation still holds, but with a relatively larger scatter.",
"The degree of this scatter clearly depends on $I_{c}$ being relatively large in a dark region (intergranular lanes) and small in a bright region (granules).",
"To determine $v_{h}$ , we process the fitting approach in the same way as introduced in section 4.2, but separately at several continuum intensity regions segmented with a spacing of $I_{c}$ =0.04.",
"The horizontal flow speed is also determined by averaging over the four best-fitted values derived with the four heliocentric angular coverages.",
"Fig.REF provides the obtained height structure of the horizontal flow amplitudes, as well as the structure of the vertical flow for a comparison with the relation of the granular upflow and intergranular donwflow.",
"The horizontal flow speeds are generally faster in the dark regions than those in the bright region because the maximum speed occurs inside the intergranular lanes, $I_{c}$ =0.92-0.96, and the amplitude decreases monotonically toward either side of the bright or even darker region.",
"With any choice of $\\mu $ ranges of the fitting (i.e., $\\mu $ = 0.6-1.0, 0.7-1.0, 0.8-1.0, and 0.9-1.0), the fastest speed is at the same location ($I_{c}$ =0.92-96).",
"Note that the range of $I_{c}=$ 0.76-1.40 displayed in Fig.REF covers almost the entire region (98%), whereas our bisector analysis cannot sample brighter granules ($I_{c}>$ 1.40) or darker intergranular lanes ($I_{c}<$ 0.76).",
"One remark is that some of granules depart from the average sense, e.g., a faint granule as displaying upflow but with $I_{c}<$ 0, and accordingly Fig.REF should be interpreted as it simply draws the average structure.",
"Table REF lists the horizontal flow amplitudes at each bisector level in the granular and intergranular regions, separately, each of which is derived from averaging over regions of $I_{c}>$ 1.00 or $I_{c}<$ 1.00, respectively.",
"For more simplicity, the horizontal flow speed averaging over the six bisector levels is 1.58 km/s in the granules and 1.79 km/s in the intergranular lanes, i.e., the flow is faster in the intergranular lanes."
],
[
"Horizontal flow amplitude",
" Our deconvolution provides a remarkably enhanced amplitudes in LOS velocity fields over the center-to-limb.",
"Fig.REF gives one example of $v_{los, std}$ as a function of the heliocentric angular distance, obtained at a bisector level of 0.55.",
"This amplitude monotonically increases from 0.82 km/s at $\\mu $ = 1.0 to 1.22 km/s at $\\mu $ = 0.6.",
"The estimated amplitude of $v_{h}$ should be 1.95 km/s to explain the mentioned center-to-limb variation ($\\mu $ =0.6-1.0).",
"Note that the estimated value becomes large when considering the variation only at a higher $\\mu $ range, e.g., the reasonable value for $\\mu $ =0.9-1.0 is 2.27 km/s.",
"First, let us compare the obtained LOS amplitude variation with those of previous studies [27], [41], [45].",
"Their highest amplitudes peak at $\\mu $ =0.6-0.8, whereas their sampling over a heliocentric angle is discrete, e.g., $\\Delta \\mu $ = 0.2, 0.4, and 0.4, respectively.",
"Our maximum amplitude of 1.22 km/s at $\\mu $ =0.6 highly exceeds their maximum amplitude of 0.4-0.6 km/s.",
"This fact implies that our spectral data after deconvolution processing provide resolved granulation even far from the disk center by mitigating the image degradation owing to the foreshortening effect.",
"By contrast, their estimations of $v_{h}$ largely differ from each other, although their LOS amplitudes are comparable among them.",
"[27] estimated the horizontal flow speed of $\\approx $ 2.0 km/s, compensating for several causes of the degradation other than the spatial smearing, e.g., an observed emergent intensity cumulatively originates from several layers along the LOS direction, making the observed signal smoothed over the physical information.",
"With a better spatial resolution achieved using a balloon-borne 30-cm telescope whose data are almost free from an atmospheric seeing degradation [42], [41] estimated a flow speed of $\\approx $ 0.7 km/s in RMS, and later [45] concluded a typical value of $\\approx $ 0.6 km/s; the above values are taken at the geometrical height corresponding to our bisector sampling levels.",
"Thus, such a threefold difference among their estimation implies that the horizontal flow speed heavily depends on their approach.",
"Our result, simply using the spatially well-resolved data, indicates confidently a large amplitude (e.g., 2.10 km/s), even without treating the other causes of degradation.",
"If we appropriately correct for the other degradation, this amplitude will be even larger.",
"This additional effort is beyond our current scope; rather, we insist here that the horizontal flow speed is at least 1.8-2.4 km/s in terms of the standard deviation, derived using only treatment for the spatial resolution.",
"Next, we compare our horizontal flow amplitude with that derived through feature tracking techniques in the past, as summarized in Table REF .",
"Our value is the highest at 2-4 times larger than some of their values.",
"We emphasize that a Doppler diagnostic approach can reach a higher amplitude than that derived through a feature tracking approach.",
"Whereas feature tracking techniques work well at a spatial scale of larger than a typical granule, the Doppler approach benefits from having the capability of access at a small-scale, such as intergranular lanes and inside the granules.",
"This is demonstrated through present study using currently available data with the aid of deconvolution processing.",
"Furthermore, by improving the spatial resolution achieved in the future, a Doppler diagnosis will directly mitigate the image degradation owing to the foreshortening effect, thereby extracting more horizontal flows at a small scale."
],
[
"Average radial structures of gas convection",
"Our study succeeded in providing horizontal flows in granules and intergranular lanes separately.",
"Past observational studies ([24] and [56]) have also drawn the height structure of horizontal flow fields, although they indirectly estimated them by referring to vertical flow fields with the mass balance imposed, i.e., calculated in accordance with div($\\rho v$ ) = 0, where $\\rho $ is the gas density.",
"Our approach is instead to derive horizontal flow fields by analyzing the center-to-limb variation at each continuum intensity in the granules and intergranular lanes.",
"There may exist a concern regarding whether the continuum intensities at different heliocentric angles represent the identical location in the granulation.",
"This concern does not deserve serious consideration, however.",
"In this study, $I_{c}$ is normalized by averaging over the entire FOV in each dataset to correct for the monotonic decrease of the continuum intensity toward the limb, which is widely known as limb darkening [66].",
"To check the validation of this normalization approach, we draw histograms of the normalized continuum intensity in each dataset at the disk center to the most off-disk center accessed in our study, $\\mu $ = 0.6.",
"These histograms (not shown) do not differ from each other, indicating that a continuum intensity normalized within each dataset represents more or less an identical location.",
"Fig.REF illustrates the average radial structure of the gas convection when viewed as a slice of a granule, summarizing simply our findings introduced in section 4.3.",
"Here, we discuss the sequential phases (a) and (b); the former represents the boundary between the up- and downflows, and the latter represents the location where the horizontal flow has the fastest speed.",
"The location of phase (a) is fairly compatible with $I_{c}$ = 1.00, which is commonly referred to as outline granules (e.g., [21], [8], [1]).",
"This consistency confirms that $I_{c}$ =1.00 is a place where the material's flow alters its traveling direction from upward to downward [29].",
"The positional relation between phases (a) and (b), i.e., at $I_{c}$ =1.00 and 0.94, respectively, implies the following convectional aspect, which has yet to be argued.",
"A flowing gas parcel, even after turning into a downflow, still experiences a pressure gradient force horizontally directed from a granule to an intergranular lane, having the fastest speed in the intergranular lanes; the parcel subsequently decelerates as it proceeds toward the vertices of the intergranular lane.",
"The enhanced pressure gradient force directing outside the granule will be locally created by the strong stratified atmosphere in the granules, created by gas excessively accumulated in the granular center, which sweeps the upwelling gas into the horizontal directions [46].",
"Thus, we identified where the fastest horizontal flow occurs slightly outside a granule or inside the intergranular lane.",
"Finally, we quantitatively compare our observational results with a synthesized three-dimensional atmosphere calculated by the recent numerical simulation code, namely, the MURaM code [77].",
"Fig.REF describes a height structure of the velocity fields in the numerical simulation in a manner analogous to the observational result (Fig.REF ).",
"In Fig.REF , $I_{c}$ is calculated through the radiative transfer code of SPINOR [19], whose detailed setup is described in [49].",
"Note that both vertical and horizontal flows are real values (not the Doppler velocity), referring directly to the physical quantities in the numerical atmosphere.",
"The velocity field is sampled at geometrical heights of 50-140 km, which are identical to those heights to which our bisector analysis refers.",
"The fastest speed of a horizontal flow is at $I_{c}$ =0.92-0.96 irrespective of the bisector level, which is identical to our observational result.",
"Therefore, the synthesized granulation is fairly consistent with our observational findings, indicating that a numerical simulation actually reproduces well horizontal gas flows in the granulation."
],
[
"Limitation to the approach",
" We should remark a validity of the several assumptions made to derive the average radial structure of gas flow, and discuss how they are related to our results.",
"First, we estimate how much the derived amplitude of horizontal flow includes flows with a spatial scale larger than the granulation, namely supergranulation [55].",
"Its horizontal flow speed is 0.3-0.4 km/s and vertical flow is much slow, 0.03 km/s, with a spatial extent of roughly 30 Mm.",
"In analogous way to the error estimation of $v_{los,h}$ (section 4.2), by distributing the amplitude of 0.3 km/s randomly on the $v_{los, std}$ variation over $\\mu $ =0.6-1.0 (0.3 km/s is multiplied by a factor of $\\sqrt{1-\\mu ^{2}}$ to be projected in the LOS direction), where 24 numbers of supergranules exist, we found the best fitted $v_{hor, std}$ with an overestimation of 0.15 km/s.",
"Consequently, the horizontal flow speeds listed in this paper are mostly the granulation alone, while approximately 8 percent is contributed from the supergranulation.",
"Second, we consider effects associated with the limb observations other than the foreshoretening effect.",
"The optical corrugation (as noted in section 3.2) is particularly expected to appear as a consequence of a radiative transfer, which is generally difficult to be taken in account.",
"This effect forces the observer at an inclined angle to see the larger projected area of granules and the smaller projected area of intergranular lanes [4], and may bias the normalization of the continuum intensity.",
"Such a difference of the normalization value leads to possible ambiguity about whether a location of $I_{c}$ =0.94 at $\\mu <$ 1.0 corresponds to intergranular lanes seen at the disk center ($\\mu $ =1.0).",
"We may think that our dataset, whose heliocentric angle extends up to $\\mu $ =0.6, still do not significantly suffer from such effect.",
"Fig.REF provides $v_{los}$ averaged over each continuum intensity separately at $\\mu $ =0.6, showing that a crossing point between upflow or downflow, i.e., a boundary between granules and intergranular lanes, falls into $I_{c}$ =0.98-1.02.",
"It should be emphasized that we did not intentionally make the null-velocity correction.",
"We may conclude that, although the optical corrugation effect leads to an ambiguity of the consistency in the continuum intensity sampling between the disk center and the off disk-center, the fast horizontal flow actually happens in the downflow regions, i.e., in intergranular lanes.",
"Finally, it should be emphasized that Fig.REF indicates a radial structure of gas convection, in which the arrows are to represent horizontal flow that degenerates both the radial and non-radial components.",
"The cartoon of Fig.REF should be interpreted to be compulsorily projected onto a two-dimensional plane for convenience, while the existence of some non-radial flows (i.e., the perpendicular direction to the paper surface) are actually indicated by Figs.REF and REF .",
"Our future interest is to depict radial and non-radial components separately, providing the three-dimensional flow fields in the granulation."
],
[
"Summary",
"We studied the solar photospheric convection, focusing on horizontal gas motions, based on spectral data observed by the Hinode spacecraft.",
"This instrument is well suited to our study; the Hinode's spacecrafts's seeing-free observation easily maintains nearly a constant image quality among all of the center-to-limb dataset, which is a key requirement in calculating the horizontal flow amplitude.",
"Its spatial resolution (0$$ 3) is the highest among all the previous studies charactering the horizontal flow speed.",
"Moreover, our deconvolution processing reasonably corrects for the instrumental degradation from stray light, aiming to address one of the issues in a limb analysis.",
"This study succeeded in providing a spatial distribution of the horizontal flows in the granules and intergranular lanes separately.",
"Our observational findings can be summarized as follows: The LOS amplitude monotonically increases from 0.82 km/s at $\\mu $ = 1.0, reaching the maximum value of 1.22 km/s at $\\mu $ = 0.6, and decreases toward the farther limb.",
"The horizontal flow speed averaging over the granules and intergranular lanes ranges from 1.8 to 2.4 km/s.",
"The horizontal flow speed is slower in granules with an amplitude of 1.58 km/s and is faster in intergranular lanes with an amplitude of 1.79 km/s.",
"The maximum speed is observed after entering the intergranular lanes.",
"The two-dimensional LOS flow field at the off-center disk indicates the existene of divergent flows radially from a granular center, and other non-radially streaming flows.",
"Intergranular lanes, previously considered as simply a downflowing region, are now revealed to have another aspect, i.e., a strong horizontal flowing region.",
"How the horizontal flow acts on the fluxtubes deserves a further investigation because such tubes have been observed to divert rapidly from the center of a convection cell and advected slowly along the cell boundary [63].",
"We would like to emphasize that a spectral data analysis with a higher spatial resolution can shed light on the intergranular convective dynamics, achieved using a future telescope or even a currently available instrument if somehow treating stray light in the instruments.",
"This would be helpful in disentangling the relation between the granulation and the phenomena produced in the photosphere, and hopefully even in the upper atmosphere.",
"Hinode is a Japanese mission developed and launched by ISAS/JAXA, collaborating with NAOJ as a domestic partner, NASA and STFC (UK) as international partners.",
"Scientific operation of the Hinode mission is conducted by the Hinode science team organized at ISAS/JAXA.",
"This team mainly consists of scientists from institutes in the partner countries.",
"Support for the post-launch operation is provided by JAXA and NAOJ (Japan), STFC (U.K.), NASA, ESA, and NSC (Norway).",
"We are grateful to the Hinode team for obtaining a series of datasets well suited to this analysis.",
"We also thank S. K. Solanki for providing helpful comments and T. L. Riethmüller for instructions regarding how to treat numerical simulation code.",
"The present study was financially supported by the Advanced Research Course Program of SOKENDAI and by JSPS KAKENHI, Grant NO.",
"JP16J07106 and 18H05878."
]
] | 2001.03575 |
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